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<title>Independent component analysis</title>

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Independent component analysis

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Machine_learning" title="Machine learning">Machine learning</a> and <a href="Data_mining" title="Data mining">data mining</a></th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Paradigms</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Supervised_learning" title="Supervised learning">Supervised learning</a></li>

<li><a href="Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li>

<li><a href="Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">Semi-supervised learning</a></li>

<li><a href="Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li>

<li><a href="Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li>

<li><a href="Meta-learning_(computer_science)" title="Meta-learning (computer science)">Meta-learning</a></li>

<li><a href="Online_machine_learning" title="Online machine learning">Online learning</a></li>

<li><a href="Batch_learning" class="mw-redirect" title="Batch learning">Batch learning</a></li>

<li><a href="Curriculum_learning" title="Curriculum learning">Curriculum learning</a></li>

<li><a href="Rule-based_machine_learning" title="Rule-based machine learning">Rule-based learning</a></li>

<li><a href="Neuro-symbolic_AI" title="Neuro-symbolic AI">Neuro-symbolic AI</a></li>

<li><a href="Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic engineering</a></li>

<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Problems</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Generative_model" title="Generative model">Generative modeling</a></li>

<li><a href="Regression_analysis" title="Regression analysis">Regression</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Clustering</a></li>

<li><a href="Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></li>

<li><a href="Density_estimation" title="Density estimation">Density estimation</a></li>

<li><a href="Anomaly_detection" title="Anomaly detection">Anomaly detection</a></li>

<li><a href="Data_cleaning" class="mw-redirect" title="Data cleaning">Data cleaning</a></li>

<li><a href="Automated_machine_learning" title="Automated machine learning">AutoML</a></li>

<li><a href="Association_rule_learning" title="Association rule learning">Association rules</a></li>

<li><a href="Semantic_analysis_(machine_learning)" title="Semantic analysis (machine learning)">Semantic analysis</a></li>

<li><a href="Structured_prediction" title="Structured prediction">Structured prediction</a></li>

<li><a href="Feature_engineering" title="Feature engineering">Feature engineering</a></li>

<li><a href="Feature_learning" title="Feature learning">Feature learning</a></li>

<li><a href="Learning_to_rank" title="Learning to rank">Learning to rank</a></li>

<li><a href="Grammar_induction" title="Grammar induction">Grammar induction</a></li>

<li><a href="Ontology_learning" title="Ontology learning">Ontology learning</a></li>

<li><a href="Multimodal_learning" title="Multimodal learning">Multimodal learning</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Supervised_learning" title="Supervised learning">Supervised learning</a> (<a href="Statistical_classification" title="Statistical classification">classification</a> • <a href="Regression_analysis" title="Regression analysis">regression</a>) </div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Apprenticeship_learning" title="Apprenticeship learning">Apprenticeship learning</a></li>

<li><a href="Decision_tree_learning" title="Decision tree learning">Decision trees</a></li>

<li><a href="Ensemble_learning" title="Ensemble learning">Ensembles</a>

<ul><li><a href="Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li>

<li><a href="Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li>

<li><a href="Random_forest" title="Random forest">Random forest</a></li></ul></li>

<li><a href="Linear_regression" title="Linear regression">Linear regression</a></li>

<li><a href="Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li>

<li><a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural networks</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic regression</a></li>

<li><a href="Perceptron" title="Perceptron">Perceptron</a></li>

<li><a href="Relevance_vector_machine" title="Relevance vector machine">Relevance vector machine (RVM)</a></li>

<li><a href="Support_vector_machine" title="Support vector machine">Support vector machine (SVM)</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Cluster_analysis" title="Cluster analysis">Clustering</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="BIRCH" title="BIRCH">BIRCH</a></li>

<li><a href="Hierarchical_clustering" title="Hierarchical clustering">Hierarchical</a></li>

<li><a href="K-means_clustering" title="K-means clustering">k-means</a></li>

<li><a href="Fuzzy_clustering" title="Fuzzy clustering">Fuzzy</a></li>

<li><a href="Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization (EM)</a></li>

<li> <a href="DBSCAN" title="DBSCAN">DBSCAN</a></li>

<li><a href="OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a></li>

<li><a href="Mean_shift" title="Mean shift">Mean shift</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Structured_prediction" title="Structured prediction">Structured prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Graphical_model" title="Graphical model">Graphical models</a>

<ul><li><a href="Bayesian_network" title="Bayesian network">Bayes net</a></li>

<li><a href="Conditional_random_field" title="Conditional random field">Conditional random field</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov</a></li></ul></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Random_sample_consensus" title="Random sample consensus">RANSAC</a></li>

<li><a href="Local_outlier_factor" title="Local outlier factor">Local outlier factor</a></li>

<li><a href="Isolation_forest" title="Isolation forest">Isolation forest</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Neural_network_(machine_learning)" title="Neural network (machine learning)">Neural networks</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Autoencoder" title="Autoencoder">Autoencoder</a></li>

<li><a href="Deep_learning" title="Deep learning">Deep learning</a></li>

<li><a href="Feedforward_neural_network" title="Feedforward neural network">Feedforward neural network</a></li>

<li><a href="Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network</a>

<li><a href="Reservoir_computing" title="Reservoir computing">reservoir computing</a></li></ul></li>

<li><a href="Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a>

<ul><li><a href="Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted</a></li></ul></li>

<li><a href="Diffusion_model" title="Diffusion model">Diffusion model</a></li>

<li><a href="Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network</a>

<li><a href="LeNet" title="LeNet">LeNet</a></li>

<li><a href="AlexNet" title="AlexNet">AlexNet</a></li>

<li><a href="DeepDream" title="DeepDream">DeepDream</a></li></ul></li>

<li><a href="Neural_field" title="Neural field">Neural field</a>

<ul><li><a href="Neural_radiance_field" title="Neural radiance field">Neural radiance field</a></li>

<li><a href="Physics-informed_neural_networks" title="Physics-informed neural networks">Physics-informed neural networks</a></li></ul></li>

<li><a href="Transformer_(deep_learning_architecture)" title="Transformer (deep learning architecture)">Transformer</a>

<ul><li><a href="Vision_transformer" title="Vision transformer">Vision</a></li></ul></li>

<li><a href="Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li>

<li><a href="Spiking_neural_network" title="Spiking neural network">Spiking neural network</a></li>

<li><a href="Memtransistor" title="Memtransistor">Memtransistor</a></li>

<li><a href="Electrochemical_RAM" title="Electrochemical RAM">Electrochemical RAM</a> (ECRAM)</li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Q-learning" title="Q-learning">Q-learning</a></li>

<li><a href="Policy_gradient_method" title="Policy gradient method">Policy gradient</a></li>

<li><a href="State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li>

<li><a href="Temporal_difference_learning" title="Temporal difference learning">Temporal difference (TD)</a></li>

<li><a href="Multi-agent_reinforcement_learning" title="Multi-agent reinforcement learning">Multi-agent</a>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Learning with humans</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Active_learning_(machine_learning)" title="Active learning (machine learning)">Active learning</a></li>

<li><a href="Crowdsourcing" title="Crowdsourcing">Crowdsourcing</a></li>

<li><a href="Human-in-the-loop" title="Human-in-the-loop">Human-in-the-loop</a></li>

<li><a href="Mechanistic_interpretability" title="Mechanistic interpretability">Mechanistic interpretability</a></li>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Model diagnostics</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li>

<li><a href="Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li>

<li><a href="Learning_curve_(machine_learning)" title="Learning curve (machine learning)">Learning curve</a></li>

<li><a href="Receiver_operating_characteristic" title="Receiver operating characteristic">ROC curve</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Mathematical foundations</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Kernel_machines" class="mw-redirect" title="Kernel machines">Kernel machines</a></li>

<li><a href="Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li>

<li><a href="Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li>

<li><a href="Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></li>

<li><a href="Occam_learning" title="Occam learning">Occam learning</a></li>

<li><a href="Probably_approximately_correct_learning" title="Probably approximately correct learning">PAC learning</a></li>

<li><a href="Statistical_learning_theory" title="Statistical learning theory">Statistical learning</a></li>

<li><a href="Vapnik%E2%80%93Chervonenkis_theory" title="Vapnik–Chervonenkis theory">VC theory</a></li>

<li><a href="Topological_deep_learning" title="Topological deep learning">Topological deep learning</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Journals and conferences</div><div class="sidebar-list-content mw-collapsible-content hlist">

<li><a href="Conference_on_Neural_Information_Processing_Systems" title="Conference on Neural Information Processing Systems">NeurIPS</a></li>

<li><a href="International_Joint_Conference_on_Artificial_Intelligence" title="International Joint Conference on Artificial Intelligence">IJCAI</a></li>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Related articles</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></li>

<li><a href="List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a>

<ul><li><a href="List_of_datasets_in_computer_vision_and_image_processing" title="List of datasets in computer vision and image processing">List of datasets in computer vision and image processing</a></li></ul></li>

<li><a href="Outline_of_machine_learning" title="Outline of machine learning">Outline of machine learning</a></li></ul></div></div></td>

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In <a href="Signal_processing" title="Signal processing">signal processing</a>, independent component analysis (ICA) is a computational method for separating a <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> signal into additive subcomponents. This is done by assuming that at most one subcomponent is Gaussian and that the subcomponents are <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">statistically independent</a> from each other.<a href="#cite_note-1">[1]</a> ICA was invented by Jeanny Hérault and Christian Jutten in 1985.<a href="#cite_note-jutten85-2">[2]</a> ICA is a special case of <a href="Blind_source_separation" class="mw-redirect" title="Blind source separation">blind source separation</a>. A common example application of ICA is the "<a href="Cocktail_party_problem" class="mw-redirect" title="Cocktail party problem">cocktail party problem</a>" of listening in on one person's speech in a noisy room.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2></div>

Independent component analysis attempts to decompose a multivariate signal into independent non-Gaussian signals. As an example, sound is usually a signal that is composed of the numerical addition, at each time t, of signals from several sources. The question then is whether it is possible to separate these contributing sources from the observed total signal. When the statistical independence assumption is correct, blind ICA separation of a mixed signal gives very good results.<a href="#cite_note-ComoJ2010-5">[5]</a> It is also used for signals that are not supposed to be generated by mixing for analysis purposes.

A simple application of ICA is the "<a href="Cocktail_party_problem" class="mw-redirect" title="Cocktail party problem">cocktail party problem</a>", where the underlying speech signals are separated from a sample data consisting of people talking simultaneously in a room. Usually the problem is simplified by assuming no time delays or echoes. Note that a filtered and delayed signal is a copy of a dependent component, and thus the statistical independence assumption is not violated.

Mixing weights for constructing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M}">

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The success of ICA separation of mixed signals relies on two assumptions and three effects of mixing source signals. Two assumptions:

<ol><li>The source signals are independent of each other.</li>

<li>The values in each source signal have non-Gaussian distributions.</li></ol>

Three effects of mixing source signals:

<ol><li>Independence: As per assumption 1, the source signals are independent; however, their signal mixtures are not. This is because the signal mixtures share the same source signals.</li>

<li>Normality: According to the <a href="Central_Limit_Theorem" class="mw-redirect" title="Central Limit Theorem">Central Limit Theorem</a>, the distribution of a sum of independent random variables with finite variance tends towards a Gaussian distribution. Loosely speaking, a sum of two independent random variables usually has a distribution that is closer to Gaussian than any of the two original variables. Here we consider the value of each signal as the random variable.</li>

<li>Complexity: The temporal complexity of any signal mixture is greater than that of its simplest constituent source signal.</li></ol>

Those principles contribute to the basic establishment of ICA. If the signals extracted from a set of mixtures are independent and have non-Gaussian distributions or have low complexity, then they must be source signals.<a href="#cite_note-Stone_2004-6">[6]</a><a href="#cite_note-7">[7]</a>

Another common example is image <a href="Steganography" title="Steganography">steganography</a>, where ICA is used to embed one image within another. For instance, two grayscale images can be linearly combined to create mixed images in which the hidden content is visually imperceptible. ICA can then be used to recover the original source images from the mixtures. This technique underlies digital watermarking, which allows the embedding of ownership information into images, as well as more covert applications such as undetected information transmission. The method has even been linked to real-world cyberespionage cases. In such applications, ICA serves to unmix the data based on statistical independence, making it possible to extract hidden components that are not apparent in the observed data.

Steganographic techniques, including those potentially involving ICA-based analysis, have been used in real-world cyberespionage cases. In 2010, the FBI uncovered a Russian spy network known as the "Illegals Program" (Operation Ghost Stories), where agents used custom-built steganography tools to conceal encrypted text messages within image files shared online.<a href="#cite_note-8">[8]</a>

In another case, a former General Electric engineer, Xiaoqing Zheng, was convicted in 2022 for economic espionage. Zheng used steganography to exfiltrate sensitive turbine technology by embedding proprietary data within image files for transfer to entities in China.<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Defining_component_independence">Defining component independence</h2></div>

ICA finds the independent components (also called factors, latent variables or sources) by maximizing the statistical independence of the estimated components. We may choose one of many ways to define a proxy for independence, and this choice governs the form of the ICA algorithm. The two broadest definitions of independence for ICA are

<ol><li>Minimization of mutual information</li>

<li>Maximization of non-Gaussianity</li></ol>

The Minimization-of-<a href="Mutual_information" title="Mutual information">Mutual information</a> (MMI) family of ICA algorithms uses measures like <a href="Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback-Leibler Divergence</a> and <a href="Principle_of_maximum_entropy" title="Principle of maximum entropy">maximum entropy</a>. The non-Gaussianity family of ICA algorithms, motivated by the <a href="Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, uses <a href="Kurtosis" title="Kurtosis">kurtosis</a> and <a href="Negentropy" title="Negentropy">negentropy</a>.<a href="#cite_note-comon94-10">[10]</a>

Typical algorithms for ICA use centering (subtract the mean to create a zero mean signal), <a href="Whitening_transformation" title="Whitening transformation">whitening</a> (usually with the <a href="Eigenvalue_decomposition" class="mw-redirect" title="Eigenvalue decomposition">eigenvalue decomposition</a>),<a href="#cite_note-Springer-11">[11]</a> and <a href="Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a> as preprocessing steps in order to simplify and reduce the complexity of the problem for the actual iterative algorithm.

<div class="mw-heading mw-heading2"><h2 id="Mathematical_definitions">Mathematical definitions</h2></div>

Linear independent component analysis can be divided into noiseless and noisy cases, where noiseless ICA is a special case of noisy ICA. Nonlinear ICA should be considered as a separate case.

<div class="mw-heading mw-heading3"><h3 id="General_Derivation">General Derivation</h3></div>

In the classical ICA model, it is assumed that the observed data <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{i}\in \mathbb {R} ^{m}}">

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<div class="mw-heading mw-heading4"><h4 id="Reduced_Mixing_Problem">Reduced Mixing Problem</h4></div>

Independent component analysis (ICA) addresses the problem of recovering a set of unobserved source signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}=(s_{i1},s_{i2},\dots ,s_{im})^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and the source signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s_{i}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{s_{i}\}}</annotation>

</semantics>

</math><img src="./dd1ae018894d935d40725639ef12ae372e2b02cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.215ex; height:2.843ex;" alt="{\displaystyle \{s_{i}\}}" loading="lazy"> solely from the observed data <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{i}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x_{i}\}}</annotation>

</semantics>

</math><img src="./c9d90fd628e1497151325d23b808b6d0296e7701.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.454ex; height:2.843ex;" alt="{\displaystyle \{x_{i}\}}" loading="lazy">.

After centering, the Gram matrix is computed as:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X^{*})^{T}X^{*}=Q\,D\,Q^{T},}</annotation>

</semantics>

</math><img src="./ccf3dc2d740f0dc7a958f0b8c0e62ab2fda741e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.81ex; height:3.176ex;" alt="{\displaystyle (X^{*})^{T}X^{*}=Q\,D\,Q^{T},}" loading="lazy">

where D is a diagonal matrix with positive entries (assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation>

</semantics>

</math><img src="./01924e6e5570e2631081fea6c6981b4872d3e04b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}" loading="lazy"> has maximum rank), and Q is an orthogonal matrix.<a href="#cite_note-Springer-11">[11]</a>

Writing the SVD of the mixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=U\Sigma V^{T}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=U\Sigma V^{T}}</annotation>

</semantics>

</math><img src="./d28c591b48aa67947c5871f76b2bc9a691c3ca41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.608ex; height:2.676ex;" alt="{\displaystyle A=U\Sigma V^{T}}" loading="lazy"> and comparing with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AA^{T}=U\Sigma ^{2}U^{T}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AA^{T}=U\Sigma ^{2}U^{T}}</annotation>

</semantics>

</math><img src="./9b2ae669c72001d9f3e44c8ca5365111d638e5bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.719ex; height:2.676ex;" alt="{\displaystyle AA^{T}=U\Sigma ^{2}U^{T}}" loading="lazy"> the mixing A has the form

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=Q\,D^{1/2}\,V^{T}.}</annotation>

</semantics>

</math><img src="./1bb3151946c46413cf411dc7f7aece530f29c205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.03ex; height:3.176ex;" alt="{\displaystyle A=Q\,D^{1/2}\,V^{T}.}" loading="lazy">

So, the normalized source values satisfy

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{i}^{*}=V\,y_{i}^{*}}</annotation>

</semantics>

</math><img src="./f66d933b01e95e7e51c52db18f60aaa757d62686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.632ex; height:2.843ex;" alt="{\displaystyle s_{i}^{*}=V\,y_{i}^{*}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{*}=D^{-{\tfrac {1}{2}}}Q^{T}x_{i}^{*}.}">

</mrow>

</mrow>

</msubsup>

<msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}^{*}=D^{-{\tfrac {1}{2}}}Q^{T}x_{i}^{*}.}</annotation>

</semantics>

</math><img src="./99a7822f7b9aaed37344b1e21b288bb64ba06a0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.665ex; height:4.676ex;" alt="{\displaystyle y_{i}^{*}=D^{-{\tfrac {1}{2}}}Q^{T}x_{i}^{*}.}" loading="lazy">

Thus, ICA reduces to finding the orthogonal matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. This matrix can be computed using optimization techniques via projection pursuit methods (see <a href="#Projection_pursuit">Projection Pursuit</a>).<a href="#cite_note-Springer-11">[11]</a>

Well-known algorithms for ICA include <a href="Infomax" title="Infomax">infomax</a>, <a href="FastICA" title="FastICA">FastICA</a>, <a href="JADE_(ICA)" class="mw-redirect" title="JADE (ICA)">JADE</a>, and <a href="Kernel-independent_component_analysis" title="Kernel-independent component analysis">kernel-independent component analysis</a>, among others. In general, ICA cannot identify the actual number of source signals, a uniquely correct ordering of the source signals, nor the proper scaling (including sign) of the source signals.

ICA is important to <a href="Blind_signal_separation" class="mw-redirect" title="Blind signal separation">blind signal separation</a> and has many practical applications. It is closely related to (or even a special case of) the search for a <a href="Factorial_code" title="Factorial code">factorial code</a> of the data, i.e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector (loss-free coding), but the code components are statistically independent.

<div class="mw-heading mw-heading4"><h4 id="Linear_noiseless_ICA">Linear noiseless ICA</h4></div>

The components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> of the observed random vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}=(x_{1},\ldots ,x_{m})^{T}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}=(x_{1},\ldots ,x_{m})^{T}}</annotation>

</semantics>

</math><img src="./82d2672fb6c006196a357a27e9d81683a54c73b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.396ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {x}}=(x_{1},\ldots ,x_{m})^{T}}" loading="lazy"> are generated as a sum of the independent components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{k}}</annotation>

</semantics>

</math><img src="./04f159343172781e7666dbc88280c91f34117c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.009ex;" alt="{\displaystyle s_{k}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\ldots ,n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1,\ldots ,n}</annotation>

</semantics>

</math><img src="./02703686f808b37fedb436806fa72ca3522e22de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.045ex; height:2.509ex;" alt="{\displaystyle k=1,\ldots ,n}" loading="lazy">:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=a_{i,1}s_{1}+\cdots +a_{i,k}s_{k}+\cdots +a_{i,n}s_{n}}</annotation>

</semantics>

</math><img src="./da1bcb4ff12b6448d5ada6576250d14ef40bd25f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.794ex; height:2.676ex;" alt="{\displaystyle x_{i}=a_{i,1}s_{1}+\cdots +a_{i,k}s_{k}+\cdots +a_{i,n}s_{n}}" loading="lazy">

weighted by the mixing weights <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{i,k}}</annotation>

</semantics>

</math><img src="./513ba16f4bf7e8e3f6eda23db04a7e66ec9f5178.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.343ex; height:2.343ex;" alt="{\displaystyle a_{i,k}}" loading="lazy">.

The same generative model can be written in vector form as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}=\sum _{k=1}^{n}s_{k}{\boldsymbol {a}}_{k}}">

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}=\sum _{k=1}^{n}s_{k}{\boldsymbol {a}}_{k}}</annotation>

</semantics>

</math><img src="./65e8b26dbc93b91dc7082139a85d2b8730ecf7d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.112ex; height:6.843ex;" alt="{\displaystyle {\boldsymbol {x}}=\sum _{k=1}^{n}s_{k}{\boldsymbol {a}}_{k}}" loading="lazy">, where the observed random vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}}</annotation>

</semantics>

</math><img src="./606b7680d510560a505937143775ea80fa958051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.532ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {x}}}" loading="lazy"> is represented by the basis vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}_{k}=({\boldsymbol {a}}_{1,k},\ldots ,{\boldsymbol {a}}_{m,k})^{T}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {a}}_{k}=({\boldsymbol {a}}_{1,k},\ldots ,{\boldsymbol {a}}_{m,k})^{T}}</annotation>

</semantics>

</math><img src="./1382429d699572ef04d42ef03855615a8de5d920.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.335ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {a}}_{k}=({\boldsymbol {a}}_{1,k},\ldots ,{\boldsymbol {a}}_{m,k})^{T}}" loading="lazy">. The basis vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {a}}_{k}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {a}}_{k}}</annotation>

</semantics>

</math><img src="./493f447b52837d1493281459664451b3f22f4c49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.56ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {a}}_{k}}" loading="lazy"> form the columns of the mixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}=({\boldsymbol {a}}_{1},\ldots ,{\boldsymbol {a}}_{n})}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}=({\boldsymbol {a}}_{1},\ldots ,{\boldsymbol {a}}_{n})}</annotation>

</semantics>

</math><img src="./d559f2a79216b312d0f0261b6a4c78979a98e6ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.321ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {A}}=({\boldsymbol {a}}_{1},\ldots ,{\boldsymbol {a}}_{n})}" loading="lazy"> and the generative formula can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}={\boldsymbol {A}}{\boldsymbol {s}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}={\boldsymbol {A}}{\boldsymbol {s}}}</annotation>

</semantics>

</math><img src="./af63472217b2ffc65407450779634f04d5096741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.884ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {x}}={\boldsymbol {A}}{\boldsymbol {s}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {s}}=(s_{1},\ldots ,s_{n})^{T}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {s}}=(s_{1},\ldots ,s_{n})^{T}}</annotation>

</semantics>

</math><img src="./4fccf9a9ef34f453f6c51137a403e61abf2afabf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.163ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {s}}=(s_{1},\ldots ,s_{n})^{T}}" loading="lazy">.

Given the model and realizations (samples) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}_{1},\ldots ,{\boldsymbol {x}}_{N}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}_{1},\ldots ,{\boldsymbol {x}}_{N}}</annotation>

</semantics>

</math><img src="./2ebed21a154ca42d867219e0005dcfebfe0f7432.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.987ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {x}}_{1},\ldots ,{\boldsymbol {x}}_{N}}" loading="lazy"> of the random vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}}</annotation>

</semantics>

</math><img src="./606b7680d510560a505937143775ea80fa958051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.532ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {x}}}" loading="lazy">, the task is to estimate both the mixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}}</annotation>

</semantics>

</math><img src="./5a8b5a6d1dbadead8b1dc48719c888e6cac5f861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {A}}}" loading="lazy"> and the sources <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {s}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {s}}}</annotation>

</semantics>

</math><img src="./adfccdf93cc6115ade3deffbbbf062d6849a1f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.234ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {s}}}" loading="lazy">. This is done by adaptively calculating the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {w}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {w}}}</annotation>

</semantics>

</math><img src="./4a36660d03c65332b9ce8b29d7869818762cb970.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {w}}}" loading="lazy"> vectors and setting up a cost function which either maximizes the non-gaussianity of the calculated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}={\boldsymbol {w}}^{T}{\boldsymbol {x}}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{k}={\boldsymbol {w}}^{T}{\boldsymbol {x}}}</annotation>

</semantics>

</math><img src="./e9809acad1f457fd5f8ef9144c7b5d10d90228a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.13ex; height:3.009ex;" alt="{\displaystyle s_{k}={\boldsymbol {w}}^{T}{\boldsymbol {x}}}" loading="lazy"> or minimizes the mutual information. In some cases, a priori knowledge of the probability distributions of the sources can be used in the cost function.

The original sources <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {s}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {s}}}</annotation>

</semantics>

</math><img src="./adfccdf93cc6115ade3deffbbbf062d6849a1f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.234ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {s}}}" loading="lazy"> can be recovered by multiplying the observed signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}}</annotation>

</semantics>

</math><img src="./606b7680d510560a505937143775ea80fa958051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.532ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {x}}}" loading="lazy"> with the inverse of the mixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {W}}={\boldsymbol {A}}^{-1}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {W}}={\boldsymbol {A}}^{-1}}</annotation>

</semantics>

</math><img src="./db9aae3ed82c9d9f658996916fbfb4fb06c4d7f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.255ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {W}}={\boldsymbol {A}}^{-1}}" loading="lazy">, also known as the unmixing matrix. Here it is assumed that the mixing matrix is square (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=m}</annotation>

</semantics>

</math><img src="./480d6131c6cb07a90f4ec18a376a59fab884b860.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle n=m}" loading="lazy">). If the number of basis vectors is greater than the dimensionality of the observed vectors, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>m}</annotation>

</semantics>

</math><img src="./e64e2a4a5b6cb58f1553c6a65551b4898bb82403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle n>m}" loading="lazy">, the task is overcomplete but is still solvable with the <a href="Pseudo_inverse" class="mw-redirect" title="Pseudo inverse">pseudo inverse</a>.

<div class="mw-heading mw-heading4"><h4 id="Linear_noisy_ICA">Linear noisy ICA</h4></div>

With the added assumption of zero-mean and uncorrelated Gaussian noise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\sim N(0,\operatorname {diag} (\Sigma ))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\sim N(0,\operatorname {diag} (\Sigma ))}</annotation>

</semantics>

</math><img src="./87c2b17211062a305e1e98fe03a58eea92d83b7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.314ex; height:2.843ex;" alt="{\displaystyle n\sim N(0,\operatorname {diag} (\Sigma ))}" loading="lazy">, the ICA model takes the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}={\boldsymbol {A}}{\boldsymbol {s}}+n}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}={\boldsymbol {A}}{\boldsymbol {s}}+n}</annotation>

</semantics>

</math><img src="./001676401b15913ff86c3c896588ae9317631a95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.119ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {x}}={\boldsymbol {A}}{\boldsymbol {s}}+n}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Nonlinear_ICA">Nonlinear ICA</h4></div>

The mixing of the sources does not need to be linear. Using a nonlinear mixing function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\cdot |\theta )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\cdot |\theta )}</annotation>

</semantics>

</math><img src="./8a938be286acbb07e4e1ad3c65652ae2665240e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.843ex;" alt="{\displaystyle f(\cdot |\theta )}" loading="lazy"> with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> the nonlinear ICA model is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=f(s|\theta )+n}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=f(s|\theta )+n}</annotation>

</semantics>

</math><img src="./61d35119cdcc37443e4f7899017f60e773dc9ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.579ex; height:2.843ex;" alt="{\displaystyle x=f(s|\theta )+n}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Identifiability">Identifiability</h3></div>

The independent components are identifiable up to a permutation and scaling of the sources.<a href="#cite_note-13">[13]</a> This identifiability requires that:

<ul><li>At most one of the sources <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{k}}</annotation>

</semantics>

</math><img src="./04f159343172781e7666dbc88280c91f34117c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.009ex;" alt="{\displaystyle s_{k}}" loading="lazy"> is Gaussian,</li>

<li>The number of observed mixtures, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">, must be at least as large as the number of estimated components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\geq n}</annotation>

</semantics>

</math><img src="./6b02f25e62da7fe3162ac80446437cdc1c0fd341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle m\geq n}" loading="lazy">. It is equivalent to say that the mixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}}</annotation>

</semantics>

</math><img src="./5a8b5a6d1dbadead8b1dc48719c888e6cac5f861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {A}}}" loading="lazy"> must be of full <a href="Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> for its inverse to exist.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Binary_ICA">Binary ICA</h2></div>

A special variant of ICA is binary ICA in which both signal sources and monitors are in binary form and observations from monitors are disjunctive mixtures of binary independent sources. The problem was shown to have applications in many domains including <a href="Medical_diagnosis" title="Medical diagnosis">medical diagnosis</a>, multi-cluster assignment, <a href="Network_tomography" title="Network tomography">network tomography</a> and internet resource management.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x_{1},x_{2},\ldots ,x_{m}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {x_{1},x_{2},\ldots ,x_{m}}}</annotation>

</semantics>

</math><img src="./b9c8f2c13280e7895c57c5f8fb14b5a422d0b361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.985ex; height:2.009ex;" alt="{\displaystyle {x_{1},x_{2},\ldots ,x_{m}}}" loading="lazy"> be the set of binary variables from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> monitors and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {y_{1},y_{2},\ldots ,y_{n}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {y_{1},y_{2},\ldots ,y_{n}}}</annotation>

</semantics>

</math><img src="./beb52c75a22b40004dbb949e9ea7f5b0379ae4c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.957ex; height:2.009ex;" alt="{\displaystyle {y_{1},y_{2},\ldots ,y_{n}}}" loading="lazy"> be the set of binary variables from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> sources. Source-monitor connections are represented by the (unknown) mixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {G}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\boldsymbol {G}}}</annotation>

</semantics>

</math><img src="./54f980b2b704f566adb9507dfa02ba5c89e9accb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.061ex; height:2.176ex;" alt="{\textstyle {\boldsymbol {G}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{ij}=1}</annotation>

</semantics>

</math><img src="./718fe03d6b108f20dd898db3779c066a02a47cf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.847ex; height:2.843ex;" alt="{\displaystyle g_{ij}=1}" loading="lazy"> indicates that signal from the i-th source can be observed by the j-th monitor. The system works as follows: at any time, if a source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> is active (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}=1}</annotation>

</semantics>

</math><img src="./13864bfc60110b9306a50b300d2d72b22fa1e371.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.2ex; height:2.509ex;" alt="{\displaystyle y_{i}=1}" loading="lazy">) and it is connected to the monitor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{ij}=1}</annotation>

</semantics>

</math><img src="./718fe03d6b108f20dd898db3779c066a02a47cf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.847ex; height:2.843ex;" alt="{\displaystyle g_{ij}=1}" loading="lazy">) then the monitor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> will observe some activity (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j}=1}</annotation>

</semantics>

</math><img src="./95c0a59667c986ecc2a3f6a84f387699c2f25b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.5ex; height:2.843ex;" alt="{\displaystyle x_{j}=1}" loading="lazy">). Formally we have:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=\bigvee _{j=1}^{n}(g_{ij}\wedge y_{j}),i=1,2,\ldots ,m,}</annotation>

</semantics>

</math><img src="./644b8d4d5dddfefc5fd3f4d5d06453006e34f053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.192ex; height:7.176ex;" alt="{\displaystyle x_{i}=\bigvee _{j=1}^{n}(g_{ij}\wedge y_{j}),i=1,2,\ldots ,m,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation>

</semantics>

</math><img src="./1caa4004cb216ef2930bb12fe805a76870caed94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }" loading="lazy"> is Boolean AND and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vee }</annotation>

</semantics>

</math><img src="./7b76220c6805c9b465d6efbc7686c624f49f3023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }" loading="lazy"> is Boolean OR. Noise is not explicitly modelled, rather, can be treated as independent sources.

The above problem can be heuristically solved<a href="#cite_note-Hyvärinen-14">[14]</a> by assuming variables are continuous and running <a href="FastICA" title="FastICA">FastICA</a> on binary observation data to get the mixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {G}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\boldsymbol {G}}}</annotation>

</semantics>

</math><img src="./54f980b2b704f566adb9507dfa02ba5c89e9accb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.061ex; height:2.176ex;" alt="{\textstyle {\boldsymbol {G}}}" loading="lazy"> (real values), then apply <a href="Round_number" title="Round number">round number</a> techniques on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {G}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\boldsymbol {G}}}</annotation>

</semantics>

</math><img src="./54f980b2b704f566adb9507dfa02ba5c89e9accb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.061ex; height:2.176ex;" alt="{\textstyle {\boldsymbol {G}}}" loading="lazy"> to obtain the binary values. This approach has been shown to produce a highly inaccurate result.

Another method is to use <a href="Dynamic_programming" title="Dynamic programming">dynamic programming</a>: recursively breaking the observation matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\boldsymbol {X}}}</annotation>

</semantics>

</math><img src="./260a0485ebd57e6c6f08ff860bffc10b52ab39a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.215ex; height:2.176ex;" alt="{\textstyle {\boldsymbol {X}}}" loading="lazy"> into its sub-matrices and run the inference algorithm on these sub-matrices. The key observation which leads to this algorithm is the sub-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {X}}^{0}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\boldsymbol {X}}^{0}}</annotation>

</semantics>

</math><img src="./ed4d42aa1e36224e82e14391c516d8ef9d2ec13d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.273ex; height:2.676ex;" alt="{\textstyle {\boldsymbol {X}}^{0}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\boldsymbol {X}}}</annotation>

</semantics>

</math><img src="./260a0485ebd57e6c6f08ff860bffc10b52ab39a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.215ex; height:2.176ex;" alt="{\textstyle {\boldsymbol {X}}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{ij}=0,\forall j}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x_{ij}=0,\forall j}</annotation>

</semantics>

</math><img src="./df46c054cf1a310f83b0925705a5c7ab370d4c8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.352ex; height:2.843ex;" alt="{\textstyle x_{ij}=0,\forall j}" loading="lazy"> corresponds to the unbiased observation matrix of hidden components that do not have connection to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">-th monitor. Experimental results from<a href="#cite_note-Huyna-15">[15]</a> show that this approach is accurate under moderate noise levels.

The Generalized Binary ICA framework<a href="#cite_note-Generalized_Binary_ICA-16">[16]</a> introduces a broader problem formulation which does not necessitate any knowledge on the generative model. In other words, this method attempts to decompose a source into its independent components (as much as possible, and without losing any information) with no prior assumption on the way it was generated. Although this problem appears quite complex, it can be accurately solved with a <a href="Branch_and_bound" title="Branch and bound">branch and bound</a> search tree algorithm or tightly upper bounded with a single multiplication of a matrix with a vector.

<div class="mw-heading mw-heading2"><h2 id="Methods_for_blind_source_separation">Methods for blind source separation</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Projection_pursuit">Projection pursuit</h3></div>

Signal mixtures tend to have Gaussian probability density functions, and source signals tend to have non-Gaussian probability density functions. Each source signal can be extracted from a set of signal mixtures by taking the inner product of a weight vector and those signal mixtures where this inner product provides an orthogonal projection of the signal mixtures. The remaining challenge is finding such a weight vector. One type of method for doing so is <a href="Projection_pursuit" title="Projection pursuit">projection pursuit</a>.<a href="#cite_note-James_V._Stone_2004-17">[17]</a><a href="#cite_note-18">[18]</a>

Projection pursuit seeks one projection at a time such that the extracted signal is as non-Gaussian as possible. This contrasts with ICA, which typically extracts M signals simultaneously from M signal mixtures, which requires estimating a M × M unmixing matrix. One practical advantage of projection pursuit over ICA is that fewer than M signals can be extracted if required, where each source signal is extracted from M signal mixtures using an M-element weight vector.

We can use <a href="Kurtosis" title="Kurtosis">kurtosis</a> to recover the multiple source signal by finding the correct weight vectors with the use of projection pursuit.

The kurtosis of the probability density function of a signal, for a finite sample, is computed as

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K={\frac {\operatorname {E} [(\mathbf {y} -\mathbf {\overline {y}} )^{4}]}{(\operatorname {E} [(\mathbf {y} -\mathbf {\overline {y}} )^{2}])^{2}}}-3}</annotation>

</semantics>

</math><img src="./f2ed7961602e0ea393c9890f132b5352d4a0cc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.384ex; height:6.676ex;" alt="{\displaystyle K={\frac {\operatorname {E} [(\mathbf {y} -\mathbf {\overline {y}} )^{4}]}{(\operatorname {E} [(\mathbf {y} -\mathbf {\overline {y}} )^{2}])^{2}}}-3}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\overline {y}} }">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\overline {y}} }</annotation>

</semantics>

</math><img src="./fc3cb6fe9e66d5391258f1ebb7f8e76c2d024b06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.526ex; height:2.676ex;" alt="{\displaystyle \mathbf {\overline {y}} }" loading="lazy"> is the <a href="Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation>

</semantics>

</math><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy">, the extracted signals. The constant 3 ensures that Gaussian signals have zero kurtosis, Super-Gaussian signals have positive kurtosis, and Sub-Gaussian signals have negative kurtosis. The denominator is the <a href="Variance" title="Variance">variance</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation>

</semantics>

</math><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy">, and ensures that the measured kurtosis takes account of signal variance. The goal of projection pursuit is to maximize the kurtosis, and make the extracted signal as non-normal as possible.

Using kurtosis as a measure of non-normality, we can now examine how the kurtosis of a signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} =\mathbf {w} ^{T}\mathbf {x} }">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =\mathbf {w} ^{T}\mathbf {x} }</annotation>

</semantics>

</math><img src="./02ba081bc727b4ebff478ad1e55c3404033fa55c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.241ex; height:3.009ex;" alt="{\displaystyle \mathbf {y} =\mathbf {w} ^{T}\mathbf {x} }" loading="lazy"> extracted from a set of M mixtures <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{M})^{T}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{M})^{T}}</annotation>

</semantics>

</math><img src="./ce0218e7a85c3af6f3e80424cc74c2818aebbf70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.977ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{M})^{T}}" loading="lazy"> varies as the weight vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation>

</semantics>

</math><img src="./20795664b5b048744a2fd88977851104cc5816f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }" loading="lazy"> is rotated around the origin. Given our assumption that each source signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation>

</semantics>

</math><img src="./644ae690160e658898a141e568a7fb0ee6040004.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }" loading="lazy"> is super-gaussian we would expect:

<ol><li>the kurtosis of the extracted signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation>

</semantics>

</math><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"> to be maximal precisely when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} =\mathbf {s} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =\mathbf {s} }</annotation>

</semantics>

</math><img src="./7bf4841987de687c9e248a8b26043b124699e233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.565ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} =\mathbf {s} }" loading="lazy">.</li>

<li>the kurtosis of the extracted signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation>

</semantics>

</math><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"> to be maximal when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation>

</semantics>

</math><img src="./20795664b5b048744a2fd88977851104cc5816f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }" loading="lazy"> is orthogonal to the projected axes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation>

</semantics>

</math><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation>

</semantics>

</math><img src="./1143e284d5f25cef778ab482edf6617a523ddd9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}" loading="lazy">, because we know the optimal weight vector should be orthogonal to a transformed axis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation>

</semantics>

</math><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation>

</semantics>

</math><img src="./1143e284d5f25cef778ab482edf6617a523ddd9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}" loading="lazy">.</li></ol>

For multiple source mixture signals, we can use kurtosis and <a href="Gram-Schmidt" class="mw-redirect" title="Gram-Schmidt">Gram-Schmidt</a> Orthogonalization (GSO) to recover the signals. Given M signal mixtures in an M-dimensional space, GSO project these data points onto an (M-1)-dimensional space by using the weight vector. We can guarantee the independence of the extracted signals with the use of GSO.

In order to find the correct value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation>

</semantics>

</math><img src="./20795664b5b048744a2fd88977851104cc5816f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }" loading="lazy">, we can use <a href="Gradient_descent" title="Gradient descent">gradient descent</a> method. We first of all whiten the data, and transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"> into a new mixture <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} }</annotation>

</semantics>

</math><img src="./82eca5d0928078d5a61b9e7e98cc73db31070909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {z} }" loading="lazy">, which has unit variance, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} =(z_{1},z_{2},\ldots ,z_{M})^{T}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} =(z_{1},z_{2},\ldots ,z_{M})^{T}}</annotation>

</semantics>

</math><img src="./e38481100377d0a6d62f23b8d65ab6f7be39c85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.008ex; height:3.176ex;" alt="{\displaystyle \mathbf {z} =(z_{1},z_{2},\ldots ,z_{M})^{T}}" loading="lazy">. This process can be achieved by applying <a href="Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy">,

</mrow>

</mrow>

</mrow>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\mathbf {U} \mathbf {D} \mathbf {V} ^{T}}</annotation>

</semantics>

</math><img src="./1db808f95e6c706ccb58d3415fbe62d752d1635e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.025ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} =\mathbf {U} \mathbf {D} \mathbf {V} ^{T}}" loading="lazy"></dd></dl>

Rescaling each vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}=U_{i}/\operatorname {E} (U_{i}^{2})}">

<msub>

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<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{i}=U_{i}/\operatorname {E} (U_{i}^{2})}</annotation>

</semantics>

</math><img src="./827944f22626dac20754c706dcd8eb0004953e35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.71ex; height:3.176ex;" alt="{\displaystyle U_{i}=U_{i}/\operatorname {E} (U_{i}^{2})}" loading="lazy">, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} =\mathbf {U} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} =\mathbf {U} }</annotation>

</semantics>

</math><img src="./90deb78e225dd7d132aa98a212c5ffd3fc0fae3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.343ex; height:2.176ex;" alt="{\displaystyle \mathbf {z} =\mathbf {U} }" loading="lazy">. The signal extracted by a weighted vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation>

</semantics>

</math><img src="./20795664b5b048744a2fd88977851104cc5816f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} =\mathbf {w} ^{T}\mathbf {z} }">

</mrow>

<msup>

</mrow>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =\mathbf {w} ^{T}\mathbf {z} }</annotation>

</semantics>

</math><img src="./347922e4ea7a3b539ff7594fd1765cb76000d811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.018ex; height:3.009ex;" alt="{\displaystyle \mathbf {y} =\mathbf {w} ^{T}\mathbf {z} }" loading="lazy">. If the weight vector w has unit length, then the variance of y is also 1, that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [(\mathbf {w} ^{T}\mathbf {z} )^{2}]=1}">

<msup>

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</msup>

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<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [(\mathbf {w} ^{T}\mathbf {z} )^{2}]=1}</annotation>

</semantics>

</math><img src="./9f29c4655db008f7c5eaa5ead02b3e2405c0a8df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.509ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [(\mathbf {w} ^{T}\mathbf {z} )^{2}]=1}" loading="lazy">. The kurtosis can thus be written as:

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

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</mfrac>

</mrow>

<msup>

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</mrow>

</msup>

</mrow>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K={\frac {\operatorname {E} [\mathbf {y} ^{4}]}{(\operatorname {E} [\mathbf {y} ^{2}])^{2}}}-3=\operatorname {E} [(\mathbf {w} ^{T}\mathbf {z} )^{4}]-3.}</annotation>

</semantics>

</math><img src="./28060dbc6e2d065bcf18f3b40c8a6bf2572df31f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.205ex; height:6.676ex;" alt="{\displaystyle K={\frac {\operatorname {E} [\mathbf {y} ^{4}]}{(\operatorname {E} [\mathbf {y} ^{2}])^{2}}}-3=\operatorname {E} [(\mathbf {w} ^{T}\mathbf {z} )^{4}]-3.}" loading="lazy"></dd></dl>

The updating process for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation>

</semantics>

</math><img src="./20795664b5b048744a2fd88977851104cc5816f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }" loading="lazy"> is:

<msub>

</mrow>

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<msub>

</mrow>

</mrow>

</msub>

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</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{new}=\mathbf {w} _{old}-\eta \operatorname {E} [\mathbf {z} (\mathbf {w} _{old}^{T}\mathbf {z} )^{3}].}</annotation>

</semantics>

</math><img src="./430c12e076a4db6617c2323ff75901ad19b805ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.973ex; height:3.343ex;" alt="{\displaystyle \mathbf {w} _{new}=\mathbf {w} _{old}-\eta \operatorname {E} [\mathbf {z} (\mathbf {w} _{old}^{T}\mathbf {z} )^{3}].}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> is a small constant to guarantee that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation>

</semantics>

</math><img src="./20795664b5b048744a2fd88977851104cc5816f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }" loading="lazy"> converges to the optimal solution. After each update, we normalize <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{new}={\frac {\mathbf {w} _{new}}{|\mathbf {w} _{new}|}}}">

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{new}={\frac {\mathbf {w} _{new}}{|\mathbf {w} _{new}|}}}</annotation>

</semantics>

</math><img src="./dbd808658c2e2e59e44ff43c8eee1c1aec441a90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.413ex; height:5.509ex;" alt="{\displaystyle \mathbf {w} _{new}={\frac {\mathbf {w} _{new}}{|\mathbf {w} _{new}|}}}" loading="lazy">, and set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} _{old}=\mathbf {w} _{new}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{old}=\mathbf {w} _{new}}</annotation>

</semantics>

</math><img src="./4ca433df12ffa468d7daac0c1c02c1f23d096b45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.502ex; height:2.009ex;" alt="{\displaystyle \mathbf {w} _{old}=\mathbf {w} _{new}}" loading="lazy">, and repeat the updating process until convergence. We can also use another algorithm to update the weight vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation>

</semantics>

</math><img src="./20795664b5b048744a2fd88977851104cc5816f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }" loading="lazy">.

Another approach is using <a href="Negentropy" title="Negentropy">negentropy</a><a href="#cite_note-comon94-10">[10]</a><a href="#cite_note-19">[19]</a> instead of kurtosis. Using negentropy is a more robust method than kurtosis, as kurtosis is very sensitive to outliers. The negentropy methods are based on an important property of Gaussian distribution: a Gaussian variable has the largest entropy among all continuous random variables of equal variance. This is also the reason why we want to find the most nongaussian variables. A simple proof can be found in <a href="Differential_entropy" title="Differential entropy">Differential entropy</a>.

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<annotation encoding="application/x-tex">{\displaystyle J(x)=S(y)-S(x)\,}</annotation>

</semantics>

</math><img src="./f9cc71286f9d45eff9eb1619ad961f458fcd6231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.039ex; height:2.843ex;" alt="{\displaystyle J(x)=S(y)-S(x)\,}" loading="lazy"></dd></dl>

y is a Gaussian random variable of the same covariance matrix as x

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<annotation encoding="application/x-tex">{\displaystyle S(x)=-\int p_{x}(u)\log p_{x}(u)du}</annotation>

</semantics>

</math><img src="./6a20a9f59da8c75cff3a1e25e7949ae46a34bb8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.766ex; height:5.676ex;" alt="{\displaystyle S(x)=-\int p_{x}(u)\log p_{x}(u)du}" loading="lazy"></dd></dl>

An approximation for negentropy is

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<annotation encoding="application/x-tex">{\displaystyle J(x)={\frac {1}{12}}(E(x^{3}))^{2}+{\frac {1}{48}}(kurt(x))^{2}}</annotation>

</semantics>

</math><img src="./a6a101bb025c20a541da26bf76ec155322a5cfa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.135ex; height:5.343ex;" alt="{\displaystyle J(x)={\frac {1}{12}}(E(x^{3}))^{2}+{\frac {1}{48}}(kurt(x))^{2}}" loading="lazy"></dd></dl>

A proof can be found in the original papers of Comon;<a href="#cite_note-pc91-20">[20]</a><a href="#cite_note-comon94-10">[10]</a> it has been reproduced in the book Independent Component Analysis by Aapo Hyvärinen, Juha Karhunen, and <a href="Erkki_Oja" title="Erkki Oja">Erkki Oja</a><a href="#cite_note-21">[21]</a> This approximation also suffers from the same problem as kurtosis (sensitivity to outliers). Other approaches have been developed.<a href="#cite_note-22">[22]</a>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle J(y)=k_{1}(E(G_{1}(y)))^{2}+k_{2}(E(G_{2}(y))-E(G_{2}(v))^{2}}</annotation>

</semantics>

</math><img src="./667499f601ac3c6166bf9f5a506c3960c93f8d6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.833ex; height:3.176ex;" alt="{\displaystyle J(y)=k_{1}(E(G_{1}(y)))^{2}+k_{2}(E(G_{2}(y))-E(G_{2}(v))^{2}}" loading="lazy"></dd></dl>

A choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle G_{1}}</annotation>

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</math><img src="./e6ea4f4668b8334c8a7d3d284b0fd22131ef5f52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"> are

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<annotation encoding="application/x-tex">{\displaystyle G_{1}={\frac {1}{a_{1}}}\log(\cosh(a_{1}u))}</annotation>

</semantics>

</math><img src="./97ce65cb6fa16d0746f42e92024debdd13bd31f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.094ex; height:5.509ex;" alt="{\displaystyle G_{1}={\frac {1}{a_{1}}}\log(\cosh(a_{1}u))}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}=-\exp(-{\frac {u^{2}}{2}})}">

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle G_{2}=-\exp(-{\frac {u^{2}}{2}})}</annotation>

</semantics>

</math><img src="./b97b221a6d712bd17df16dd970bb9dd3c4a22de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.565ex; height:5.676ex;" alt="{\displaystyle G_{2}=-\exp(-{\frac {u^{2}}{2}})}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Based_on_infomax">Based on infomax</h3></div>

Infomax ICA<a href="#cite_note-Bell-Sejnowski-23">[23]</a> is essentially a multivariate, parallel version of projection pursuit. Whereas projection pursuit extracts a series of signals one at a time from a set of M signal mixtures, ICA extracts M signals in parallel. This tends to make ICA more robust than projection pursuit.<a href="#cite_note-ReferenceA-24">[24]</a>

The projection pursuit method uses <a href="Gram-Schmidt" class="mw-redirect" title="Gram-Schmidt">Gram-Schmidt</a> orthogonalization to ensure the independence of the extracted signal, while ICA use <a href="Infomax" title="Infomax">infomax</a> and <a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimate to ensure the independence of the extracted signal. The Non-Normality of the extracted signal is achieved by assigning an appropriate model, or prior, for the signal.

The process of ICA based on <a href="Infomax" title="Infomax">infomax</a> in short is: given a set of signal mixtures <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"> and a set of identical independent model <a href="Cumulative_distribution_functions" class="mw-redirect" title="Cumulative distribution functions">cumulative distribution functions</a>(cdfs) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">, we seek the unmixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy"> which maximizes the joint <a href="Entropy" title="Entropy">entropy</a> of the signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Y} =g(\mathbf {y} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} =g(\mathbf {y} )}</annotation>

</semantics>

</math><img src="./c1010d76e2b7090cb682027e2307997ce6fc8fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.454ex; height:2.843ex;" alt="{\displaystyle \mathbf {Y} =g(\mathbf {y} )}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} =\mathbf {Wx} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =\mathbf {Wx} }</annotation>

</semantics>

</math><img src="./8c06d543d236c28ba1fda9a651e98b307ed69a3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.683ex; height:2.509ex;" alt="{\displaystyle \mathbf {y} =\mathbf {Wx} }" loading="lazy"> are the signals extracted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy">. Given the optimal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy">, the signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>

</semantics>

</math><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"> have maximum entropy and are therefore independent, which ensures that the extracted signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} =g^{-1}(\mathbf {Y} )}">

</mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =g^{-1}(\mathbf {Y} )}</annotation>

</semantics>

</math><img src="./d6f8572c8c7ea66c571b633986a8f7bc9af8ae4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.789ex; height:3.176ex;" alt="{\displaystyle \mathbf {y} =g^{-1}(\mathbf {Y} )}" loading="lazy"> are also independent. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> is an invertible function, and is the signal model. Note that if the source signal model <a href="Probability_density_function" title="Probability density function">probability density function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation>

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</math><img src="./0a8201eb39454219ddec719cdb8b3ffd2b3be2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}" loading="lazy"> matches the <a href="Probability_density_function" title="Probability density function">probability density function</a> of the extracted signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {y} }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {y} }}</annotation>

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</math><img src="./49187b03d71eef32ad351b464fda39afc8e7f62d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.489ex; height:2.343ex;" alt="{\displaystyle p_{\mathbf {y} }}" loading="lazy">, then maximizing the joint entropy of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> also maximizes the amount of <a href="Mutual_information" title="Mutual information">mutual information</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

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</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Y} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>

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</math><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy">. For this reason, using entropy to extract independent signals is known as <a href="Infomax" title="Infomax">infomax</a>.

Consider the entropy of the vector variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Y} =g(\mathbf {y} )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} =g(\mathbf {y} )}</annotation>

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</math><img src="./c1010d76e2b7090cb682027e2307997ce6fc8fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.454ex; height:2.843ex;" alt="{\displaystyle \mathbf {Y} =g(\mathbf {y} )}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} =\mathbf {Wx} }">

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =\mathbf {Wx} }</annotation>

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</math><img src="./8c06d543d236c28ba1fda9a651e98b307ed69a3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.683ex; height:2.509ex;" alt="{\displaystyle \mathbf {y} =\mathbf {Wx} }" loading="lazy"> is the set of signals extracted by the unmixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

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</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy">. For a finite set of values sampled from a distribution with pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {y} }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {y} }}</annotation>

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</math><img src="./49187b03d71eef32ad351b464fda39afc8e7f62d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.489ex; height:2.343ex;" alt="{\displaystyle p_{\mathbf {y} }}" loading="lazy">, the entropy of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Y} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Y} }</annotation>

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</math><img src="./c92a7716a99fadda050469747fce1e475e0ec549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {Y} }" loading="lazy"> can be estimated as:

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<mfrac>

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</munderover>

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</msub>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle H(\mathbf {Y} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {Y} }(\mathbf {Y} ^{t})}</annotation>

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</math><img src="./483ff847396226d804870a32573d1272d965f01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.639ex; height:7.343ex;" alt="{\displaystyle H(\mathbf {Y} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {Y} }(\mathbf {Y} ^{t})}" loading="lazy"></dd></dl>

The joint pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {Y} }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {Y} }}</annotation>

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</math><img src="./e5dd335e5d2d3135b0039024906af24ff1882b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.919ex; height:2.009ex;" alt="{\displaystyle p_{\mathbf {Y} }}" loading="lazy"> can be shown to be related to the joint pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {y} }}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {y} }}</annotation>

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</math><img src="./49187b03d71eef32ad351b464fda39afc8e7f62d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.489ex; height:2.343ex;" alt="{\displaystyle p_{\mathbf {y} }}" loading="lazy"> of the extracted signals by the multivariate form:

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</msub>

<mfrac>

<mrow>

<msub>

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</msub>

</mrow>

</mrow>

<mrow>

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<mfrac>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {Y} }(Y)={\frac {p_{\mathbf {y} }(\mathbf {y} )}{|{\frac {\partial \mathbf {Y} }{\partial \mathbf {y} }}|}}}</annotation>

</semantics>

</math><img src="./03875faadce148117edb4bb261a9d1274c3348da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:16.056ex; height:7.843ex;" alt="{\displaystyle p_{\mathbf {Y} }(Y)={\frac {p_{\mathbf {y} }(\mathbf {y} )}{|{\frac {\partial \mathbf {Y} }{\partial \mathbf {y} }}|}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} ={\frac {\partial \mathbf {Y} }{\partial \mathbf {y} }}}">

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<mfrac>

<mrow>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\frac {\partial \mathbf {Y} }{\partial \mathbf {y} }}}</annotation>

</semantics>

</math><img src="./e83472d62fd5e856a6415ca413be04e966246603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.653ex; height:6.009ex;" alt="{\displaystyle \mathbf {J} ={\frac {\partial \mathbf {Y} }{\partial \mathbf {y} }}}" loading="lazy"> is the <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>. We have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {J} |=g'(\mathbf {y} )}">

</mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\mathbf {J} |=g'(\mathbf {y} )}</annotation>

</semantics>

</math><img src="./846fd72fac82a216656cb3a642afe3932e6c7d26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.796ex; height:3.009ex;" alt="{\displaystyle |\mathbf {J} |=g'(\mathbf {y} )}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g'}</annotation>

</semantics>

</math><img src="./e7a53c0df5d85b36e3fd327c74db998f679f4f55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.803ex; height:2.843ex;" alt="{\displaystyle g'}" loading="lazy"> is the pdf assumed for source signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g'=p_{s}}">

<msup>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g'=p_{s}}</annotation>

</semantics>

</math><img src="./5eb54379ce533d4e4234a6c274bf8579e7eaaf1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.074ex; height:2.843ex;" alt="{\displaystyle g'=p_{s}}" loading="lazy">, therefore,

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {Y} }(Y)={\frac {p_{\mathbf {y} }(\mathbf {y} )}{|{\frac {\partial \mathbf {Y} }{\partial \mathbf {y} }}|}}={\frac {p_{\mathbf {y} }(\mathbf {y} )}{p_{\mathbf {s} }(\mathbf {y} )}}}</annotation>

</semantics>

</math><img src="./59e418f7a33b3b69e25568e7fe4f1737691a50af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:25.61ex; height:7.843ex;" alt="{\displaystyle p_{\mathbf {Y} }(Y)={\frac {p_{\mathbf {y} }(\mathbf {y} )}{|{\frac {\partial \mathbf {Y} }{\partial \mathbf {y} }}|}}={\frac {p_{\mathbf {y} }(\mathbf {y} )}{p_{\mathbf {s} }(\mathbf {y} )}}}" loading="lazy"></dd></dl>

therefore,

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(\mathbf {Y} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln {\frac {p_{\mathbf {y} }(\mathbf {y} )}{p_{\mathbf {s} }(\mathbf {y} )}}}</annotation>

</semantics>

</math><img src="./f4d1ff33a631c247fee6ce3f4f868a56a65447be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.61ex; height:7.343ex;" alt="{\displaystyle H(\mathbf {Y} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln {\frac {p_{\mathbf {y} }(\mathbf {y} )}{p_{\mathbf {s} }(\mathbf {y} )}}}" loading="lazy"></dd></dl>

We know that when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {y} }=p_{s}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {y} }=p_{s}}</annotation>

</semantics>

</math><img src="./3a87db1dbed603526af8176cef39841ade8a1d4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:7.76ex; height:2.343ex;" alt="{\displaystyle p_{\mathbf {y} }=p_{s}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {Y} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {Y} }}</annotation>

</semantics>

</math><img src="./e5dd335e5d2d3135b0039024906af24ff1882b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.919ex; height:2.009ex;" alt="{\displaystyle p_{\mathbf {Y} }}" loading="lazy"> is of uniform distribution, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H({\mathbf {Y} })}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H({\mathbf {Y} })}</annotation>

</semantics>

</math><img src="./003bd89c05fe56091ef15cfb1bd5aa898e0956bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.892ex; height:2.843ex;" alt="{\displaystyle H({\mathbf {Y} })}" loading="lazy"> is maximized. Since

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {y} }(\mathbf {y} )={\frac {p_{\mathbf {x} }(\mathbf {x} )}{|{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}|}}={\frac {p_{\mathbf {x} }(\mathbf {x} )}{|\mathbf {W} |}}}</annotation>

</semantics>

</math><img src="./4d4f19473b1bd95ae14b67400ab1a53a7ba33ce8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:24.818ex; height:7.843ex;" alt="{\displaystyle p_{\mathbf {y} }(\mathbf {y} )={\frac {p_{\mathbf {x} }(\mathbf {x} )}{|{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}|}}={\frac {p_{\mathbf {x} }(\mathbf {x} )}{|\mathbf {W} |}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {W} |}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\mathbf {W} |}</annotation>

</semantics>

</math><img src="./d147816a89807397bce55f38c14a49878535b574.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.056ex; height:2.843ex;" alt="{\displaystyle |\mathbf {W} |}" loading="lazy"> is the absolute value of the determinant of the unmixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy">. Therefore,

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(\mathbf {Y} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln {\frac {p_{\mathbf {x} }(\mathbf {x} ^{t})}{|\mathbf {W} |p_{\mathbf {s} }(\mathbf {y} ^{t})}}}</annotation>

</semantics>

</math><img src="./65082630e288aaa774ef20d16b98594fc499a23d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.241ex; height:7.343ex;" alt="{\displaystyle H(\mathbf {Y} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln {\frac {p_{\mathbf {x} }(\mathbf {x} ^{t})}{|\mathbf {W} |p_{\mathbf {s} }(\mathbf {y} ^{t})}}}" loading="lazy"></dd></dl>

so,

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(\mathbf {Y} )={\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {s} }(\mathbf {y} ^{t})+\ln |\mathbf {W} |+H(\mathbf {x} )}</annotation>

</semantics>

</math><img src="./c1d46622a526d1de2236490ddae6993c9062c5e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.888ex; height:7.343ex;" alt="{\displaystyle H(\mathbf {Y} )={\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {s} }(\mathbf {y} ^{t})+\ln |\mathbf {W} |+H(\mathbf {x} )}" loading="lazy"></dd></dl>

since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\mathbf {x} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {x} }(\mathbf {x} ^{t})}">

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(\mathbf {x} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {x} }(\mathbf {x} ^{t})}</annotation>

</semantics>

</math><img src="./f1b683f5777d9bd53bc93653f2f3cdad164501e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.991ex; height:7.343ex;" alt="{\displaystyle H(\mathbf {x} )=-{\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {x} }(\mathbf {x} ^{t})}" loading="lazy">, and maximizing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy"> does not affect <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathbf {x} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\mathbf {x} }}</annotation>

</semantics>

</math><img src="./49439f720f3917a3654449f6c679abb79021afaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.161ex; height:2.509ex;" alt="{\displaystyle H_{\mathbf {x} }}" loading="lazy">, so we can maximize the function

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(\mathbf {Y} )={\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {s} }(\mathbf {y} ^{t})+\ln |\mathbf {W} |}</annotation>

</semantics>

</math><img src="./b37f823ab06cf6ca693f9c0cca6265ed7da90e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.039ex; height:7.343ex;" alt="{\displaystyle h(\mathbf {Y} )={\frac {1}{N}}\sum _{t=1}^{N}\ln p_{\mathbf {s} }(\mathbf {y} ^{t})+\ln |\mathbf {W} |}" loading="lazy"></dd></dl>

to achieve the independence of the extracted signal.

If there are M marginal pdfs of the model joint pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {s} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {s} }}</annotation>

</semantics>

</math><img src="./f7e243b5acdacb573b96ae9c1aadfd58ee74ace0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.238ex; height:2.009ex;" alt="{\displaystyle p_{\mathbf {s} }}" loading="lazy"> are independent and use the commonly super-gaussian model pdf for the source signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {s} }=(1-\tanh(\mathbf {s} )^{2})}">

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {s} }=(1-\tanh(\mathbf {s} )^{2})}</annotation>

</semantics>

</math><img src="./6c5e0ac87eddfb410ae5a3a1853072ea48d30eeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:19.719ex; height:3.176ex;" alt="{\displaystyle p_{\mathbf {s} }=(1-\tanh(\mathbf {s} )^{2})}" loading="lazy">, then we have

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(\mathbf {Y} )={\frac {1}{N}}\sum _{i=1}^{M}\sum _{t=1}^{N}\ln(1-\tanh(\mathbf {w} _{i}^{\mathsf {T}}\mathbf {x} ^{t})^{2})+\ln |\mathbf {W} |}</annotation>

</semantics>

</math><img src="./e13a6e9b7b4a19a19df9a3433c8fdeecbb6d95c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.047ex; height:7.343ex;" alt="{\displaystyle h(\mathbf {Y} )={\frac {1}{N}}\sum _{i=1}^{M}\sum _{t=1}^{N}\ln(1-\tanh(\mathbf {w} _{i}^{\mathsf {T}}\mathbf {x} ^{t})^{2})+\ln |\mathbf {W} |}" loading="lazy"></dd></dl>

In the sum, given an observed signal mixture <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy">, the corresponding set of extracted signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation>

</semantics>

</math><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"> and source signal model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathbf {s} }=g'}">

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{\mathbf {s} }=g'}</annotation>

</semantics>

</math><img src="./303c45374417f36e27b3ab9e4af00cb46d97d8c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.139ex; height:2.843ex;" alt="{\displaystyle p_{\mathbf {s} }=g'}" loading="lazy">, we can find the optimal unmixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy">, and make the extracted signals independent and non-gaussian. Like the projection pursuit situation, we can use gradient descent method to find the optimal solution of the unmixing matrix.

<div class="mw-heading mw-heading3"><h3 id="Based_on_maximum_likelihood_estimation">Based on maximum likelihood estimation</h3></div>

<a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a> estimation (MLE) is a standard statistical tool for finding parameter values (e.g. the unmixing matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy">) that provide the best fit of some data (e.g., the extracted signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">) to a given a model (e.g., the assumed joint probability density function (pdf) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation>

</semantics>

</math><img src="./0a8201eb39454219ddec719cdb8b3ffd2b3be2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}" loading="lazy"> of source signals).<a href="#cite_note-ReferenceA-24">[24]</a>

The ML "model" includes a specification of a pdf, which in this case is the pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation>

</semantics>

</math><img src="./0a8201eb39454219ddec719cdb8b3ffd2b3be2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}" loading="lazy"> of the unknown source signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">. Using ML ICA, the objective is to find an unmixing matrix that yields extracted signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\mathbf {W} x}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=\mathbf {W} x}</annotation>

</semantics>

</math><img src="./feb18d915a7ba39ef5d7df45932823c0da9b7199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.346ex; height:2.509ex;" alt="{\displaystyle y=\mathbf {W} x}" loading="lazy"> with a joint pdf as similar as possible to the joint pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation>

</semantics>

</math><img src="./0a8201eb39454219ddec719cdb8b3ffd2b3be2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}" loading="lazy"> of the unknown source signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">.

MLE is thus based on the assumption that if the model pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation>

</semantics>

</math><img src="./0a8201eb39454219ddec719cdb8b3ffd2b3be2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}" loading="lazy"> and the model parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>

</semantics>

</math><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> are correct then a high probability should be obtained for the data <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> that were actually observed. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>

</semantics>

</math><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"> is far from the correct parameter values then a low probability of the observed data would be expected.

Using MLE, we call the probability of the observed data for a given set of model parameter values (e.g., a pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation>

</semantics>

</math><img src="./0a8201eb39454219ddec719cdb8b3ffd2b3be2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}" loading="lazy"> and a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>

</semantics>

</math><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy">) the likelihood of the model parameter values given the observed data.

We define a likelihood function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L(W)} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L(W)} }</annotation>

</semantics>

</math><img src="./39abcd84ff46a03908b91a8e4c2ff3de9e657c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.45ex; height:2.843ex;" alt="{\displaystyle \mathbf {L(W)} }" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy">:

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L(W)} =p_{s}(\mathbf {W} x)|\det \mathbf {W} |.}</annotation>

</semantics>

</math><img src="./166ac074ab26e79eb41db1d8957c54d7e02bebbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.33ex; height:2.843ex;" alt="{\displaystyle \mathbf {L(W)} =p_{s}(\mathbf {W} x)|\det \mathbf {W} |.}" loading="lazy">

This equals to the probability density at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\mathbf {W} x}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=\mathbf {W} x}</annotation>

</semantics>

</math><img src="./397fdfd842fc34e4a2a2cd6abe322771e77c6ad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.281ex; height:2.176ex;" alt="{\displaystyle s=\mathbf {W} x}" loading="lazy">.

Thus, if we wish to find a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy"> that is most likely to have generated the observed mixtures <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> from the unknown source signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> with pdf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{s}}</annotation>

</semantics>

</math><img src="./0a8201eb39454219ddec719cdb8b3ffd2b3be2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{s}}" loading="lazy"> then we need only find that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy"> which maximizes the likelihood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L(W)} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L(W)} }</annotation>

</semantics>

</math><img src="./39abcd84ff46a03908b91a8e4c2ff3de9e657c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.45ex; height:2.843ex;" alt="{\displaystyle \mathbf {L(W)} }" loading="lazy">. The unmixing matrix that maximizes equation is known as the MLE of the optimal unmixing matrix.

It is common practice to use the log likelihood, because this is easier to evaluate. As the logarithm is a monotonic function, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy"> that maximizes the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L(W)} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {L(W)} }</annotation>

</semantics>

</math><img src="./39abcd84ff46a03908b91a8e4c2ff3de9e657c99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.45ex; height:2.843ex;" alt="{\displaystyle \mathbf {L(W)} }" loading="lazy"> also maximizes its logarithm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \mathbf {L(W)} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln \mathbf {L(W)} }</annotation>

</semantics>

</math><img src="./779a90354b828dc9eca16083700540c4c39d3514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.776ex; height:2.843ex;" alt="{\displaystyle \ln \mathbf {L(W)} }" loading="lazy">. This allows us to take the logarithm of equation above, which yields the log likelihood function

</mrow>

</mrow>

</munder>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln \mathbf {L(W)} =\sum _{i}\sum _{t}\ln p_{s}(w_{i}^{T}x_{t})+N\ln |\det \mathbf {W} |}</annotation>

</semantics>

</math><img src="./75afb362d28516fe88eadc4a49c630eea8395547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.554ex; height:5.509ex;" alt="{\displaystyle \ln \mathbf {L(W)} =\sum _{i}\sum _{t}\ln p_{s}(w_{i}^{T}x_{t})+N\ln |\det \mathbf {W} |}" loading="lazy">

If we substitute a commonly used high-<a href="Kurtosis" title="Kurtosis">Kurtosis</a> model pdf for the source signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}=(1-\tanh(s)^{2})}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{s}=(1-\tanh(s)^{2})}</annotation>

</semantics>

</math><img src="./14fbaee5c2b6141d10094e16a9bb04720b4eff6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:19.779ex; height:3.176ex;" alt="{\displaystyle p_{s}=(1-\tanh(s)^{2})}" loading="lazy"> then we have

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln \mathbf {L(W)} ={1 \over N}\sum _{i}^{M}\sum _{t}^{N}\ln(1-\tanh(w_{i}^{T}x_{t})^{2})+\ln |\det \mathbf {W} |}</annotation>

</semantics>

</math><img src="./010baf0a5137a7733e33a37baebb0e3d6dc5d13f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.349ex; height:7.343ex;" alt="{\displaystyle \ln \mathbf {L(W)} ={1 \over N}\sum _{i}^{M}\sum _{t}^{N}\ln(1-\tanh(w_{i}^{T}x_{t})^{2})+\ln |\det \mathbf {W} |}" loading="lazy">

This matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} }</annotation>

</semantics>

</math><img src="./04749f1e87cca59c094da23c79cc64b085b0df12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.176ex;" alt="{\displaystyle \mathbf {W} }" loading="lazy"> that maximizes this function is the <a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimation.

<div class="mw-heading mw-heading2"><h2 id="History_and_background">History and background</h2></div>

The early general framework for independent component analysis was introduced by Jeanny Hérault and Bernard Ans from 1984,<a href="#cite_note-25">[25]</a> further developed by Christian Jutten in 1985 and 1986,<a href="#cite_note-jutten85-2">[2]</a><a href="#cite_note-26">[26]</a><a href="#cite_note-27">[27]</a> and refined by Pierre Comon in 1991,<a href="#cite_note-pc91-20">[20]</a> and popularized in his paper of 1994.<a href="#cite_note-comon94-10">[10]</a> In 1995, Tony Bell and <a href="Terry_Sejnowski" title="Terry Sejnowski">Terry Sejnowski</a> introduced a fast and efficient ICA algorithm based on <a href="Infomax" title="Infomax">infomax</a>, a principle introduced by Ralph Linsker in 1987. A link exists between maximum-likelihood estimation and Infomax approaches.<a href="#cite_note-card97-28">[28]</a> A quite comprehensive tutorial on the maximum-likelihood approach to ICA has been published by J-F. Cardoso in 1998.<a href="#cite_note-card98-29">[29]</a>

There are many algorithms available in the literature which do ICA. A largely used one, including in industrial applications, is the FastICA algorithm, developed by Hyvärinen and Oja,<a href="#cite_note-30">[30]</a> which uses the <a href="Negentropy" title="Negentropy">negentropy</a> as cost function, already proposed 7 years before by Pierre Comon in this context.<a href="#cite_note-comon94-10">[10]</a> Other examples are rather related to <a href="Blind_source_separation" class="mw-redirect" title="Blind source separation">blind source separation</a> where a more general approach is used. For example, one can drop the independence assumption and separate mutually correlated signals, thus, statistically "dependent" signals. Sepp Hochreiter and <a href="J%C3%BCrgen_Schmidhuber" title="Jürgen Schmidhuber">Jürgen Schmidhuber</a> showed how to obtain non-linear ICA or source separation as a by-product of <a href="Regularization_(mathematics)" title="Regularization (mathematics)">regularization</a> (1999).<a href="#cite_note-HochreiterSchmidhuber1999-31">[31]</a> Their method does not require a priori knowledge about the number of independent sources.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

ICA can be extended to analyze non-physical signals. For instance, ICA has been applied to discover discussion topics on a bag of news list archives.

Some ICA applications are listed below:<a href="#cite_note-Stone_2004-6">[6]</a>

<ul><li>image <a href="Steganography" title="Steganography">steganography</a><a href="#cite_note-32">[32]</a></li>

<li>optical Imaging of neurons<a href="#cite_note-33">[33]</a></li>

<li>neuronal spike sorting<a href="#cite_note-34">[34]</a></li>

<li>face recognition<a href="#cite_note-35">[35]</a></li>

<li>modelling receptive fields of primary visual neurons<a href="#cite_note-36">[36]</a></li>

<li>predicting stock market prices<a href="#cite_note-37">[37]</a></li>

<li>mobile phone communications<a href="#cite_note-38">[38]</a></li>

<li>colour based detection of the ripeness of tomatoes<a href="#cite_note-39">[39]</a></li>

<li>removing artifacts, such as eye blinks, from <a href="EEG" class="mw-redirect" title="EEG">EEG</a> data.<a href="#cite_note-40">[40]</a></li>

<li>predicting decision-making using EEG<a href="#cite_note-41">[41]</a></li>

<li>analysis of changes in gene expression over time in single <a href="Cell_(biology)" title="Cell (biology)">cell</a> <a href="RNA-sequencing" class="mw-redirect" title="RNA-sequencing">RNA-sequencing</a> experiments.<a href="#cite_note-42">[42]</a></li>

<li>studies of the <a href="Resting_state_fMRI" title="Resting state fMRI">resting state network</a> of the brain.<a href="#cite_note-Kiviniemi2003-43">[43]</a></li>

<li>astronomy and cosmology<a href="#cite_note-44">[44]</a></li>

<li>finance<a href="#cite_note-45">[45]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Availability">Availability</h2></div>

ICA can be applied through the following software:

<li><a href="R_(programming_language)" title="R (programming language)">R</a> ICA package</li>

<li><a href="Scikit-learn" title="Scikit-learn">scikit-learn</a> Python implementation <a rel="nofollow" class="external text" href="https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.FastICA.html">sklearn.decomposition.FastICA</a></li>

<li><a href="Mlpack" title="Mlpack">mlpack</a> C++ implementation of RADICAL (The Robust Accurate, Direct ICA aLgorithm (RADICAL).) <a rel="nofollow" class="external autonumber" href="https://github.com/mlpack/mlpack/tree/master/src/mlpack/methods/radical">[1]</a></li></ul>

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<ul><li><a href="Blind_deconvolution" title="Blind deconvolution">Blind deconvolution</a></li>

<li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li>

<li><a href="Hilbert_spectrum" title="Hilbert spectrum">Hilbert spectrum</a></li>

<li><a href="Image_processing" class="mw-redirect" title="Image processing">Image processing</a></li>

<li><a href="Non-negative_matrix_factorization" title="Non-negative matrix factorization">Non-negative matrix factorization (NMF)</a></li>

<li><a href="Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">Nonlinear dimensionality reduction</a></li>

<li><a href="Projection_pursuit" title="Projection pursuit">Projection pursuit</a></li>

<li><a href="Varimax_rotation" title="Varimax rotation">Varimax rotation</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite id="CITEREFHyvarinen2001a" class="citation book cs1">Hyvarinen, A, Karhunen, J & Oja, E (2001a). Independent component analysis. New York: John Wiley and Sons.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)

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<li id="cite_note-40"><a href="#cite_ref-40">^</a> <cite id="CITEREFDelormeSejnowski,_TMakeig,_S2007" class="citation journal cs1">Delorme, A; Sejnowski, T; Makeig, S (2007). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2895624">"Enhanced detection of artifacts in EEG data using higher-order statistics and independent component analysis"</a>. NeuroImage. 34 (4): 1443–1449. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.neuroimage.2006.11.004">10.1016/j.neuroimage.2006.11.004</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2895624">2895624</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17188898">17188898</a>.</cite>

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<li id="cite_note-41"><a href="#cite_ref-41">^</a> <cite id="CITEREFDouglas2013" class="citation journal cs1">Douglas, P (2013). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3728485">"Single trial decoding of belief decision making from EEG and fMRI data using independent components features"</a>. Frontiers in Human Neuroscience. 7: 392. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3389%2Ffnhum.2013.00392">10.3389/fnhum.2013.00392</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3728485">3728485</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23914164">23914164</a>.</cite>

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<li id="cite_note-42"><a href="#cite_ref-42">^</a> <cite id="CITEREFTrapnellCacchiarelli,_DGrimsby,_J2014" class="citation journal cs1">Trapnell, C; Cacchiarelli, D; Grimsby, J (2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4122333">"The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells"</a>. Nature Biotechnology. 32 (4): 381–386. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnbt.2859">10.1038/nbt.2859</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4122333">4122333</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24658644">24658644</a>.</cite>

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<li id="cite_note-Kiviniemi2003-43"><a href="#cite_ref-Kiviniemi2003_43-0">^</a> <cite id="CITEREFKiviniemiKantolaJauhiainenHyvärinen2003" class="citation journal cs1">Kiviniemi, Vesa J.; Kantola, Juha-Heikki; Jauhiainen, Jukka; Hyvärinen, Aapo; Tervonen, Osmo (2003). "Independent component analysis of nondeterministic fMRI signal sources". NeuroImage. 19 (2): 253–260. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS1053-8119%2803%2900097-1">10.1016/S1053-8119(03)00097-1</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12814576">12814576</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17110486">17110486</a>.</cite>

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<li id="cite_note-44"><a href="#cite_ref-44">^</a> <cite id="CITEREFWangXuGuAn2010" class="citation journal cs1">Wang, Jingying; Xu, Haiguang; Gu, Junhua; An, Tao; Cui, Haijuan; Li, Jianxun; Zhang, Zhongli; Zheng, Qian; Wu, Xiang-Ping (2010-11-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0004-637X%2F723%2F1%2F620">"How to Identify and Separate Bright Galaxy Clusters from the Low-frequency Radio Sky?"</a>. The Astrophysical Journal. 723 (1): 620–633. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1008.3391">1008.3391</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010ApJ...723..620W">2010ApJ...723..620W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0004-637X%2F723%2F1%2F620">10.1088/0004-637X/723/1/620</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0004-637X">0004-637X</a>.</cite>

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<li id="cite_note-45"><a href="#cite_ref-45">^</a> <cite id="CITEREFMorauxVilla2003" class="citation book cs1">Moraux, Franck; Villa, Christophe (2003). "The dynamics of the term structure of interest rates: An Independent Component Analysis". Connectionist Approaches in Economics and Management Sciences. Advances in Computational Management Science. Vol. 6. pp. 215–232. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-3722-6_11">10.1007/978-1-4757-3722-6_11</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4757-3722-6</bdi>.</cite>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>Comon, Pierre (1994): <a rel="nofollow" class="external text" href="http://mlsp.cs.cmu.edu/courses/fall2012/lectures/ICA.pdf">"Independent Component Analysis: a new concept?"</a>, Signal Processing, 36(3):287–314 (The original paper describing the concept of ICA)</li>

<li>Hyvärinen, A.; Karhunen, J.; Oja, E. (2001): <a rel="nofollow" class="external text" href="http://www.cis.hut.fi/projects/ica/book/">Independent Component Analysis</a>, New York: Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-40540-5</bdi> ( <a rel="nofollow" class="external text" href="http://www.cis.hut.fi/projects/ica/book/intro.pdf">Introductory chapter</a> )</li>

<li>Hyvärinen, A.; Oja, E. (2000): <a rel="nofollow" class="external text" href="http://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf">"Independent Component Analysis: Algorithms and Application"</a>, Neural Networks, 13(4-5):411-430. (Technical but pedagogical introduction).</li>

<li>Comon, P.; Jutten C., (2010): Handbook of Blind Source Separation, Independent Component Analysis and Applications. Academic Press, Oxford UK. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-374726-6</bdi></li>

<li>Lee, T.-W. (1998): Independent component analysis: Theory and applications, Boston, Mass: Kluwer Academic Publishers, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-8261-7</bdi></li>

<li>Acharyya, Ranjan (2008): A New Approach for Blind Source Separation of Convolutive Sources - Wavelet Based Separation Using Shrinkage Function <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-639-07797-0</bdi> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3639077971</bdi> (this book focuses on unsupervised learning with Blind Source Separation)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.cs.helsinki.fi/u/ahyvarin/whatisica.shtml">What is independent component analysis?</a> by Aapo Hyvärinen</li>

<li><a rel="nofollow" class="external text" href="http://www.cis.hut.fi/aapo/papers/IJCNN99_tutorialweb/IJCNN99_tutorial3.html">Independent Component Analysis: A Tutorial</a> by Aapo Hyvärinen</li>

<li><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1404.2986">A Tutorial on Independent Component Analysis</a></li>

<li><a rel="nofollow" class="external text" href="http://www.cis.hut.fi/projects/ica/fastica/">FastICA as a package for Matlab, in R language, C++</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060630205321/http://www.bsp.brain.riken.go.jp/ICALAB/">ICALAB Toolboxes</a> for Matlab, developed at <a href="RIKEN" class="mw-redirect" title="RIKEN">RIKEN</a></li>

<li><a rel="nofollow" class="external text" href="http://nic.uoregon.edu/projects/hipersat/index.php">High Performance Signal Analysis Toolkit</a> provides C++ implementations of FastICA and Infomax</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070613191045/http://isp.imm.dtu.dk/toolbox/">ICA toolbox</a> Matlab tools for ICA with Bell-Sejnowski, Molgedey-Schuster and mean field ICA. Developed at DTU.</li>

<li><a rel="nofollow" class="external text" href="http://www.cis.hut.fi/projects/ica/cocktail/cocktail_en.cgi">Demonstration of the cocktail party problem</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100313152128/http://www.cis.hut.fi/projects/ica/cocktail/cocktail_en.cgi">Archived</a> 2010-03-13 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="http://sccn.ucsd.edu/eeglab/">EEGLAB Toolbox</a> ICA of <a href="Electroencephalogram" class="mw-redirect" title="Electroencephalogram">EEG</a> for Matlab, developed at UCSD.</li>

<li><a rel="nofollow" class="external text" href="http://sccn.ucsd.edu/fmrlab/">FMRLAB Toolbox</a> ICA of <a href="FMRI" class="mw-redirect" title="FMRI">fMRI</a> for Matlab, developed at UCSD</li>

<li><a rel="nofollow" class="external text" href="https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/MELODIC">MELODIC</a>, part of the <a href="FMRIB_Software_Library" title="FMRIB Software Library">FMRIB Software Library</a>.</li>

<li><a rel="nofollow" class="external text" href="http://brandon-merkl.blogspot.com/2005/12/independent-component-analysis.html">Discussion of ICA used in a biomedical shape-representation context</a></li>

<li><a rel="nofollow" class="external text" href="http://mdp-toolkit.sourceforge.net/">FastICA, CuBICA, JADE and TDSEP algorithm for Python and more...</a></li>

<li><a rel="nofollow" class="external text" href="http://icatb.sourceforge.net/">Group ICA Toolbox and Fusion ICA Toolbox</a></li>

<li><a rel="nofollow" class="external text" href="http://www.nbtwiki.net/doku.php?id=tutorial:compute_independent_component_analysis">Tutorial: Using ICA for cleaning EEG signals</a></li></ul></div><div><div class="zim-footer">

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