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<title>Pattern recognition</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Pattern recognition</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about pattern recognition as a branch of engineering. For the cognitive process, see <a href="Pattern_recognition_(psychology)" title="Pattern recognition (psychology)">Pattern recognition (psychology)</a>. For other uses, see <a href="Pattern_recognition_(disambiguation)" class="mw-disambig" title="Pattern recognition (disambiguation)">Pattern recognition (disambiguation)</a>.</div>

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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><br>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><br><span class="nobold"><span style="font-size: 85%;">(<b><a href="Statistical_classification" title="Statistical classification">classification</a></b> • <b><a href="Regression_analysis" title="Regression analysis">regression</a></b>)</span></span> </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"><i>k</i>-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><br><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 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="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">

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<p><b>Pattern recognition</b> is the task of assigning a <a href="Categorical_variable" title="Categorical variable">class</a> to an observation based on patterns extracted from data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess PR capabilities but their primary function is to distinguish and create emergent patterns. PR has applications in statistical <a href="Data_analysis" title="Data analysis">data analysis</a>, <a href="Signal_processing" title="Signal processing">signal processing</a>, <a href="Image_analysis" title="Image analysis">image analysis</a>, <a href="Information_retrieval" title="Information retrieval">information retrieval</a>, <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a>, <a href="Data_compression" title="Data compression">data compression</a>, <a href="Computer_graphics" title="Computer graphics">computer graphics</a> and <a href="Machine_learning" title="Machine learning">machine learning</a>. Pattern recognition has its origins in statistics and engineering; some modern approaches to pattern recognition include the use of <a href="Machine_learning" title="Machine learning">machine learning</a>, due to the increased availability of <a href="Big_data" title="Big data">big data</a> and a new abundance of <a href="Processing_power" class="mw-redirect" title="Processing power">processing power</a>.

</p><p>Pattern recognition systems are commonly trained from labeled "training" data. When no <a href="Labeled_data" title="Labeled data">labeled data</a> are available, other algorithms can be used to discover previously unknown patterns. <a href="Data_mining" title="Data mining">KDD</a> and data mining have a larger focus on unsupervised methods and stronger connection to business use. Pattern recognition focuses more on the signal and also takes acquisition and <a href="Signal_processing" title="Signal processing">signal processing</a> into consideration. It originated in <a href="Engineering" title="Engineering">engineering</a>, and the term is popular in the context of <a href="Computer_vision" title="Computer vision">computer vision</a>: a leading computer vision conference is named <a href="Conference_on_Computer_Vision_and_Pattern_Recognition" title="Conference on Computer Vision and Pattern Recognition">Conference on Computer Vision and Pattern Recognition</a>.

</p><p>In <a href="Machine_learning" title="Machine learning">machine learning</a>, pattern recognition is the assignment of a label to a given input value. In statistics, <a href="Linear_discriminant_analysis" title="Linear discriminant analysis">discriminant analysis</a> was introduced for this same purpose in 1936. An example of pattern recognition is <a href="Classification_(machine_learning)" class="mw-redirect" title="Classification (machine learning)">classification</a>, which attempts to assign each input value to one of a given set of <i>classes</i> (for example, determine whether a given email is "spam"). Pattern recognition is a more general problem that encompasses other types of output as well. Other examples are <a href="Regression_analysis" title="Regression analysis">regression</a>, which assigns a <a href="Real_number" title="Real number">real-valued</a> output to each input;<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="Sequence_labeling" title="Sequence labeling">sequence labeling</a>, which assigns a class to each member of a sequence of values<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (for example, <a href="Part_of_speech_tagging" class="mw-redirect" title="Part of speech tagging">part of speech tagging</a>, which assigns a <a href="Part_of_speech" title="Part of speech">part of speech</a> to each word in an input sentence); and <a href="Parsing" title="Parsing">parsing</a>, which assigns a <a href="Parse_tree" title="Parse tree">parse tree</a> to an input sentence, describing the <a href="Syntactic_structure" class="mw-redirect" title="Syntactic structure">syntactic structure</a> of the sentence.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Pattern recognition algorithms generally aim to provide a reasonable answer for all possible inputs and to perform "most likely" matching of the inputs, taking into account their statistical variation. This is opposed to <i><a href="Pattern_matching" title="Pattern matching">pattern matching</a></i> algorithms, which look for exact matches in the input with pre-existing patterns. A common example of a pattern-matching algorithm is <a href="Regular_expression" title="Regular expression">regular expression</a> matching, which looks for patterns of a given sort in textual data and is included in the search capabilities of many <a href="Text_editor" title="Text editor">text editors</a> and <a href="Word_processor" title="Word processor">word processors</a>.

</p>

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

<div role="note" class="hatnote navigation-not-searchable">Further information on Combination Of Shifted FIlter REsponses: <a href="COSFIRE" title="COSFIRE">COSFIRE</a></div>

<p>A modern definition of pattern recognition is:

</p>

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</style><blockquote class="templatequote"><p>The field of pattern recognition is concerned with the automatic discovery of regularities in data through the use of computer algorithms and with the use of these regularities to take actions such as classifying the data into different categories.<sup id="cite_ref-Bishop2006_4-0" class="reference"><a href="#cite_note-Bishop2006-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></p></blockquote>

<p>Pattern recognition is generally categorized according to the type of learning procedure used to generate the output value. <i><a href="Supervised_learning" title="Supervised learning">Supervised learning</a></i> assumes that a set of training data (the <a href="Training_set" class="mw-redirect" title="Training set">training set</a>) has been provided, consisting of a set of instances that have been properly labeled by hand with the correct output. A learning procedure then generates a model that attempts to meet two sometimes conflicting objectives: Perform as well as possible on the training data, and generalize as well as possible to new data (usually, this means being as simple as possible, for some technical definition of "simple", in accordance with <a href="Occam's_Razor" class="mw-redirect" title="Occam's Razor">Occam's Razor</a>, discussed below). <a href="Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a>, on the other hand, assumes training data that has not been hand-labeled, and attempts to find inherent patterns in the data that can then be used to determine the correct output value for new data instances.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A combination of the two that has been explored is <a href="Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">semi-supervised learning</a>, which uses a combination of labeled and unlabeled data (typically a small set of labeled data combined with a large amount of unlabeled data). In cases of unsupervised learning, there may be no training data at all.

</p><p>Sometimes different terms are used to describe the corresponding supervised and unsupervised learning procedures for the same type of output. The unsupervised equivalent of classification is normally known as <i><a href="Data_clustering" class="mw-redirect" title="Data clustering">clustering</a></i>, based on the common perception of the task as involving no training data to speak of, and of grouping the input data into clusters based on some inherent <a href="Similarity_measure" title="Similarity measure">similarity measure</a> (e.g. the <a href="Distance" title="Distance">distance</a> between instances, considered as vectors in a multi-dimensional <a href="Vector_space" title="Vector space">vector space</a>), rather than assigning each input instance into one of a set of pre-defined classes. In some fields, the terminology is different. In <a href="Community_ecology" class="mw-redirect" title="Community ecology">community ecology</a>, the term <i>classification</i> is used to refer to what is commonly known as "clustering".

</p><p>The piece of input data for which an output value is generated is formally termed an <i>instance</i>. The instance is formally described by a <a href="Feature_vector" class="mw-redirect" title="Feature vector">vector</a> of features, which together constitute a description of all known characteristics of the instance. These feature vectors can be seen as defining points in an appropriate <a href="Space_(mathematics)" title="Space (mathematics)">multidimensional space</a>, and methods for manipulating vectors in <a href="Vector_space" title="Vector space">vector spaces</a> can be correspondingly applied to them, such as computing the <a href="Dot_product" title="Dot product">dot product</a> or the angle between two vectors. Features typically are either <a href="Categorical_data" class="mw-redirect" title="Categorical data">categorical</a> (also known as <a href="Nominal_data" class="mw-redirect" title="Nominal data">nominal</a>, i.e., consisting of one of a set of unordered items, such as a gender of "male" or "female", or a blood type of "A", "B", "AB" or "O"), <a href="Ordinal_data" title="Ordinal data">ordinal</a> (consisting of one of a set of ordered items, e.g., "large", "medium" or "small"), <a href="Integer" title="Integer">integer-valued</a> (e.g., a count of the number of occurrences of a particular word in an email) or <a href="Real_number" title="Real number">real-valued</a> (e.g., a measurement of blood pressure). Often, categorical and ordinal data are grouped together, and this is also the case for integer-valued and real-valued data. Many algorithms work only in terms of categorical data and require that real-valued or integer-valued data be <i>discretized</i> into groups (e.g., less than 5, between 5 and 10, or greater than 10).

</p>

<div class="mw-heading mw-heading3"><h3 id="Probabilistic_classifiers">Probabilistic classifiers</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Probabilistic_classifier" class="mw-redirect" title="Probabilistic classifier">Probabilistic classifier</a></div>

<p>Many common pattern recognition algorithms are <i>probabilistic</i> in nature, in that they use <a href="Statistical_inference" title="Statistical inference">statistical inference</a> to find the best label for a given instance. Unlike other algorithms, which simply output a "best" label, often probabilistic algorithms also output a <a href="Probability" title="Probability">probability</a> of the instance being described by the given label. In addition, many probabilistic algorithms output a list of the <i>N</i>-best labels with associated probabilities, for some value of <i>N</i>, instead of simply a single best label. When the number of possible labels is fairly small (e.g., in the case of <a href="Classification_(machine_learning)" class="mw-redirect" title="Classification (machine learning)">classification</a>), <i>N</i> may be set so that the probability of all possible labels is output. Probabilistic algorithms have many advantages over non-probabilistic algorithms:

</p>

<ul><li>They output a confidence value associated with their choice. (Note that some other algorithms may also output confidence values, but in general, only for probabilistic algorithms is this value mathematically grounded in <a href="Probability_theory" title="Probability theory">probability theory</a>. Non-probabilistic confidence values can in general not be given any specific meaning, and only used to compare against other confidence values output by the same algorithm.)</li>

<li>Correspondingly, they can <i>abstain</i> when the confidence of choosing any particular output is too low.</li>

<li>Because of the probabilities output, probabilistic pattern-recognition algorithms can be more effectively incorporated into larger machine-learning tasks, in a way that partially or completely avoids the problem of <i>error propagation</i>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Number_of_important_feature_variables">Number of important feature variables</h3></div>

<p><a href="Feature_selection" title="Feature selection">Feature selection</a> algorithms attempt to directly prune out redundant or irrelevant features. A general introduction to <a href="Feature_selection" title="Feature selection">feature selection</a> which summarizes approaches and challenges, has been given.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The complexity of feature-selection is, because of its non-monotonous character, an <a href="Optimization_problem" title="Optimization problem">optimization problem</a> where given a total of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> features the <a href="Powerset" class="mw-redirect" title="Powerset">powerset</a> consisting of all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./51e4bd4ef2f9549d026cbf643a91c0d12a8c6794.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.384ex; height:2.509ex;" alt="{\displaystyle 2^{n}-1}" loading="lazy"></span> subsets of features need to be explored. The <a href="Branch_and_bound" title="Branch and bound">Branch-and-Bound algorithm</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> does reduce this complexity but is intractable for medium to large values of the number of available features <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>

</p><p>Techniques to transform the raw feature vectors (<b>feature extraction</b>) are sometimes used prior to application of the pattern-matching algorithm. <a href="Feature_extraction" class="mw-redirect" title="Feature extraction">Feature extraction</a> algorithms attempt to reduce a large-dimensionality feature vector into a smaller-dimensionality vector that is easier to work with and encodes less redundancy, using mathematical techniques such as <a href="Principal_components_analysis" class="mw-redirect" title="Principal components analysis">principal components analysis</a> (PCA). The distinction between <b>feature selection</b> and <b>feature extraction</b> is that the resulting features after feature extraction has taken place are of a different sort than the original features and may not easily be interpretable, while the features left after feature selection are simply a subset of the original features.

</p>

<div class="mw-heading mw-heading2"><h2 id="Problem_statement">Problem statement</h2></div>

<p>The problem of pattern recognition can be stated as follows: Given an unknown function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}}</annotation>

</semantics>

</math></span><img src="./bfa821cd6baa6f015e66552f920d0f02d736a2eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.202ex; height:2.509ex;" alt="{\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}}" loading="lazy"></span> (the <i>ground truth</i>) that maps input instances <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}\in {\mathcal {X}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c53d1037e98c2475a2f6f153e6f64c1295ef74c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.248ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {x}}\in {\mathcal {X}}}" loading="lazy"></span> to output labels <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in {\mathcal {Y}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2fca6d018a308729c9ef591e40b7744f1c752c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.656ex; height:2.509ex;" alt="{\displaystyle y\in {\mathcal {Y}}}" loading="lazy"></span>, along with training data <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} =\{({\boldsymbol {x}}_{1},y_{1}),\dots ,({\boldsymbol {x}}_{n},y_{n})\}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {D} =\{({\boldsymbol {x}}_{1},y_{1}),\dots ,({\boldsymbol {x}}_{n},y_{n})\}}</annotation>

</semantics>

</math></span><img src="./e21a4a7da1373f38043e6ba97fc0af333a4b5bd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.225ex; height:2.843ex;" alt="{\displaystyle \mathbf {D} =\{({\boldsymbol {x}}_{1},y_{1}),\dots ,({\boldsymbol {x}}_{n},y_{n})\}}" loading="lazy"></span> assumed to represent accurate examples of the mapping, produce a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:{\mathcal {X}}\rightarrow {\mathcal {Y}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h:{\mathcal {X}}\rightarrow {\mathcal {Y}}}</annotation>

</semantics>

</math></span><img src="./675325ba6d28649da683e24b8759d828fef70059.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.425ex; height:2.509ex;" alt="{\displaystyle h:{\mathcal {X}}\rightarrow {\mathcal {Y}}}" loading="lazy"></span> that approximates as closely as possible the correct mapping <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

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

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</math></span><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"></span>. (For example, if the problem is filtering spam, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}_{i}}">

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

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</math></span><img src="./6dfbd99e446ac6b35c8b37e9ff3c360be9089f82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.331ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {x}}_{i}}" loading="lazy"></span> is some representation of an email and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math></span><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"></span> is either "spam" or "non-spam"). In order for this to be a well-defined problem, "approximates as closely as possible" needs to be defined rigorously. In <a href="Decision_theory" title="Decision theory">decision theory</a>, this is defined by specifying a <a href="Loss_function" title="Loss function">loss function</a> or cost function that assigns a specific value to "loss" resulting from producing an incorrect label. The goal then is to minimize the <a href="Expected_value" title="Expected value">expected</a> loss, with the expectation taken over the <a href="Probability_distribution" title="Probability distribution">probability distribution</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {X}}}">

</mrow>

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</mrow>

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

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</math></span><img src="./8c7e5461c5286852df4ef652fca7e4b0b63030e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:2.176ex;" alt="{\displaystyle {\mathcal {X}}}" loading="lazy"></span>. In practice, neither the distribution of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {X}}}">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./8c7e5461c5286852df4ef652fca7e4b0b63030e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:2.176ex;" alt="{\displaystyle {\mathcal {X}}}" loading="lazy"></span> nor the ground truth function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}}</annotation>

</semantics>

</math></span><img src="./bfa821cd6baa6f015e66552f920d0f02d736a2eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.202ex; height:2.509ex;" alt="{\displaystyle g:{\mathcal {X}}\rightarrow {\mathcal {Y}}}" loading="lazy"></span> are known exactly, but can be computed only empirically by collecting a large number of samples of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {X}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c7e5461c5286852df4ef652fca7e4b0b63030e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:2.176ex;" alt="{\displaystyle {\mathcal {X}}}" loading="lazy"></span> and hand-labeling them using the correct value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Y}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8935ef8cf454efa0b58363386c33f16a48ec36ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.659ex; height:2.509ex;" alt="{\displaystyle {\mathcal {Y}}}" loading="lazy"></span> (a time-consuming process, which is typically the limiting factor in the amount of data of this sort that can be collected). The particular loss function depends on the type of label being predicted. For example, in the case of <a href="Classification_(machine_learning)" class="mw-redirect" title="Classification (machine learning)">classification</a>, the simple <a href="Zero-one_loss_function" class="mw-redirect" title="Zero-one loss function">zero-one loss function</a> is often sufficient. This corresponds simply to assigning a loss of 1 to any incorrect labeling and implies that the optimal classifier minimizes the <a href="Bayes_error_rate" title="Bayes error rate">error rate</a> on independent test data (i.e. counting up the fraction of instances that the learned function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:{\mathcal {X}}\rightarrow {\mathcal {Y}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h:{\mathcal {X}}\rightarrow {\mathcal {Y}}}</annotation>

</semantics>

</math></span><img src="./675325ba6d28649da683e24b8759d828fef70059.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.425ex; height:2.509ex;" alt="{\displaystyle h:{\mathcal {X}}\rightarrow {\mathcal {Y}}}" loading="lazy"></span> labels wrongly, which is equivalent to maximizing the number of correctly classified instances). The goal of the learning procedure is then to minimize the error rate (maximize the <a href="Correctness_(computer_science)" title="Correctness (computer science)">correctness</a>) on a "typical" test set.

</p><p>For a probabilistic pattern recognizer, the problem is instead to estimate the probability of each possible output label given a particular input instance, i.e., to estimate a function of the form

</p>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})=f\left({\boldsymbol {x}};{\boldsymbol {\theta }}\right)}</annotation>

</semantics>

</math></span><img src="./abe5bfeaa59f17bf0e31b205359b80bce991071b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:22.814ex; height:2.843ex;" alt="{\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})=f\left({\boldsymbol {x}};{\boldsymbol {\theta }}\right)}" loading="lazy"></span></dd></dl>

<p>where the <a href="Feature_vector" class="mw-redirect" title="Feature vector">feature vector</a> input is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>, and the function <i>f</i> is typically parameterized by some parameters <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In a <a href="Discriminative_model" title="Discriminative model">discriminative</a> approach to the problem, <i>f</i> is estimated directly. In a <a href="Generative_model" title="Generative model">generative</a> approach, however, the inverse probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({{\boldsymbol {x}}|{\rm {label}}})}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({{\boldsymbol {x}}|{\rm {label}}})}</annotation>

</semantics>

</math></span><img src="./94771a84bc6f22f9829bb8c4cd42dd55a2ecef54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.028ex; height:2.843ex;" alt="{\displaystyle p({{\boldsymbol {x}}|{\rm {label}}})}" loading="lazy"></span> is instead estimated and combined with the <a href="Prior_probability" title="Prior probability">prior probability</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}</annotation>

</semantics>

</math></span><img src="./bca4582ca8644ed7f4cfa780d02a8b878c3310ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.802ex; height:2.843ex;" alt="{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}" loading="lazy"></span> using <a href="Bayes'_rule" class="mw-redirect" title="Bayes' rule">Bayes' rule</a>, as follows:

</p>

</mrow>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mtext>all labels</mtext>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})={\frac {p({{\boldsymbol {x}}|{\rm {label,{\boldsymbol {\theta }}}}})p({\rm {label|{\boldsymbol {\theta }}}})}{\sum _{L\in {\text{all labels}}}p({\boldsymbol {x}}|L)p(L|{\boldsymbol {\theta }})}}.}</annotation>

</semantics>

</math></span><img src="./25bbdc4c11f79d738d6d375c9c160ec597c30256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:41.839ex; height:6.676ex;" alt="{\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})={\frac {p({{\boldsymbol {x}}|{\rm {label,{\boldsymbol {\theta }}}}})p({\rm {label|{\boldsymbol {\theta }}}})}{\sum _{L\in {\text{all labels}}}p({\boldsymbol {x}}|L)p(L|{\boldsymbol {\theta }})}}.}" loading="lazy"></span></dd></dl>

<p>When the labels are <a href="Continuous_distribution" class="mw-redirect" title="Continuous distribution">continuously distributed</a> (e.g., in <a href="Regression_analysis" title="Regression analysis">regression analysis</a>), the denominator involves <a href="Integral" title="Integral">integration</a> rather than summation:

</p>

</mrow>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<msub>

<mtext>all labels</mtext>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})={\frac {p({{\boldsymbol {x}}|{\rm {label,{\boldsymbol {\theta }}}}})p({\rm {label|{\boldsymbol {\theta }}}})}{\int _{L\in {\text{all labels}}}p({\boldsymbol {x}}|L)p(L|{\boldsymbol {\theta }})\operatorname {d} L}}.}</annotation>

</semantics>

</math></span><img src="./2b6c6e7db2cc15c6351c82ae9f3ae1d52f739355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:44.132ex; height:6.843ex;" alt="{\displaystyle p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})={\frac {p({{\boldsymbol {x}}|{\rm {label,{\boldsymbol {\theta }}}}})p({\rm {label|{\boldsymbol {\theta }}}})}{\int _{L\in {\text{all labels}}}p({\boldsymbol {x}}|L)p(L|{\boldsymbol {\theta }})\operatorname {d} L}}.}" loading="lazy"></span></dd></dl>

<p>The value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span> is typically learned using <a href="Maximum_a_posteriori" class="mw-redirect" title="Maximum a posteriori">maximum a posteriori</a> (MAP) estimation. This finds the best value that simultaneously meets two conflicting objects: To perform as well as possible on the training data (smallest <a href="Bayes_error_rate" title="Bayes error rate">error-rate</a>) and to find the simplest possible model. Essentially, this combines <a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimation with a <a href="Regularization_(mathematics)" title="Regularization (mathematics)">regularization</a> procedure that favors simpler models over more complex models. In a <a href="Bayesian_inference" title="Bayesian inference">Bayesian</a> context, the regularization procedure can be viewed as placing a <a href="Prior_probability" title="Prior probability">prior probability</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\boldsymbol {\theta }})}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d1586925e0dcb8ca509076b61227be8773a90d2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.375ex; height:2.843ex;" alt="{\displaystyle p({\boldsymbol {\theta }})}" loading="lazy"></span> on different values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span>. Mathematically:

</p>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\theta }}^{*}=\arg \max _{\boldsymbol {\theta }}p({\boldsymbol {\theta }}|\mathbf {D} )}</annotation>

</semantics>

</math></span><img src="./514b1450226a39e29de4a2ddb8d495bb99fb3c57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.777ex; height:4.009ex;" alt="{\displaystyle {\boldsymbol {\theta }}^{*}=\arg \max _{\boldsymbol {\theta }}p({\boldsymbol {\theta }}|\mathbf {D} )}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3959de75cbeb5f25ac5b070c3f25d722d86c7f2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.361ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {\theta }}^{*}}" loading="lazy"></span> is the value used for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span> in the subsequent evaluation procedure, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\boldsymbol {\theta }}|\mathbf {D} )}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\boldsymbol {\theta }}|\mathbf {D} )}</annotation>

</semantics>

</math></span><img src="./18767832fef9c552e5194ccb90e8eb32e1824e1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.071ex; height:2.843ex;" alt="{\displaystyle p({\boldsymbol {\theta }}|\mathbf {D} )}" loading="lazy"></span>, the <a href="Posterior_probability" title="Posterior probability">posterior probability</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span>, is given by

</p>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\boldsymbol {\theta }}|\mathbf {D} )=\left[\prod _{i=1}^{n}p(y_{i}|{\boldsymbol {x}}_{i},{\boldsymbol {\theta }})\right]p({\boldsymbol {\theta }}).}</annotation>

</semantics>

</math></span><img src="./3750170b737eed276b09e05fe3a6365ee985e3c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:31.792ex; height:7.509ex;" alt="{\displaystyle p({\boldsymbol {\theta }}|\mathbf {D} )=\left[\prod _{i=1}^{n}p(y_{i}|{\boldsymbol {x}}_{i},{\boldsymbol {\theta }})\right]p({\boldsymbol {\theta }}).}" loading="lazy"></span></dd></dl>

<p>In the <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian</a> approach to this problem, instead of choosing a single parameter vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3959de75cbeb5f25ac5b070c3f25d722d86c7f2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.361ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {\theta }}^{*}}" loading="lazy"></span>, the probability of a given label for a new instance <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span> is computed by integrating over all possible values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span>, weighted according to the posterior probability:

</p>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\rm {label}}|{\boldsymbol {x}})=\int p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})p({\boldsymbol {\theta }}|\mathbf {D} )\operatorname {d} {\boldsymbol {\theta }}.}</annotation>

</semantics>

</math></span><img src="./a492783cae9cd0bdd661aa0b6a81d7b6723d6c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:38.987ex; height:5.676ex;" alt="{\displaystyle p({\rm {label}}|{\boldsymbol {x}})=\int p({\rm {label}}|{\boldsymbol {x}},{\boldsymbol {\theta }})p({\boldsymbol {\theta }}|\mathbf {D} )\operatorname {d} {\boldsymbol {\theta }}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Frequentist_or_Bayesian_approach_to_pattern_recognition">Frequentist or Bayesian approach to pattern recognition</h3></div>

<p>The first pattern classifier – the linear discriminant presented by <a href="Fisher_discriminant_analysis" class="mw-redirect" title="Fisher discriminant analysis">Fisher</a> – was developed in the <a href="Frequentist_inference" title="Frequentist inference">frequentist</a> tradition. The frequentist approach entails that the model parameters are considered unknown, but objective. The parameters are then computed (estimated) from the collected data. For the linear discriminant, these parameters are precisely the mean vectors and the <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a>. Also the probability of each class <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}</annotation>

</semantics>

</math></span><img src="./bca4582ca8644ed7f4cfa780d02a8b878c3310ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.802ex; height:2.843ex;" alt="{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}" loading="lazy"></span> is estimated from the collected dataset. Note that the usage of '<a href="Bayes_rule" class="mw-redirect" title="Bayes rule">Bayes rule</a>' in a pattern classifier does not make the classification approach Bayesian.

</p><p><a href="Bayesian_inference" title="Bayesian inference">Bayesian statistics</a> has its origin in Greek philosophy where a distinction was already made between the '<a href="A_priori_and_a_posteriori" title="A priori and a posteriori">a priori</a>' and the '<a href="A_priori_and_a_posteriori" title="A priori and a posteriori">a posteriori</a>' knowledge. Later <a href="A_priori_and_a_posteriori#Immanuel_Kant" title="A priori and a posteriori">Kant</a> defined his distinction between what is a priori known – before observation – and the empirical knowledge gained from observations. In a Bayesian pattern classifier, the class probabilities <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}</annotation>

</semantics>

</math></span><img src="./bca4582ca8644ed7f4cfa780d02a8b878c3310ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.802ex; height:2.843ex;" alt="{\displaystyle p({\rm {label}}|{\boldsymbol {\theta }})}" loading="lazy"></span> can be chosen by the user, which are then a priori. Moreover, experience quantified as a priori parameter values can be weighted with empirical observations – using e.g., the <a href="Beta_distribution" title="Beta distribution">Beta-</a> (<a href="Conjugate_prior_distribution" class="mw-redirect" title="Conjugate prior distribution">conjugate prior</a>) and <a href="Dirichlet_distribution" title="Dirichlet distribution">Dirichlet-distributions</a>. The Bayesian approach facilitates a seamless intermixing between expert knowledge in the form of subjective probabilities, and objective observations.

</p><p>Probabilistic pattern classifiers can be used according to a frequentist or a Bayesian approach.

</p>

<p>Within medical science, pattern recognition is the basis for <a href="Computer-aided_diagnosis" title="Computer-aided diagnosis">computer-aided diagnosis</a> (CAD) systems. CAD describes a procedure that supports the doctor's interpretations and findings. Other typical applications of pattern recognition techniques are automatic <a href="Speech_recognition" title="Speech recognition">speech recognition</a>, <a href="Speaker_identification" class="mw-redirect" title="Speaker identification">speaker identification</a>, <a href="Document_classification" title="Document classification">classification of text into several categories</a> (e.g., spam or non-spam email messages), the <a href="Handwriting_recognition" title="Handwriting recognition">automatic recognition of handwriting</a> on postal envelopes, automatic <a href="Image_recognition" class="mw-redirect" title="Image recognition">recognition of images</a> of human faces, or handwriting image extraction from medical forms.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The last two examples form the subtopic <a href="Image_analysis" title="Image analysis">image analysis</a> of pattern recognition that deals with digital images as input to pattern recognition systems.<sup id="cite_ref-duda2001_11-0" class="reference"><a href="#cite_note-duda2001-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p><p>Optical character recognition is an example of the application of a pattern classifier. The method of signing one's name was captured with stylus and overlay starting in 1990. The strokes, speed, relative min, relative max, acceleration and pressure is used to uniquely identify and confirm identity. Banks were first offered this technology, but were content to collect from the FDIC for any bank fraud and did not want to inconvenience customers.

</p><p>Pattern recognition has many real-world applications in image processing. Some examples include:

</p>

<ul><li>identification and authentication: e.g., <a href="License_plate_recognition" class="mw-redirect" title="License plate recognition">license plate recognition</a>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> fingerprint analysis, <a href="Face_detection" title="Face detection">face detection</a>/verification,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and <a href="Voice-based_authentication" class="mw-redirect" title="Voice-based authentication">voice-based authentication</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li>

<li>medical diagnosis: e.g., screening for cervical cancer (Papnet),<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> breast tumors or heart sounds;</li>

<li>defense: various navigation and guidance systems, <a href="Automatic_target_recognition" title="Automatic target recognition">target recognition</a> systems, shape recognition technology etc.</li>

<li>mobility: <a href="Advanced_driver-assistance_systems" class="mw-redirect" title="Advanced driver-assistance systems">advanced driver assistance systems</a>, <a href="Self-driving_car" title="Self-driving car">autonomous vehicle technology</a>, etc.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li></ul>

<p>In psychology, <a href="Pattern_recognition_(psychology)" title="Pattern recognition (psychology)">pattern recognition</a> is used to make sense of and identify objects, and is closely related to perception. This explains how the sensory inputs humans receive are made meaningful. Pattern recognition can be thought of in two different ways. The first concerns template matching and the second concerns feature detection. A template is a pattern used to produce items of the same proportions. The template-matching hypothesis suggests that incoming stimuli are compared with templates in the long-term memory. If there is a match, the stimulus is identified. Feature detection models, such as the Pandemonium system for classifying letters (Selfridge, 1959), suggest that the stimuli are broken down into their component parts for identification. One observation is a capital E having three horizontal lines and one vertical line.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>Algorithms for pattern recognition depend on the type of label output, on whether learning is supervised or unsupervised, and on whether the algorithm is statistical or non-statistical in nature. Statistical algorithms can further be categorized as <a href="Generative_model" title="Generative model">generative</a> or <a href="Discriminative_model" title="Discriminative model">discriminative</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Classification_methods_(methods_predicting_categorical_labels)">Classification methods (methods predicting categorical labels)</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Statistical_classification" title="Statistical classification">Statistical classification</a></div>

<p>Parametric:<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Linear discriminant analysis</a></li>

<li><a href="Quadratic_classifier" title="Quadratic classifier">Quadratic discriminant analysis</a></li>

<li><a href="Maximum_entropy_classifier" class="mw-redirect" title="Maximum entropy classifier">Maximum entropy classifier</a> (aka <a href="Logistic_regression" title="Logistic regression">logistic regression</a>, <a href="Multinomial_logistic_regression" title="Multinomial logistic regression">multinomial logistic regression</a>): Note that logistic regression is an algorithm for classification, despite its name. (The name comes from the fact that logistic regression uses an extension of a linear regression model to model the probability of an input being in a particular class.)</li></ul>

<p>Nonparametric:<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Decision_tree" title="Decision tree">Decision trees</a>, <a href="Decision_list" title="Decision list">decision lists</a></li>

<li><a href="Variable_kernel_density_estimation#Use_for_statistical_classification" title="Variable kernel density estimation">Kernel estimation</a> and <a href="K-nearest-neighbor" class="mw-redirect" title="K-nearest-neighbor">K-nearest-neighbor</a> algorithms</li>

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

<li><a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Neural networks</a> (multi-layer perceptrons)</li>

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

<li><a href="Support_vector_machine" title="Support vector machine">Support vector machines</a></li>

<li><a href="Gene_expression_programming" title="Gene expression programming">Gene expression programming</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Clustering_methods_(methods_for_classifying_and_predicting_categorical_labels)">Clustering methods (methods for classifying and predicting categorical labels)</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></div>

<ul><li>Categorical <a href="Mixture_model" title="Mixture model">mixture models</a></li>

<li><a href="Hierarchical_clustering" title="Hierarchical clustering">Hierarchical clustering</a> (agglomerative or divisive)</li>

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

<li><a href="Correlation_clustering" title="Correlation clustering">Correlation clustering</a></li>

<li><a href="Kernel_principal_component_analysis" title="Kernel principal component analysis">Kernel principal component analysis</a> (Kernel PCA)</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Ensemble_learning_algorithms_(supervised_meta-algorithms_for_combining_multiple_learning_algorithms_together)">Ensemble learning algorithms (supervised meta-algorithms for combining multiple learning algorithms together)</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Ensemble_learning" title="Ensemble learning">Ensemble learning</a></div>

<ul><li><a href="Boosting_(meta-algorithm)" class="mw-redirect" title="Boosting (meta-algorithm)">Boosting (meta-algorithm)</a></li>

<li><a href="Bootstrap_aggregating" title="Bootstrap aggregating">Bootstrap aggregating</a> ("bagging")</li>

<li><a href="Ensemble_averaging" class="mw-redirect" title="Ensemble averaging">Ensemble averaging</a></li>

<li><a href="Mixture_of_experts" title="Mixture of experts">Mixture of experts</a>, <a href="Hierarchical_mixture_of_experts" class="mw-redirect" title="Hierarchical mixture of experts">hierarchical mixture of experts</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="General_methods_for_predicting_arbitrarily-structured_(sets_of)_labels">General methods for predicting arbitrarily-structured (sets of) labels</h3></div>

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

<li><a href="Markov_random_field" title="Markov random field">Markov random fields</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Multilinear_subspace_learning_algorithms_(predicting_labels_of_multidimensional_data_using_tensor_representations)">Multilinear subspace learning algorithms (predicting labels of multidimensional data using tensor representations)</h3></div>

<p>Unsupervised:

</p>

<ul><li><a href="Multilinear_principal_component_analysis" title="Multilinear principal component analysis">Multilinear principal component analysis</a> (MPCA)</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Real-valued_sequence_labeling_methods_(predicting_sequences_of_real-valued_labels)">Real-valued sequence labeling methods (predicting sequences of real-valued labels)</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Sequence_labeling" title="Sequence labeling">sequence labeling</a></div>

<ul><li><a href="Kalman_filter" title="Kalman filter">Kalman filters</a></li>

<li><a href="Particle_filter" title="Particle filter">Particle filters</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Regression_methods_(predicting_real-valued_labels)">Regression methods (predicting real-valued labels)</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Regression_analysis" title="Regression analysis">Regression analysis</a></div>

<ul><li><a href="Gaussian_process_regression" class="mw-redirect" title="Gaussian process regression">Gaussian process regression</a> (kriging)</li>

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

<li><a href="Independent_component_analysis" title="Independent component analysis">Independent component analysis</a> (ICA)</li>

<li><a href="Principal_components_analysis" class="mw-redirect" title="Principal components analysis">Principal components analysis</a> (PCA)</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Sequence_labeling_methods_(predicting_sequences_of_categorical_labels)">Sequence labeling methods (predicting sequences of categorical labels)</h3></div>

<ul><li><a href="Conditional_random_field" title="Conditional random field">Conditional random fields</a> (CRFs)</li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov models</a> (HMMs)</li>

<li><a href="Maximum_entropy_Markov_model" class="mw-redirect" title="Maximum entropy Markov model">Maximum entropy Markov models</a> (MEMMs)</li>

<li><a href="Recurrent_neural_networks" class="mw-redirect" title="Recurrent neural networks">Recurrent neural networks</a> (RNNs)</li>

<li><a href="Dynamic_time_warping" title="Dynamic time warping">Dynamic time warping</a> (DTW)</li></ul>

<p><br>

</p>

<ul><li><a href="Adaptive_resonance_theory" title="Adaptive resonance theory">Adaptive resonance theory</a> – Theory in neuropsychology</li>

<li><a href="Black_box" title="Black box">Black box</a> – System where only the inputs and outputs can be viewed, and not its implementation</li>

<li><a href="Cache_language_model" title="Cache language model">Cache language model</a></li>

<li><a href="Compound-term_processing" title="Compound-term processing">Compound-term processing</a></li>

<li><a href="Computer-aided_diagnosis" title="Computer-aided diagnosis">Computer-aided diagnosis</a> – Type of diagnosis assisted by computers</li>

<li><a href="Contextual_image_classification" title="Contextual image classification">Contextual image classification</a></li>

<li><a href="Data_mining" title="Data mining">Data mining</a> – Process of extracting and discovering patterns in large data sets</li>

<li><a href="Deep_learning" title="Deep learning">Deep learning</a> – Branch of machine learning</li>

<li><a href="Grey_box_model" title="Grey box model">Grey box model</a> – Mathematical data production model with limited structure</li>

<li><a href="Information_theory" title="Information theory">Information theory</a> – Scientific study of digital information</li>

<li><a href="List_of_datasets_for_machine_learning_research" class="mw-redirect" title="List of datasets for machine learning research">List of datasets for machine learning research</a></li>

<li><a href="List_of_numerical-analysis_software" title="List of numerical-analysis software">List of numerical-analysis software</a></li>

<li><a href="List_of_numerical_libraries" title="List of numerical libraries">List of numerical libraries</a></li>

<li><a href="Neocognitron" title="Neocognitron">Neocognitron</a> – Type of artificial neural network</li>

<li><a href="Perception" title="Perception">Perception</a> – Interpretation of sensory information</li>

<li><a href="Perceptual_learning" title="Perceptual learning">Perceptual learning</a> – Process of learning better perception skills</li>

<li><a href="Predictive_analytics" title="Predictive analytics">Predictive analytics</a> – Statistical techniques analyzing facts to make predictions about unknown events</li>

<li><a href="Prior_knowledge_for_pattern_recognition" title="Prior knowledge for pattern recognition">Prior knowledge for pattern recognition</a></li>

<li><a href="Sequence_mining" class="mw-redirect" title="Sequence mining">Sequence mining</a> – Data mining technique<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Template_matching" title="Template matching">Template matching</a> – Technique in digital image processing</li></ul>

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

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<cite id="CITEREFIman_ForoutanJack_Sklansky1987" class="citation journal cs1">Iman Foroutan; Jack Sklansky (1987). "Feature Selection for Automatic Classification of Non-Gaussian Data". <i>IEEE Transactions on Systems, Man, and Cybernetics</i>. <b>17</b> (2): <span class="nowrap">187–</span>198. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTSMC.1987.4309029">10.1109/TSMC.1987.4309029</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:9871395">9871395</a>.</cite>.</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">For <a href="Linear_discriminant_analysis" title="Linear discriminant analysis">linear discriminant analysis</a> the parameter vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"></span> consists of the two mean vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mu }}_{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3225ecfd1bfdbfd80a9df95e857ce15ef0102c9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.7ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\mu }}_{1}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mu }}_{2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./02c7c5a478c611ea5c840e7e2d89cd10ad771bc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.7ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\mu }}_{2}}" loading="lazy"></span> and the common <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Sigma }}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8532511177f5a2d2dc2b8c1ea37d483c70266911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Sigma }}}" loading="lazy"></span>.</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFMilewskiGovindaraju,_Venu2008" class="citation journal cs1">Milewski, Robert; Govindaraju, Venu (31 March 2008). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://dl.acm.org/citation.cfm?id=1324656">"Binarization and cleanup of handwritten text from carbon copy medical form images"</a></span>. <i>Pattern Recognition</i>. <b>41</b> (4): <span class="nowrap">1308–</span>1315. <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/2008PatRe..41.1308M">2008PatRe..41.1308M</a>. <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.patcog.2007.08.018">10.1016/j.patcog.2007.08.018</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200910174840/https://dl.acm.org/doi/10.1016/j.patcog.2007.08.018">Archived</a> from the original on 10 September 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">26 October</span> 2011</span>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFSarangiSahidullah,_MdSaha,_Goutam2020" class="citation journal cs1">Sarangi, Susanta; Sahidullah, Md; Saha, Goutam (September 2020). "Optimization of data-driven filterbank for automatic speaker verification". <i>Digital Signal Processing</i>. <b>104</b>: 102795. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2007.10729">2007.10729</a></span>. <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/2020DSP...10402795S">2020DSP...10402795S</a>. <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.dsp.2020.102795">10.1016/j.dsp.2020.102795</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:220665533">220665533</a>.</cite></span>

</li>

<li id="cite_note-duda2001-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-duda2001_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRichard_O._Duda,_Peter_E._Hart,_David_G._Stork2001" class="citation book cs1"><a href="Richard_O._Duda" title="Richard O. Duda">Richard O. Duda</a>, <a href="Peter_E._Hart" title="Peter E. Hart">Peter E. Hart</a>, <a href="David_G._Stork" title="David G. Stork">David G. Stork</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Br33IRC3PkQC"><i>Pattern classification</i></a> (2nd ed.). Wiley, New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-05669-0</bdi>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200819004737/https://books.google.com/books?id=Br33IRC3PkQC">Archived</a> from the original on 2020-08-19<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-26</span></span>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">R. Brunelli, <i>Template Matching Techniques in Computer Vision: Theory and Practice</i>, Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-51706-2</bdi>, 2009</span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://anpr-tutorial.com/">The Automatic Number Plate Recognition Tutorial</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060820175245/http://www.anpr-tutorial.com/">Archived</a> 2006-08-20 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> <a rel="nofollow" class="external free" href="http://anpr-tutorial.com/">http://anpr-tutorial.com/</a> </span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.cs.cmu.edu/afs/cs.cmu.edu/usr/mitchell/ftp/faces.html">Neural Networks for Face Recognition</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304065030/http://www.cs.cmu.edu/afs/cs.cmu.edu/usr/mitchell/ftp/faces.html">Archived</a> 2016-03-04 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Companion to Chapter 4 of the textbook Machine Learning.</span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFPoddarSahidullah,_MdSaha,_Goutam2018" class="citation journal cs1">Poddar, Arnab; Sahidullah, Md; Saha, Goutam (March 2018). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190903174139/https://ieeexplore.ieee.org/document/8302747/">"Speaker Verification with Short Utterances: A Review of Challenges, Trends and Opportunities"</a>. <i>IET Biometrics</i>. <b>7</b> (2): <span class="nowrap">91–</span>101. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fiet-bmt.2017.0065">10.1049/iet-bmt.2017.0065</a>. Archived from <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/8302747">the original</a> on 2019-09-03<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-27</span></span>.</cite></span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://health-asia.org/papnet-for-cervical-screening/">PAPNET For Cervical Screening</a> <a rel="nofollow" class="external text" href="https://archive.today/20120708211332/http://health-asia.org/papnet-for-cervical-screening/">Archived</a> 2012-07-08 at <a href="Archive.today" title="Archive.today">archive.today</a></span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://saemobilus.sae.org/content/2018-01-0035">"Development of an Autonomous Vehicle Control Strategy Using a Single Camera and Deep Neural Networks (2018-01-0035 Technical Paper)- SAE Mobilus"</a>. <i>saemobilus.sae.org</i>. 3 April 2018. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4271%2F2018-01-0035">10.4271/2018-01-0035</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190906084436/https://saemobilus.sae.org/content/2018-01-0035">Archived</a> from the original on 2019-09-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-09-06</span></span>.</cite></span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFGerdesKegelmanKapaniaBrown2019" class="citation journal cs1">Gerdes, J. Christian; Kegelman, John C.; Kapania, Nitin R.; Brown, Matthew; Spielberg, Nathan A. (2019-03-27). <a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscirobotics.aaw1975">"Neural network vehicle models for high-performance automated driving"</a>. <i>Science Robotics</i>. <b>4</b> (28): eaaw1975. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscirobotics.aaw1975">10.1126/scirobotics.aaw1975</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2470-9476">2470-9476</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/33137751">33137751</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:89616974">89616974</a>.</cite></span>

</li>

<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFPickering2017" class="citation web cs1">Pickering, Chris (2017-08-15). <a rel="nofollow" class="external text" href="https://www.theengineer.co.uk/ai-autonomous-cars/">"How AI is paving the way for fully autonomous cars"</a>. <i>The Engineer</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190906084433/https://www.theengineer.co.uk/ai-autonomous-cars/">Archived</a> from the original on 2019-09-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-09-06</span></span>.</cite></span>

</li>

<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFRayJanaPeiTian2017" class="citation journal cs1">Ray, Baishakhi; Jana, Suman; Pei, Kexin; Tian, Yuchi (2017-08-28). "DeepTest: Automated Testing of Deep-Neural-Network-driven Autonomous Cars". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1708.08559">1708.08559</a></span>. <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/2017arXiv170808559T">2017arXiv170808559T</a>.</cite> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{cite journal}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (help)</span></span>

</li>

<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><cite id="CITEREFSinhaHadjiiskiMutib1993" class="citation journal cs1">Sinha, P. K.; Hadjiiski, L. M.; Mutib, K. (1993-04-01). "Neural Networks in Autonomous Vehicle Control". <i>IFAC Proceedings Volumes</i>. 1st IFAC International Workshop on Intelligent Autonomous Vehicles, Hampshire, UK, 18–21 April. <b>26</b> (1): <span class="nowrap">335–</span>340. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS1474-6670%2817%2949322-0">10.1016/S1474-6670(17)49322-0</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/1474-6670">1474-6670</a>.</cite></span>

</li>

<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.s-cool.co.uk/a-level/psychology/attention/revise-it/pattern-recognition">"A-level Psychology Attention Revision - Pattern recognition | S-cool, the revision website"</a>. S-cool.co.uk. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130622023719/http://www.s-cool.co.uk/a-level/psychology/attention/revise-it/pattern-recognition">Archived</a> from the original on 2013-06-22<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-09-17</span></span>.</cite></span>

</li>

<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Assuming known distributional shape of feature distributions per class, such as the <a href="Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian</a> shape.</span>

</li>

<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">No distributional assumption regarding shape of feature distributions per class.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFFukunaga1990" class="citation book cs1">Fukunaga, Keinosuke (1990). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontost1990fuku"><i>Introduction to Statistical Pattern Recognition</i></a></span> (2nd ed.). Boston: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-269851-4</bdi>.</cite></li>

<li><cite id="CITEREFHorneggerPaulus1999" class="citation book cs1">Hornegger, Joachim; Paulus, Dietrich W. R. (1999). <i>Applied Pattern Recognition: A Practical Introduction to Image and Speech Processing in C++</i> (2nd ed.). San Francisco: Morgan Kaufmann Publishers. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-528-15558-2</bdi>.</cite></li>

<li><cite id="CITEREFSchuermann1996" class="citation book cs1">Schuermann, Juergen (1996). <i>Pattern Classification: A Unified View of Statistical and Neural Approaches</i>. New York: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-13534-0</bdi>.</cite></li>

<li><cite id="CITEREFGodfried_T._Toussaint1988" class="citation book cs1">Godfried T. Toussaint, ed. (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ObOjBQAAQBAJ"><i>Computational Morphology</i></a>. Amsterdam: North-Holland Publishing Company. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781483296722</bdi>.</cite></li>

<li><cite id="CITEREFKulikowskiWeiss1991" class="citation book cs1">Kulikowski, Casimir A.; Weiss, Sholom M. (1991). <i>Computer Systems That Learn: Classification and Prediction Methods from Statistics, Neural Nets, Machine Learning, and Expert Systems</i>. San Francisco: Morgan Kaufmann Publishers. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-55860-065-2</bdi>.</cite></li>

<li><cite id="CITEREFDudaHartStork2000" class="citation book cs1">Duda, Richard O.; Hart, Peter E.; Stork, David G. (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Br33IRC3PkQC"><i>Pattern Classification</i></a> (2nd ed.). Wiley-Interscience. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0471056690</bdi>.</cite></li>

<li><cite id="CITEREFJainDuinMao2000" class="citation journal cs1">Jain, Anil.K.; Duin, Robert.P.W.; Mao, Jianchang (2000). "Statistical pattern recognition: a review". <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>. <b>22</b> (1): <span class="nowrap">4–</span>37. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.123.8151">10.1.1.123.8151</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F34.824819">10.1109/34.824819</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:192934">192934</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140911114525/http://egmont-petersen.nl/classifiers.htm">An introductory tutorial to classifiers (introducing the basic terms, with numeric example)</a></li>

<li><cite id="CITEREFKovalevsky1980" class="citation book cs1">Kovalevsky, V. A. (1980). <i>Image Pattern Recognition</i>. New York, NY: Springer New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-6033-2</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/852790446">852790446</a>.</cite></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.iapr.org">The International Association for Pattern Recognition</a></li>

<li><a rel="nofollow" class="external text" href="http://cgm.cs.mcgill.ca/~godfried/teaching/pr-web.html">List of Pattern Recognition web sites</a></li>

<li><a rel="nofollow" class="external text" href="http://www.jprr.org">Journal of Pattern Recognition Research</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080908110041/http://www.jprr.org/">Archived</a> 2008-09-08 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120302040520/http://www.docentes.unal.edu.co/morozcoa/docs/pr.php">Pattern Recognition Info</a></li>

<li><a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/journal/00313203">Pattern Recognition</a> (Journal of the Pattern Recognition Society)</li>

<li><a rel="nofollow" class="external text" href="http://www.worldscinet.com/ijprai/mkt/archive.shtml">International Journal of Pattern Recognition and Artificial Intelligence</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041211192151/http://www.worldscinet.com/ijprai/mkt/archive.shtml">Archived</a> 2004-12-11 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="http://www.inderscience.com/ijapr">International Journal of Applied Pattern Recognition</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150215163124/http://www.openpr.org.cn/">Open Pattern Recognition Project</a>, intended to be an open source platform for sharing algorithms of pattern recognition</li>

<li><a rel="nofollow" class="external text" href="https://www.academia.edu/31957815/Improved_Pattern_Matching_Applied_to_Surface_Mounting_Devices_Components_Localization_on_Automated_Optical_Inspection">Improved Fast Pattern Matching</a> Improved Fast Pattern Matching</li></ul>

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