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Granular computing

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Granular computing is an emerging <a href="Computing" title="Computing">computing</a> paradigm of <a href="Data_processing" title="Data processing">information processing</a> that concerns the processing of complex information entities called "information <a href="Granulation" title="Granulation">granules</a>", which arise in the process of data abstraction and <a href="Knowledge_extraction" title="Knowledge extraction">derivation of knowledge</a> from information or data. Generally speaking, information granules are collections of entities that usually originate at the numeric level and are arranged together due to their <a href="Similarity_measure" title="Similarity measure">similarity</a>, functional or physical adjacency, indistinguishability, coherency, or the like.

At present, granular computing is more a theoretical perspective than a coherent set of methods or principles. As a theoretical perspective, it encourages an approach to data that recognizes and exploits the knowledge present in data at various levels of resolution or scales. In this sense, it encompasses all methods which provide flexibility and adaptability in the resolution at which knowledge or information is extracted and represented.

<div class="mw-heading mw-heading2"><h2 id="Types_of_granulation">Types of granulation</h2></div>

As mentioned above, granular computing is not an algorithm or process; there is no particular method that is called "granular computing". It is rather an approach to looking at data that recognizes how different and interesting regularities in the data can appear at different levels of granularity, much as different features become salient in <a href="Satellite_images" class="mw-redirect" title="Satellite images">satellite images</a> of greater or lesser resolution. On a low-resolution satellite image, for example, one might notice interesting cloud patterns representing <a href="Cyclones" class="mw-redirect" title="Cyclones">cyclones</a> or other large-scale weather phenomena, while in a higher-resolution image, one misses these large-scale atmospheric phenomena but instead notices smaller-scale phenomena, such as the interesting pattern that is the streets of <a href="Manhattan" title="Manhattan">Manhattan</a>. The same is generally true of all data: At different resolutions or granularities, different features and relationships emerge. The aim of granular computing is to try to take advantage of this fact in designing more effective machine-learning and reasoning systems.

There are several types of granularity that are often encountered in <a href="Data_mining" title="Data mining">data mining</a> and <a href="Machine_learning" title="Machine learning">machine learning</a>, and we review them below:

<div class="mw-heading mw-heading3"><h3 id="Value_granulation_(discretization/quantization)">Value granulation (discretization/quantization)</h3></div>

One type of granulation is the <a href="Quantization_(signal_processing)" title="Quantization (signal processing)">quantization</a> of variables. It is very common that in data mining or machine-learning applications the resolution of variables needs to be decreased in order to extract meaningful regularities. An example of this would be a variable such as "outside temperature" (temp), which in a given application might be recorded to several decimal places of <a href="Arithmetic_precision" class="mw-redirect" title="Arithmetic precision">precision</a> (depending on the sensing apparatus). However, for purposes of extracting relationships between "outside temperature" and, say, "number of health-club applications" (club), it will generally be advantageous to quantize "outside temperature" into a smaller number of intervals.

<div class="mw-heading mw-heading4"><h4 id="Motivations">Motivations</h4></div>

There are several interrelated reasons for granulating variables in this fashion:

<ul><li>Based on prior <a href="Domain_knowledge" title="Domain knowledge">domain knowledge</a>, there is no expectation that minute variations in temperature (e.g., the difference between 80–80.7 °F (26.7–27.1 °C)) could have an influence on behaviors driving the number of health-club applications. For this reason, any "regularity" which our learning algorithms might detect at this level of resolution would have to be spurious, as an artifact of overfitting. By coarsening the temperature variable into intervals the difference between which we do anticipate (based on prior domain knowledge) might influence number of health-club applications, we eliminate the possibility of detecting these spurious patterns. Thus, in this case, reducing resolution is a method of controlling <a href="Overfitting" title="Overfitting">overfitting</a>.</li>

<li>By reducing the number of intervals in the temperature variable (i.e., increasing its grain size), we increase the amount of sample data indexed by each interval designation. Thus, by coarsening the variable, we increase sample sizes and achieve better statistical estimation. In this sense, increasing granularity provides an antidote to the so-called <a href="Curse_of_dimensionality" title="Curse of dimensionality">curse of dimensionality</a>, which relates to the exponential decrease in statistical power with increase in number of dimensions or variable cardinality.</li>

<li>Independent of prior domain knowledge, it is often the case that meaningful regularities (i.e., which can be detected by a given learning methodology, representational language, etc.) may exist at one level of resolution and not at another.</li></ul>

For example, a simple learner or pattern recognition system may seek to extract regularities satisfying a <a href="Conditional_probability" title="Conditional probability">conditional probability</a> threshold such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(Y=y_{j}|X=x_{i})\geq \alpha .}">

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<annotation encoding="application/x-tex">{\displaystyle p(Y=y_{j}|X=x_{i})\geq \alpha .}</annotation>

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</math><img src="./3d5808fda4455bfeaf1fc9cdb67f7481ea886788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:23.076ex; height:3.009ex;" alt="{\displaystyle p(Y=y_{j}|X=x_{i})\geq \alpha .}" loading="lazy"> In the special case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1,}">

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

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</math><img src="./b99d217bfa3a15fc9dbe0a37586a63db0b75990a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.395ex; height:2.509ex;" alt="{\displaystyle \alpha =1,}" loading="lazy"> this recognition system is essentially detecting <a href="Logical_implication" class="mw-redirect" title="Logical implication">logical implication</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=x_{i}\rightarrow Y=y_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle X=x_{i}\rightarrow Y=y_{j}}</annotation>

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</math><img src="./c6e6547abea84000c3a5c81697454265922b9c9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.743ex; height:2.843ex;" alt="{\displaystyle X=x_{i}\rightarrow Y=y_{j}}" loading="lazy"> or, in words, "if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=x_{i},}">

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

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</math><img src="./352564ce502516b5b11216d56e896f077f48b9c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.855ex; height:2.509ex;" alt="{\displaystyle X=x_{i},}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=y_{j}}">

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

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</math><img src="./5f23394ebacd10d6994e3a2d6ad0b914128a7f29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.921ex; height:2.843ex;" alt="{\displaystyle Y=y_{j}}" loading="lazy">". The system's ability to recognize such implications (or, in general, conditional probabilities exceeding threshold) is partially contingent on the resolution with which the system analyzes the variables.

As an example of this last point, consider the feature space shown to the right. The variables may each be regarded at two different resolutions. Variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> may be regarded at a high (quaternary) resolution wherein it takes on the four values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{1},x_{2},x_{3},x_{4}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x_{1},x_{2},x_{3},x_{4}\}}</annotation>

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</math><img src="./610554a4014fd1ead9f53c7ebfe15c34b0a50842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.962ex; height:2.843ex;" alt="{\displaystyle \{x_{1},x_{2},x_{3},x_{4}\}}" loading="lazy"> or at a lower (binary) resolution wherein it takes on the two values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{1},X_{2}\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \{X_{1},X_{2}\}.}</annotation>

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</math><img src="./4d6f4606573fece1b2d91373bdff7c584af618d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.963ex; height:2.843ex;" alt="{\displaystyle \{X_{1},X_{2}\}.}" loading="lazy"> Similarly, variable <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><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> may be regarded at a high (quaternary) resolution or at a lower (binary) resolution, where it takes on the values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{1},y_{2},y_{3},y_{4}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{y_{1},y_{2},y_{3},y_{4}\}}</annotation>

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</math><img src="./b000d161969cde313ae5b0cfd9d32cadc249353d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.201ex; height:2.843ex;" alt="{\displaystyle \{y_{1},y_{2},y_{3},y_{4}\}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{Y_{1},Y_{2}\},}">

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

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</math><img src="./7d810185a85fb4b57a9fee5fcdd034e36e2fb559.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.815ex; height:2.843ex;" alt="{\displaystyle \{Y_{1},Y_{2}\},}" loading="lazy"> respectively. At the high resolution, there are no detectable implications of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=x_{i}\rightarrow Y=y_{j},}">

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<annotation encoding="application/x-tex">{\displaystyle X=x_{i}\rightarrow Y=y_{j},}</annotation>

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</math><img src="./a67985d97398929ba0dbf7267b5efae6a993d46c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.389ex; height:2.843ex;" alt="{\displaystyle X=x_{i}\rightarrow Y=y_{j},}" loading="lazy"> since every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> is associated with more than one <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{j},}">

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

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</math><img src="./5080e0d3456dd22b4cbbfd3d6ecab4efdf504dcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.696ex; height:2.343ex;" alt="{\displaystyle y_{j},}" loading="lazy"> and thus, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i},}">

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

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</math><img src="./bb5a9bebf6a6fd218e4cf7f5e63dac56d42bcba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.009ex;" alt="{\displaystyle x_{i},}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(Y=y_{j}|X=x_{i})<1.}">

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<annotation encoding="application/x-tex">{\displaystyle p(Y=y_{j}|X=x_{i})<1.}</annotation>

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</math><img src="./f486bc6a751c696627244056826d2a9f8dee12be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:22.751ex; height:3.009ex;" alt="{\displaystyle p(Y=y_{j}|X=x_{i})<1.}" loading="lazy"> However, at the low (binary) variable resolution, two bilateral implications become detectable: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=X_{1}\leftrightarrow Y=Y_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle X=X_{1}\leftrightarrow Y=Y_{1}}</annotation>

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</math><img src="./1c4a88736d7dbd80b8893ff91ddbf9ab7d8dcb7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.948ex; height:2.509ex;" alt="{\displaystyle X=X_{1}\leftrightarrow Y=Y_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=X_{2}\leftrightarrow Y=Y_{2}}">

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</math><img src="./a2e755833e8685783c0c92ed4b54e04ee1bacd98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.948ex; height:2.509ex;" alt="{\displaystyle X=X_{2}\leftrightarrow Y=Y_{2}}" loading="lazy">, since every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">

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</math><img src="./f70b2694445a5901b24338a2e7a7e58f02a72a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}" loading="lazy"> occurs iff <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{1}}">

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

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</math><img src="./c909ab9869a6ca440f1244dc0e494f42b741b937.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle Y_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">

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</math><img src="./2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy"> occurs iff <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{2}.}">

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

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</math><img src="./958f9bab47e3bd3a7a200ace38850f9814642514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.052ex; height:2.509ex;" alt="{\displaystyle Y_{2}.}" loading="lazy"> Thus, a pattern recognition system scanning for implications of this kind would find them at the binary variable resolution, but would fail to find them at the higher quaternary variable resolution.

<div class="mw-heading mw-heading4"><h4 id="Issues_and_methods">Issues and methods</h4></div>

It is not feasible to exhaustively test all possible discretization resolutions on all variables in order to see which combination of resolutions yields interesting or significant results. Instead, the feature space must be preprocessed (often by an <a href="Information_entropy" class="mw-redirect" title="Information entropy">entropy</a> analysis of some kind) so that some guidance can be given as to how the discretization process should proceed. Moreover, one cannot generally achieve good results by naively analyzing and discretizing each variable independently, since this may obliterate the very interactions that we had hoped to discover.

A sample of papers that address the problem of variable discretization in general, and multiple-variable discretization in particular, is as follows: <a href="#CITEREFChiuWongCheung1991">Chiu, Wong & Cheung (1991)</a>, <a href="#CITEREFBay2001">Bay (2001)</a>, <a href="#CITEREFLiuHussainTanDasii2002">Liu et al. (2002)</a>, <a href="#CITEREFWangLiu1998">Wang & Liu (1998)</a>, <a href="#CITEREFZighedRabasédaRakotomalala1998">Zighed, Rabaséda & Rakotomalala (1998)</a>, <a href="#CITEREFCatlett1991">Catlett (1991)</a>, <a href="#CITEREFDoughertyKohaviSahami1995">Dougherty, Kohavi & Sahami (1995)</a>, <a href="#CITEREFMontiCooper1999">Monti & Cooper (1999)</a>, <a href="#CITEREFFayyadIrani1993">Fayyad & Irani (1993)</a>, <a href="#CITEREFChiuCheungWong1990">Chiu, Cheung & Wong (1990)</a>, <a href="#CITEREFNguyenNguyen1998">Nguyen & Nguyen (1998)</a>, <a href="#CITEREFGrzymala-BusseStefanowski2001">Grzymala-Busse & Stefanowski (2001)</a>, <a href="#CITEREFTing1994">Ting (1994)</a>, <a href="#CITEREFLudlWidmer2000">Ludl & Widmer (2000)</a>, <a href="#CITEREFPfahringer1995">Pfahringer (1995)</a>, <a href="#CITEREFAnCercone1999">An & Cercone (1999)</a>,

<a href="#CITEREFChiuCheung1989">Chiu & Cheung (1989)</a>, <a href="#CITEREFChmielewskiGrzymala-Busse1996">Chmielewski & Grzymala-Busse (1996)</a>, <a href="#CITEREFLeeShin1994">Lee & Shin (1994)</a>, <a href="#CITEREFLiuWellman2002">Liu & Wellman (2002)</a>, <a href="#CITEREFLiuWellman2004">Liu & Wellman (2004)</a>.

<div class="mw-heading mw-heading3"><h3 id="Variable_granulation_(clustering/aggregation/transformation)">Variable granulation (clustering/aggregation/transformation)</h3></div>

Variable granulation is a term that could describe a variety of techniques, most of which are aimed at reducing dimensionality, redundancy, and storage requirements. We briefly describe some of the ideas here, and present pointers to the literature.

<div class="mw-heading mw-heading4"><h4 id="Variable_transformation">Variable transformation</h4></div>

A number of classical methods, such as <a href="Principal_component_analysis" title="Principal component analysis">principal component analysis</a>, <a href="Multidimensional_scaling" title="Multidimensional scaling">multidimensional scaling</a>, <a href="Factor_analysis" title="Factor analysis">factor analysis</a>, and <a href="Structural_equation_modeling" title="Structural equation modeling">structural equation modeling</a>, and their relatives, fall under the genus of "variable transformation." Also in this category are more modern areas of study such as <a href="Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a>, <a href="Projection_pursuit" title="Projection pursuit">projection pursuit</a>, and <a href="Independent_component_analysis" title="Independent component analysis">independent component analysis</a>. The common goal of these methods in general is to find a representation of the data in terms of new variables, which are a linear or nonlinear transformation of the original variables, and in which important statistical relationships emerge. The resulting variable sets are almost always smaller than the original variable set, and hence these methods can be loosely said to impose a granulation on the feature space. These dimensionality reduction methods are all reviewed in the standard texts, such as <a href="#CITEREFDudaHartStork2001">Duda, Hart & Stork (2001)</a>, <a href="#CITEREFWittenFrank2005">Witten & Frank (2005)</a>, and <a href="#CITEREFHastieTibshiraniFriedman2001">Hastie, Tibshirani & Friedman (2001)</a>.

<div class="mw-heading mw-heading4"><h4 id="Variable_aggregation">Variable aggregation</h4></div>

A different class of variable granulation methods derive more from <a href="Data_clustering" class="mw-redirect" title="Data clustering">data clustering</a> methodologies than from the linear systems theory informing the above methods. It was noted fairly early that one may consider "clustering" related variables in just the same way that one considers clustering related data. In data clustering, one identifies a group of similar entities (using a "<a href="Measure_of_similarity" class="mw-redirect" title="Measure of similarity">measure of similarity</a>" suitable to the domain — <a href="#CITEREFMartinoGiulianiRizzi2018">Martino, Giuliani & Rizzi (2018)</a>), and then in some sense replaces those entities with a prototype of some kind. The prototype may be the simple average of the data in the identified cluster, or some other representative measure. But the key idea is that in subsequent operations, we may be able to use the single prototype for the data cluster (along with perhaps a statistical model describing how exemplars are derived from the prototype) to stand in for the much larger set of exemplars. These prototypes are generally such as to capture most of the information of interest concerning the entities.

Similarly, it is reasonable to ask whether a large set of variables might be aggregated into a smaller set of prototype variables that capture the most salient relationships between the variables. Although variable clustering methods based on <a href="Linear_correlation" class="mw-redirect" title="Linear correlation">linear correlation</a> have been proposed (<a href="#CITEREFDudaHartStork2001">Duda, Hart & Stork 2001</a>;<a href="#CITEREFRencher2002">Rencher 2002</a>), more powerful methods of variable clustering are based on the <a href="Mutual_information" title="Mutual information">mutual information</a> between variables. Watanabe has shown (<a href="#CITEREFWatanabe1960">Watanabe 1960</a>;<a href="#CITEREFWatanabe1969">Watanabe 1969</a>) that for any set of variables one can construct a <a href="Polytomy" title="Polytomy">polytomic</a> (i.e., n-ary) tree representing a series of variable agglomerations in which the ultimate "total" correlation among the complete variable set is the sum of the "partial" correlations exhibited by each agglomerating subset (see figure). Watanabe suggests that an observer might seek to thus partition a system in such a way as to minimize the interdependence between the parts "... as if they were looking for a natural division or a hidden crack."

One practical approach to building such a tree is to successively choose for agglomeration the two variables (either atomic variables or previously agglomerated variables) which have the highest pairwise mutual information (<a href="#CITEREFKraskovStögbauerAndrzejakGrassberger2003">Kraskov et al. 2003</a>). The product of each agglomeration is a new (constructed) variable that reflects the local <a href="Joint_distribution" class="mw-redirect" title="Joint distribution">joint distribution</a> of the two agglomerating variables, and thus possesses an entropy equal to their <a href="Joint_entropy" title="Joint entropy">joint entropy</a>.

(From a procedural standpoint, this agglomeration step involves replacing two columns in the attribute-value table—representing the two agglomerating variables—with a single column that has a unique value for every unique combination of values in the replaced columns (<a href="#CITEREFKraskovStögbauerAndrzejakGrassberger2003">Kraskov et al. 2003</a>). No information is lost by such an operation; however, if one is exploring the data for inter-variable relationships, it would generally not be desirable to merge redundant variables in this way, since in such a context it is likely to be precisely the redundancy or dependency between variables that is of interest; and once redundant variables are merged, their relationship to one another can no longer be studied.

<div class="mw-heading mw-heading3"><h3 id="System_granulation_(aggregation)">System granulation (aggregation)</h3></div>

In <a href="Database_systems" class="mw-redirect" title="Database systems">database systems</a>, aggregations (see e.g. <a href="OLAP" class="mw-redirect" title="OLAP">OLAP aggregation</a> and <a href="Business_intelligence" title="Business intelligence">Business intelligence</a> systems) result in transforming original data tables (often called information systems) into the tables with different semantics of rows and columns, wherein the rows correspond to the groups (granules) of original tuples and the columns express aggregated information about original values within each of the groups. Such aggregations are usually based on SQL and its extensions. The resulting granules usually correspond to the groups of original tuples with the same values (or ranges) over some pre-selected original columns.

There are also other approaches wherein the groups are defined basing on, e.g., physical adjacency of rows. For example, <a href="Infobright" title="Infobright">Infobright</a> implemented a database engine wherein data was partitioned onto rough rows, each consisting of 64K of physically consecutive (or almost consecutive) rows. Rough rows were automatically labeled with compact information about their values on data columns, often involving multi-column and multi-table relationships. It resulted in a higher layer of granulated information where objects corresponded to rough rows and attributes - to various aspects of rough information. Database operations could be efficiently supported within such a new framework, with an access to the original data pieces still available (<a href="#CITEREFSlezakSynakWojnaWroblewski2013">Slezak et al. 2013</a>).

<div class="mw-heading mw-heading3"><h3 id="Concept_granulation_(component_analysis)">Concept granulation (component analysis)</h3></div>

The origins of the granular computing ideology are to be found in the <a href="Rough_sets" class="mw-redirect" title="Rough sets">rough sets</a> and <a href="Fuzzy_sets" class="mw-redirect" title="Fuzzy sets">fuzzy sets</a> literatures. One of the key insights of rough set research—although by no means unique to it—is that, in general, the selection of different sets of features or variables will yield different concept granulations. Here, as in elementary rough set theory, by "concept" we mean a set of entities that are indistinguishable or indiscernible to the observer (i.e., a simple concept), or a set of entities that is composed from such simple concepts (i.e., a complex concept). To put it in other words, by projecting a data set (<a href="Value-attribute_system" class="mw-redirect" title="Value-attribute system">value-attribute system</a>) onto different sets of variables, we recognize alternative sets of equivalence-class "concepts" in the data, and these different sets of concepts will in general be conducive to the extraction of different relationships and regularities.

<div class="mw-heading mw-heading4"><h4 id="Equivalence_class_granulation">Equivalence class granulation</h4></div>

We illustrate with an example. Consider the attribute-value system below:

<caption>Sample Information System

</caption>

<th>Object</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./398f438d75434e6fbf48dc232c1ad7228a738568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87858df7457aa93caaef5a316db87a7240cc8c29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation>

</semantics>

</math><img src="./f92db8e65bb75b79799f0f3a29e975b37e227069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation>

</semantics>

</math><img src="./48763cb9875de7b4125c40e95e542e775c1cbc4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation>

</semantics>

</math><img src="./9446d54bf4a29799ff14169ffc7ba4423102fd08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}" loading="lazy">

</th></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92ef134d3a8ed900181455e6b569683132371e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{1}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fa32f5095ddebd2d52739d978bf85f274f900c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{2}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{3}}</annotation>

</semantics>

</math><img src="./224804b26ffc73e496a43debc9015fc42db37876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{3}}" loading="lazy">

</th>

<td>0

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{4}}</annotation>

</semantics>

</math><img src="./ea5c6d29abe52e2bee85d14e5cff906b80f6a885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{4}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{5}}</annotation>

</semantics>

</math><img src="./ef2313d78f5b4423b9f00c809486e238d96eb943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{5}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{6}}</annotation>

</semantics>

</math><img src="./7afccb28ba07863033e9a3fbd99fba4a0455f3c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{6}}" loading="lazy">

</th>

<td>2

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{7}}</annotation>

</semantics>

</math><img src="./a51d680a0fa5a53888840d3d3782f0198150a5df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{7}}" loading="lazy">

</th>

<td>0

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{8}}</annotation>

</semantics>

</math><img src="./449a91d122d110a52deccf0da4309fea311433a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{8}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{9}}</annotation>

</semantics>

</math><img src="./956785d7604633901fb9384647ec154accefaffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{9}}" loading="lazy">

</th>

<td>2

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{10}}</annotation>

</semantics>

</math><img src="./641b8924fd702d194ddbf0a4f58579d3d4a4c3a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.65ex; height:2.509ex;" alt="{\displaystyle O_{10}}" loading="lazy">

</th>

<td>0

</td></tr></tbody></table></dd></dl>

When the full set of attributes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\{P_{1},P_{2},P_{3},P_{4},P_{5}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=\{P_{1},P_{2},P_{3},P_{4},P_{5}\}}</annotation>

</semantics>

</math><img src="./d64ffc3851f42b81501375f04d3f5cc1398a1bfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.037ex; height:2.843ex;" alt="{\displaystyle P=\{P_{1},P_{2},P_{3},P_{4},P_{5}\}}" loading="lazy"> is considered, we see that we have the following seven equivalence classes or primitive (simple) concepts:

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{7},O_{10}\}\\\{O_{4}\}\\\{O_{5}\}\\\{O_{6}\}\\\{O_{8}\}\\\{O_{9}\}\end{cases}}}</annotation>

</semantics>

</math><img src="./a162f977b5a6e2cd678bbe5636a1fb2fb097e41a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:16.515ex; height:20.509ex;" alt="{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{7},O_{10}\}\\\{O_{4}\}\\\{O_{5}\}\\\{O_{6}\}\\\{O_{8}\}\\\{O_{9}\}\end{cases}}}" loading="lazy"></dd></dl>

Thus, the two objects within the first equivalence class, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{O_{1},O_{2}\},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6adc31f23a82e93cf34c95f82b86581e835162ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.661ex; height:2.843ex;" alt="{\displaystyle \{O_{1},O_{2}\},}" loading="lazy"> cannot be distinguished from one another based on the available attributes, and the three objects within the second equivalence class, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{O_{3},O_{7},O_{10}\},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{O_{3},O_{7},O_{10}\},}</annotation>

</semantics>

</math><img src="./cf464bba9e91f7cf3bd3a5b187e87af52abca0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.344ex; height:2.843ex;" alt="{\displaystyle \{O_{3},O_{7},O_{10}\},}" loading="lazy"> cannot be distinguished from one another based on the available attributes. The remaining five objects are each discernible from all other objects. Now, let us imagine a projection of the attribute value system onto attribute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./398f438d75434e6fbf48dc232c1ad7228a738568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}" loading="lazy"> alone, which would represent, for example, the view from an observer which is only capable of detecting this single attribute. Then we obtain the following much coarser equivalence class structure.

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{5},O_{7},O_{9},O_{10}\}\\\{O_{4},O_{6},O_{8}\}\end{cases}}}</annotation>

</semantics>

</math><img src="./9152e7cc845a4d2c5c708d53284bf3a684c4a40b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:24.238ex; height:8.843ex;" alt="{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{5},O_{7},O_{9},O_{10}\}\\\{O_{4},O_{6},O_{8}\}\end{cases}}}" loading="lazy"></dd></dl>

This is in a certain regard the same structure as before, but at a lower degree of resolution (larger grain size). Just as in the case of <a href="#Value_granulation_(discretization/quantization)">value granulation (discretization/quantization)</a>, it is possible that relationships (dependencies) may emerge at one level of granularity that are not present at another. As an example of this, we can consider the effect of concept granulation on the measure known as attribute dependency (a simpler relative of the <a href="Mutual_information" title="Mutual information">mutual information</a>).

To establish this notion of dependency (see also <a href="Rough_sets" class="mw-redirect" title="Rough sets">rough sets</a>), let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}=\{Q_{1},Q_{2},Q_{3},\dots ,Q_{N}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}=\{Q_{1},Q_{2},Q_{3},\dots ,Q_{N}\}}</annotation>

</semantics>

</math><img src="./0e19eec12492f20183cb16c787d108dea4832947.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.032ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}=\{Q_{1},Q_{2},Q_{3},\dots ,Q_{N}\}}" loading="lazy"> represent a particular concept granulation, where each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b9f7193081d440425e522698e80817b5d558df03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}" loading="lazy"> is an equivalence class from the concept structure induced by attribute set Q. For example, if the attribute set Q consists of attribute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./398f438d75434e6fbf48dc232c1ad7228a738568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}" loading="lazy"> alone, as above, then the concept structure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> will be composed of

<mtr>

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<mtd>

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</msub>

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<mtr>

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</mtd>

<mtd>

<msub>

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<msub>

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<mtr>

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</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Q_{1}&=\{O_{1},O_{2}\},\\Q_{2}&=\{O_{3},O_{5},O_{7},O_{9},O_{10}\},\\Q_{3}&=\{O_{4},O_{6},O_{8}\}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./00b3c171aab742634d51613b6416d4b27b41e7fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:28.81ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}Q_{1}&=\{O_{1},O_{2}\},\\Q_{2}&=\{O_{3},O_{5},O_{7},O_{9},O_{10}\},\\Q_{3}&=\{O_{4},O_{6},O_{8}\}.\end{aligned}}}" loading="lazy"></dd></dl>

The dependency of attribute set Q on another attribute set P, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{P}(Q),}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{P}(Q),}</annotation>

</semantics>

</math><img src="./3ed7a1e7fe4bbd4af638c9d97beda04822356010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.965ex; height:2.843ex;" alt="{\displaystyle \gamma _{P}(Q),}" loading="lazy"> is given by

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{P}(Q)={\frac {\left|\sum _{i=1}^{N}{\underline {P}}Q_{i}\right|}{\left|\mathbb {U} \right|}}\leq 1}</annotation>

</semantics>

</math><img src="./81d01c21274cc231d51b567b705de84b3e2c690a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.933ex; height:7.509ex;" alt="{\displaystyle \gamma _{P}(Q)={\frac {\left|\sum _{i=1}^{N}{\underline {P}}Q_{i}\right|}{\left|\mathbb {U} \right|}}\leq 1}" loading="lazy"></dd></dl>

That is, for each equivalence class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b9f7193081d440425e522698e80817b5d558df03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q},}</annotation>

</semantics>

</math><img src="./149ce4aff96b452e9ea53866fe6f4e1aab732847.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.802ex; height:3.009ex;" alt="{\displaystyle [x]_{Q},}" loading="lazy"> we add up the size of its "lower approximation" (see <a href="Rough_sets" class="mw-redirect" title="Rough sets">rough sets</a>) by the attributes in P, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {P}}Q_{i}.}">

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b4b652e4a64016014b95adba42d6d715b3cc35e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.165ex; margin-bottom: -0.827ex; width:5.196ex; height:3.176ex;" alt="{\displaystyle {\underline {P}}Q_{i}.}" loading="lazy"> More simply, this approximation is the number of objects which on attribute set P can be positively identified as belonging to target set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./120a263ce95ecd7e0616775b010579bf2421b6ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.285ex; height:2.509ex;" alt="{\displaystyle Q_{i}.}" loading="lazy"> Added across all equivalence classes in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q},}</annotation>

</semantics>

</math><img src="./149ce4aff96b452e9ea53866fe6f4e1aab732847.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.802ex; height:3.009ex;" alt="{\displaystyle [x]_{Q},}" loading="lazy"> the numerator above represents the total number of objects which—based on attribute set P—can be positively categorized according to the classification induced by attributes Q. The dependency ratio therefore expresses the proportion (within the entire universe) of such classifiable objects, in a sense capturing the "synchronization" of the two concept structures <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{P}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{P}.}</annotation>

</semantics>

</math><img src="./ee7b9ee7603d1e8c6d6d3196e16695f8df490c23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.737ex; height:2.843ex;" alt="{\displaystyle [x]_{P}.}" loading="lazy"> The dependency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{P}(Q)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{P}(Q)}</annotation>

</semantics>

</math><img src="./e8e102ac774612fb4fad3414f094a7017f77d8ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.318ex; height:2.843ex;" alt="{\displaystyle \gamma _{P}(Q)}" loading="lazy"> "can be interpreted as a proportion of such objects in the information system for which it suffices to know the values of attributes in P to determine the values of attributes in Q" (Ziarko & Shan 1995).

Having gotten definitions now out of the way, we can make the simple observation that the choice of concept granularity (i.e., choice of attributes) will influence the detected dependencies among attributes. Consider again the attribute value table from above:

<caption>Sample Information System

</caption>

<th>Object</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./398f438d75434e6fbf48dc232c1ad7228a738568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87858df7457aa93caaef5a316db87a7240cc8c29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation>

</semantics>

</math><img src="./f92db8e65bb75b79799f0f3a29e975b37e227069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation>

</semantics>

</math><img src="./48763cb9875de7b4125c40e95e542e775c1cbc4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}" loading="lazy"></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation>

</semantics>

</math><img src="./9446d54bf4a29799ff14169ffc7ba4423102fd08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}" loading="lazy">

</th></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92ef134d3a8ed900181455e6b569683132371e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{1}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fa32f5095ddebd2d52739d978bf85f274f900c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{2}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{3}}</annotation>

</semantics>

</math><img src="./224804b26ffc73e496a43debc9015fc42db37876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{3}}" loading="lazy">

</th>

<td>0

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{4}}</annotation>

</semantics>

</math><img src="./ea5c6d29abe52e2bee85d14e5cff906b80f6a885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{4}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{5}}</annotation>

</semantics>

</math><img src="./ef2313d78f5b4423b9f00c809486e238d96eb943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{5}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{6}}</annotation>

</semantics>

</math><img src="./7afccb28ba07863033e9a3fbd99fba4a0455f3c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{6}}" loading="lazy">

</th>

<td>2

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{7}}</annotation>

</semantics>

</math><img src="./a51d680a0fa5a53888840d3d3782f0198150a5df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{7}}" loading="lazy">

</th>

<td>0

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{8}}</annotation>

</semantics>

</math><img src="./449a91d122d110a52deccf0da4309fea311433a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{8}}" loading="lazy">

</th>

<td>1

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{9}}</annotation>

</semantics>

</math><img src="./956785d7604633901fb9384647ec154accefaffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.509ex;" alt="{\displaystyle O_{9}}" loading="lazy">

</th>

<td>2

</td></tr>

<tr>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{10}}</annotation>

</semantics>

</math><img src="./641b8924fd702d194ddbf0a4f58579d3d4a4c3a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.65ex; height:2.509ex;" alt="{\displaystyle O_{10}}" loading="lazy">

</th>

<td>0

</td></tr></tbody></table></dd></dl>

Consider the dependency of attribute set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\{P_{4},P_{5}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q=\{P_{4},P_{5}\}}</annotation>

</semantics>

</math><img src="./d6ee8a14b58210942289ca1d3acccf97b04fd38f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.389ex; height:2.843ex;" alt="{\displaystyle Q=\{P_{4},P_{5}\}}" loading="lazy"> on attribute set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\{P_{2},P_{3}\}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=\{P_{2},P_{3}\}.}</annotation>

</semantics>

</math><img src="./cc963462337a78a8a12bf8efe7cba5156bbe6313.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.943ex; height:2.843ex;" alt="{\displaystyle P=\{P_{2},P_{3}\}.}" loading="lazy"> That is, we wish to know what proportion of objects can be correctly classified into classes of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> based on knowledge of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{P}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{P}.}</annotation>

</semantics>

</math><img src="./ee7b9ee7603d1e8c6d6d3196e16695f8df490c23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.737ex; height:2.843ex;" alt="{\displaystyle [x]_{P}.}" loading="lazy"> The equivalence classes of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> and of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{P}}</annotation>

</semantics>

</math><img src="./643078ef0c81ff3460fd4f91aa5dd7163e401bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:2.843ex;" alt="{\displaystyle [x]_{P}}" loading="lazy"> are shown below.

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy">

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{P}}</annotation>

</semantics>

</math><img src="./643078ef0c81ff3460fd4f91aa5dd7163e401bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:2.843ex;" alt="{\displaystyle [x]_{P}}" loading="lazy">

</th></tr>

<tr>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{7},O_{10}\}\\\{O_{4},O_{5},O_{8}\}\\\{O_{6},O_{9}\}\end{cases}}}</annotation>

</semantics>

</math><img src="./92e3ce1f252fe3529cee1c5704441bc3c7ea74d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:16.515ex; height:11.509ex;" alt="{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{7},O_{10}\}\\\{O_{4},O_{5},O_{8}\}\\\{O_{6},O_{9}\}\end{cases}}}" loading="lazy">

</td>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{7},O_{10}\}\\\{O_{4},O_{6}\}\\\{O_{5},O_{9}\}\\\{O_{8}\}\end{cases}}}</annotation>

</semantics>

</math><img src="./dd14b0a10597a927cb90680c5a097b6bb258c84d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:16.515ex; height:14.509ex;" alt="{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{7},O_{10}\}\\\{O_{4},O_{6}\}\\\{O_{5},O_{9}\}\\\{O_{8}\}\end{cases}}}" loading="lazy">

</td></tr></tbody></table></dd></dl>

The objects that can be definitively categorized according to concept structure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

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<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

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</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> based on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{P}}">

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<annotation encoding="application/x-tex">{\displaystyle [x]_{P}}</annotation>

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</math><img src="./643078ef0c81ff3460fd4f91aa5dd7163e401bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:2.843ex;" alt="{\displaystyle [x]_{P}}" loading="lazy"> are those in the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{O_{1},O_{2},O_{3},O_{7},O_{8},O_{10}\},}">

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<annotation encoding="application/x-tex">{\displaystyle \{O_{1},O_{2},O_{3},O_{7},O_{8},O_{10}\},}</annotation>

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</math><img src="./80993dbcfd7ca4758744e3851d69eb6ef396a0f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.929ex; height:2.843ex;" alt="{\displaystyle \{O_{1},O_{2},O_{3},O_{7},O_{8},O_{10}\},}" loading="lazy"> and since there are six of these, the dependency of Q on P, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{P}(Q)=6/10.}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{P}(Q)=6/10.}</annotation>

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</math><img src="./4e9ecf20281fcd1fb2ea2f7e49f0b046035fdd02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.713ex; height:2.843ex;" alt="{\displaystyle \gamma _{P}(Q)=6/10.}" loading="lazy"> This might be considered an interesting dependency in its own right, but perhaps in a particular data mining application only stronger dependencies are desired.

We might then consider the dependency of the smaller attribute set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\{P_{4}\}}">

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<annotation encoding="application/x-tex">{\displaystyle Q=\{P_{4}\}}</annotation>

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</math><img src="./b0e8368be037303759ce5b8232c3f10c41581af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.808ex; height:2.843ex;" alt="{\displaystyle Q=\{P_{4}\}}" loading="lazy"> on the attribute set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\{P_{2},P_{3}\}.}">

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<annotation encoding="application/x-tex">{\displaystyle P=\{P_{2},P_{3}\}.}</annotation>

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</math><img src="./cc963462337a78a8a12bf8efe7cba5156bbe6313.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.943ex; height:2.843ex;" alt="{\displaystyle P=\{P_{2},P_{3}\}.}" loading="lazy"> The move from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\{P_{4},P_{5}\}}">

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<annotation encoding="application/x-tex">{\displaystyle Q=\{P_{4},P_{5}\}}</annotation>

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</math><img src="./d6ee8a14b58210942289ca1d3acccf97b04fd38f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.389ex; height:2.843ex;" alt="{\displaystyle Q=\{P_{4},P_{5}\}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\{P_{4}\}}">

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<annotation encoding="application/x-tex">{\displaystyle Q=\{P_{4}\}}</annotation>

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</math><img src="./b0e8368be037303759ce5b8232c3f10c41581af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.808ex; height:2.843ex;" alt="{\displaystyle Q=\{P_{4}\}}" loading="lazy"> induces a coarsening of the class structure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q},}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle [x]_{Q},}</annotation>

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</math><img src="./149ce4aff96b452e9ea53866fe6f4e1aab732847.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.802ex; height:3.009ex;" alt="{\displaystyle [x]_{Q},}" loading="lazy"> as will be seen shortly. We wish again to know what proportion of objects can be correctly classified into the (now larger) classes of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

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</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> based on knowledge of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{P}.}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{P}.}</annotation>

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</math><img src="./ee7b9ee7603d1e8c6d6d3196e16695f8df490c23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.737ex; height:2.843ex;" alt="{\displaystyle [x]_{P}.}" loading="lazy"> The equivalence classes of the new <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

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<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

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</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> and of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{P}}</annotation>

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</math><img src="./643078ef0c81ff3460fd4f91aa5dd7163e401bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:2.843ex;" alt="{\displaystyle [x]_{P}}" loading="lazy"> are shown below.

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<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

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</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy">

</th>

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle [x]_{P}}</annotation>

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</math><img src="./643078ef0c81ff3460fd4f91aa5dd7163e401bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:2.843ex;" alt="{\displaystyle [x]_{P}}" loading="lazy">

</th></tr>

<tr>

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</mtd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\{O_{1},O_{2},O_{3},O_{7},O_{10}\}\\\{O_{4},O_{5},O_{6},O_{8},O_{9}\}\end{cases}}}</annotation>

</semantics>

</math><img src="./131f956c7dcc191d445ac7a1d435e465dfe34380.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.915ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\{O_{1},O_{2},O_{3},O_{7},O_{10}\}\\\{O_{4},O_{5},O_{6},O_{8},O_{9}\}\end{cases}}}" loading="lazy">

</td>

<mrow>

<mtr>

<mtd>

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<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{7},O_{10}\}\\\{O_{4},O_{6}\}\\\{O_{5},O_{9}\}\\\{O_{8}\}\end{cases}}}</annotation>

</semantics>

</math><img src="./dd14b0a10597a927cb90680c5a097b6bb258c84d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:16.515ex; height:14.509ex;" alt="{\displaystyle {\begin{cases}\{O_{1},O_{2}\}\\\{O_{3},O_{7},O_{10}\}\\\{O_{4},O_{6}\}\\\{O_{5},O_{9}\}\\\{O_{8}\}\end{cases}}}" loading="lazy">

</td></tr></tbody></table></dd></dl>

Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> has a coarser granularity than it did earlier. The objects that can now be definitively categorized according to the concept structure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> based on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{P}}</annotation>

</semantics>

</math><img src="./643078ef0c81ff3460fd4f91aa5dd7163e401bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:2.843ex;" alt="{\displaystyle [x]_{P}}" loading="lazy"> constitute the complete universe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{O_{1},O_{2},\ldots ,O_{10}\}}">

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \{O_{1},O_{2},\ldots ,O_{10}\}}</annotation>

</semantics>

</math><img src="./188a02ef75002f193f7f9d7c9bfc6dbb95417eaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.842ex; height:2.843ex;" alt="{\displaystyle \{O_{1},O_{2},\ldots ,O_{10}\}}" loading="lazy">, and thus the dependency of Q on P, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{P}(Q)=1.}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{P}(Q)=1.}</annotation>

</semantics>

</math><img src="./5ae739d77285d0989ac161ef65c843a3f20acda6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.226ex; height:2.843ex;" alt="{\displaystyle \gamma _{P}(Q)=1.}" loading="lazy"> That is, knowledge of membership according to category set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{P}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{P}}</annotation>

</semantics>

</math><img src="./643078ef0c81ff3460fd4f91aa5dd7163e401bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.09ex; height:2.843ex;" alt="{\displaystyle [x]_{P}}" loading="lazy"> is adequate to determine category membership in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> with complete certainty; In this case we might say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q.}</annotation>

</semantics>

</math><img src="./aa7c8cf211b4c79cba13d1e762d35234398dbcc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.845ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q.}" loading="lazy"> Thus, by coarsening the concept structure, we were able to find a stronger (deterministic) dependency. However, we also note that the classes induced in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}}</annotation>

</semantics>

</math><img src="./f9bf03a25dd6b9c94c846d76d4673e6fea036d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.156ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}}" loading="lazy"> from the reduction in resolution necessary to obtain this deterministic dependency are now themselves large and few in number; as a result, the dependency we found, while strong, may be less valuable to us than the weaker dependency found earlier under the higher resolution view of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]_{Q}.}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle [x]_{Q}.}</annotation>

</semantics>

</math><img src="./a45d9660228bcf067a8ef4438ea0440841015617.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.802ex; height:3.009ex;" alt="{\displaystyle [x]_{Q}.}" loading="lazy">

In general it is not possible to test all sets of attributes to see which induced concept structures yield the strongest dependencies, and this search must be therefore be guided with some intelligence. Papers which discuss this issue, and others relating to intelligent use of granulation, are those by Y.Y. Yao and <a href="Lotfi_Zadeh" class="mw-redirect" title="Lotfi Zadeh">Lotfi Zadeh</a> listed in the <a href="#References">#References</a> below.

<div class="mw-heading mw-heading4"><h4 id="Component_granulation">Component granulation</h4></div>

Another perspective on concept granulation may be obtained from work on parametric models of categories. In <a href="Mixture_model" title="Mixture model">mixture model</a> learning, for example, a set of data is explained as a mixture of distinct <a href="Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian</a> (or other) distributions. Thus, a large amount of data is "replaced" by a small number of distributions. The choice of the number of these distributions, and their size, can again be viewed as a problem of concept granulation. In general, a better fit to the data is obtained by a larger number of distributions or parameters, but in order to extract meaningful patterns, it is necessary to constrain the number of distributions, thus deliberately coarsening the concept resolution. Finding the "right" concept resolution is a tricky problem for which many methods have been proposed (e.g., <a href="Akaike_information_criterion" title="Akaike information criterion">AIC</a>, <a href="Bayesian_information_criterion" title="Bayesian information criterion">BIC</a>, <a href="Minimum_description_length" title="Minimum description length">MDL</a>, etc.), and these are frequently considered under the rubric of "<a href="Model_regularization" class="mw-redirect" title="Model regularization">model regularization</a>".

<div class="mw-heading mw-heading2"><h2 id="Different_interpretations_of_granular_computing">Different interpretations of granular computing</h2></div>

Granular computing can be conceived as a framework of theories, methodologies, techniques, and tools that make use of information granules in the process of problem solving. In this sense, granular computing is used as an umbrella term to cover topics that have been studied in various fields in isolation. By examining all of these existing studies in light of the unified framework of granular computing and extracting their commonalities, it may be possible to develop a general theory for problem solving.

In a more philosophical sense, granular computing can describe a way of thinking that relies on the human ability to perceive the real world under various levels of granularity (i.e., abstraction) in order to abstract and consider only those things that serve a specific interest and to switch among different granularities. By focusing on different levels of granularity, one can obtain different levels of knowledge, as well as a greater understanding of the inherent knowledge structure. Granular computing is thus essential in human problem solving and hence has a very significant impact on the design and implementation of intelligent systems.

<ul><li><a href="Rough_set" title="Rough set">Rough Sets</a>, <a href="Discretization" title="Discretization">Discretization</a></li>

<li><a href="Type-2_Fuzzy_Sets_and_Systems" class="mw-redirect" title="Type-2 Fuzzy Sets and Systems">Type-2 Fuzzy Sets and Systems</a></li></ul>

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

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