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Granular Computing: . . . How to Combine Data . . . Which Specificity . . . Shift-Invariance: . . . Measures of Specificity Used Additional Natural . . . in the Principle of Justifiable Scale-Invariance: . . . Full Scale-Invariance Is . . . Granularity: A Theoretical Let Us Combine Shift- . . . Result Explanation of Empirically Home Page Optimal Selections Title Page ◭◭ ◮◮ Olga Kosheleva and Vladik Kreinovich ◭ ◮ University of Texas at El Paso Page 1 of 44 El Paso, Texas 79968, USA olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit

Granular Computing: . . . How to Combine Data . . . 1. Granular Computing: a Brief Reminder Which Specificity . . . • In many practical situations, it is difficult to deal with Shift-Invariance: . . . the whole amount of data. Additional Natural . . . Scale-Invariance: . . . • It may be that we have too much data. Full Scale-Invariance Is . . . • Then, it is not feasible to apply the usual data process- Let Us Combine Shift- . . . ing algorithms to the data as a whole. Result Home Page • This is the situation known as big data . Title Page • It may be that: ◭◭ ◮◮ – while in principle, it is possible to eventually pro- cess all the data points, ◭ ◮ – this would take longer time than we have, Page 2 of 44 – e.g., when we need to make a decision right away. Go Back Full Screen Close Quit

Granular Computing: . . . How to Combine Data . . . 2. Granular Computing (cont-d) Which Specificity . . . • It may also be that we want to use our intuition to Shift-Invariance: . . . better process the data. Additional Natural . . . Scale-Invariance: . . . • And to use our intuition, we need to present the data Full Scale-Invariance Is . . . in presentable form. Let Us Combine Shift- . . . • There may be other cases when we have too much data. Result Home Page • To deal with such cases, a natural idea is compress the original data into a smaller set. Title Page • The overall amount of available data can be estimated ◭◭ ◮◮ by multiplying: ◭ ◮ – the overall number of data points Page 3 of 44 – by the average amount of bits in each data point. Go Back • In general, each data point does not carry too much Full Screen information. Close Quit

Granular Computing: . . . How to Combine Data . . . 3. Granular Computing (cont-d) Which Specificity . . . • So the main way to decrease the overall amount of Shift-Invariance: . . . information is to decrease the number of data points. Additional Natural . . . Scale-Invariance: . . . • Of course, we could simply take a sample from the Full Scale-Invariance Is . . . original data set. Let Us Combine Shift- . . . • However, this would deprive us of all the information Result provided by the un-used data points. Home Page • A much better idea is to each each new “data point” Title Page correspond to several original ones. ◭◭ ◮◮ • This “combined” data point is known as a granule . ◭ ◮ • The resulting technique is known as granular comput- Page 4 of 44 ing . Go Back • The general idea of granular computing can be traced Full Screen to Lotfi Zadeh. Close Quit

Granular Computing: . . . How to Combine Data . . . 4. Granular Computing (cont-d) Which Specificity . . . • There are many possible types of granules. Shift-Invariance: . . . Additional Natural . . . • The most widely used type of granular computing is Scale-Invariance: . . . clustering , when we: Full Scale-Invariance Is . . . – divide all possible objects Let Us Combine Shift- . . . – into several reasonable groups (clusters). Result • Another widely used type of granularity is histograms , Home Page when: Title Page – we visualize the data ◭◭ ◮◮ – by describing the number of data points in different ◭ ◮ intervals. Page 5 of 44 • Also, instead of several numerical values, Go Back – we can consider intervals (or, more generally, sets ) Full Screen – that contain all – or at least most – of the data points. Close Quit

Granular Computing: . . . How to Combine Data . . . 5. Granular Computing (cont-d) Which Specificity . . . • This is done in histograms, this is done in clustering, Shift-Invariance: . . . and this is done in many other practical situations. Additional Natural . . . Scale-Invariance: . . . • We can consider fuzzy sets , that describe: Full Scale-Invariance Is . . . – not only which values are possible, Let Us Combine Shift- . . . – but also to what degree different data points are Result possible. Home Page • We can consider type-2 fuzzy or probabilistic granules. Title Page ◭◭ ◮◮ • We can consider rough sets , etc. ◭ ◮ Page 6 of 44 Go Back Full Screen Close Quit

Granular Computing: . . . How to Combine Data . . . 6. How to Combine Data Points into a Granule: Which Specificity . . . the Principle of Justifiable Granularity Shift-Invariance: . . . • Suppose that we have selected a group of data points Additional Natural . . . that we want to compress into a granule. Scale-Invariance: . . . Full Scale-Invariance Is . . . • Then, the question is which granule to select based on Let Us Combine Shift- . . . these data points. Result • If we include all data points into a granule, the granule Home Page often becomes too wide to be useful. Title Page • On the other hand: ◭◭ ◮◮ – if the granule is too narrow, ◭ ◮ – it includes only a few of the corresponding points, Page 7 of 44 – and is, thus, also rather useless. Go Back • We thus need to achieve a trade-off between coverage Full Screen and specificity. Close Quit

Granular Computing: . . . How to Combine Data . . . 7. Principle of Justifiable Granularity (cont-d) Which Specificity . . . • In some cases – e.g., in histogram analysis – there are Shift-Invariance: . . . known methods for selecting the optimal granule. Additional Natural . . . Scale-Invariance: . . . • However, in the general case, we have to use semi- Full Scale-Invariance Is . . . empirical rules. Let Us Combine Shift- . . . • Most of these rules are in good accordance with the Result decision theory: Home Page – decisions of a rational decision maker can be de- Title Page scribed as ◭◭ ◮◮ – optimizing the expected value of a utility function ◭ ◮ u ( s ) – that describes the corresponding preference. Page 8 of 44 Go Back Full Screen Close Quit

Granular Computing: . . . How to Combine Data . . . 8. Principle of Justifiable Granularity (cont-d) Which Specificity . . . • In other words, if Shift-Invariance: . . . Additional Natural . . . – after making a selection a , we get situations s 1 , . . . , s n Scale-Invariance: . . . with probabilities p 1 ( a ) , . . . , p n ( a ), Full Scale-Invariance Is . . . – then we should select a for which the expected value Let Us Combine Shift- . . . p 1 ( a ) · u ( s 1 ) + . . . + p n ( a ) · u ( s n ) is the largest. Result • One can easily check that: Home Page – if we replace the utility function u ( a ) by a re-scaled Title Page one u 1 ( s ) = k · u ( s ) + ℓ, ◭◭ ◮◮ – then we get the same order between selections. ◭ ◮ • Vice versa: Page 9 of 44 – if two utility functions u ( s ) and u 1 ( s ) always lead Go Back to the same decisions, Full Screen – then u 1 ( s ) = k · u ( s ) + ℓ for some k > 0 and ℓ . Close Quit

Granular Computing: . . . How to Combine Data . . . 9. Principle of Justifiable Granularity (cont-d) Which Specificity . . . • In this sense, utility is similar to physical quantities Shift-Invariance: . . . like time or temperature. Additional Natural . . . Scale-Invariance: . . . • Their numerical values can change if we select: Full Scale-Invariance Is . . . – a different measuring unit and/or Let Us Combine Shift- . . . – a different starting point. Result Home Page • In our case, when we replace several data points, we lose information. Title Page ◭◭ ◮◮ • So in this case, the utility is negative. ◭ ◮ • In our problem, we have two situations. Page 10 of 44 • For some points, we replace these points with a granule. Go Back Full Screen Close Quit

Granular Computing: . . . How to Combine Data . . . 10. Principle of Justifiable Granularity (cont-d) Which Specificity . . . • The probability P of this replacement can be naturally Shift-Invariance: . . . computed as: Additional Natural . . . Scale-Invariance: . . . – the proportion of data points Full Scale-Invariance Is . . . – that fit into the corresponding granule. Let Us Combine Shift- . . . • This proportion depends on the size ε of the granule: Result Home Page P = P ( ε ) . Title Page • The larger the size, the higher the proportion. ◭◭ ◮◮ • The utility of this replacement also depends on the size ◭ ◮ ε of the granule: u = u ( ε ). Page 11 of 44 • The larger the size, the smaller the utility. Go Back • Other points do not fit into the granule and are dis- Full Screen missed (or processed in a more complex way). Close Quit