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Focused Reducts

Focused Reducts. Janusz A. Starzyk and Dale Nelson. ASSUMPTION: This is ALL we know. Sampled Data. Model. What Do We Know? Major Assumption. Real World. … 1024. . . . 1602. Problem Size Dilemma. Rough Set Tutorial. Difference between rough sets and fuzzy sets Labeling data

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Focused Reducts

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  1. Focused Reducts Janusz A. Starzyk and Dale Nelson

  2. ASSUMPTION: This is ALL we know Sampled Data Model What Do We Know?Major Assumption Real World

  3. …1024 . . . 1602 Problem Size Dilemma

  4. Rough Set Tutorial • Difference between rough sets and fuzzy sets • Labeling data • Remove duplicates/ambiguities • What is a core? • What is a reduct?

  5. Rough Sets vs Fuzzy Sets Fuzzy Sets - How gray is the pixel Rough Sets - How big is the pixel

  6. ExampleSample HRR Data

  7. Labeling can be different for different columns/attributes Ranges can be different for different columns/attributes ExampleLabel Data Label 1 < .25 .25 >= Label 2 <=.45 Label 3 > .45

  8. Remove Ambiguities & Duplicates

  9. Equivalence Classes E1={1, 2, 3} E2={4, 5} E3={6} E4={7} E5={8}

  10. Definitions • Reduct - A reduct is a reduction of an information system which results in no loss of information (classification ability) by removing attributes (range bins). There may be one or many for a given information system) • Core - A core is the set of attributes (range bins) which are common to all reducts.

  11. Compute Core Signals 6 and 8 are ambiguous upon removal of Range Bin 1. Therefore, Range Bin 1 is part of core. Core - The range bins common to ALL reducts - The most essential range bins without which signals cannot be classified

  12. Compute Core No ambiguous signals therefore, Range Bin 2 is NOT part of core.

  13. Compute Core No ambiguous signals therefore, Range Bin 3 is NOT part of core.

  14. Compute Core No ambiguous signals therefore, Range Bin 4 is NOT part of core.

  15. Compute ReductsRange Bin 1 + Range Bin 2 Range Bin 1 and Range Bin 2 classify therefore, they belong to a reduct

  16. Compute ReductsRange Bin 1 + Range Bin 3 Range Bin 1 and Range Bin 3 do not classify therefore, they do NOT belong to a reduct

  17. Compute ReductsRange Bin 1 + Range Bin 4 Range Bin 1 and Range Bin 4 classify therefore, they belong to a reduct

  18. Reduct Summary • Range bins 1 and 2 are a reduct • Sufficient to classify all signals • Range bins 1 and 4 are a reduct • Sufficient to classify all signals • Range bins 1 and 3 are NOT a reduct • Cannot distinguish target classes 2 and 3 • No need to try • Range bins 1, 2, 3 • Range bins 1, 2, 4

  19. Did You Notice? • Calculating a reduct is time consuming! • n = 29 value = 536,870,911 • We are interested in n  50 • This is a BIG NUMBER requiring a lot of time to compute reduct which is a f (# signals), too

  20. Why Haven’t Rough Sets Been Used Before?

  21. The Procedure • Normalize signal • Partition signal • Block • Interleave • Wavelet transform • Binary multi-class entropy labeling • Entropy based range bin selection • Determine minimal reducts • Fuse marginal reducts for classification

  22. Data • Synthetic generated by XPATCH • Six targets • 1071 Signals per target • 128 Range bins/signal • Azimuth -25o to +25o • Elevation -20o to 0o

  23. Normalize the Data • Ensures all data is range normalized • Use the 2 Norm • Divide each signal bin value by N

  24. 1 1 128 128 1 64 65 1 2 1 32 33 64 65 96 97 128 1 2 3 4 1 16 17 32 33 48 48 64 65 80 81 96 97 112 113 128 1 2 3 4 5 6 7 8 Partition the Signal Block Partitioning

  25. 1 1 128 128 1st 2nd 3rd 4th 5th 6th 7th 8th Partition the Signal 1 Piece 2 Pieces 1 128 1 4 Pieces 1 128 8 Pieces Interleave Partitioning

  26. Best Wavelet 50/60 Signals Classified!! Original Signal Best- 20/60 signals Classified Many features are better than the best from original signal Why Use a Wavelet Transform?

  27. HRR Signal and Its Haar Transform

  28. Multi-Class Information Entropy Using this definition we define two other probabilities Let xi be range bin values across all signals for a target class Define where Without assuming any particular distribution we can define the probability as: Then multi-class entropy is defined as:

  29. Binary Multi-Class Labeling

  30. Range Bin Selection • Total range bins available depends on partition size • We chose 50 bins per reduct • Time considerations • Implications • Based on maximum relative entropy

  31. Compute Core • Computation of core is easy and fast • Eliminate one range bin at a time and see if the training set is ambiguous - only that range bin can discriminate between the ambiguous signals • Accumulate the bins resulting in ambiguous data - that is the core • These range bins MUST be in every reduct • O(n) process

  32. Compute Minimal Reducts • To the core add one range bin at a time and compute the number of ambiguities • Select the range bin(s) with the fewest ambiguities-there may be several-save these as we will use them to compute the reduct • Add that range bin to the core and repeat previous step until there are no ambiguities - this is a reduct • Calculate reducts for all bins with equivalent number of ambiguities-yields multiple reducts • O(n2) process

  33. Need 50 Time Complexity Training Set Size 50 to 400 Attributes (Range Bins) 1602 Signals Test Set Size 4823 Signals

  34. Fuzzy Rough Set Classification • Test signals may have a range bin value very close to labeling division point • If this happens we define a distance  where this is considered a “don’t care” region • Classification process proceeds without the “don’t care” range bin

  35. Weighting FormulaRequirements • We desire the following for combining classifications • All Pcc(s) = 0  weight = 0 • All Pcc(s) = 1  weight = 1 • Several low Pcc(s)  weight higher than any of the Pcc(s) • One high Pcc and several low Pcc(s)  weight higher than the highest Pcc

  36. Weighting Formula

  37. Fusing Marginal Reducts • Each signal is marked with the classification by each reduct along with the reduct’s performance (Pcc) on the training set • A weight is computed for each target class for each signal • A signal is assigned the target class with the highest weight

  38. Results - Training

  39. Results Testing

  40. Conjectures • Robust in the presence of noise • Due to binary labeling • Due to fuzzification • Robust to signal registration • Due to binary labeling • Due to averaging effect of wavelets on interleaved partitions • Due to fuzzification

  41. 0 1 TIME Rough Set Theoretic HRR ATR - Summary APPLICATIONS -1-D Signals -HRR -LADAR vibration -Sonar -Medical -Stock market -Data Mining BREAKTHROUGHS -Reduct (classifier) generation time from exponential to quadratic ! -Fusion of marginal (poor performing) reducts -Wavelet Transform Aiding -Multi partition to increase number of range bins considered -Use of binary multi-class entropy labeling -Entropy based range bin selection -Performance within 1% of theoretic best -Max problem size increased by 2 orders of magnitude METHOD -Normalize Signal -Partition Signal - Block - Interleave -Wavelet Transform -Binary Multi-class Entropy Labeling -Entropy based Range Bin Selection -Determine Minimal Reducts -Fuse marginal reducts for classification Exponential Quadratic

  42. Future Directions • Fuzz factor sensitivity study • Sensitivity to signal alignment • Sensitivity to noise • Iterated wavelet transform performance study • Effectiveness on air to ground targets • Other application areas

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