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# Multivariate Coarse Classing of Nominal Variables

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1. Multivariate Coarse Classing of Nominal Variables Geraldine E. Rosario Talk given at Fair Isaac on July 14, 2003 Based on paper “Mapping Nominal Values to Numbers for Effective Visualization”, InfoVis 2003.

2. Outline • Motivation • Overview of Distance-Quantification-Classing approach • Algorithmic Details • Experimental Evaluation • Wrap-Up

3. Those pesky nominal variables • Nominal variable: variables whose values do not have a natural ordering or distance • High cardinality nominal variable: has large number of distinct values • Examples? • Examples of business applications using nominal variables? • Why do you usually pre-process/transform them before doing data analysis?

4. Visualizing Nominal Variables • Most data visualization • tools are designed for • numeric variables. • What if variable is • nominal? • Most tools which are • designed for nominal • variables cannot handle • large # of values.

5. Quantified Nominal Variables Are the order and spacing of values within each variable believable?

6. Coarse Classing Nominal Variables • Possible ways of classing nominal variables with high cardinality: • Domain expertise • Univariate: using information about the variable itself. e.g. based on frequency of occurrence of the attributes • Bivariate: using information from one other variable. e.g. relationship with predictor variable • Multivariate: based on the profile across several other variables. e.g. using cluster analysis • Is multivariate coarse classing better?

7. The approach

8. Proposed Approach Pre-process nominal variables using a Distance-Quantification-Classing (DQC) approach Steps: • Distance – transform the data so that the distance between 2 nominal values can be calculated (based on the variable’s relationship with other variables) • Quantification– assign order and spacing to the nominal values • Classing or intra-dimension clustering – determine which values are similar to each other and can be grouped together Each step can be done by more than one technique.

9. Target variable & data set with nominal variables DISTANCE STEP Transformed data for distance calculation QUANTIFICATION STEP CLASSING STEP Nominal-to-numeric mapping Classing tree Distance-Quantification-Classing Approach

10. Observed Counts COLOR by QUALITY Good Ok Bad Total Blue 187 727 546 1460 Green 267 538 356 1161 Orange 276 411 191 878 Purple 155 436 361 952 Red 283 307 357 947 White 459 366 327 1152 Total 1627 2785 2138 6550 blue purple green red orange white -0.02 0 -0.54 -0.5 0.55 0.57 Example Input to Output Task: Pre-process color based on its patterns across quality and size. Data: Quality (3): good,ok,bad Color (6) : blue,green,orange, purple,red,white Size (10) : a to j

11. Other Potential Uses of DQC as Pre-Processor • For techniques that require numeric inputs: linear regression, some clustering algorithms (can speed up calculations but with some loss of accuracy) • For techniques that require low cardinality nominal variables: scorecards, neural networks, association rules • FICO-specific: • Multivariate coarse classing • ClusterBots – nominal variables could be quantified and distance calculations would be simpler. Could be applied to mixed variables? • Product groups, merchant groups • Can you think of other uses?

12. Details … Details …

13. Distance Step: Correspondence Analysis • Used for analyzing n-way tables containing some measure of association between rows and columns • Simple Correspondence Analysis (SCA) – for 2 variables • Multiple Correspondence Analysis (MCA) – for > 2 variables. Uses SCA. • Focused Correspondence Analysis (FCA) – proposed alternative to MCA when memory is limited. Uses SCA. • Reinvented as Dual Scaling, Reciprocal Averaging, Homogeneity Analysis, etc. • Similar to PCA but for nominal variables

14. Observed Counts COLOR by QUALITY Good Ok Bad Total Blue 187 727 546 1460 Green 267 538 356 1161 Orange 276 411 191 878 Purple 155 436 361 952 Red 283 307 357 947 White 459 366 327 1152 Total 1627 2785 2138 6550 Row Percentages Good Ok Bad Blue 13 50 37 100 Green 23 46 31 100 Orange 31 47 22 100 Purple 16 46 38 100 Red 30 32 38 100 White 40 32 28 100 Simple Correspondence Analysis – The Basic Idea Calculate c2 statistic (measures the strength of association between COLOR and QUALITY based on assumption of independence). Any deviation from independence will increase the c2 value. Can we find similar COLORs based on its association with QUALITY? Similar profiles

15. Simple Correspondence Analysis – Steps Row percentage matrix Column percentage matrix Normalize counts table Similar row profiles: (blue,purple), … Similar column profiles: (ok,bad), … Eigenvalues Identify a few independent dimensions which can reconstruct the c2 value. (SVD, EigenAnalysis). Coordinates for Independent Dimensions Dim1 Dim2 Blue - 0.02 - 0.28 Green - 0.54 0.14 Orange 0.55 0.10 Purple 0 - 0.25 Red - 0.50 0.20 White 0.57 0.19 Scale the new dimensions such that c2 distances between row points is maximized.

16. Simple Correspondence Analysis – The Output • Coordinates Matrix • Set of independent dimensions • Dimensions ordered by diminishing importance • Total # of independent dimensions = min(r,c)-1 • Similar to principal components from PCA • Eigenvalues • Indicates the importance of each independent dimension

17. Distance Step Alternative: Multiple Correspondence Analysis • Steps: • BurtTable(rawdataMatrix)  burtMatrix • SCA(burtMatrix)  coordMatrix, evaluesVector • ReduceNDim(coordMatrix, evaluesVector)  coordMatrixSubset • Input to SCA - Burt Table: crosses all variables by all variables X1 X2 X3 … X1 by X1 counts table X1 by X2 counts table X1 X2 X3 …

18. Multiple Correspondence Analysis • Features: • For a given variable, determines which values are similar to each other by comparing value profiles across all other variables • multivariate • maximizes usage of information • memory-intensive • Simultaneously analyzes of all variables • efficient calculations

19. Reduce Number of Dimensions to Keep • Reduce the number of independent dimensions to keep for subsequent analysis (due to large # of analysis variables and high cardinality) eigenvalue 1 2 3 4 5 dimension #

20. Distance Step Alternative:Focused Correspondence Analysis • Proposed alternative to MCA when memory space is limited • Core idea: instead of comparing value profiles across all other nominal variables, just compare value profiles across the nominal variables which are most correlated with the target variable • Input to Simple CA: … X3 X1 X9 target variable Xi Xi by X3 counts table Xi by X1 counts table

21. Focused Correspondence Analysis • Steps: • PairwiseAssociate(rawdataMatrix)  assocMatrix • Set k (# analysis variables to use) • FCATable(rawdataMatrix, k, assocMatrix)  fcaInputMatrix • SCA(fcaInputMatrix)  coordMatrix, evaluesVector • ReduceNDim(coordMatrix, evaluesVector)  coordMatrixSubset

22. U(R|C) Quality Color Size Quality 1.0 0.0287 0.0028 Color 0.0173 1.0 0.1234 Size 0.0017 0.1267 1.0 FCA: Calculate Pairwise Association • Used Uncertainty Coefficient U(R|C) to measure strength of nominal association • Bounded [0,1] • U(R|C)=1  value of row variable R can be known precisely given value of column variable C • Example: U(R|C) association matrix

23. FCA: Determine top k associated variables for each nominal variable • Set k >= 2 to ensure use of at least one analysis variable per target variable • Cannot use a threshold on the association measure

24. Focused Correspondence Analysis • Features: • One-at-a-time analysis • Less/controllable memory usage • Sub-optimal quantification compared to MCA • Requires pre-processing step to determine top correlated variables per target variable • longer run time

25. Nominal Numeric Blue -0.02 Green -0.54 Orange 0.55 Purple 0 Red -0.50 White 0.57 • Rec Q1 Q2 ... Score • 0.5 -0.3 … 0.4 • -0.6 0.1 … -0.02 • … Quantification Step: Modified Optimal Scaling Nominal-to-numeric mapping Coordinates for Independent Dimensions Dim1 Dim2 Blue - 0.02 - 0.28 Green - 0.54 0.14 Orange 0.55 0.10 Purple 0 - 0.25 Red - 0.50 0.20 White 0.57 0.19 Optimal Scaling Optimal Scaling goal: maximize the variance of the scores of the records, where score = average(qij)

26. Quantification Step: Modified Optimal Scaling • Problem with Optimal Scaling: perfect associations between variables are not recreated in the quantified versions • Modified Optimal Scaling: • Let p = # of eigenvalues = 1.0 • If p >= 1 then set • Else set

27. Coordinates for Independent Dimensions Dim1 Dim2 Counts Blue - 0.02 - 0.28 1460 Green - 0.54 0.14 1161 Orange 0.55 0.10 878 Purple 0 - 0.25 952 Red - 0.50 0.20 947 White 0.57 0.19 1152 blue purple green red orange white Classing Step: Hierarchical Cluster Analysis Cluster Analysis weighted by counts [from FCA]

28. 100 Observed Counts COLOR by SIZE U(R|C) = 0.1234 a b … j Total Blue 0 8 … 1460 Green 0 2 … 1161 Orange 7 49 … 878 Purple 0 5 … 952 Red 0 0 … 947 White 6 70 … 1152 Total 13 134 … 6550 50 0 Info loss blue purple green red orange white Loss of Information due to Classing • Determine variable V with highest association with target X. • Create X by V counts table. • Calculate total table measure of association (eg, U(X|V)). • Starting from bottom of tree, for every pair of nodes merged, • calculate cumulative information loss:

29. Target variable & data set with nominal variables DISTANCE STEP Transformed data for distance calculation QUANTIFICATION STEP CLASSING STEP Nominal-to-numeric mapping Classing tree Distance-Quantification-Classing Approach

30. Does this approach work?

31. Experimental Evaluation • Wrong quantification and classing will introduce artificial patterns and cause errors in interpretation • Evaluation measures: • Believability • Quality of Visual Display • Quality of classing • Quality of quantification • Space – FCA less space • Run time – MCA faster perception statistical computational

32. Test Data Sets

33. Believability and Quality of Visual Display • Given two displays resulting from different nominal-to-numeric mappings: • Which mapping gives a more believable ordering and spacing? • Based on your domain knowledge, are the values that are positioned close together similar to each other? • Are the values that are positioned far from the rest of the values really outliers? • Which display has less clutter?

34. Automobile Data: Alphabetical

35. Automobile Data: MCA Are these patterns believable?

36. Automobile Data: FCA Are these patterns believable?

37. PERF Data: Alphabetical Region-Country: 1-many Country-Product: many-many Are these associations preserved and revealed?

38. PERF Data: FCA Region-Country: 1-many Country-Product: many-many Are these associations preserved and revealed?

39. Quality of Classing • Classing A is better than classing B if, given a classing tree, the rate of information loss with each merging is slower Information loss due to classing for one variable  [The lower the line, the slower the info loss, the better the classing.] Calculate difference between the lines. 

40. Which classing is better … depends on dataset Distribution of difference between the lines.

41. Quality of Quantification • A quantification is good if … • If data points that are close together in nominal space are also close together in numeric space • If two variables are highly associated with each other, then their quantified versions should also have high correlation.

42. MCA gives better quantification  Average Squared Correlation [higher value = better quantification]  Correlation between MCA and FCA scales [how close are FCA scales to MCA scales]

44. Going back to Multivariate Coarse Classing • Other issues: • Missing values • Mixed or numeric variables as analysis variables • Nominal values with small counts • Robustness of quantification and classing

45. Can you think of other uses of DQC at FICO? • For techniques that require numeric inputs: linear regression, some clustering algorithms (can speed up calculations but with some loss of accuracy) • For techniques that require low cardinality nominal variables: scorecards, neural networks, association rules • FICO-specific: • Multivariate coarse classing • ClusterBots – nominal variables could be quantified and distance calculations would be simpler. Could be applied to mixed variables? • Product groups, merchant groups • ???????

46. Implementation • SAS version exists • PROC CORRESP, PROC CLUSTER, PROC FREQ • C++ version in development

47. Summary • DQC is a general-purpose approach for pre-processing nominal variables for data analysis techniques requiring numeric variables or low cardinality nominal variables • DQC – multivariate, data-driven, scalable, distance-preserving, association-preserving • FCA is a viable alternative to MCA when memory space is limited • Quality of classing and quantification • depends on strength of associations within the data set. • is in the eye of the user

48. Yippee, it’s over! Original InfoVis2003 paper: Mapping Nominal Values to Numbers for Effective Visualization. http://davis.wpi.edu/~xmdv/documents.html XmdvTool Homepage: http://davis.wpi.edu/~xmdv xmdv@cs.wpi.edu Code is free for research and education.

49. References • [Gre93] GREENACRE, M.J., 1993, Correspondence Analysis in Practice, London :Academic Press • [Gre84] Greenacre, M. (1984), Theory and Applications of Correspondence Analysis, London: Academic Press • [Sta] StatSoft Inc. Correspondence Analysis. http://www.statsoftinc.com/textbook/stcoran.html • [Fri99] Friendly, Michael. 1999. "visualizing Categorical Cata." In Sirken, Monroe G. et. al. (eds). Cognition and Survey Research. New York: John Wiley & Sons. • [Kei97] Keim D. A.: Visual Techniques for Exploring Databases, Invited Tutorial, Int. Conference on Knowledge Discovery in Databases (KDD'97), Newport Beach, CA, 1997. • [Hua97b] Zhexue Huang. A Fast Clustering Algorithm to Cluster Very Large Categorical Data Sets in Data Mining (1997) • SAS Manuals (PROC CORRESP, PROC CLUSTER, PROC FREQ)

50. What input tables can SCA accept? • In general, SCA can use as input any table that has the properties: • The table must use the same physical units or measurements, and • The values in the table must be non-negative. The FCA input table satisfies these properties.