1. Dimension Reduction in Workers Compensation
CAS predictive Modeling Seminar
Louise Francis, FCAS, MAAA
Francis Analytics and Actuarial Data Mining, Inc.
www,data-mines.com

2. Objectives
Answer questions: What is dimension reduction and why use it?
Introduce key methods of dimension reduction
Illustrate with examples in Workers Compensation
There will be some formulas, but emphasis is on insight into basic mechanisms of the procedures

3. Introduction “How do mere observations become data for analysis?”
“Specific variable values are never immutable characteristics of the data”
Jacoby, Data Theory and Dimension Analysis, Sage Publications
Many of the dimension reduction/measurement techniques originated in the social sciences and dealt with how to create scales from responses on attitusional and opinion surveys

4. Unsupervised learning
Dimension reduction methods generally unsupervised learning
Supervised Learning
A dependent or target variable
Unsupervised learning
No target variable
Group like variables or like records together

5. The Data BLS Economic indexes
Components of inflation
Employment data
Health insurance inflation
Texas Department of Insurance closed claim data for 2002 and 2003
Employment related injury
Excludes small claims
About 1800 records

6. What is a dimension?
Jacoby – any information that adds significant variability
In many studies each variable is a dimension
However,we can also view each record in a database as a dimension

7. Dimensions

8. The Two Major Categories of Dimension Reduction
Variable reduction
Factor Analysis
Principal Components Analysis
Record reduction
Clustering
Other methods tend to be developments on these

9. Principal Components Analysis
A form of dimension (variable) reduction
Suppose we want to combine all the information related to the “inflation” dimension of insurance costs
Medical care costs
Employment (wage) costs
Other
Energy
Transportation
Services

10. Principal Components
These variables are correlated but not perfectly correlated
We replace many variables with a weighted sum of the variables
These are then used as independent variables in a predictive model

11. Factor Analysis: A Latent Factor

12. Factor/Principal Components Analysis
Linear methods – use linear correlation matrix
Correlation matrix decomposed to find smaller number of factors the are related to the same underlying drivers
Highly correlated variables tend to have high load on the same factor

13. Factor/Principal Components Analysis

14. Factor/Principal Components Analysis
Uses eignevectors and eigenvalues
R is correlation matrix, V eigenvectors, lambda eigenvalues

15. Inflation Data

16. Factor Rotation Find simpler more easily interpretable factors
Use notion of factor complexity

17. Factor Rotation Quartimax Rotation Varimax Rotation Maximize q
Maximizes the variance of squared loadings for each factor rather than for each variable

18. Varimax Rotation

19. Plot of Loadings on Factors

20. How Many Factors to Keep?
Eigenvalues provide information on how much variance is explained
Proportion explained by a given component=corresponding eigenvalue/n
Use Scree Plot
Rule of thumb: keep all factors with eigenvalues>1

22. WC Severity vs Factor 1

23. WC Severity vs Factor 2

24. What About Categorical Data?
Factor analysis is performed on numeric data
You could code data as binary dummy variables
Categorical Variables from Texas data
Injury
Cause of loss
Business Class
Health Insurance (Y/N)

25. Optimal Scaling A method of dealing with categorical variables
Uses regression to
Assign numbers to categories
Fit regression coefficients
Y*=f(X*)
In each round of fitting, a new Y* and X* is created

26. Variable Correlations

27. Visualizations of Scaled Variables

28. Can we use scaled variables in prediction?

29. Row Reduction: Cluster Analysis
Records are grouped in categories that have similar values on the variables
Examples
Marketing: People with similar values on demographic variables (i.e., age, gender, income) may be grouped together for marketing
Text analysis: Use words that tend to occur together to classify documents
Fraud modeling
Note: no dependent variable used in analysis

30. Clustering Common Method: k-means, hierarchical
No dependent variable – records are grouped into classes with similar values on the variable
Start with a measure of similarity or dissimilarity
Maximize dissimilarity between members of different clusters

31. Dissimilarity (Distance) Measure – Continuous Variables
Euclidian Distance
Manhattan Distance

32. Binary Variables

33. Binary Variables
Sample Matching
Rogers and Tanimoto

34. Example: Texas Data
Data from 2002 and 200 3closed claim database by Texas Ins Dept
Only claims over a threshold included
Variables used for clustering:
Report Lag
Settlement Lag
County (ranked by how often in data)
Injury
Cause of Loss
Business class

35. Results Using Only Numeric Variables
Used squared distance measure

36. Two Stage Clustering With Categorical Variables
First compute distances
Then get clusters
Find optimum number of clusters

38. Loadings of Injuries on Cluster

39. Age and Cluster

40. County vs Cluster

41. Means of Financial Variables by Cluster

42. Modern dimension reduction
Hidden layer in neural networks like a nonlinear principle components
Projection Pursuit Regression – a nonlinear PCA
Kahonen self-organizing maps – a kind of neural network that does clustering
These can be understood as enhancements factor analysys or clusterini

44. Kahonen SOM for Fraud