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Variable Reduction for Predictive Modeling with Clustering Casualty Actuarial Society Seminar on Ratemaking Bob Sanche March 14, 2006 © 2006 Towers Perrin Contents Data Storage and Amount of Predictive Variables Predictive Modeling and Model Generalization Dimension Reduction Goal of Variable Clustering What Is Clustering? Variable Clustering When Does Variable Clustering Occur During the Predictive Modeling Process? Example © 2006 Towers Perrin 2 Data Storage and Predictive Variables Data storage economics “In 1956, IBM sold its first magnetic disk system, RAMAC (Random Access Method of Accounting and Control). It used 50 24-inch metal disks, with 100 tracks per side. It could store 5 megabytes of data and cost $10,000 per megabyte. (As of 2005, disk storage costs less than $1 per gigabyte).” http://en.wikipedia.org/wiki/History_of_computing_hardware 1 gigabyte = 130 numeric characteristics for 1 million policies for $1 © 2006 Towers Perrin 3 Data Storage and Predictive Variables New data sources Data warehousing External sources (demographics, meteorological, etc.) Policyholder, household or company information Agency Other Data storage economics and availability of data Increase the number of predictive variables Data mining paradigm Additional inputs add lift to the model © 2006 Towers Perrin 4 Predictive Modeling and Model Generalization A predictive model is created from a number of predictors that are likely to influence future results Y = α1X1 + … + αnXn + β n is universe of all available predictors Goal of predictive modeling Obtain coefficients for α’s and β Predictive of future results — Model generalizes well over time Model complexity → Overfitting © 2006 Towers Perrin 5 Dimension Reduction Need to reduce model complexity Dimension reduction Clustering (K-Means) — Rows Variable clustering — Columns — Alternatives to variable clustering – PCA and factor analysis – Difficult to interpret and deploy © 2006 Towers Perrin 6 Goal of Variable Clustering Reduce the number of variables More difficult to identify irrelevant variables than redundant variables Y = α1X1 + … + αmXm + β where m<n Why do we want to reduce the number of variables? Improve efficiency of predictive modeling process — Time to develop the model — Interpretation of the results — Reduce variance of the model estimates Demographics example Average household size, median household size, proportion of families, median vehicles per household — could be replaced by only one variable © 2006 Towers Perrin 7 What Is Clustering? “Cluster Analysis is a set of methods for constructing a sensible and informative classification of an initially unclassified set of data, using the variable values observed on each individual” B.S. Everitt , The Cambridge Dictionary of Statistics, 1998 Divide set of data (variable) into groups of similar characteristics Unsupervised learning technique Useful only when there is redundancy in the data © 2006 Towers Perrin 8 What Is Clustering? Similarity measured by distance or correlation metrics Types of clustering Hierarchical clustering — Agglomerative — Divisive Partitive (optimization) clustering © 2006 Towers Perrin 9 Variable Clustering Variable clustering divides a set of numeric* variables into clusters. A cluster representing a large set of variables can be replaced by a single member (cluster representative). * Hamming distance for categorical variables Selection of the cluster representative 1-R 2 ratio ( 1-R 2 own 2 nearest )/( 1-R ) Intuitively, we want the cluster representative to be as closely correlated to its own cluster (R2own1) and as uncorrelated to the nearest cluster (R2nearest0). Therefore, the optimal representative of a cluster is a variable where 1-R2 ratio tends to zero © 2006 Towers Perrin 10 When Does Variable Clustering Occur During the Predictive Modeling Process? SEMMA process for data mining Sample Explore Modify Model Assess © 2006 Towers Perrin 11 Example 3 CLUSTERS Cluster Cluster 1 Cluster 2 Cluster 3 © 2006 Towers Perrin R-SQUARED WITH 1-R2 Ratio Own Cluster Next Closest Rain Days 0.5995 0.0426 0.4183 Snow Days 0.8976 0.0317 0.1095 Annual Snow 0.8940 0.0314 0.1095 Population Density 0.9804 0.0228 0.0201 Car Density 0.9804 0.0113 0.0199 Population Growth 0.6459 0.0911 0.3896 Legal Expenditures 0.6459 0.0013 0.3546 Variable 12 Example Name of Variable or Cluster Population Density Legal Expenditures Snow Days Annual Snow Accumulation Rain Days Population Growth Car Density 1.00 0.95 0.90 0.85 0.80 0.75 0.70 Proportion of Variance Explained © 2006 Towers Perrin 13 Conclusion Need for dimension reduction for model generalization Variable clustering reduces the amount of variables available for predictive modeling (GLM, etc.) The predictive modeling process using variable clustering Avoid overfitting Increases interpretability Reduces time for modeling © 2006 Towers Perrin 14