Principal Components Analysis

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Principal Components Analysis

• Eric Vaagen, FCAS
• Assistant Actuary
• September 5, 2008

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Agenda

• Motivation
• What is PCA?
• Background
• Simple example
• Is PCA right for you?

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Motivation

• Forecast average premium by coverage
• Explanatory variables
• Vehicle use, territory, driving record
• Breakdown of change in average premium
• Multicollinearity exists

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Average Premium 2002-2006
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Modeling Procedure
Explanatory Variables
Variable Selection
Response Variable
Model
Chosen Variables
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Modeling Procedure
Vehicle Use Territory Drv. Record
Variable Selection
Average Premium
Multiple Regression
Chosen Variables
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Variable Selection Methods
Variable Selection

• Stepwise regression
• Forward, backward
• PCA
• Unsupervised
• Partial least squares
• Supervised
• GLM

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Background

• First described in 1901 by Karl Pearson
• Find the best lines and planes to fit a set of
  points
• What else did he discover?
• Pearsons ?²
• Linear regression
• Classification of distributions (exponential
  family)

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PCA Example
Explanatory Variables

• Vehicle use
• Pleasure
• Commute
• Business

• Territory
• Rural
• Suburban
• Urban

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Vehicle Use 2002-2006
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Territory 2002-2006
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Example Average Premium
Response Variable
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Modeling Procedure
Vehicle Use Territory
PCA
Average Premium
Multiple Regression
Chosen PCs
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PCA Procedure

• PCs
• No multicollinearity
• The 1st PC has the most variance
• Output
• Weights to create the PCs
• Variability of each PC

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Modeling Procedure
Vehicle Use Territory
5 years x 6 variables
Weights
PCA
5 years x 6 variables
Variability
Chosen PCs
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Example Scree Plot
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PC Calculation
Chosen Variables
PC 3 -0.55 0.36 0.23 -0.02 0.47 -0.55

• Pleasure
• Commute
• Business
• Rural
• Suburban
• Urban

PC 2 -0.54 0.14 0.48 -0.20 -0.31 0.58

• PC 1
• -0.19
• 0.54
• -0.40
• 0.56
• -0.45
• -0.03

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PC Calculation

• PC1 - 0.19P 0.54C - 0.40B
• 0.56R - 0.45S - 0.03U
• PC12002 -0.19(30)0.54(50)-0.40(20)
• 0.56(20)-0.45(30)-0.03(50)

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Example - Modeling Procedure
Vehicle Use Territory
PCA
Average Premium
Multiple Regression
Chosen PCs
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Example Results
Multiple Regression
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ICBC Personal TPB
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Advantages

• Eliminates multicollinearity
• Most of the original variance is captured in a
  few principal components
• More refined selection method

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Disadvantages

• Can be hard to interpret the PCs
• PC weights may not be stable from year to year
• Difficult to explain

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Is PCA Right For You?

• Concerned about multicollinearity?
• Confident in the set of explanatory variables?
• Want to reduce dimensionality, without throwing
  away variables?

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For More Information

• 2008 Discussion Paper
• PCA and Partial Least Squares Two Dimension
  Reduction Techniques for Regression
• http//www.casact.org/pubs/dpp/dpp08/08dpp76.pdf
• Predictive modeling seminar
• Oct 6-7, 2008 in San Diego, CA
• PCA and Partial Least Squares