1. Principal Components Analysis
Get gifts for the audience.
Eric Vaagen, FCAS
Assistant Actuary
September 5, 2008

2. Agenda Motivation What is PCA? Background Simple example
Is PCA right for you?

3. Motivation Forecast average premium by coverage Explanatory variables
Vehicle use, territory, driving record
Breakdown of change in average premium
Multicollinearity exists
Forecast of average premium by coverage is done as a part of our ratemaking exercise.

4. Average Premium
We needed to understand what is driving average premium and come up with an accurate forecast for 2007.
A $1 difference in premium corresponds to $3M which is x% on our rate indication.
A model exercise is undertaken -> next slide.

5. Modeling Procedure Explanatory Variables Variable Selection Response
This is a generic model flowchart -> taking into account average premium and vehicle use, territory, and driving experience, the flowchart looks like…
Chosen
Variables
Model

6. Modeling Procedure Vehicle Use Territory Drv. Record Variable
Selection
Average
Premium
Chosen
Variables
Multiple
Regression

7. Variable Selection Methods
Stepwise regression
Forward, backward
PCA
Unsupervised
Partial least squares
Supervised
GLM

8. 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?
Pearson’s χ²
Linear regression
Classification of distributions (exponential family)
Regression of sons heights upon that of their fathers.

9. PCA Example Vehicle use Territory Pleasure Commute Business Rural
Explanatory
Variables
Vehicle use
Pleasure
Commute
Business
Territory
Rural
Suburban
Urban

10. Vehicle Use 2002-2006 These are the explanatory variables.
Example: in 2002, there was 30% Pleasure, …

11. Territory
Explanatory variable proportions, along with the low, medium, and high rates, produce the following average premium graph…

12. Example – Average Premium
Response
Variable
Does this look familiar? It is the same as the 2007 rate filing.

13. Modeling Procedure Vehicle Use Territory PCA Average Premium Multiple
Here is the modeling flow chart with PCA as the variable selection method.
We will now focus on the PCA method itself.
Chosen PCs
Multiple
Regression

14. PCA Procedure PCs Output No multicollinearity
The 1st PC has the most variance
Output
Weights to create the PCs
Variability of each PC
The 1st PC has the most variance of any linear combination of the explanatory variables.
The 2nd PC has the most of the remaining variance.
Technical:
The PCs are an orthogonal linear transformation of the original variables to a new coordinate system. An eigenvalue decomposition is done on the covariance matrix of the explanatory variables -> the eigenvectors are the weights for the PCs and the eigenvalues are the variability of each PC.
The variability of each PC is used to choose which variables to keep in the model.

15. Modeling Procedure Vehicle Use Territory 5 years x 6 variables Weights
PCA
5 years x 6 variables
Here is the modeling flow chart with PCA as the variable selection method.
We will now focus on the PCA method itself.
Variability
Chosen PCs

16. Example – Scree Plot
Scree – broken rock that appears at the bottom of crags and cliff faces.
Keep PCs with more variance than average (16.66%)
Keep a certain % of total variance (90% or 95%)
Now that we have decided to keep the first 3 PCs, let’s see how each one is calculated.

17. PC Calculation Pleasure Commute Business Rural Suburban Urban PC #1
Chosen
Variables
PC Calculation
Pleasure
Commute
Business
Rural
Suburban
Urban
PC #1
-0.19
0.54
-0.40
0.56
-0.45
-0.03
PC #2
-0.54
0.14
0.48
-0.20
-0.31
0.58
PC #3
-0.55
0.36
0.23
-0.02
0.47

18. PC Calculation Pleasure P Rural R Commute C Suburban S Business B
PC1 = P C B
+ 0.56R S U
PC12002 = -0.19(30%)+0.54(50%)-0.40(20%)
+0.56(20%)-0.45(30%)-0.03(50%)

19. Example - Modeling Procedure
Vehicle Use
Territory
PCA
Average
Premium
I have explained how to get from the explanatory variables to the chosen PCs.
Now, let’s look at the modeling results. The modeling part is outside the scope of this presentation, but I am including it so that we can see results.
Chosen PCs
Multiple
Regression

20. Example – Results Multiple Regression
Stepwise is worse – because it doesn’t capture the changes from variables not selected in the forecast period.
Outside the scope - economic variables are included in the forecasts of the explanatory variables.

21. ICBC Personal TPB

22. Advantages Eliminates multicollinearity
Most of the original variance is captured in a few principal components
More refined selection method
Also, you can determine the impact of each rating variable on the overall change in average premium because all of the explanatory variables are included.

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

24. Is PCA Right For You? Concerned about multicollinearity?
Confident in the set of explanatory variables?
Want to reduce dimensionality, without throwing away variables?

25. For More Information 2008 Discussion Paper Predictive modeling seminar
PCA and Partial Least Squares: Two Dimension Reduction Techniques for Regression
Predictive modeling seminar
Oct 6-7, 2008 in San Diego, CA
PCA and Partial Least Squares