Combining GLM and data mining techniques

1. Combining GLM and data mining techniques Greg Taylor Taylor Fry Consulting Actuaries University of Melbourne University of New South Wales Casualty Actuarial Society Special Interest Seminar on Predictive Modeling Boston, October 4-5 2006

2. Overview • Examine general form of model of claims data • Examine the specific case of a GLM to represent the data • Consider how the GLM structure is chosen • Introduce and discuss Artificial Neural Networks (ANNs) • Consider how these may assist in formulating a GLM • Presentation draws heavily on work of colleague Dr Peter Mulquiney

3. Model of claims data • General form of claims data model Yi = f(Xi; β) + εi • Yi = some observation on claims experience • β = vector of parameters that apply to all observations • Xi = vector of attributes (covariates) of i-th observation • εi = vector of centred stochastic error terms

4. Model of claims data • General form of claims data model Yi = f(Xi; β) + εi • Yi = some observation on claims experience • β = vector of parameters that apply to all observations • Xi = vector of attributes (covariates) of i-th observation • εi = vector of centred stochastic error terms • Examples • Yi = Yad = paid losses in (a,d) cell • a = accident period • d = development period • Yi = cost of i-th completed claim

5. Examples (cont’d) • Yad = paid losses in (a,d) cell • E[Yad] = βdΣr=1d-1Yar (chain ladder)

6. Examples (cont’d) • Yad = paid losses in (a,d) cell • E[Yad] = βdΣr=1d-1Yar (chain ladder) • E[Yad] = A db exp(-cd) = exp [α+β ln d - γd] (Hoerl curve for each accident period’s payments)

7. Examples (cont’d) • Yad = paid losses in (a,d) cell • E[Yad] = βdΣr=1d-1Yar (chain ladder) • E[Yad] = A db exp(-cd) = exp [α+β ln d - γd] (Hoerl curve for each accident period’s payments) • Yi = cost of i-th completed claim • Yi ~ Gamma • E[Yi] = exp [α+β ti] where • ai = accident period to which i-th claim belongs • ti = operational time at completion of i-th claim = proportion of claims from the accident period ai completed before i-th claim

8. Examples of individual claim models More generally E[Yi] = exp {function of operational time}

9. Examples of individual claim models (cont’d) More generally E[Yi] = exp {function of operational time + function of accident period (legislative change)}

10. Examples of individual claim models (cont’d) More generally E[Yi] = exp {function of operational time + function of accident period (legislative change) + function of completion period (superimposed inflation)}

11. Examples of individual claim models (cont’d) More generally E[Yi] = exp {function of operational time + function of accident period (legislative change) + function of completion period (superimposed inflation) + joint function (interaction) of operational time & accident period (change in payment pattern attributable to legislative change)}

12. Examples of individual claim models (cont’d) • Models of this type may be very detailed • May include • Operational time effect (payment pattern) • Seasonality • Creeping change in payment pattern • Abrupt change in payment pattern • Accident period effect (legislative change) • Completion quarter effect (superimposed inflation) • Variations in superimposed inflation over time • Variations of superimposed inflation with operational time • etc

13. Identification of data features • Typically largely ad hoc, using • Trial and error regressions • Diagnostics, e.g. residual plots

14. Modelling about 60,000 Auto Bodily Injury claims First fitting just an operational time effect Identification of data features - illustration

15. But there appear to be unmodelled trends by Accident quarter Completion (finalisation) quarter Identification of data features - illustration

16. Identification of data features - illustration • Final model includes terms for: • Operational time • Seasonality • Claim frequency • Decrease induces increased claim sizes • Accident quarter • Change in Scheme rules • Change in operational time effect with change in Scheme rules • Superimposed inflation • Varying with operational time

17. Identification of data features – alternative approach • Final model is complex in structure • Structure identified in ad hoc manner • More rigorous approach desirable • Try Artificial Neural Network (ANN) • Essentially a form of non-linear regression

18. (Feed-forward) ANN for regression problem Y = f(X) • Start with vector of P inputs X = {xp} • Create hidden layer with M hidden units • Make M linear combinations of inputs • Linear combinations then passed through layer of activation functions g(hm)

19. ANN for Regression problem Y = f(X) • Activation function • Commonly a sigmoidal curve • Function  introduces non-linearity to model  keeps response bounded

20. ANN for Regression problem Y = f(X) • Y is then given by a linear combination of the outputs from the hidden layer • This function can describe any continuous function • 2 hidden layers  ANN can describe any function

21. Y Illustration of ANN Wm g Zm hm wm Xi

22. Training of ANN • Weights are usually determined by minimising the least-squares error • Weight decay penalty function stops overfitting • Larger   smaller weights • Smaller weights  smoother fit

23. Training of ANN - example • Training data set: 70% of available data • Test data set: 30% of available data • Network structure: • Single hidden layer • 20 units • Weight decay λ=0.05 • These tuning parameters determined by cross-validation • Prediction error in test data set

24. GLM Average absolute error = $33,777 ANN Average absolute error = $33,559 Comparison of GLM and ANN

25. Both by simple extrapolation of trends here ANN case Development quarter 10: red Development quarter 20: green Development quarter 30: yellow Development quarter 40: blue Note negative superimposed inflation May be undesirable GLM and ANN forecasts ANN extrapolation

26. Note negative superimposed inflation May be undesirable But ANN useful in searching out general form of past superimposed inflation Which can then be modelled explicitly in GLM GLM and ANN forecasts ANN extrapolation

27. Application of ANN • Generalisation of preceding remark • ANN may be most useful as an automated tool for seeking out detailed trends in data • Apply ANN to data set • Study trends in fitted model against a range of predictors or pairs of predictors • Use this knowledge to choose the functional forms of included in the linear predictor of the GLM

28. Ultimate test of the GLM is to apply ANN to its residuals, seeking structure There should be none The example indicates that the chosen GLM structure may: Over-estimate the more recent experience at the mid-ages of claim Under-estimate it at the older ages Application of ANN (cont’d)

29. Conclusions • GLMs provide a powerful and flexible family of models for claims data • Complex GLM structures may be required for adequate representation of the data • The identification of these may be difficult • The identification procedures are likely to be ad hoc • ANNs provide an alternative form of non-linear regression • These are likely to involve their own shortcomings if left to stand on their own • They may, however, provide considerable assistance if used in parallel with GLMs to identify GLM structure