Combining GLM and data mining techniques

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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

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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

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Model of claims data

• General form of claims data model
• Yi f(Xi ß) ei
• Yi some observation on claims experience
• ß vector of parameters that apply to all
  observations
• Xi vector of attributes (covariates) of i-th
  observation
• ei vector of centred stochastic error terms

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Model of claims data

• General form of claims data model
• Yi f(Xi ß) ei
• Yi some observation on claims experience
• ß vector of parameters that apply to all
  observations
• Xi vector of attributes (covariates) of i-th
  observation
• ei 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

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Examples (contd)

• Yad paid losses in (a,d) cell
• EYad ßd Sr1d-1 Yar (chain ladder)

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Examples (contd)

• Yad paid losses in (a,d) cell
• EYad ßd Sr1d-1 Yar (chain ladder)
• EYad A db exp(-cd) exp aß ln d - ?d
  (Hoerl curve for each accident periods payments)

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Examples (contd)

• Yad paid losses in (a,d) cell
• EYad ßd Sr1d-1 Yar (chain ladder)
• EYad A db exp(-cd) exp aß ln d - ?d
  (Hoerl curve for each accident periods payments)
• Yi cost of i-th completed claim
• Yi Gamma
• EYi exp aß 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

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Examples of individual claim models

• More generally
• EYi
• exp function of operational time

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Examples of individual claim models (contd)

• More generally
• EYi
• exp function of operational time
• function of accident period (legislative
  change)

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Examples of individual claim models (contd)

• More generally
• EYi
• exp function of operational time
• function of accident period (legislative
  change)
• function of completion period (superimposed
  inflation)

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Examples of individual claim models (contd)

• More generally
• EYi
• 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)

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Examples of individual claim models (contd)

• 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

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Identification of data features

• Typically largely ad hoc, using
• Trial and error regressions
• Diagnostics, e.g. residual plots

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Identification of data features - illustration

• Modelling about 60,000 Auto Bodily Injury claims
• First fitting just an operational time effect

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Identification of data features - illustration

• But there appear to be unmodelled trends by
• Accident quarter
• Completion (finalisation) quarter

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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

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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

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(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)

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ANN for Regression problem Y f(X)

• Activation function
• Commonly a sigmoidal curve
• Function ? introduces non-linearity to model
• ? keeps response bounded

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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

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Illustration of ANN

Wm
g
Zm
hm
wm
Xi
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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

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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 ?0.05
• These tuning parameters determined by
  cross-validation
• Prediction error in test data set

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Comparison of GLM and ANN

• GLM
• Average absolute error
• 33,777

• ANN
• Average absolute error
• 33,559

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GLM and ANN forecasts

• 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

ANN extrapolation
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GLM and ANN forecasts

• 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

ANN extrapolation
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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

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Application of ANN (contd)

• 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

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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