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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; 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

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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

• Examples1. Yi = Yad = paid losses in (a,d) cell

• a = accident period• d = development period

2. Yi = cost of i-th completed claim

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Examples (cont’d)

• Yad = paid losses in (a,d) cell

1. E[Yad] = βd Σr=1d-1 Yar (chain ladder)

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Examples (cont’d)

• Yad = paid losses in (a,d) cell

1. E[Yad] = βd Σr=1d-1 Yar (chain ladder)

2. E[Yad] = A db exp(-cd) = exp [α+β ln d - γd] (Hoerl curve for each accident period’s payments)

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Examples (cont’d)

• Yad = paid losses in (a,d) cell1. E[Yad] = βd Σr=1

d-1 Yar (chain ladder)2. 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]where1. ai = accident period to which i-th claim belongs2. 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

E[Yi]

= exp {function of operational time}

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Examples of individual claim models (cont’d)

More generally

E[Yi]

= exp {function of operational time

+ function of accident period (legislative change)}

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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)}

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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)}

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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

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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

Linear Predictor

8.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

optime

optime

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Identification of data features - illustration

• But there appear to be unmodelled trends by• Accident quarter

• Completion (finalisation) quarter

Studentized Standardized Deviance Residuals by finalisation quarter

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Dec-94

Mar-95

Jun-95

Sep-95

Dec-95

Mar-96

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Dec-99

Mar-00

Jun-00

Sep-00

Dec-00

Mar-01

Jun-01

Sep-01

Dec-01

Mar-02

Jun-02

Sep-02

Dec-02

Mar-03

Jun-03

Sep-03

Studentized Standardized Deviance Residuals by accident quarter

-8

-6

-4

-2

0

2

4

6

8

10

Sep-94

Dec-94

Mar-95

Jun-95

Sep-95

Dec-95

Mar-96

Jun-96

Sep-96

Dec-96

Mar-97

Jun-97

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Dec-97

Mar-98

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Dec-99

Mar-00

Jun-00

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Dec-00

Mar-01

Jun-01

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Dec-01

Mar-02

Jun-02

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Dec-02

Mar-03

Jun-03

Sep-03

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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)

p

pmpm xwh

)()( p

pmpmm xwghgZ

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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

hehg

1

1)(

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-5 -4 -3 -2 -1 0 1 2 3 4 5

h

g(h)

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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

)( m p

pmpmm

mm xwgWZWY )( m p

pmpmm

mm xwgWZWY

)( m p

pmpmm

mm xwgWZWY

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Illustration of ANN

Xi

g

Zm

hm

Y

Wm

wm

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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

N

iii XfyErr

1

2))((2

1

)( 22 m p

mpm

m wWErr

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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 λ=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 (cont’d)

• 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