using risk factors in insurance analytics data driven strategies...
TRANSCRIPT
Using risk factors in insurance analyticsdata driven strategies
Chaire DAMI workshop, Louvain-la-Neuve
Katrien AntonioLRisk - KU Leuven and ASE - University of Amsterdam
September 2017
K. Antonio, KU Leuven & UvA 1 / 28
Motivation
Claim frequency and claim severity
as function of
nominal / numeric ∼ ordinal / spatial
features
K. Antonio, KU Leuven & UvA Motivation 2 / 28
Research questions
I Generalized Linear Models (GLMs) for frequency (∼ Poisson) andseverity (∼ gamma).
I How to:
(1) select risk factors or features?
(2) cluster (or bin or fuse) levels within a risk factor?
age groups / postal code clusters / clusters of car models
I Procedure should be data driven, scalable to large (big) data.
I End product is interpretable, within actuarial comfort zone.
K. Antonio, KU Leuven & UvA Motivation 3 / 28
A data driven strategyfor the construction of insurance tariff classes
Henckaerts, Antonio, et al., 2017 (online)
GAM as a starting point
I Generalized Additive Model with predictor:
ηi = g(µi ) = β0 +
p∑j=1
βjxdij +
q∑j=1
fj(xcij ) +
r∑j=1
fj(xsij , y
sij).
I Information criteria:
AIC = −2 · logL+ 2 · EDF
BIC = −2 · logL+ log(n) · EDF,
balancing goodness of fit and complexity.
K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 5 / 28
GAM as a starting point
I MTPL data set from Denuit & Lang (2004), 163 231 records.
I Lowest BIC among exhaustive search with 1 024 fitted models:
log(E(nclaims)) =log(exp) + β0 + β1coveragePO + β2coverageFO + β3fueldiesel+
f1(ageph) + f2(power) + f3(bm) + f4(ageph, power) + f5(long, lat).
which combines offset and
categorical ∼ nominal continuous ∼ ordinal
interactions spatial
risk factors.
K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 6 / 28
GAM as a starting point
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K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 7 / 28
Bin smooth GAM effects - spatial
I Bin or cluster f̂5(longi , lati ) for i ∈ {1, ..., 1 146}.
I We use: (also see classint package in R)
• equal intervals;
• quantile binning;
• complete linkage (see Kaufman & Rousseeuw, 1990)
• Fisher’s natural breaks (see Fisher, 1958 and Slocum et al., 2005).
I Tune number of clusters towards lowest AIC for the GAM with abinned spatial effect
K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 8 / 28
Bin smooth GAM effects - spatial
Equal
[−0.48,−0.32)
[−0.32,−0.15)
[−0.15,0.012)
[0.012,0.18)
[0.18,0.34]
Quantile
[−0.48,−0.18)
[−0.18,−0.092)
[−0.092,−0.019)
[−0.019,0.067)
[0.067,0.34]
Complete
[−0.48,−0.32)
[−0.32,−0.079)
[−0.079,0.047)
[0.047,0.23)
[0.23,0.34]
Fisher
[−0.48,−0.27)
[−0.27,−0.14)
[−0.14,−0.036)
[−0.036,0.11)
[0.11,0.34]
K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 9 / 28
Bin smooth GAM effects - continuous
I Bin f̂1(ageph), f̂2(power), f̂3(bm), f̂4(ageph, power).
I Use evolutionary tree (evtree, Grubinger et al, 2014) e.g. on:
Covariate: ageph Response: f̂1(ageph) Weight: w
18 0.495 1619 0.459 11620 0.424 393
I Evaluate tree with MSE + α · complexity penalty,
MSE =
∑max(ageph)i=min(ageph)
wagephi(f̂1(agephi )− f̂ b1 (agephi ))
2∑max(ageph)i=min(ageph)
wagephi
and α tuning parameter.
K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 10 / 28
Bin smooth GAM effects - continuous
K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 11 / 28
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GLMcoefficients
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f̂5GLMcoefficients
−0.329
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0
0.199
K. Antonio, KU Leuven & UvA Henckaerts, Antonio, et al., 2017 12 / 28
Sparse modeling of risk factors in insurance analytics
Devriendt, Antonio, et al., 2017 (in progress)
Lasso
I Less is more: (Hastie, Tibshirani & Wainwright, 2015)
a sparse model is easier to estimate and interpret than a dense model.
I Regularize (with budget constraint t, or regularization parameter λ):
minβ0,β{− logL} subject to ‖β‖1 ≤ t,
or equivalenty
minβ0,β
− logL+ λ ·p∑
j=1
|βj |
.
Shrinks coefficients and even sets some to zero.
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 14 / 28
Lasso and friends
I Adjust lasso regularization to the type of risk factor:
• Determine type (nominal / numeric ∼ ordinal / spatial);
• Allocate logical penalty.
I Thus, for J risk factors, each with regularization term Pj(.), we wantto optimize:
− logL (β1, . . . ,βJ) + λ ·J∑
j=1
Pj (βj).
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 15 / 28
Matching regularization to type of risk factor
I Ordinal risk factors: fused lasso∑i
wi |βi+1 − βi |.
I Nominal risk factors: generalized fused lasso∑i>k
wi ,k |βi − βk |.
I Spatial risk factor: graph guided fused lasso∑(i ,k)∈G
wi ,k |βi − βk |.
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 16 / 28
Unified GLM framework with multiple type of penalties
I Gertheiss & Tutz (2010) and Oelker & Gertheiss (2017):
• GLMs with various penalties.
• R package available: gvcm.cat (not maintained).
I Uses local quadratic approximations of penalties and PIRLS:
• non-exact selection or fusion;
• computationally intensive.
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 17 / 28
Unified GLM framework with multiple type of penalties
I Our contribution:
• implements an efficient algorithm (with proximal operators);
• scalable to big data (splits into smaller sub-problems);
• flexibility of regularization
penalty takes type of risk factor into account;
works for all popular penalties;
• unifies penalty-specific (machine learning) literature with statistical (or:actuarial) literature.
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 18 / 28
MTPL claim frequency with multiple type of penalties
I Fit Poisson GLM with different penalties.
I Settings:
• Incorporate adaptive and standardization weights for better consistencyand predictive performance.
• Tune λ with BIC (λ̂BIC = 784).
I Re-estimate the final sparse GLM with standard GLM routines (from387 to 30 parms.).
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 19 / 28
MTPL claim frequency with multiple type of penalties
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period
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BICAICGCV BIC.reestAIC.reestGCV.reest
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 20 / 28
MTPL claim frequency with multiple type of penalties
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 21 / 28
MTPL claim frequency with multiple type of penalties
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carage
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GAM pGLM pGLM refit
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 22 / 28
MTPL claim frequency with multiple type of penalties
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fleet2 fourbyfour1 fuel2 monovolume1 sex2 sport2 use2
Binary variables
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coverage2 coverage3 period2 period3 period4
Coverage & Period variables
β̂
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A4
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K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 23 / 28
Wrap-up
I From multi-step to less is more.
I To do:
• further improve algorithm efficiency;
• implement penalties for spatial information, interaction effects;
• R package and paper in progress.
K. Antonio, KU Leuven & UvA Devriendt, Antonio, et al., 2017 24 / 28
More information
For more information, please visit:
LRisk website, www.lrisk.be;
my homepage, www.econ.kuleuven.be/katrien.antonio.
Thanks to
Ageas Continental Europe Argenta
K. Antonio, KU Leuven & UvA More information 25 / 28
References
Henckaerts, R., Antonio, K., Clijsters, M. and Verbelen, R. (2017)A data driven strategy for the construction of insurance tariff classes.Submitted.
Wood, S. (2006)Generalized additive models: an introduction with R.Chapman and Hall/CRC Press.
Gertheiss, J. and Tutz, G. (2010).Sparse modeling of categorial explanatory variables.The Annals of Applied Statistics, 4(4), 2150-2180.
Oelker, M. and Gertheiss, J. (2017).A uniform framework for the combination of penalties in generalizedstructured models.Advances in Data Analysis and Classification, 11(1),97-120.
K. Antonio, KU Leuven & UvA List of references 26 / 28
References
Grubinger, T., Zeileis, A., and Pfeiffer, K.-P. (2014).evtree: Evolutionary learning of globally optimal classification andregression trees in R.Journal of Statistical Software, 61(1), 1-29.
Bivand, R. (2015).classInt: Choose Univariate Class Intervals.R package version 0.1-23.
Parikh, N. and Boyd, S. (2013).Proximal algorithms.Foundations and Trends in Optimization, 1(3):123-231.
Hastie, T., Tibshirani, R. and Wainwright, M. (2015)Statistical learning with sparsity: the Lasso and generalizations.Chapman and Hall/CRC Press.
K. Antonio, KU Leuven & UvA List of references 27 / 28
Extra plots
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K. Antonio, KU Leuven & UvA Extra 28 / 28