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Statistical analysis of weather-related propertyinsurance claims
Christian Rohrbeck1
Joint work with Emma Eastoe, Arnoldo Frigessi and Jonathan Tawn
Department of Mathematics and Statistics & Data Science Institute
July 16, 2018
1Beneficiary of an AXA Research Fund postdoctoral grantChristian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Motivation
Weather event Claims?
Hazards:
Severe rainfall
Thunderstorm
Snow-melt
Events occur rarelyand differ in length
Risks:
Localized flooding
Sewage back-flow
Blocked pipes
Lag in recording process
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Motivation
Weather event Claims?
Hazards:
Severe rainfall
Thunderstorm
Snow-melt
Events occur rarelyand differ in length
Risks:
Localized flooding
Sewage back-flow
Blocked pipes
Lag in recording process
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Motivation
Weather event Claims?
Hazards:
Severe rainfall
Thunderstorm
Snow-melt
Events occur rarelyand differ in length
Risks:
Localized flooding
Sewage back-flow
Blocked pipes
Lag in recording process
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Motivation
Weather event Claims?
Hazards:
Severe rainfall
Thunderstorm
Snow-melt
Events occur rarelyand differ in length
Risks:
Localized flooding
Sewage back-flow
Blocked pipes
Lag in recording process
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Data I
Daily records for Norwegianmunicipalities for 1997-2006 on
Reported number ofwater-related claims N
Amount of precipitation R
Amount of snow S
Surface run-off D
Mean-temperature T
We aim to model N independence on X = (R,S ,D,T ).
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Data II
Oslo Bergen
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0 50 100 150
050
100
150
Rain previous day
Rai
n cu
rren
t day
4●4
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6●
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4●
11●
50●
5●
51●
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Previous research
Scheel et al. (2013) propose a model of the form
P (N = n | X) =
{α(X) if n = 0,
[1− α(X)]P(Y = n | X,Y > 0) if n > 0,
where Y is a Poisson random variable.
But Table 2 in Scheel et al. (2013) shows that their modelunderpredicts the highest claim numbers:
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Previous research
Scheel et al. (2013) propose a model of the form
P (N = n | X) =
{α(X) if n = 0,
[1− α(X)]P(Y = n | X,Y > 0) if n > 0,
where Y is a Poisson random variable.
But Table 2 in Scheel et al. (2013) shows that their modelunderpredicts the highest claim numbers:
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Our approach
Rohrbeck et al. (2018) extend Scheel et al. (2013) by
Introducing a more flexible model using methodology fromextreme value theory to handle the highest numbers of claims,
Deriving additional predictors to incorporate the spatial andtemporal behaviour of the rainfall and snow-melt,
Proposing a clustering algorithm to merge days whose claimsare probably related to the same severe weather event.
In the following, we will focus on the third point.
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Our approach
Rohrbeck et al. (2018) extend Scheel et al. (2013) by
Introducing a more flexible model using methodology fromextreme value theory to handle the highest numbers of claims,
Deriving additional predictors to incorporate the spatial andtemporal behaviour of the rainfall and snow-melt,
Proposing a clustering algorithm to merge days whose claimsare probably related to the same severe weather event.
In the following, we will focus on the third point.
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Our approach
Rohrbeck et al. (2018) extend Scheel et al. (2013) by
Introducing a more flexible model using methodology fromextreme value theory to handle the highest numbers of claims,
Deriving additional predictors to incorporate the spatial andtemporal behaviour of the rainfall and snow-melt,
Proposing a clustering algorithm to merge days whose claimsare probably related to the same severe weather event.
In the following, we will focus on the third point.
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm I - Concept
Motivation Potential lag in the recording process
Event may effect claim dynamics on several days
Trigger Heavy rain Rt > c
Snow-melt St−1 − St > 0
Cluster end Small change in surface run-off Dt − Dt−1 ≤ d
No snow left on the ground St = 0
Predictors Aggregated snow-melt ∆S
Aggregated rainfall RΣ
Maximum daily rainfall Rmax
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm I - Concept
Motivation Potential lag in the recording process
Event may effect claim dynamics on several days
Trigger Heavy rain Rt > c
Snow-melt St−1 − St > 0
Cluster end Small change in surface run-off Dt − Dt−1 ≤ d
No snow left on the ground St = 0
Predictors Aggregated snow-melt ∆S
Aggregated rainfall RΣ
Maximum daily rainfall Rmax
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm I - Concept
Motivation Potential lag in the recording process
Event may effect claim dynamics on several days
Trigger Heavy rain Rt > c
Snow-melt St−1 − St > 0
Cluster end Small change in surface run-off Dt − Dt−1 ≤ d
No snow left on the ground St = 0
Predictors Aggregated snow-melt ∆S
Aggregated rainfall RΣ
Maximum daily rainfall Rmax
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm I - Concept
Motivation Potential lag in the recording process
Event may effect claim dynamics on several days
Trigger Heavy rain Rt > c
Snow-melt St−1 − St > 0
Cluster end Small change in surface run-off Dt − Dt−1 ≤ d
No snow left on the ground St = 0
Predictors Aggregated snow-melt ∆S
Aggregated rainfall RΣ
Maximum daily rainfall Rmax
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm II - Example
Original Data
N D S T Rt
N1 0.4 20.4 -8.3 11.2N2 0.4 31.6 -2.8 3.5N3 0.7 28.1 2.0 0.0N4 1.3 18.8 3.1 1.0N5 2.0 8.8 3.3 9.0N6 2.4 4.6 1.6 2.0N7 2.4 4.6 -0.1 1.9
Clustered Data
N ∆S RΣ Rmax
N1 0.0 0.0 0.0N2 0.0 0.0 0.0∑7i=3 Ni 27.0 3.0 9.0
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm II - Example
Original Data
N D S T Rt
N1 0.4 20.4 -8.3 11.2N2 0.4 31.6 -2.8 3.5N3 0.7 28.1 2.0 0.0N4 1.3 18.8 3.1 1.0N5 2.0 8.8 3.3 9.0N6 2.4 4.6 1.6 2.0N7 2.4 4.6 -0.1 1.9
Clustered Data
N ∆S RΣ Rmax
N1 0.0 0.0 0.0
N2 0.0 0.0 0.0∑7i=3 Ni 27.0 3.0 9.0
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm II - Example
Original Data
N D S T Rt
N1 0.4 20.4 -8.3 11.2N2 0.4 31.6 -2.8 3.5N3 0.7 28.1 2.0 0.0N4 1.3 18.8 3.1 1.0N5 2.0 8.8 3.3 9.0N6 2.4 4.6 1.6 2.0N7 2.4 4.6 -0.1 1.9
Clustered Data
N ∆S RΣ Rmax
N1 0.0 0.0 0.0N2 0.0 0.0 0.0
∑7i=3 Ni 27.0 3.0 9.0
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm II - Example
Original Data
N D S T Rt
N1 0.4 20.4 -8.3 11.2N2 0.4 31.6 -2.8 3.5N3 0.7 28.1 2.0 0.0N4 1.3 18.8 3.1 1.0N5 2.0 8.8 3.3 9.0N6 2.4 4.6 1.6 2.0N7 2.4 4.6 -0.1 1.9
Clustered Data
N ∆S RΣ Rmax
N1 0.0 0.0 0.0N2 0.0 0.0 0.0
∑7i=3 Ni 27.0 3.0 9.0
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm II - Example
Original Data
N D S T RN1 0.4 20.4 -8.3 11.2N2 0.4 31.6 -2.8 3.5N3 0.7 28.1 2.0 0.0N4 1.3 18.8 3.1 1.0N5 2.0 8.8 3.3 9.0N6 2.4 4.6 1.6 2.0N7 2.4 4.6 -0.1 1.9
Clustered Data
N ∆S RΣ Rmax
N1 0.0 0.0 0.0N2 0.0 0.0 0.0∑7i=3 Ni 27.0 3.0 9.0
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm III - Results
We set the thresholds c and d in the clustering algorithm as
c = q0.8 (Rt | Rt > 0)
andd = q0.8 (Dt − Dt−1) .
This gave the following frequency of cluster lengths:
Cluster length in days 1 2 3 4 5 6 > 6
Oslo 2091 254 57 98 43 23 17Bærum 2453 105 43 92 46 19 18Bergen 1868 340 55 131 39 23 11
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm IV - Results
Oslo Bergen
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Rain previous day
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Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Clustering algorithm IV - Results
Oslo Bergen
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Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Statistical model
Model N | (X,N > 0) via a two-component mixture
N | (X,N > 0) ∼
{Y | (X,Y > 0) with probability p,
Z | Z > 0 with probability 1− p..
Y corresponds to claims related to the observed rainfall andsnow-melt
Z represents claims due to unobserved processes or a high lag.
Both Y and Z are Poisson distributed, but we replace the tailof Y with a distribution used in extreme value analysis.
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Statistical model
Model N | (X,N > 0) via a two-component mixture
N | (X,N > 0) ∼
{Y | (X,Y > 0) with probability p,
Z | Z > 0 with probability 1− p..
Y corresponds to claims related to the observed rainfall andsnow-melt
Z represents claims due to unobserved processes or a high lag.
Both Y and Z are Poisson distributed, but we replace the tailof Y with a distribution used in extreme value analysis.
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Results - Oslo
Original Data Clustered Data
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0 20 40 60 800
2040
6080
RΣ
Rm
ax
10●
7●
5●
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13●
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25●
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Data plot Data plot
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Results - Oslo
Original Data Clustered Data
0 10 20 30 40 50 60
010
2030
4050
60
Theoretical quantiles
Em
piric
al q
uant
iles
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0 20 40 60 800
2040
6080
RΣ
Rm
ax
10●
7●
5●
6●
9●
13●
5●
6●
25●
71●
6●
22●
5●
12●
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8●
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6●
7●
10●
5●
7●
QQ plot Data plot
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Results - Oslo
Original Data Clustered Data
0 10 20 30 40 50 60
010
2030
4050
60
Theoretical quantiles
Em
piric
al q
uant
iles
0 20 40 60 80
020
4060
80
Theoretical quantiles
Em
piric
al q
uant
iles
QQ plot QQ plot
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Results - Bergen
Original Data Clustered Data
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0 50 100 150
050
100
150
Rain previous day
Rai
n cu
rren
t day
4●4
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0 50 100 1500
5010
015
0
RΣ
Rm
ax
5● 5
●
32●12
●
5●
10●
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10●
6●
6● 8
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5●6
● 5●
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63●
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53●
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6●7
●
5●
Data plot Data plot
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Results - Bergen
Original Data Clustered Data
0 10 20 30 40 50 60
010
2030
4050
60
Theoretical quantiles
Em
piric
al q
uant
iles
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0 50 100 1500
5010
015
0
RΣ
Rm
ax
5● 5
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32●12
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5●
10●
11●
9●
7●
9●
7●
5●
11●
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6●
6● 8
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63●
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QQ plot Data plot
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Results - Bergen
Original Data Clustered Data
0 10 20 30 40 50 60
010
2030
4050
60
Theoretical quantiles
Em
piric
al q
uant
iles
0 20 40 60
020
4060
Theoretical quantiles
Em
piric
al q
uant
iles
QQ plot QQ plot
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Results - Bærum
Original Data Clustered Data
0 50 100 150
050
100
150
Theoretical quantiles
Em
piric
al q
uant
iles
0 50 100 150
050
100
150
Theoretical quantiles
Em
piric
al q
uant
iles
QQ plot QQ plot
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
Current Research
We want to consider all 430municipalities.
But: Days with a highernumber of claims are veryrare for rural municipalities.
Idea: Share statisticalinformation acrossmunicipalities.
First step: Detect clustersof municipalities with similarsevere rainfall pattern.
Christian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims
References
Rohrbeck, C., Eastoe, E. F., Frigessi, A., and Tawn, J.A. (2018).
Extreme-value modelling of water-related insurance claims. Annals of
Applied Statistics, 12(1):246-282.
Rohrbeck, C. and Tawn, J.A. (2018). Spatial clustering of extremal
behaviour for hydrological variables. In preparation.
Scheel, I., Ferkingstad, E., Frigessi, A., Haug, O., Hinnerichsen, M.,
and Meze-Hausken, E. (2013). A Bayesian hierarchical model with
spatial variable selection: the effect of weather on insurance claims.
Journal of the Royal Statistical Society: Series C, 62(1):85-100.
Thank youChristian Rohrbeck Data Science Institute, Lancaster University
Statistical analysis of weather-related property insurance claims