1 general iteration algorithms by luyang fu, ph. d., state auto insurance company cheng-sheng peter...
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1
General Iteration Algorithms
by
Luyang Fu, Ph. D., State Auto Insurance Company
Cheng-sheng Peter Wu, FCAS, ASA, MAAA, Deloitte Consulting LLP
2007 CAS Predictive Modeling Seminar
Las Vegas, Oct 11-12, 2007
2
Agenda History and Overview of Minimum Bias Method
General Iteration Algorithms (GIA)
Conclusions
Demonstration of a GIA Tool
Q&A
3
History on Minimum Bias A technique with long history for actuaries:
Bailey and Simon (1960) Bailey (1963) Brown (1988) Feldblum and Brosius (2002) A topic in CAS Exam 9
Concepts: Derive multivariate class plan parameters by minimizing a
specified “bias” function Use an “iterative” method in finding the parameters
4
History on Minimum Bias
Various bias functions proposed in the past for minimization
Examples of multiplicative bias functions proposed in the past:
ji jiji
jijiji
jijijiji
jijijiji
yxw
yxrwBiasSquaredChi
yxrwBiasSquared
yxrwBiasBalanced
, ,
2,,
2
,,,
,,,
)(
)(
)(
5
History on Minimum Bias
Then, how to determine the class plan parameters by minimizing the bias function?
One simple way is the commonly used an “iterative” methodology for root finding: Start with a random guess for the values of xi and yj
Calculate the next set of values for xi and yj using the root finding formula for the bias function
Repeat the steps until the values converge
Easy to understand and can be programmed in almost any tool
6
History on Minimum Bias For example, using the balanced bias functions for
the multiplicative model:
itiji
ijiji
tj
jtjji
jjiji
ti
jijijiji
xw
rwy
yw
rw
x
Then
yxrwBiasBalanced
1,,
,,
,
1,,
,,
,
,,,
ˆˆ
ˆˆ
,
0)(
7
History on Minimum Bias
Past minimum bias models with the iterative method:
j tj
jiti
jtjji
jtjjiji
ti
jtjji
jjiji
ti
y
r
nx
yw
yrw
x
yw
rw
x
1,
,,
2/1
1,,
11,
2,,
,
1,,
,,
,
ˆ1
ˆ
ˆ
ˆ
ˆ
ˆˆ
jtjji
jtjjiji
ti
jtjji
jtjjiji
ti
yw
yrw
x
yw
yrw
x
21,,
1,,,
,
21,
2,
1,,2,
,
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
8
Iteration Algorithm for Minimum Bias
Theoretically, is not “bias”. Bias is defined as the difference between an estimator and the
true value. For example, is bias. If , then xhat is an unbiased estimator of x.
To be consistent with statistical terminology, we name our approach as General Iteration Algorithm.
)(,
,, ji
jijiji yxrw
ii xx ˆ 0ˆ ii xx
9
Issues with the Iterative Method Two questions regarding the “iterative” method:
How do we know that it will converge? How fast/efficient that it will converge?
Answers: Numerical Analysis or Optimization textbooks Mildenhall (1999)
Efficiency is a less important issue due to the modern computation power
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Other Issues with Minimum Bias
What is the statistical meaning behind these models? More models to try? Which models to choose?
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Summary on Historical Minimum Bias
A numerical method, not a statistical approach Best answers when bias functions are minimized Use of an “iterative” methodology for root finding in
determining parameters Easy to understand and can be programmed in many
tools
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Connection Between Minimum Bias and Statistical Models
Brown (1988) Show that some minimum bias functions can be derived
by maximizing the likelihood functions of corresponding distributions
Propose several more minimum bias models Mildenhall (1999)
Prove that minimum bias models with linear bias functions are essentially the same as those from Generalized Linear Models (GLM)
Propose two more minimum bias models
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Connection Between Minimum Bias and Statistical Models
Past minimum bias models and their corresponding statistical models
lExponentiay
r
nx
yw
yrw
x
Poissonyw
rw
x
j tj
jiti
jtjji
jtjjiji
ti
jtjji
jjiji
ti
ˆ
1ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
1,
,,
2
2/1
1,,
11,
2,,
,
1,,
,,
,
SquaredLeastyw
yrw
x
Normalyw
yrw
x
jtjji
jtjjiji
ti
jtjji
jtjjiji
ti
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
21,,
1,,,
,
21,
2,
1,,2,
,
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Statistical Models - GLM
Advantages include: Commercial software and built-in procedures available Characteristics well determined, such as confidence level Computation efficiency compared to the iterative procedure
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Statistical Models - GLM
Issues include: Requires more advanced knowledge of statistics for GLM
models Lack of flexibility:
Reliance on commercial software / built-in procedures.
Cannot do the mixed model.
Assumes a pre-determined distribution of exponential families.
Limited distribution selections in popular statistical software.
Difficult to program from scratch.
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Motivations for GIA
Can we unify all the past minimum bias models? Can we completely represent the wide range of GLM and
statistical models using Minimum Bias Models? Can we expand the model selection options that go beyond all
the currently used GLM and minimum bias models? Can we fit mixed models or constraint models?
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General Iteration Algorithm
Starting with the basic multiplicative formula
The alternative estimates of x and y:
The next question is – how to roll up xi,j to xi, and yj,i to yj ?
jiji yxr ,
,,2,1,/ˆ
,2,1,/ˆ
,,
,,
mtoixry
ntojyrx
ijiij
jjiji
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Possible Weighting Functions First and the obvious option - straight average to roll
up
Using the straight average results in the Exponential model by Brown (1988)
i i
jiij
ij
j j
jiji
ji
x
r
my
my
y
r
nx
nx
ˆ1
ˆ1
ˆ
ˆ1
ˆ1
ˆ
,,
,,
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Possible Weighting Functions
Another option is to use the relativity-adjusted exposure as weight function
This is Bailey (1963) model, or Poisson model by Brown (1988).
iiji
ijiji
i
ji
ii
iji
ijiij
ii
iji
ijij
jjji
jjiji
j
ji
jj
jji
jjiji
jj
jji
jjii
xw
rw
x
r
xw
xwy
xw
xwy
yw
rw
y
r
yw
ywx
yw
ywx
ˆˆˆ
ˆˆ
ˆ
ˆˆ
ˆˆˆ
ˆˆ
ˆ
ˆˆ
,
,,,
,
,,
,
,
,
,,,
,
,,
,
,
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Possible Weighting Functions
Another option: using the square of relativity-adjusted exposure
This is the normal model by Brown (1988).
iiji
iijiji
iji
iiji
ijij
jjji
jjjiji
jij
jjji
jjii
xw
xrwy
xw
xwy
yw
yrw
xyw
ywx
22,
,2,
,22,
22,
22,
,2,
,22,
22,
ˆ
ˆˆ
ˆ
ˆˆ
ˆ
ˆ
ˆˆ
ˆˆ
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Possible Weighting Functions
Another option: using relativity-square-adjusted exposure
This is the least-square model by Brown (1988).
iiji
iijiji
iji
iiji
ijij
jjji
jjjiji
jij
jjji
jjii
xw
xrwy
xw
xwy
yw
yrw
xyw
ywx
2,
,,
,2,
2,
2,
,,
,2,
2,
ˆ
ˆˆ
ˆ
ˆˆ
ˆ
ˆ
ˆˆ
ˆˆ
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General Iteration Algorithms
So, the key for generalization is to apply different “weighting functions” to roll up xi,j to xi and yj,i to yj
Propose a general weighting function of two factors, exposure and relativity: WpXq and WpYq
Almost all published to date minimum bias models are special cases of GMBM(p,q)
Also, there are more modeling options to choose since there is no limitation, in theory, on (p,q) values to try in fitting data – comprehensive and flexible
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2-parameter GIA 2-parameter GIA with exposure and relativity adjusted
weighting function are:
i
qi
pji
i
qiji
pji
iji
i
qi
pji
qi
pji
j
j
qj
pji
j
qjji
pji
jij
j
qj
pji
qj
pji
i
xw
xrwy
xw
xwy
yw
yrw
xyw
ywx
1,
1,,
,,
,
,
1,,
,,
,
ˆ
ˆˆ
ˆ
ˆˆ
ˆ
ˆ
ˆˆ
ˆˆ
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2-parameter GIA vs. GLM
p q GLM
1 -1 Inverse Gaussian
1 0 Gamma
1 1 Poisson
1 2 Normal
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2-parameter GIA and GLM GMBM with p=1 is the same as GLM model with the
variance function of Additional special models:
0<q<1, the distribution is Tweedie, for pure premium models 1<q<2, not exponential family -1<q<0, the distribution is between gamma and inverse Gaussian
After years of technical development in GLM and minimum bias, at the end of day, all of these models are connected through the game of “weighted average”.
qV 2)(
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3-parameter GIA
One model published to date not covered by the 2-parameter GMBM: Chi-squared model by Bailey and Simon (1960)
Further generalization using a similar concept of link function in GLM, f(x) and f(y)
Estimate f(x) and f(y) through the iterative method Calculate x and y by inverting f(x) and f(y)
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3-parameter GIA
i
qi
pji
i i
jiqi
pji
iji
i
qi
pji
qi
pji
j
j
qj
pji
j j
jiqj
pji
jij
j
qj
pji
qj
pji
i
xw
x
rfxw
yfxw
xwyf
yw
y
rfyw
xfyw
ywxf
ˆ
)ˆ
(ˆ
)ˆ(ˆ
ˆ)ˆ(
ˆ
)ˆ
(ˆ
)ˆ(ˆ
ˆ)ˆ(
,
,,
,,
,
,
,,
,,
,
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3-parameter GIA Propose 3-parameter GMBM by using the power link
function f(x)=xk
k
i
qi
pji
i
kqi
kji
pji
j
k
j
qj
pji
j
kqj
kji
pji
i
xw
xrwy
yw
yrw
x
/1
,
,,
/1
,
,,
ˆ
ˆˆ
ˆ
ˆ
ˆ
29
3-parameter GIA When k=2, p=1 and q=1
This is the Chi-Square model by Bailey and Simon (1960) The underlying assumption of Chi-Square model is that r2
follows a Tweedie distribution with a variance function
2/1
,
12,,
2/1
,
12,,
ˆ
ˆˆ
ˆ
ˆ
ˆ
iiji
iijiji
j
jjji
jjjiji
i
xw
xrwy
yw
yrw
x
5.1)( V
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Additive GIA
i
pji
iiji
pji
j
j
pji
jjji
pji
i
w
xrwy
w
yrw
x
,
,,
,
,,
)(ˆ
)(
ˆ
31
Mixed GIA
For commonly used personal line rating structures, the formula is typically a mixed multiplicative and additive model: Price = Base*(X + Y) * Z
ji
hjihji
ih
hjihji
jh
hjihji
yx
rz
xz
ry
yz
rx
,,,,
,,,,
,,,,
ˆ
ˆ
ˆ
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Constraint GIA
In real world, for most of the pricing factors, the range of their values are capped due to market and regulatory constraints
)),ˆ95.0min(,ˆ75.0max(ˆ
ˆ
/1
,2
,2,2
112
/1
,1
,1,1
1
k
j
qj
pj
j
kqj
kj
pj
k
j
qj
pj
j
kqj
kj
pj
yw
yrw
xxx
yw
yrw
x
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Numerical Methodology for GIA
For all algorithms: Use the mean of the response variable as the base Starting points:1 for multiplicative factors; 0 for additive factors Use the latest relativities in the iterations All the reported GIAs converge within 8 steps for our test examples
For mixed models: In each step, adjust multiplicative factors from one rating variable
proportionally so that its weighted average is one. For the last multiplicative variable, adjust its factors so that the
weighted average of the product of all multiplicative variables is one.
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Conclusions 2 and 3 Parameter GIA can completely represent GLM and
minimum bias models Can fit mixed models and models with constraints Provide additional model options for data fitting Easy to understand and does not require advanced statistical
knowledge Can program in many different tools Calculation efficiency is not an issue because of modern
computer power.
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Demonstration of a GIA Tool Written in VB.NET and runs on Windows PCs Approximately 200 hours for tool development Efficiency statistics:Efficiency for different test cases
Excel Data CSV Data
# of Records # of Variables Loading Time Model Time Loading Time Model Time
508 3 0.7 sec 0.5 sec 0.1 sec 0.5 sec
25,904 6 2.8 sec 6 sec 1 sec 5 sec
48,517 13 9 sec 50 sec 1.5 sec 50 sec
642,128 49 N/A 40 sec
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Q & A