Fuzzy Mortality Model Based on Banach Algebra · Fuzzy Mortality Model Based on Banach Algebra 247...

InternatIonal Journal of IntellIgent technologIes and applIed statIstIcs

Vol.7, no.3 (2014) pp.241-265, DOI: 10.6148/IJITAS.2014.0703.04

© Airiti Press

Fuzzy Mortality Model Based on Banach AlgebraAndrzej Szymański* and Agnieszka Rossa

Department of Demogra phy and Social Gerontology, University of Lodz, Lodz, Poland

ABSTRACTThe problem of the determining the best mortality models is one of the basic fields in the forecasting strategy of insurance companies. For a given age group x at year t the mortality rate, mx(t) can be expressed in the form of so-called Lee-Carter stochastic mortality model (LC). However, the LC-model assumes the homoscedasticity of error terms which is not adequate to the real life. Koissi and Shapiro have formulated the fuzzy version of the LC-model (FLC), where the model coefficients are assumed to be fuzzy numbers with the symmetric triangular membership function (STMF). To make the inference based on improved FLC more precise and elegant, we apply the Banach algebra of fuzzy numbers, i.e., OFN-algebra introduced by Kosiński et al. [8].

Keywords: Lee-Carter mortality model; Singular Value Decomposition Method; Fuzzy version of Lee-Carter model; Membership function; Triangular fuzzy number; Representation theorem; Diamond distance; Fuzzification and defuzzification; Oriented Fuzzy Numbers (OFN); Banach Algebra of OFN

1. Introduction

The Lee-Carter model (LC) was proposed by Lee and Carter [10]. It includes two risk factors, i.e., age and time model and uses matrix decomposition to extract a single time-varying index of the level of mortality which is then projected using a time series model.

The mortality mx(t) for an individual at age x in year t is estimated by three parameters ax, bx, k(t). Parameter ax can be interpreted as the mean mortality at age x, k(t) is treated as the time trend parameter, and bx reflects the effect of age x on mortality decline.

There are different estimation ways of ax, bx, k(t). One method is Singular Value Decomposition (SVD) used by Lee and Carter [10]. Another one is the Least Square Method, as described in Brouhns et al. [1].

* Corresponding author: [email protected]

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242 Szymański and Rossa

The standard Lee-Carter model (LC) which uses Singular Value Decomposition assumes that the errors have a constant variance over all ages and years. This assumption is not often satisfied and valid.

In general, two groups of mortality models are considered in the literature. The first group which is the widest one are static or stationary models, i.e., the log-odds function of the death probability or mortality rates are expressed in analytical forms that could be linear or nonlinear functions of individuals age, time periods and parameters that have to be estimated.

The second group are dynamic models, where the probability of death or mortality rates are expressed as nonlinear functions of solutions of stochastic differential equations without jumps or with jumps. The classical LC-model belongs to the first group of mortality models.

Koissi and Shapiro [6] proposed fuzzy formulation of the Lee Carter method (FLC), where the parameters of the LC model are fuzzy numbers. The advantage of such an approach is that the errors are viewed as fuzziness of the model structure and the homoscedasticity is not important. What is more, they assumed that the fuzzy numbers are symmetric triangular numbers (STFN) and fuzzy arithmetic operations are based on min and max operators. For STFN these operators have known forms containing max operation and absolute values of the so-called central values of STFN.

2. Lee-Carter model

2.1 Notation

Let mx(t) denote an age-specific crude mortality rate in the year t, defined as

(1)

where:Dx(t) -- number of deaths observed at age x in the year t,Nx(t) -- risk exposure estimated as an average population size in the middle of year t,x = 0, 1, 2, …, w -- one-year age groups,t = 1, 2, …, T -- years of observation period.

2.2 Lee-Carter model formulation

The model structure proposed by Lee and Carter [10] is given by

(2)

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243Fuzzy Mortality Model Based on Banach Algebra

whereax -- the age-specific parameter averaged across years and indexed by x,bx -- first principal component reflecting change in the log mortality rate at each age x,k(t) -- time parameter indexed by t, the first set of principal component scores by year t,e x(t) -- independent random errors, e x(t) ~ N(0, s2

e ) the residual at age x and year t.In order to ensure the unique solution of (2), Lee and Carter [10] imposed two

additional constraints, given as

(3)

where T is the number of years and w is the age limit in the observed data set.

2.3 Forecasts by means of the Lee-Carter model

Estimates of k(t) can be treated as a realization of a stochastic random walk with a drift

(4)

where c is a constant drift, and e(t) independent random terms. An estimator of c is given by the formula

(5)

and a variance estimator of the error term is

(6)

Estimates of ax, bx together with forecasts of k(t) for t > T , allow making forecasts of crude mortality rates mx(t) using (2).

Forecasts of other life-table characteristics are also possible. For instance, probability qx(t) of death in a year t, giving survival to an age x, is obtained from mx(t) by means of the following approximation

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(7)

and the expected remaining life-time ex(t) for t > T can be received from forecasts of mx(t) as

(8)

3. Estimation of the Lee-Carter model for Poland

Results of estimation of model parameters ax, bx and k(t), based on the crude death rates for Poland for the time period 1958 ~ 2000 together with forecasts of k(t) up to 2025, are shown on the Figures 1 ~ 3.

4. Fuzzy f ormulation of the LC-model according to Koissi-Shapiro concept

4.1 Theoretical backgrounds of the fuzzy analysis

Definition 1 [5]We define a fuzzy number A as a set of the form

(9)

where mA(t) : R → [0, 1], called a membership function of A, satisfies the following conditions

1. A is normal set, i.e., there is such an element t0 ∈ R that mA(t0) = 1.2. m is fuzzy-convex, i.e, for all t, s ∈ R and l ∈ [0, 1] there is

3. m is upper continuous, i.e., mA-1([l, 1]) is closed for all l ∈ [0, 1].

4. supp (A) is a compact, where supp (A) = cl{t ∈ R: m(t) > 0}, and cl is a closure operator.

Definition 2The l-cut of the fuzzy number A is a set

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Figure 1. Estimates of ax, x ∈ [0, 100].

Figure 2. Estimates of bx, x ∈ [0, 100].

Age x

a x

Age x

b x

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(10)

where

(11)

The above notions are illustrated on Figure 4.

Definition 3A triangular fuzzy number A = (a, lA, rA), with a center a ∈ R, and left and right

spreads lA, rA, is defined by its triangular membership function (see the Figure 4).

Definition 4A symmetric triangular fuzzy number (STFN) A = (a, lA) with a center a ∈ R

and the spread lA is defined by its triangular membership function with rA = lA = a.

Figure 3. Estimates of k(t) for years 1958 ~ 2000 and forecasts up to 2025.Years y

k (t )

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Property 1 Representation theorem for arbitrary fuzzy number [4].An arbitrary fuzzy number A with compact support can be represented by a pair

of functions (A_(l ), A_

(l ), 0 ≤ l ≤ 1), which satisfy the following requirements.1. A_(l ) is a bounded left continuous nondecreasing function over [0: 1],2. A

_(l ) is a bounded left continuous nonincreasing function over [0: 1],

3. A_(l ) ≤ A_

(l ), 0 ≤ l ≤ 1.

Example 1 (Fuzzy triangular number representation)If A is a fuzzy triangular number then it can be written in a form A = (a, lA, rA)

and its l-cuts can be denoted by

where

In a special case when a triangular fuzzy number is symmetric, i.e., lA = rA = sA, the l-cuts of A can be shown in the form

Figure 4. Membership function of a symmetric triangular fuzzy number.

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Definition 5For two fuzzy numbers A and B with the l-cuts Al = [AL

l, ARl ] and Bl = [BL

l, BRl ]

the Diamond metric is expressed as

(12)

or

(13)

Property 2 (Diamond metric for triangular numbers)Let A = (a, lA, rA) and B = (b, lB, rB) be two triangular fuzzy numbers. Then the

Diamond metric takes the form

(14)

Property 3 Let A = (a, sA) and B = (b, sB) be two triangular symmetric fuzzy numbers. Then

the Diamond metric can be written in the form

(15)

4.2 Fuzzy LC mortality model description [6]

Koissi and Shapiro [6] gave the following arguments toward the fuzzy version of the LC model.

Two main reasons suggest a fuzzy formulation of the Lee-Carter model:1. In the standard LC model (which uses Singular Value Decomposition, SVD),

the error terms e x(t) are assumed Gaussian with constant variance s2e . But, this

homoscedasticity assumption is violated in many applications of the model [1, 2, 7, 11]. Fuzzy logic can be used to solve the LC equation […]. The advantage of such approach is that the errors are viewed as fuzziness of the model structure [3], hence the homoscedasticity is not an issue.

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2. In addition, the magnitude of the variance in the error term suffers from precision. In effect, Lee wrote: “It is our hope, of course, that the error term is well-behaved and of relatively small variance.” How small should the variance be? The vagueness in the definition of s 2

e and the violation of the homoscedasticity assumption suggest the use of a fuzzy formulation of the LC model [9].

They expressed the fuzzy version of LC mortality model in the following form

(16)

whereY x(t) -- known fuzzy log-central death rates, Ax, Bx K(t) -- unknown fuzzy parameters of the model,

The interpretation of the fuzzy variables is the following:Ax -- the fuzzy representation of the average age-specific pattern of mortality, K(t) -- represents the fuzzy formulation of the general mortality level,Bx -- captures the decline in mortality at age x.

Koissi and Shapiro [6] assumed that the fuzzy numbers are symmetric triangular numbers (STFN) and may be expressed in the form

(17)

(18)

(19)

The assumption about triangular character of the fuzzy parameters in the Koissi-Shapiro model has been justif ied in our considerations by means of the simulation giving rise to the fuzzy version of LC-model.

Fuzzy arithmetic operations are based on min and max operators. For STFN they have known forms containing max operation and absolute values of central values of the STFN fuzzy numbers Ax, Bx K(t).

Let A = (a, a ) and B = (b, b ) then

(20)

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(21)

Then the Koissi-Shapiro model takes the form

(22)

4.3 Fuzzification of log-central death rates

Koissi and Shapiro [6] used a fuzzy least-squares regression based on minimum fuzziness criterion to fuzzify the crisp data Y x(t) available to the researchers. For simplicity, symmetric fuzzy numbers are considered. Given the log-central death rates Y x(t) for age x in year t, the task is to find STFN’s

(23)

(24)

(25)

with centers c0x, c1x, yx(t), and spreads s0x, s1x, ex(t), such that

for each age-group x.To find fuzzy numbers A0 and A1 two methods are used. First, ordinary least-

squares regression is used to f ind the center values c0x, c1x, by f inding regression function

where yx(t) = ln mx(t) are the observed log-central death rates. Explicitly, for each age-group x in the year t the estimates

(26)

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and

(27)

are obtained.The spreads s0x and s1x are derived from the minimum fuzziness criterion, by

minimizing the spreads of the estimated Y(t) = (c0x, s0x) + (c1x, s1x) × t, and requiring each death rate Y x(t) to fall within the estimated Yx(t) at a level h.

The minimum fuzziness criterion leads to s0x, s1x which minimise the sum

(28)

for each age x and for a fixed h ∈ [0, 1), subject to the constraints

(29)

(30)

(31)

Greater values of h lead to greater spreads s0x, s1x, so we may assume in the sequel that h = 0.

On the basis of the obtained values of s0x, s1x the values of Yx(t) = (yx(t), ex(t)) are determined, where yx(t) = ln mx(t), ex(t) = max(s0x, s1xt).

4.4 Estimation of fuzzy parameters in the Koissi-Shapiro model

Koissi and Shapiro applied the Diamond distance to obtain least-square estimates of the model parameters, by minimizing the square of the distance between Ax ⊕ (Bx ⊗ K(t)) and Y x(t), i.e.,

(32)

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The Diamond distance D2(Ax ⊕ (Bx ⊗ K(t)), Y x(t)) can be rewritten in the form

The task is to find such coefficients ax, bx, k(t), sAx, sBx, sK(t), which minimize the sum of Diamond distances. However, there exists no effective optimization algorithm for minimizing this sum, due to the expression max (sAx, |bx| sK(t). |k(t)|sBx). Therefore, we propose Extended Fuzzy Mortality Model based on Oriented Fuzzy Numbers (OFN).

5. Description of extended fuzzy Lee-Carter model using Banach algebra of OFN

5.1 Oriented fuzzy numbers

Definition 6 [8]Oriented fuzzy number (OFN) A

→

is an ordered pair

where f , g: [0, 1] → R are continuous functions.Functions f and g are termed a part up and a part down of the oriented fuzzy

number, respectively.Due to Property 1, we know that an arbitrary fuzzy number with compact

support can be represented by a pair functions (A_(l ), A_

(l )) where 0 ≤ l ≤ 1 and the functions A_(l ), A

_(l ) satisfy the requirements (a)-(c) of the Property 1.

Let us denote by F the fuzzy number space with compact support. As it has been shown by Congxin and Ming [4], the space F can be imbedded into the Banach space B = C[0, 1] × C[0, 1], where the metric is defined as

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for an arbitrary (u, v) = C[0, 1] × C[0, 1].If a fuzzy number A has a continuous membership function mA, then taking

we get the OFN A→

= (f , g). Using the result shown in the Example 1, it can be shown that any triangular fuzzy number A = (a, lA, rA) generates an oriented fuzzy number A

→

= (f , g) with

It follows, that a symmetric triangular fuzzy number (STFN) A = (a, sA) generates an oriented fuzzy number (OFN) A

→

= (f , g), with

A triangular fuzzy number STFN and corresponding oriented fuzzy number OFN are illustrated on Figure 5.

Figure 5. Membership function for STFN and Oriented Fuzzy Number corresponding with STFN.

x

f (x )

, g(x

), f-1 (x

), g-1 (x

)

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5.2 Addition and multiplication of oriented fuzzy numbers

Definition 7Let be given three OFN’s

The fuzzy number C→

is a sum of A→

and B→

, denoted by

if

(33)

Definition 8Let be given three OFN’s

The fuzzy number C→

is a product of A→

and B→

, denoted by

if

(34)

5.3 Extended fuzzy Lee-Carter model (EFLC) based on the Banach algebra of OFN

Let A = (a, sA) and B = (b, sB), then for z ∈ [0, 1] we have

(35)

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(36)

and

(37)

It follows, that

(38)

(39)

Consider the fuzzy numbers Ax, Bx, K(t) and Yx(t). Assume that they are STFN, i.e.,

then the oriented fuzzy numbers A→

x, B→

x, K→

(t) take the form

where

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The EFLC model can be expressed in the form

(40)

5.4 The Diamond metric for fuzzy numbers

For two fuzzy numbers A and B with l-cuts Al = [ALl, AR

l ] and Bl = [BLl, BR

l] the Diamond metric is defined as follows

(41)

Let A = (a, lA, rA) and B = (b, lB, rB) be two triangular fuzzy numbers. Then the Diamond metric takes the form

(42)

5.5 Parameter estimation of the EFLC model

where Y→

x(t) = ( fYx(z), gYx

(z)) and

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Let us rearrange the terms under the double sum, then we have

Let us denote

(43)

(44)

(45)

We get

Squaring both sides, summing over x and t, and denoting by y x,t(z), we have

(46)

Integrating the function y x,t(z) over interval [0, 1], we get the expression for the Diamond metric

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From the formula

we get

Now, we have to minimize the functional

(47)

One way of solution is deriving the partial derivatives of F with respect to ax, bx, k(t), sAX, sBx, sK(t), and next equaling them to 0. Thus, for fixed x = 0, 1, …, w or t = 1, 2, …, T we have

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Hence, we have

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Unfortunately, the solution is not unique. Therefore we assume additional constraints for parameters bx, sBx and k(t), sK(t), similar to constraints in the Lee-Carter model. Thus, we assume

(48)

and

(49)

In order to minimize the sum (47), under constraints (48) and (49), any non-linear optimization algorithms can be applied, e.g. available in Excel Solver Package.

5.6 Estimation results of the EFLC model for Poland

Figures 6 ~ 11 illustrate estimates of the parameters of the proposed EFLC mortality model for Poland in time period 1990~2007. Thay can be compared with estimates of respective parameters of the LC model.

Figure 6. Estimates of ax, x ∈ [0, 100] together with the intervals [ax - sAx, ax + sAx] (males).

Age x

a x

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Figure 7. Estimates of ax, x ∈ [0, 100] together with the intervals [ax - sAx, ax + sAx] (females).

Figure 8. Estimates of bx, x ∈ [0, 100] together with the intervals [bx - sBx, bx + sBx] (males).

Age x

a x

Age x

b x

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Figure 9. Estimates of bx, x ∈ [0, 100] together with the intervals [bx - sBx, bx + sBx] (females).

Figure 10. Estimates of k(t) together with the intervals [k(t) - sK(t), k(t) + sK(t)] (males).

Age x

b x

Years t

k (t )

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Figure 11. Estimates of k(t) together with the intervals [k(t) - sK(t), k(t) + sK(t)] (females).

Table 1. Mean Squared Errors for LC and EFLC models.

YearsMales Females

LC EFLC LC EFLC1990 2.9167 1.2294 4.8895 11.02411991 3.2089 7.4789 3.3874 0.83071992 1.2044 2.2453 3.2487 2.32791993 0.8163 1.8831 4.5104 2.26491994 0.3027 0.5893 3.2546 0.28461995 0.7809 2.5923 3.1060 0.54671996 0.3015 2.6022 6.2683 6.53391997 3.9263 1.5995 4.0959 2.89231998 10.7352 9.1075 3.3494 1.18141999 9.4397 4.4450 3.6485 3.39932000 7.8916 7.5526 3.0495 0.57862001 10.1886 11.2413 3.5312 0.61712002 2.8288 4.9652 3.1009 2.02272003 5.5230 4.8609 3.4547 0.94542004 3.0865 2.5681 4.2723 1.18782005 3.3528 3.4876 3.2259 1.28252006 3.0632 3.3946 3.3469 5.01402007 2.1491 4.2089 3.8928 4.0404

1990 ~ 2007 5.1685 5.0817 3.8402 3.7072

Years t

k (t )

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6. Conclusions

In the paper a fuzzy mortality model, termed Extended Fuzzy Lee-Carter Model (EFLC), is presented. It is an alternative to the standard Lee-Carter mortality model (LC) proposed to cope with heteroscedasticity of error terms. Both EFLC and LC were applied to model age-specific mortality rates in Poland. The results have shown the EFLC model to perform usually better than the standard LC model regarding the predictive accuracy. Further investigations will be focused on fuzzy forecasting of life-table parameters, i.e., expected remaining life-time.

Acknowledgement

This research was supported for both authors by a grant from the National Science Center under contract DEC-2011/01/B/HS4/02882, for which the authors are indebted.

Reference

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[8] W. Kosiński, P. Prokopowicz and D. Ślęzak (2003). Ordered fuzzy numbers, Bulletin of the Polish Academy of Sciences Mathematics, 51, 327-338.

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[9] R. D. Lee (2000). The Lee-Carter method for forecasting mortality, with various extensions and applications, North American Actuarial Journal, 1, 80-91.

[10] R. D. Lee and L. R. Carter (1992). Modeling and forecasting U.S. mortality, Journal of the American Statistical Association, 87, 659-671.

[11] A. E. Renshaw and S. Haberman (2003). On the forecasting of mortality reduction factors, Insurance: Mathematics and Economics, 32, 379-401.

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Fuzzy Mortality Model Based on Banach Algebra · Fuzzy Mortality Model Based on Banach Algebra 247...

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