MODELING OF VEHICLE INSURANCE CLAIM USING EXPONENTIAL HIDDEN MARKOV. Bogor Agricultural University Meranti Street, Bogor, 16680, INDONESIA

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1 International Journal of Pure and Applied Mathematics Volume 118 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpamv118i215 PAijpameu MODELING OF VEHICLE INSURANCE CLAIM USING EXPONENIAL HIDDEN MARKOV Irfani Azis 1, Berlian Setiawaty 2, I Gusti Putu Purnaba 3 1,2,3 Department of Mathematics Bogor Agricultural University Meranti Street, Bogor, 16680, INDONESIA Abstract: he purpose of this research is to model the vehicle insurance claim using exponential hidden Markov model (EHMM) he maximum likelihood method, forward-backward algorithm and expectation maximization (EM) algorithm are used to determine the EHMM parameter he best model was selected based on Bayesian information criterion (BIC) AMS Subject Classification: 60J22, 60J28 Key Words: claim, exponential, hidden Markov, vehicle insurance 1 Introduction Insurance is an agreement in which the insurer binds itself to an insured company, by receiving a premium to provide a replacement due to loss, damage or loss of expected profit, which may occur due to an uncertain event Vehicle insurance is one type of insurance that protects the vehicle from loss he cause of loss is unpredictable, so the fund which is paid for a claim by an insurance company is unknown Received: Revised: Published: March 17, 2018 c 2018 Academic Publications, Ltd url: wwwacadpubleu Correspondence author

2 310 I Azis, B Setiawaty, IGP Purnaba Figure 1: Histogram frequency of vehicle insurance claim of P XX in 2014 he greater loss that happened the greater claim would be paid he loss that happened can be caused by several factors, such as weather condition, traffic condition, driver condition and other factors Figure 1 shows that the small loss often occurred but huge loss rarely occurred he distribution of claim can be seen as an exponential distribution or a mix of the exponential distribution If the cause of loss is assumed to form a Markov chain and is not observed directly, and only data of claim is observed, then the pair of claim and cause of loss can be modeled by hidden Markov model he purpose of this research is to model of vehicle insurance claim using exponential hidden Markov model 2 Exponential Hidden Markov Model Exponential hidden Markov model (EHMM) is a model which consists of pair of stochastic process {X t,y t } t N {X t } t N is unobserved and form a Markov chain {Y t } t N is an observation process which depends on {X t } t N Y t X t has the exponential distribution, for all t Let {X t } t N be the Markov chain with state space S X = {1,2,3,,m} and transition probability matrix A = [a ij ] m m, where a ij = P(X t = j X t 1 = i) = P(X 2 = j X 1 = i),a ij 0 and j=1 a ij = 1 Let φ = [φ i ] m 1 is the initial state probability vector, where φ i = P(X 1 = i), j=1 φ i = 1 he distribution of Y t X t = i,i S X ;t N has an exponential distribution with parameter λ i he probability density function of Y t X t = i is γ yi =

3 MODELING OF VEHICLE INSURANCE CLAIM 311 f(y) = λ i e λ iy,y > 0 Suppose that λ = [λ i ] m 1 So EHMM can be identified by parameter φ = (m,a,λ) 3 Parameter Estimation of Exponential Hidden Markov Model he parameter φ = (m,a,λ) is estimated using maximum likelihood method (see [1]) Suppose that is the number of observation, y = (y 1,y 2,, y ) is the observation data of {Y t } t N, x = (i 1,i 2,,i ) is the unobserved data of {X t } t N, (x,y) = (i 1,y 1,i 2,y 2,,i,y ) is the data of {X,Y t } t N, and Φ m = {φ = (m,a,λ) : A = [0,1] m2,λ [ǫ,1/ǫ] m } is the parameter space of EHMM Assume that a ij, λ i, and ϕ i are continuous function from R m to R with a ij (φ) = a ij, λ i (φ) = λ i and ϕ i (φ) = ϕ i, for all i,j S X Let L (m) (φ) be an incomplete likelihood function, that is a probability density function of y 1,y 2,,y L (m) (φ) = f φ(y) = f φ (y 1,,y ) = m i 1 =1 m ϕ i1 γ y1 i 1 a it 1 i t γ yti t i =1 Let L C(m) (φ) be a complete likelihood function Where L C(m) (φ) = f φ (x,y) = f φ (i 1,y 1,,i,y ) = ϕ i1 γ y1 i 1 We obtain L (m) (φ) = x L C(m) (φ) t=2 a it 1 i t γ yti t Calculating L (m) (φ) directly when m and are big, is a complex process Because of that, we use the forward-backward algorithm to calculate L (m) (φ) he forward-backward algorithm consists of two sections, those are forward algorithm, and backward algorithm Define forward variabel α t (i,φ,) for t = 1,2,, and i S X as follows t=2 α t (i,φ,) = P φ (Y 1 = y 1,Y 2 = y 2,,Y t = y t,x t = i) Define backward variable β t (i,φ,) for t =, 1,,1;i S X as follows β t (i,φ,) = P φ (Y t+1 = y t+1,y t+2 = y t+2,,y = y X t = i)

4 312 I Azis, B Setiawaty, IGP Purnaba heorem 1 (see [5]) For any t = 1,2,, then m (φ) = α t (i,φ,)β t (i,φ,) L (m) i=1 Now, the problem is to find φ argmax ln(l (m) (φ)) In this case, Expectation Maximization (EM) algorithm is used EM algorithm is an iterative algorithm which consists of two steps for every iteration, those are E step, and M step EM algorithm is as follows 1 For k = 0, the initial value φ (k) is given 2 (E-step) : Calculate Q(φ,φ (k) ) = E φ (k) ( ln(l C(m) Use heorem 2 for calculating Q(φ,φ (k) ) heorem 2 (see [2]) + m m i=1 j=1 + = Q(φ,φ (k) ) = E φ(k) ( m i=1 1 m i=1 ln(l C(m) ) (φ)) y ) (φ)) y α 1 (i,φ (k),)β 1 (i,φ (k),) l=1 α t(l,φ (k),)β t (l,φ (k),) ln(ϕ i) t=1 α t(j,φ (k),)γ (k) y t+1,j a(k) ij β t+1(j,φ (k),) l=1 α t(l,φ (k),)β t (l,φ (k) ln(a ij ),) t=1 α t(j,φ (k),)β t (j,φ (k),) l=1 α t(l,φ (k),)β t (l,φ (k),) ln(γ y ti) heorem 3 (see [6]) Suppose that Φ m = {φ = (m,a,λ) : A = [0,1] m2,λ [ǫ,1/ǫ] m} and ǫ > 0 is very small near zero and EHMM parameter continuous assumption is fulfilled, then (a) Φ m is compact set in R m2 +m

5 MODELING OF VEHICLE INSURANCE CLAIM 313 (b) ln(l (m) (φ)) is a continuous function in Φ m and differentiable in interior Φ m (c) Q(ˆφ,φ) is a continuous function in Φ m Φ m 3 (M-step) : Find φ (k+1) which maximizes Q(φ,φ (k) ) hat is Q(φ (k+1),φ (k) ) Q(φ,φ (k) ), φ Φ m he existence of φ (k+1) is guaranteed by heorem 3 Furthermore, using heorem 4 we have the parameter φ (k+1) heorem 4 (see [2]) Suppose that φ (k) Φ m with φ (k) = (m,a k,λ k ), thentheparameterφ (k+1) = (m,a (k+1),λ (k+1) )whichmaximizesq(φ,φ (k) ) in Φ m can be written as a k+1 ij = 1 t=1 α t(j,φ (k),)γ (k) y t+1,j a(k) ij β t+1(j,φ (k),) l=1 α t(l,φ (k),)β t (l,φ (k),) and λ (k+1) i = ( ) 1 t=1 α t(j,φ (k),)β t (j,φ (k),)(y t ) l=1 α t(l,φ (k),)β t (l,φ (k),) 4 Replace k with k +1 and if ln(l (m) (φ(k+1) )) ln(l (m) (φ(k) )) > ǫ then go back to 2 nd until 4 th step Iteration is stopped when ln(l (m) (φ(k+1) )) ln(l (m) (φ(k) )) < ǫ In other words {ln(l (m) (φ(k+1) ))} converges to {ln(l (m) (φ ))} Its existence is guaranteed by heorem 5 heorem 5 (see [6]) Suppose that Q(ˆφ,φ) is continuous function in Φ m Φ m and {φ (k) } is the set of EHMM estimator which is obtained from EM algorithm, then (a) {ln(l (m) (φ(k) ))} is monotonically increasing sequence (b) φ Φ m lim k ln(l (m) (φ(k) )) = ln(l (m) (φ )) (c) If lim k φ (k) = φ, then φ is the stasionary point of ( L (m) (φ) ) Supposethat ˆφ = φ (k+1) isobtainedinthelast iteration ofemalgorithm, so ˆφ is EHMM parameter estimation, where ˆφ = (ˆm,Â,ˆλ) Bayesian information criterion (BIC) is used for finding state ˆm (see [4]) he selection of ˆm is based

6 314 I Azis, B Setiawaty, IGP Purnaba on ˆm which maximizes ln(l (m) t (φ)) (ln)d m, where ln(l (m) t (φ)) is the value of likelihood function and d m is the number of estimation parameter, so BIC = lnl (m) (φ) (ln)d m 2 he number of estimation parameter in EHMM is (m 2 +m), so the formula of BIC as follows BIC = lnl (m) (φ) (ln)(m2 +m) 2 Furthermore to validate of the model, a set of estimation data ŷ 1,ŷ 2,,ŷ, is generated, based on ˆφ, where ŷ t = Eˆφ ( Yt = y t Y1 t 1 ) 1 = ) L m t 1(ˆφ 0 y t L m t (ˆφ) dy t, and error is calculated by mean absolute percentage error (MAPE), where MAPE = 1 Y t Ŷt Y t 100% t=1 Interpretation of MAPE is as follows able 1: Interpretation of MAPE [3] MAPE% Interpretation 10 Highly accurate forecasting Good forecasting Reasonable forecasting 4 Programming Algorithm of Exponential Hidden Markov Model Suppose that y = {y 1,y 2,,y } is an observation set and EHMM parameter φ = (m,a,λ) will be estimated he algorithm which is used is as follows 1 i) Input the number of data and data y 1,y 2,,y ii) Set m = 2,k = 0,l = 1

7 MODELING OF VEHICLE INSURANCE CLAIM i) For k = 0, choose the initial value of φ (k) = (m,a (k),λ (k) ), where A (k) and λ (k) are generated randomly, A (k) = [a (k) ij ] (m m),λ (k) = [λ (k) i ] (m 1) and fulfill j=1 a(k) ij = 1,λ (k) i 0 and i = 1,2,,m ii) Generate ϕ (k) value which fulfill (ϕ (k) ) = (ϕ (k) ) A (k) iii) Generate γ (k) y l i l from Y l X l = i l 3 Determine ǫ = 10 ( 1) as bound of the likelihood iteration, that is ln(l (m) (φ(k+1) )) ln(l (m) (φ(k) )) < ǫ 4 Calculate Forward and Backward function 5 (E-Step) Calculate L l (φ (k) ) L l (φ (k) ) = m α l (i,φ (k),l)β l (i,φ (k),l) i=1 6 (M-Step)Calculateparameterestimationvalueφ (k+1) = (m,a (k+1),λ (k+1) ), where 1 a k+1 t=1 ij = α t(j,φ (k),)γ (k) y t+1,j a(k) ij β t+1(j,φ (k),) l=1 α t(l,φ (k),)β t (l,φ (k),) ( ) 1 λ (k+1) t=1 i = α t(j,φ (k),)β t (j,φ (k),)(y t ) l=1 α t(l,φ (k),)β t (l,φ (k),) 7 i) If ln(l (m) (φ(k) )) > ǫ, go back to 4 th until 6 th step and replace k by k+1 Otherwise go to the next step Replace l by l+1 and return to 2 nd step, where l = 1,2,, 1 ii) Generate set of estimation data Ŷl iii) Verify that Ŷl Y l Y l < 7%, otherwise replace l to l + 1, where l = 1,2,, 1 and return to 1st step iv) Calculate BIC value v) Calculate error value of estimation data by MAPE If MAPE> 30% then return to 2nd step (φ(k+1) )) ln(l (m) 8 i) For determining optimum m, do the 1st step until 8th step for every state m = 2,3,,6 with different ii) Chose m which maximize BIC

8 316 I Azis, B Setiawaty, IGP Purnaba Figure 2: Vehicle Insurance Claim for each Day in Application of EHMM on Vehicle Insurance Claim his research uses data of vehicle insurance claim in one of Indonesia insurance company in 2014 he data can be seen in Figure 2 he data of claim is called observation data Suppose that Y t is the observation data, X t is the unobserved data, and t is t th day, where t = 1,2,, and = 298 days Parameter estimation of the model is exercised for 2-states until 6-states he last 10 data are used for testing estimation data or prediction data Based on the programming result, log-likelihood value and BIC value is obtained for every state with 288 days of observation data which can be seen in able 2 able 2: Log-likelihood value and BIC value for every state with 288 days observation data m - state Loglikelihood value BIC value 2 - state state state state state Based on able 2, if the number of states increase, then the BIC value will decrease Afterwards interval observation will be determined starting from a

9 MODELING OF VEHICLE INSURANCE CLAIM 317 week, a month, and 3 months Where assumption of a week is 6 day and a month is 26 days Based on programming log-likelihood value and BIC value is obtained for every observation interval and state can be seen in able 3 and able 4 able 3: Log-likelihood value for every state and observation interval m - state Loglikelihood value 1 week 1 month 3 month 2 - state state state state state able 4: BIC value for every state and observation interval m - state BIC value 1 week 1 month 3 month 2 - state state state state state he best model is chosen by the highest BIC value Based on able 2, EHMM with state m = 2 has the highest BIC value, so the optimum parameter estimation is state m = 2 Based on able 3 and able 4, if the observation interval increase then log-likelihood value and BIC value will decrease So the optimum number of state and observation interval is state m = 2 and 1 week observation interval Parameter estimation value which is obtained from 1 week observation interval with state m = 2 are transition probability matrix A, initial transition probability vector ϕ, and exponential parameter estimation vector λ, where the number of 1 week observation data is 283rd until 288th day of claim data

10 318 I Azis, B Setiawaty, IGP Purnaba ransition probability matrix A is obtained as follows ( ) A = Based on matrix A, if the previous claim is caused by state 1, then the claim would happen again by state 1 with probability value If the previous claim is caused by state 2, then the claim would happen again by state 1 with probability value If the previous claim is caused by state 1, then the claim would happen again by state 2 with probability value If the previous claim is caused by state 2, then the claim would happen again by state 2 with probability value Initial transition probability matrix was obtained as follows ϕ = ( ϕ shows that, the claim which is caused by state 1 has parobability value and the claim which is caused by state 2 has probability value Exponential parameter estimation vector λ as follows λ = ( λ shows that, the claim that will be happened by state 1 has rate million rupiahs per day Afterward, the claim that will be happened by state 2 has rate million rupiahs per day Furthermore the set of estimation data is generated by the best obtained parameter Estimation data and Observation data can be seen in Figure 3 he MAPE value is obtained is 1955%, which interprets that estimation data is a good forecasting Observation data and estimation data of Figure 3 are presented in able 5 ) )

11 MODELING OF VEHICLE INSURANCE CLAIM 319 Figure 3: Comparison of observation data and estimation data able 5: Comparison of observation data and estimation data for every single day Day Observation data Estimation data (Million rupiah) (Million rupiah) References [1] AP Dempster, NM Laird, DB Rubin, Maximum likelihood from incomplete data via the EM algorithm, Series B: Journal of the Royal Statistical Society, 39, No 1 (1977), 1-38 [2] M Firmansyah, B Setiawaty, IGP Purnaba, Parameter estimation of exponential hidden Markov model and convergence of its parameter estimator sequence, St Petersburg Polytechnic University Journal (2017), Submitted [3] Y Hu, WM Ho, ravel time prediction for urban networks: he comparisons of simulation-based and time-series models, In: proceedings 17th IS World Congress, Busan, South Korea, 2010 [4] D Lam, S Li, DC Wunsch, Hidden Markov model with information criteria clustering and extreme learning machine regression for wind forecasting, Journal of Computer Science and Cybernetics, 30, No 4 (2014), [5] LR Rabiner, BH Juang, An introduction of hidden Markov models, IEEE Assp Magazine, doi: /86/ (1986)

12 320 I Azis, B Setiawaty, IGP Purnaba [6] CFJ Wu, On the convergence properties of the EM algorithm, he Annals of Statistics, 11, No 1 (1983),