A Stochastic EM algorithm for construction of Mortality Tables A Stochastic EM algorithm for construction of Mortality Tables Luz Judith Rodriguez Esparza, F . Baltazar-Larios MAM-9 Budapest, June 28 - 30, 2016
A Stochastic EM algorithm for construction of Mortality Tables Aim We propose to use the concept of physiological age to modelling the aging process by using phase-type distributions to calculate the probability of death. For the estimation part, we will use the EM algorithm which in turn uses the Bisection method.
A Stochastic EM algorithm for construction of Mortality Tables Outline BACKGROUND: 1 MORTALITY MODELS PHYSIOLOGICAL AGE PHASE-TYPE DISTRIBUTIONS MODEL AND DATA 2 ESTIMATION: STOCHASTIC EM ALGORITHM 3 APPLICATION: REAL DATA 4
A Stochastic EM algorithm for construction of Mortality Tables MORTALITY TABLE Mortality risk A. de Moivre in 1725 proposed the first mathematical mortality model. In 1825 B. Gompertz proposed that a law of geometric progression pervades in mortality after a certain age. He obtained the following expression: µ x = α e β x . In 1860 , W. M. Makeham extended the Gompertz model by adding a constant: µ x = σ + α e β x , where σ represents all random factors with no willingness to death, for example accidents, epidemics, etc.
A Stochastic EM algorithm for construction of Mortality Tables MORTALITY TABLE Motivation Several factors can alter the probability of death, the more considered factor is the age but there are other important characteristics such as sex, clinical history, smoking, etc. In most mortality models we cannot determine the distribution of the time of death explicitly. Lin and Liu (2008) used phase-type distributions to model human mortality. We consider a more general case, giving another interpretation of the states and using a stochastic EM algorithm for the estimation part.
A Stochastic EM algorithm for construction of Mortality Tables PHYSIOLOGICAL AGE PHYSIOLOGICAL AGE Aging process: “the progressive, and essentially irreversible diminution with the passage of time of the ability of an organism or one of its parts to adapt to its environment, manifested as diminution of its capacity....". To model the aging process, we consider the concept of physiological age: relative health index representing the degree of aging on the individual. It is natural assume that the time spent in each state has an exponential behavior.
A Stochastic EM algorithm for construction of Mortality Tables PHASE-TYPE DISTRIBUTIONS PHASE-TYPE DISTRIBUTIONS Dimension n . Infinitesimal matrix: � Q � r Λ = . (1) 0 0 Let α be the initial distribution of the process.
A Stochastic EM algorithm for construction of Mortality Tables MODEL AND DATA MODEL AND DATA Let consider a finite-state Markov jump process to model the hypothetical aging process. If a person has physiological age i for i ∈ { 1 , . . . , n − 1 } , we consider three possible cases: Natural development of the aging process: the person eventually 1 transits to the next physiological age i + 1. Intensity of this transition: λ i , i + 1 . The aging process is affected by an unusual incident: the person 2 transits to some physiological age j , with j ∈ { i + 2 , i + 3 , . . . , n } . Intensity of this transition: λ ij . The possibility of death for the person at that physiological status. 3 Intensity of this transition: r i .
A Stochastic EM algorithm for construction of Mortality Tables MODEL AND DATA Thus, the sub-intensity matrix is given by − λ 1 λ 12 λ 13 . . . λ 1 n − λ 2 0 λ 23 . . . λ 2 n 0 0 − λ 3 . . . λ 3 n Q = . . . . ... ... . . . . . . 0 0 0 . . . − λ n We denote by q i ( t ) the probability of death for a person at physiological age i ∈ { 1 , 2 . . . , n } , in the interval [ 0 , t ] , which is given by q i ( t ) = P ( τ i ≤ t ) = 1 − e i exp ( t Q ) e , (2) where τ i ∼ PH ( e i , Q ) .
A Stochastic EM algorithm for construction of Mortality Tables MODEL AND DATA Considering a population of size M and the time interval [ 0 , T ] , let X m = { X m t } T t ≥ 0 , m = 1 , . . . , M , be independent Markov jump processes with the same finite state space E and the same intensity matrix. Each X m represents the aging process for each person in the population. We will use a stochastic EM algorithm for finding maximum likelihood estimators of ( α , Q ) considering the two scenarios and using the Bisection method.
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM ESTIMATION Cases: Continuous time information of the aging process of the 1 population. (See [2]). There are reports of the development process only at 2 determined moments, i.e., there are only discrete time observations of a Markov jump process. (Discrete case).
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM Continuous case Considering that the X m ’s have been observed continuously in the time interval [ 0 , T ] . Complete likelihood function: n n n n N ij ( T ) r N i ( T ) � α B i � � � L c e − λ i Z i ( T ) T ( θ ) = λ , (3) i ij i i = 1 i = 1 j � = i i = 1 Log-likelihood function: n n n � � � ℓ c T ( θ ) = B i log ( α i ) + N ij ( T ) log ( λ ij ) i = 1 i = 1 j � = i n n n n � � � � − λ ij Z i ( T ) + N i ( T ) log ( r i ) − r i Z i ( T ) . (4) i = 1 j � = i i = 1 i = 1
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM Discrete Case We are interested in the inference about the intensity matrix Λ based on samples of observations of X m ’s at discrete times points. Suppose that all processes have been observed only at K time points 0 = t 1 < . . . < t K = T denoted by Y m k = X m t k . t k + 1 − t k = ∆ , then Y m = { Y m k : k = 1 , . . . , K } is the discrete time Markov chain associated with X m with transition matrix P (∆) = exp (∆Λ) . Observed values by y = { y 1 , . . . , y M } where y m = { Y m 1 = y m 1 , . . . , Y m K = y m K } .
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM EM algorithm Let θ 0 = ( α 0 , Q 0 ) denote any initial value of parameters. (E-step) Calculate the function 1 h ( θ ) = E θ 0 ( ℓ c T ( θ ) | Y = y ); (M-step) 2 θ 0 = argmax θ h ( θ ); Go to 1 3
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM E-step n � E θ 0 ( ℓ c T ( θ ) | Y = y ) = log ( α i ) E θ 0 ( B i | Y = y ) i = 1 n n n n � � � � + log ( λ ij ) E θ 0 ( N ij ( T ) | Y = y ) − λ ij E θ 0 ( Z i ( T ) | Y = y ) i = 1 j � = i i = 1 j � = i n n � � + log ( r i ) E θ 0 ( N i ( T ) | Y = y ) − r i E θ 0 ( Z i ( T ) | Y = y ) . (5) i = 1 i = 1
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM Conditional expectations of the statistics: M � E θ 0 ( B i | Y = y ) = 1 { Y m 1 = i } ; i = 1 , . . . , n , (6) m = 1 where 1 {·} is the indicator function, M K � � N mij ˜ E θ 0 ( N ij ( T ) | Y = y ) = k ( t k − t k − 1 ) , (7) y m k − 1 y m m = 1 k = 2 M K � � ˜ Z mi E θ 0 ( Z i ( T ) | Y = y ) = k ( t k − t k − 1 ) , (8) y m k − 1 y m m = 1 k = 2 and M K � � ˜ N mi E θ 0 ( N i ( T ) | Y = y ) = k − 1 ( t k − t k − 1 ) . (9) y m m = 1 k = 2
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM Markov Bridges To calculate these expectations we propose to generate L sample paths of the Markov bridge X m ( r , s 1 , s , s 2 ) using the parameter value θ 0 = ( α 0 , Q 0 ) . I.e., a stochastic process defined on [ s 1 , s 2 ] and having the same distribution of the Markov jump process { X m t } t ∈ [ s 1 , s 2 ] conditioned on X m s 1 = r and X m s 2 = s for m = 1 , . . . , M .
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM Now, based on bridges we approximate the conditional expectations by L 1 N mij ˜ � N mij ( l ) k ( t k − t k − 1 ) ≈ k ( t k − t k − 1 ) , (10) y m k − 1 y m y m k − 1 y m L l = 1 L 1 ˜ � Z mi ( l ) Z mi k ( t k − t k − 1 ) ≈ k ( t k − t k − 1 ) , (11) y m k − 1 y m y m k − 1 y m L l = 1 L 1 ˜ � N mi ( l ) N mi k − 1 ( t k − t k − 1 ) ≈ k − 1 ( t k − t k − 1 ) , (12) y m y m L l = 1 respectively, for i , j = 1 , . . . , n ; m = 1 . . . , M ; and k = 1 , . . . , K .
A Stochastic EM algorithm for construction of Mortality Tables STOCHASTIC EM ALGORITHM Stochastic EM algorithm initial value of parameters: θ 0 = ( α 0 , Q 0 ) . Let θ = θ 0 . Given M , K 1 for m = 1 , . . . , M , k = 2 , . . . , K : Generate L paths of the Markov bridge X m ( y k − 1 , t k − 1 , y k , t k ) 2 using θ . E-step. 3 Using (10), (11), and (12) calculate ˜ N mij k ( t k − t k − 1 ) , y m k − 1 y m ˜ k ( t k − t k − 1 ) , and ˜ Z mi N mi k − 1 ( t k − t k − 1 ) . y m k − 1 y m y m Using (6), (7), (8), and (9), calculate E θ ( B i | Y = y ) , E θ ( N ij ( T ) | Y = y ) , E θ ( Z i ( T ) | Y = y ) , and E θ ( N i ( T ) | Y = y ) . M-step Calculate ˆ α , ˆ θ = ( ˆ Q ) by 4 α i = E θ ( B i | Y = y ) λ ij = E θ ( N ij ( T ) | Y = y ) r i = E θ ( N i ( T ) | Y = y ) ˆ ˆ ˆ ; E θ ( Z i ( T ) | Y = y ) ; E θ ( Z i ( T ) | Y = y ) . M θ = ˆ θ go to 2. 5
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