Iterated filtering

Overview

The data are a time series y 1 , … , y N collected at times t 1 < t 2 < ⋯ < t N . The dynamic system is modeled by a Markov process X ( t ) which is generated by a function f ( x , s , t , θ , W ) in the sense that

X ( t n ) = f ( X ( t n − 1 ) , t n − 1 , t n , θ , W )

where θ is a vector of unknown parameters and W is some random quantity that is drawn independently each time f ( . ) is evaluated. An initial condition X ( t 0 ) at some time t 0 < t 1 is specified by an initialization function, X ( t 0 ) = h ( θ ) . A measurement density g ( y n | X n , t n , θ ) completes the specification of a partially observed Markov process. We present a basic iterated filtering algorithm (IF1) followed by an iterated filtering algorithm implementing an iterated, perturbed Bayes map (IF2).

Procedure: Iterated filtering (IF1)

Input: A partially observed Markov model specified as above; Monte Carlo sample size J ; number of iterations M ; cooling parameters 0 < a < 1 and b ; covariance matrix Φ ; initial parameter vector θ ( 1 ) for m = 1 to M draw Θ F ( t 0 , j ) ∼ N o r m a l ( θ ( m ) , b a m − 1 Φ ) for j = 1 , … , J set X F ( t 0 , j ) = h ( Θ F ( t 0 , j ) ) for j = 1 , … , J set θ ¯ ( t 0 ) = θ ( m ) for n = 1 to N draw Θ P ( t n , j ) ∼ N o r m a l ( Θ F ( t n − 1 , j ) , a m − 1 Φ ) for j = 1 , … , J set X P ( t n , j ) = f ( X F ( t n − 1 , j ) , t n − 1 , t n , Θ P ( t n , j ) , W ) for j = 1 , … , J set w ( n , j ) = g ( y n | X P ( t n , j ) , t n , Θ P ( t n , j ) ) for j = 1 , … , J draw k 1 , … , k J such that P ( k j = i ) = w ( n , i ) / ∑ ℓ w ( n , ℓ ) set X F ( t n , j ) = X P ( t n , k j ) and Θ F ( t n , j ) = Θ P ( t n , k j ) for j = 1 , … , J set θ ¯ i ( t n ) to the sample mean of { Θ F , i ( t n , j ) , j = 1 , … , J } , where the vector Θ F has components { Θ F , i } set V i ( t n ) to the sample variance of { Θ P , i ( t n , j ) , j = 1 , … , J } set θ i ( m + 1 ) = θ i ( m ) + V i ( t 1 ) ∑ n = 1 N V i − 1 ( t n ) ( θ ¯ i ( t n ) − θ ¯ i ( t n − 1 ) ) Output: Maximum likelihood estimate θ ^ = θ ( M + 1 )

Variations

1. For IF1, parameters which enter the model only in the specification of the initial condition, X ( t 0 ) , warrant some special algorithmic attention since information about them in the data may be concentrated in a small part of the time series.
2. Theoretically, any distribution with the requisite mean and variance could be used in place of the normal distribution. It is standard to use the normal distribution and to reparameterise to remove constraints on the possible values of the parameters.
3. Modifications to the IF1 algorithm have been proposed to give superior asymptotic performance.

Procedure: Iterated filtering (IF2)

Input: A partially observed Markov model specified as above; Monte Carlo sample size J ; number of iterations M ; cooling parameter 0 < a < 1 ; covariance matrix Φ ; initial parameter vectors { Θ j , j = 1 , … , J } for m = 1 to M set Θ F ( t 0 , j ) ∼ N o r m a l ( Θ j , a m − 1 Φ ) for j = 1 , … , J set X F ( t 0 , j ) = h ( Θ F ( t 0 , j ) ) for j = 1 , … , J for n = 1 to N draw Θ P ( t n , j ) ∼ N o r m a l ( Θ F ( t n − 1 , k j ) , a m − 1 Φ ) for j = 1 , … , J set X P ( t n , j ) = f ( X F ( t n − 1 , j ) , t n − 1 , t n , Θ P ( t n , j ) , W ) for j = 1 , … , J set w ( n , j ) = g ( y n | X P ( t n , j ) , t n , Θ P ( t n , j ) ) for j = 1 , … , J draw k 1 , … , k J such that P ( k j = i ) = w ( n , i ) / ∑ ℓ w ( n , ℓ ) set X F ( t n , j ) = X P ( t n , k j ) and Θ F ( t n , j ) = Θ P ( t n , k j ) for j = 1 , … , J set Θ j = Θ F ( t N , j ) for j = 1 , … , J Output: Parameter vectors approximating the maximum likelihood estimate, { Θ j , j = 1 , … , J }

Software

"pomp: statistical inference for partially-observed Markov processes" : R package.