tokens: 3.3 . 3.3 Expectation–maximization algorithm 3.3.1 Overview The measurements, , can be used to solve for τ c and τ s in , N c and N s in , and σ c and σ s in in equations ( 16 )–( 22 ) using any number of maximum-likelihood techniques. In this paper, we primarily discuss the expectation–maximization algorithm (Dempster et al. 1977 ; Shumway Stoffer 1982 ), which naturally extends to situations where observations are missing or Δ t is not constant (Shumway Stoffer 1982 ; Digalakis et al. 1993 ; Gibson Ninness 2005 ). The method cleanly separates process noise in the physical variables, characterized by , and measurement noise , characterized by . Two other common techniques include using Newton’s method to maximize the log-likelihood, log L , and the score vector method (Durbin Koopman 2012 ). The former method relies on numerical estimates of the inverse of the Hessian matrix, while the latter method calculates the gradient of log L with respect to the parameters and steps towards a maximum of log L (Durbin Koopman 2012 ). In this paper, we estimate the elements of , , and , as opposed to τ c , τ s , N c , N s , σ c , and σ s . It is possible to extend this method to directly estimating the parameters in equations ( 1 ) and ( 2 ) using ‘grey-box estimation’ (Ljung 1986 , 2013 ; Kristensen, Madsen Jørgensen 2004 ). 3.3.2 Algorithm and its logical basis The expectation–maximization algorithm is used to find stationary points of the posterior distribution in the absence of a direct observation of the states, . As in the previous section, we use as compact notation for the elements of , , and . We use the product rule to separate the posterior in terms of the states The first term on the right-hand side of equation ( 27 ) is a multivariate Gaussian with a straightforward interpretation, which we discuss in Appendix B . We will use this term to find stationary points of . To start, we treat as a random variable and take an expectation value of both sides, assuming some fixed value for the parameters, , leaving with The second term on the right-hand side of equation ( 28 ) is maximized at (Dempster et al. 1977 ; Gelman et al. 2013 ), while the first term on the right-hand side can be maximized analytically by a new choice of , which increases the marginal posterior distribution on the left-hand side of equation ( 28 ). This lends itself to the following iterative algorithm (where p now defines how many times we have iterated): Start with a guess of . Calculate (‘E-step’) using current estimate of the parameters . Maximize from step (ii) over (‘M-step’). Set to the maximum. Repeat steps (ii) and (iii) for a fixed number of iterations or until converges. In Appendix B , we discuss the application of this method to our specific case. We first present the full form of . We then discuss the E-step and M-step, along with the implementation used in carrying out the simulations discussed in Section 4 . In Appendix C , we show the detailed formulae used
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