Decomposition optimization method for switching models using EM algorithm

Abstract

This study proposes a decomposition optimization-based expectation maximization algorithm for switching models. The identities of each sub-model are estimated in the expectation step, while the parameters are updated using the decomposition optimization method in the maximization step. Compared with the traditional expectation maximization algorithm and the gradient descent expectation maximization algorithm, the decomposition optimization-based expectation maximization algorithm avoids the matrix inversion and eigenvalue calculation; thus, it can be extended to complex nonlinear models and large-scale models. Convergence analysis and simulation examples are given to show the effectiveness of the proposed algorithm.

Appendix

Proof of of Theorem 1

In iteration $k-1$, the Q function can be written by

$$\begin{aligned} Q({\textbf{W}})&=\sum \limits _{t=1}^{N}\log p(y(t)|{\textbf{W}})\\&=\sum \limits _{t=1}^{N}\log \sum \limits _{i=1}^{l} p(y(t),I(t)\\&=i|Y(t-1),U(t-1),{\varvec{\omega }}^{k-1}_i)\\&=\sum \limits _{t=1}^{N}\log \sum \limits _{i=1}^{l} p^k_p(I(t)=i)\\&\quad \times \frac{p(y(t),I(t)=i|Y(t-1),U(t-1),{\varvec{\omega }}^{k-1}_i)}{p^k_p(I(t)=i)}. \end{aligned}$$

According to Jensen’s inequality, the right side of the above equation satisfies

$$\begin{aligned}&\sum \limits _{t=1}^{N} \log \sum \limits _{i=1}^{l} p^k_p(I(t)=i)\\&\qquad \times \frac{p(y(t),I(t)=i|Y(t-1),U(t-1),{\varvec{\omega }}^{k-1}_i)}{p^k_p(I(t)=i)}\\&\quad \geqslant \sum \limits _{t=1}^{N} \sum \limits _{i=1}^{l} p^k_p(I(t)=i)\log \\&\qquad \times \frac{p(y(t),I(t)=i|Y(t-1),U(t-1),{\varvec{\omega }}^{k-1}_i)}{p^k_p(I(t)=i)}. \end{aligned}$$

Based on the Kullback–Leibler divergence [44], when

$$\begin{aligned}&p^k_p(I(t)=i)\\&\quad =\frac{p(y(t)|u(t),\ldots ,u(1),{\varvec{\omega }}^{k-1}_i,I(t)=i)p(I(t)=i)}{\sum _{i=1}^{l}p(y(t)|u(t), \ldots ,u(1),{\varvec{\omega }}^{k-1}_i,I(t)=i)p(I(t)=i)}, \end{aligned}$$

the Q function is

$$\begin{aligned}&Q({\textbf{W}}|{\textbf{W}}_{k-1},p^k_p(I(t)=i))\\&\quad =\arg \max _{p(I(t)=i)} Q({\textbf{W}}|{\textbf{W}}_{k-1},p(I(t)=i)), \end{aligned}$$

that is

$$\begin{aligned}&Q({\textbf{W}}|{\textbf{W}}_{k-1},p^k_p(I(t)=i))\\&\quad =Q({\textbf{W}}_{k-1},p^k_p(I(t)=i))\\&\quad \geqslant Q({\textbf{W}}_{k-1},p^{k-1}_p(I(t)=i))\\&\quad =Q({\textbf{W}}|{\textbf{W}}_{k-1},p^{k-1}_p(I(t)=i)). \end{aligned}$$

When $p^k_p(I(t)=i)$ has been obtained, the parameter estimates ${\textbf{W}}_{k}$ of the LS algorithm satisfy

$$\begin{aligned} Q({\textbf{W}}_{k},p^k_p(I(t){=}i))=\arg \max _{{\textbf{W}}} Q({\textbf{W}},p^k_p(I(t){=}i)), \end{aligned}$$

(17)

while the estimates ${\textbf{W}}_{k}$ by using the GD-EM and DO-EM (AA-DO-EM) algorithms can guarantee

$$\begin{aligned} Q({\textbf{W}}_{k},p^k_p(I(t)=i))\geqslant Q({\textbf{W}}_{k-1},p^k_p(I(t)=i)). \end{aligned}$$

(18)

Therefore, it follows that

$$\begin{aligned} Q({\textbf{W}}_{k-1})= & {} Q({\textbf{W}}_{k-1},p^{k-1}_p(I(t)=i))\\\leqslant & {} Q({\textbf{W}}_{k-1},p^{k}_p(I(t)=i))\\\leqslant & {} Q({\textbf{W}}_{k},p^k_p(I(t)=i))=Q({\textbf{W}}_{k}). \end{aligned}$$

$\square$