Improvements Of EM Algorithm And Its Applications In Gene Series Analysis

EM Algorithm proposed by Dempster, Laird and Rudin in 1977 is an iterative procedure which provides an effective method in calculating the Maximum-Likehood (ML) estimation. EM Algorithm also establishes a universal framework in solving the ML problem with missing data. Due to two essential properties of EM Algorithm of monotonic increase in the ML and stable convergence with respect to arbitrary initial values, it is accepted by many academicians. While the low speed of the convergence seriously restricts its applications. In order to overcome this disadvantage, so many improvements have been proposed. The key conception is that those improvements should greatly share the essential properties of EM Algorithm (the monotonic increase in the ML and the stable convergence) and accelerate the speed of the convergence. Based on the same conception, we bring forward the Aitken-ECM (A-ECM) Algorithm which combine the Aitken acceleration and the ECM Algorithm and make them compensate mutually. The Aitken numerical acceleration takes effect when the estimated parameters approaching the convergent values, otherwise it will gets bad results. The ECM Algorithm accelerates the speed of convergence in former steps and shares with EM the stable convergence property. So the A-ECM Algorithm implements the acceleration in the whole procedure, and converges in the end by the ECM. Considering the system with time-varying noise correlation matrices, instead of standard Kalman filtering we use the improved adaptive Kalman filtering (AKF) to estimate the hidden system states. The standard EM Algorithm with Kalman filtering cannot solve the estimation problem when time-varying noise correlation matrices exist. If we neglect the condition of the time-varying noise, the estimated parameters will deviate the true value. Since the Sage-Husa AKF can estimate the noise correlation matrices, but not the time-varying case, so we propose the improved Sage-Husa AKF for the time-varying case. The more precisely we estimate the noise correlation matrices, the more accurate hidden system states we will obtain, which guarantees the precision of the system parameter estimation. In the end, we apply EM Algorithm with those improvements (A-ECM and AKF) in identifying the high dimension gene regulatory network and solving the time-varying noise correlation matrices case.