ICA: model order selection and dynamic source models
In this chapter we investigate ICA models in which the number of sources, M , may be less than the number of sensors, N ; so-called non-square mixing. The 'extra' sensor observations are explained as observation noise. This general approach may be called Probabilistic Independent Component Analysis (PICA) by analogy with the Probabilistic Principal Component Analysis (PPCA) model of (11); ICA and PCA don't have observation noise, PICA and PPCA do. Non-square ICA models give rise to a likelihood model for the data involving an integral which is intractable. In this chapter we build on previous work in which the integral is estimated using a Laplace approximation. By making the further assumption that the unmixing matrix lies on the decorrelating manifold we are able to make a number of simplifications. Firstly, the observation noise can be estimated using PCA methods, and, secondly, optimisation takes place in a space having a much reduced dimensionality; having order M 2 parameters rather than M × N. Again, building on previous work, we derive a model order selection criterion for selecting the appropriate number of sources. This is based on the Laplace approximation as applied to the decorrelating manifold and is compared with PCA model order selection methods. Standard ICA, if there is such a thing, is not a proper time series model, as each source is considered to be Independent and Identically Distributed (IID). But with dynamic source models, temporal information is used and, as we show, this can lead to much improved source estimation. The second part of this chapter looks at the use of such dynamic source models, where the sources are modelled using a generalised autoregressive (GAR) process. This is the usual autoregressive process 1