Node for Gaussian Markov chain random variables with time-varying dynamics.
The node models a sequence of Gaussian variables
with linear Markovian dynamics.
The time variability of the dynamics is obtained by modelling the state
dynamics matrix as a linear combination of a set of matrices with
time-varying linear combination weights. The
graphical model can be presented as:
where and are the mean and
the precision matrix of the initial state, is the
precision of the innovation noise, and are the state
dynamics matrix obtained by mixing matrices with
weights .
The probability distribution is
where
Parameters:
mu (Gaussian-like node or (...,D)-array) – , mean of , -dimensional
with plates (…)
Lambda (Wishart-like node or (...,D,D)-array) – , precision matrix of ,
-dimensional with plates (…)
B (Gaussian-like node or (...,D,D,K)-array) – , a set of state dynamics matrix,
-dimensional with plates (…,D)
S (Gaussian-like node or (...,N-1,K)-array) – , time-varying weights
of the linear combination, -dimensional with plates (…,N-1)
nu (gamma-like node or (...,D)-array) – , diagonal elements of the precision of the
innovation process, plates (…,D)
n (int, optional) – , the length of the chain. Must be given if
does not have plates over the time domain (which would not make sense).
Equivalent model block can be constructed with GaussianMarkovChain
by explicitly using SumMultiply to compute the linear combination.
However, that approach is not very efficient for large datasets because it
does not utilize the structure of , thus it explicitly
computes huge moment arrays.