In many cases, observations form a sequence in time, and it is useful to extend the factor analysis model to take into account the dynamical behaviour of the factors. In physics and signal processing, these models are in wide use and are called state space models, that is, the factors in this context are called states.
In the general nonlinear form, the model describes the sequence of
observations
x(t) which have been caused by a sequence of
states
s(t) through a mapping
f. In physics, the
state dynamics is often presented as partial differential equations.
For our purposes, discrete-time difference-equations are more
appropriate. According to the model, the state vector is assumed to
be mapped nonlinearly on the consecutive state vector through the
function
g. Since the model does not aim at representing the
complete physical state of the universe, the state is certainly
affected by some other factors besides the previous state. These
external influences are summarised in a noise model
m(t),
which is also called the innovation process. The dynamic mapping thus
has a form very similar to the mapping from states to observations:
x(t) = f(s(t)) + n(t) | (34) |
s(t+1) = g(s(t)) + m(t). | (35) |