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## Dynamic factors

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)

The distribution of the innovation process m(t) determines whether the model resembles ordinary or independent factor analysis; the distribution of m(t) has to be non-Gaussian in order for the model to be able to reveal the possible underlying independent innovation processes. The mappings are also often assumed to be functions of observed exogenous inputs u(t): f(s(t), u(t)) and g(s(t), u(t)). This corresponds to having a mixture of supervised and unsupervised learning.

Next: Algorithms Up: LINEAR FACTOR ANALYSIS AND Previous: Algorithms
Harri Valpola
2000-10-31