Introduction to Spectral Analysis of Non-Stationary Random Signals - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 237

In the adaptive and recursive methods, the parameters are enriched in a recursive
way over time, which often gives them the property to “adapt” themselves to a non-
stationary context. The modeling is called adaptive if its parameters are modified
using a given criterion as soon as a new signal value is known. These techniques are
often used for real time applications. Essentially, there are two types of stationary
modeling that are made adaptive: first, state space model the autoregressive moving
average (ARMA) modeling via the state space model and Kalman modeling, with
the particular case of adaptive AR [FAR 80] and second, Prony modeling [CAS 88,
LAM 88]. The framework of the adaptive ARMA is based more particularly in the
framework on a stochastic approach by using its state space model and the Kalman
filtering [NAJ 88]; these aspects are not tackled in this topic.

In a context of non-stationary parametric modeling, the classic adaptive methods
have memories with exponential forgeting strategies and can be regrouped in two
families of algorithms: the gradient methods and the least-squares methods. For
more information on adaptive algorithms, see [BEL 89, HAY 90].

9.3.2. Elimination of a stationary condition

In order to build a usable parametric model in a non-stationary context, we can
modify the parametric models known and used in a stationary context by eliminating
one of the stationarity condition. The model then becomes intrinsically non-
stationary and can be applied in the framework of a spectral analysis.

The unstable models and the models with variable or evolutive parameters can
be mentioned amongst all the models built this way.

In unstable models, a part of the poles is on the unit circle, making them unstable
and likely to model certain classes of particular non-stationarities. The ARIMA
models and the seasonal models [BOX 70] are amongst the most well known. Box
and Jenkins [BOX 70] have largely contributed to the popularization of the ARIMA
models ( p , d , q ): AutoRegressive Integrated Moving Average. This is a category of
non-stationary models which make it possible to describe only one type of non-
stationarity of the “polynomial” type. The ARIMA model can be seen as a particular
case of ARMA model ( p + d , q ) in which the polynomial of the autoregressive part
is of the form:

() (

) ()

d

− 1

az =− 1 z a z

p