The Linear Prediction Model

  • Peter Strobach
Part of the Springer Series in Information Sciences book series (SSINF, volume 21)

Abstract

One of the most powerful techniques for describing time-varying processes arises from the assumption that an observed process {x(t)}, represented by an ensemble of L independent measurements x0(t), x1(t),…, xL−1(t) at time step t
$$ {\mathbf{x}}{\text{(t)}}{\mkern 1mu} {\text{ = }}{{[{{{\text{x}}}_{{\text{0}}}}{\text{(t),}}{\mkern 1mu} {{{\text{x}}}_{{\text{1}}}}{\text{(t),}}{\mkern 1mu} \ldots {\mkern 1mu} ,{\mkern 1mu} {{{\text{x}}}_{{{\text{L - 1}}}}}({\text{t}})]}^{{\text{T}}}}, $$
(2.1)
can be predicted by a weighted linear combination of previous measurements available in the data vectors x(t−1), x(t−2),…, x(t−p), where
$$ x(t - m) = {{\left[ {{{x}_{O}}(t - m),{{x}_{1}}(t - m), \ldots ,{{x}_{{L - 1}}}(t - m)} \right]}^{T}};\quad 1 \leqslant m \leqslant p $$
(2.2)
and p denotes the model order. Introducing the associated weighting coefficients a1(t), a2(t), …, ap(t), one can write the linear combination as
$$ {{\hat{x}}^{f}}(t) = \sum\limits_{{m = 1}}^{p} {{{a}_{m}}(t)x(t - m)} $$
(2.3)
where f (t) is the resulting forward predicted process vector. The parameter vector
$$ {\mathbf{a}}{\text{(t)}} = {{[{{{\text{a}}}_{{\text{1}}}}{\text{(t),}}{\mkern 1mu} {{{\text{a}}}_{{\text{2}}}}{\text{(t),}}{\mkern 1mu} \ldots {\mkern 1mu} {\text{,}}{\mkern 1mu} {{{\text{a}}}_{{\text{P}}}}({\text{t}})]}^{{\text{T}}}} $$
(2.4)
should be determined in a sense such that the forward predicted process vector f (t) becomes as “close” as possible to the true process vector x(t). Introducing the matrix of past observations X(t)
$$ {\mathbf{X}}{\text{(t)}} = [{\mathbf{x}}{\text{(t - 1)}},{\mkern 1mu} {\mathbf{x}}{\text{(t - 2),}}{\mkern 1mu} \ldots {\mkern 1mu} {\text{,}}{\mkern 1mu} {\mathbf{x}}{\text{(t - p)}}] $$
(2.5)
we may express the linear combination or predictor equation (2.3) as
$$ {{{\mathbf{\hat{x}}}}^{{\text{f}}}}{\text{(t)}}{\mkern 1mu} {\text{ = }}{\mkern 1mu} {\mathbf{X}}{\text{(t)}}{\mkern 1mu} {\mkern 1mu} {\mathbf{a}}{\text{(t)}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\text{.}} $$
(2.6)
The predictor determined by the expressions (2.3) or (2.6) can be interpreted as an all-zero or moving average (MA) filter structure as shown in Fig. 2.1.

Keywords

Covariance Assimilation Autocorrelation Convolution 

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References

Chapter 2

  1. 2.1
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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Peter Strobach
    • 1
  1. 1.ZFE IS — Forschung für Informatik und SoftwareSIEMENS AG, Zentralabteilung Forschung und EntwicklungMünchen 83Fed. Rep. of Germany

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