The Linear Prediction Model

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


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
$$ {\mathbf{x}}{\text{(t)}}{\mkern 1mu} {\text{ = }}{{[{{{\text{x}}}_{{\text{0}}}}{\text{(t),}}{\mkern 1mu} {{{\text{x}}}_{{\text{1}}}}{\text{(t),}}{\mkern 1mu} \ldots {\mkern 1mu} ,{\mkern 1mu} {{{\text{x}}}_{{{\text{L - 1}}}}}({\text{t}})]}^{{\text{T}}}}, $$
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
$$ x(t - m) = {{\left[ {{{x}_{O}}(t - m),{{x}_{1}}(t - m), \ldots ,{{x}_{{L - 1}}}(t - m)} \right]}^{T}};\quad 1 \leqslant m \leqslant p $$
and p denotes the model order. Introducing the associated weighting coefficients a1(t), a2(t), …, ap(t), one can write the linear combination as
$$ {{\hat{x}}^{f}}(t) = \sum\limits_{{m = 1}}^{p} {{{a}_{m}}(t)x(t - m)} $$
where f (t) is the resulting forward predicted process vector. The parameter vector
$$ {\mathbf{a}}{\text{(t)}} = {{[{{{\text{a}}}_{{\text{1}}}}{\text{(t),}}{\mkern 1mu} {{{\text{a}}}_{{\text{2}}}}{\text{(t),}}{\mkern 1mu} \ldots {\mkern 1mu} {\text{,}}{\mkern 1mu} {{{\text{a}}}_{{\text{P}}}}({\text{t}})]}^{{\text{T}}}} $$
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)
$$ {\mathbf{X}}{\text{(t)}} = [{\mathbf{x}}{\text{(t - 1)}},{\mkern 1mu} {\mathbf{x}}{\text{(t - 2),}}{\mkern 1mu} \ldots {\mkern 1mu} {\text{,}}{\mkern 1mu} {\mathbf{x}}{\text{(t - p)}}] $$
we may express the linear combination or predictor equation (2.3) as
$$ {{{\mathbf{\hat{x}}}}^{{\text{f}}}}{\text{(t)}}{\mkern 1mu} {\text{ = }}{\mkern 1mu} {\mathbf{X}}{\text{(t)}}{\mkern 1mu} {\mkern 1mu} {\mathbf{a}}{\text{(t)}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\text{.}} $$
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.


Normal Equation Window Function Linear Prediction Past Observation Process Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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