The Linear Prediction Model

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),…, x_L−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), …, a_p(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 x̂ ^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 x̂ ^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.