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'''Linear prediction''' is a mathematical operation where future values of a [[discrete time|discrete-time]] [[Signal processing|signal]] are estimated as a [[linear transformation|linear function]] of previous samples.
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In [[digital signal processing]], linear prediction is often called [[linear predictive coding]] (LPC) and can thus be viewed as a subset of [[filter theory]]. In [[system analysis]] (a subfield of [[mathematics]]), linear prediction can be viewed as a part of [[mathematical model]]ling or [[Optimization (mathematics)|optimization]].
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== The prediction model ==
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:<math>\widehat{x}(n) = \sum_{i=1}^p a_i x(n-i)\,</math>
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where <math>\widehat{x}(n)</math> is the predicted signal value, <math>x(n-i)</math> the previous observed values, and <math>a_i</math> the predictor coefficients. The error generated by this estimate is
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:<math>e(n) = x(n) - \widehat{x}(n)\,</math>
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where <math>x(n)</math> is the true signal value.
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These equations are valid for all types of (one-dimensional) linear prediction. The differences are found in the way the parameters <math>a_i</math> are chosen.
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For multi-dimensional signals the error metric is often defined as
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:<math>e(n) = \|x(n) - \widehat{x}(n)\|\,</math>
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where <math>\|\cdot\|</math> is a suitable chosen vector [[norm (mathematics)|norm]]. Predictions such as <math>\widehat{x}(n)</math> are routinely used within [[Kalman filter]]s and smoothers <ref>{{cite book | author = Einicke, G.A.
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| year = 2012
 
| title = Smoothing, Filtering and Prediction: Estimating the Past, Present and Future
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| url = http://www.intechopen.com/books/smoothing-filtering-and-prediction-estimating-the-past-present-and-future}}</ref> to estimate current and past signal values, respectively.
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===Estimating the parameters===
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The most common choice in optimization of parameters <math>a_i</math> is the [[root mean square]] criterion which is also called the [[autocorrelation]] criterion. In this method we minimize the expected value of the squared error E[e<sup>2</sup>(n)], which yields the equation
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:<math>\sum_{i=1}^p a_i R(j-i) = -R(j),</math>
</ul>
 
for 1 ≤ ''j'' &le; ''p'', where ''R'' is the [[autocorrelation]] of signal ''x''<sub>''n''</sub>, defined as
 
:<math>\ R(i) = E\{x(n)x(n-i)\}\,</math>,
 
and ''E'' is the [[expected value]]. In the multi-dimensional case this corresponds to minimizing the [[Lp space|L<sub>2</sub> norm]].
 
The above equations are called the [[normal equations]] or [[Autoregressive model#Yule-Walker_equations|Yule-Walker equations]]. In matrix form the equations can be equivalently written as
 
:<math>Ra = -r,\,</math>
 
where the autocorrelation matrix ''R'' is a symmetric, p×p [[Toeplitz matrix]] with elements ''r''<sub>''i'',''j''</sub> = ''R''(''i'' &minus; ''j''), 0≤i,j<p, the vector ''r'' is the autocorrelation vector ''r''<sub>''j''</sub> = ''R''(''j''), 0<j≤p, and the vector ''a'' is the parameter vector.
 
Another, more general, approach is to minimize the sum of squares of the errors defined in the form
 
:<math>e(n) = x(n) - \widehat{x}(n) = x(n) - \sum_{i=1}^p a_i x(n-i) = - \sum_{i=0}^p a_i x(n-i)</math>
 
where the optimisation problem searching over all <math>a_i</math> must now be constrained with <math>a_0=-1</math>.
 
On the other hand, if the mean square prediction error is constrained to be unity and the prediction error equation is included on top of the normal equations, the augmented set of equations is obtained as
 
:<math>\ Ra = [1, 0, ... , 0]^{\mathrm{T}}</math>
 
where the index ''i'' ranges from 0 to ''p'', and ''R'' is a (''p''&nbsp;+&nbsp;1) &times; (''p''&nbsp;+&nbsp;1) matrix.
 
Specification of the parameters of the linear predictor is a wide topic and a large number of other approaches have been proposed.{{Citation needed|date=October 2010}} In fact, the autocorrelation method is the most common and it is used, for example, for [[speech coding]] in the [[Global System for Mobile Communications|GSM]] standard.
 
Solution of the matrix equation ''Ra'' = ''r'' is computationally a relatively expensive process. The [[Gauss algorithm]] for matrix inversion is probably the oldest solution but this approach does not efficiently use the symmetry of ''R'' and ''r''. A faster algorithm is the [[Levinson recursion]] proposed by [[Norman Levinson]] in 1947, which recursively calculates the solution.{{Citation needed|date=October 2010}} In particular, the autocorrelation equations above may be more efficiently solved by the Durbin algorithm.<ref>M. A. Ramirez (2008) "A Levinson Algorithm Based on an Isometric Transformation of Durbin's," IEEE Signal Processing Lett., vol. 15, pp. 99-102.</ref>
 
Later, [[Philippe Delsarte|Delsarte]] et al.&nbsp;proposed an improvement to this algorithm called the [[split Levinson recursion]] which requires about half the number of multiplications and divisions.{{Citation needed|date=October 2010}} It uses a special symmetrical property of parameter vectors on subsequent recursion levels. That is, calculations for the optimal predictor containing ''p'' terms make use of similar calculations for the optimal predictor containing ''p''&nbsp;&minus;&nbsp;1 terms.
 
Another way of identifying model parameters is to iteratively calculate state estimates using [[Kalman filter]]s and obtaining [[maximum likelihood]] estimates within [[Expectation–maximization algorithm]]s.
 
== See also ==
* [[Autoregressive model]]
* [[Prediction interval]]
* [[Rasta filtering]]
* [[Minimum mean square error]]
 
== References ==
{{reflist}}
 
{{More footnotes|date=November 2010}}
 
== Further reading ==
* G. U. Yule. "On a method of investigating periodicities in disturbed series, with special reference to wolfer’s sunspot numbers". ''Phil. Trans. Roy. Soc.'', 226-A:267–298, 1927.
* N. Levinson, “The Wiener RMS (root mean square) error criterion in filter design and prediction,” ''Journal of Mathematics and Physics'', vol. 25, no. 4, pp.&nbsp;261–278, January 1947.
* J. Makhoul. "Linear prediction: A tutorial review". ''Proceedings of the IEEE'', 63 (5):561–580, April 1975.
* M. H. Hayes. ''Statistical Digital Signal Processing and Modeling''. J. Wiley & Sons, Inc., New York, 1996.
 
== External links ==
* [http://labrosa.ee.columbia.edu/matlab/rastamat/ PLP and RASTA (and MFCC, and inversion) in Matlab]
 
{{DEFAULTSORT:Linear Prediction}}
[[Category:Time series analysis]]
[[Category:Signal processing]]
[[Category:Estimation theory]]

Latest revision as of 14:42, 2 December 2014

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