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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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| The most common representation is
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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
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| | title = Smoothing, Filtering and Prediction: Estimating the Past, Present and Future
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| | publisher = Intech
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| | location = Rijeka, Croatia
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| | isbn = 978-953-307-752-9
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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>
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| for 1 ≤ ''j'' ≤ ''p'', where ''R'' is the [[autocorrelation]] of signal ''x''<sub>''n''</sub>, defined as
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| :<math>\ R(i) = E\{x(n)x(n-i)\}\,</math>,
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| and ''E'' is the [[expected value]]. In the multi-dimensional case this corresponds to minimizing the [[Lp space|L<sub>2</sub> norm]].
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| 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
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| :<math>Ra = -r,\,</math>
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| where the autocorrelation matrix ''R'' is a symmetric, p×p [[Toeplitz matrix]] with elements ''r''<sub>''i'',''j''</sub> = ''R''(''i'' − ''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.
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| Another, more general, approach is to minimize the sum of squares of the errors defined in the form
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| :<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>
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| where the optimisation problem searching over all <math>a_i</math> must now be constrained with <math>a_0=-1</math>.
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| 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
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| :<math>\ Ra = [1, 0, ... , 0]^{\mathrm{T}}</math>
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| where the index ''i'' ranges from 0 to ''p'', and ''R'' is a (''p'' + 1) × (''p'' + 1) matrix.
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| 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.
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| 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>
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| Later, [[Philippe Delsarte|Delsarte]] et al. 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'' − 1 terms.
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| 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.
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| == See also ==
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| * [[Autoregressive model]]
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| * [[Prediction interval]]
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| * [[Rasta filtering]]
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| * [[Minimum mean square error]]
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| == References ==
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| {{reflist}}
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| {{More footnotes|date=November 2010}}
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| == Further reading ==
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| * 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.
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| * 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. 261–278, January 1947.
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| * J. Makhoul. "Linear prediction: A tutorial review". ''Proceedings of the IEEE'', 63 (5):561–580, April 1975.
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| * M. H. Hayes. ''Statistical Digital Signal Processing and Modeling''. J. Wiley & Sons, Inc., New York, 1996.
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| == External links ==
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| * [http://labrosa.ee.columbia.edu/matlab/rastamat/ PLP and RASTA (and MFCC, and inversion) in Matlab]
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| {{DEFAULTSORT:Linear Prediction}}
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| [[Category:Time series analysis]]
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| [[Category:Signal processing]]
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| [[Category:Estimation theory]]
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Though many moms are gainfully employed, yet a good number of them in order to stay at home in order to look after their offsprings. The internet technology has opened new ways through which such moms can make enough money against the comfort of their homes. If you make a mom and that appeals to you staying at home in order to think about after your children, you can still make enough money at the convenience of your home through any of these five ways.
Be apt to always check the labels of cosmetic products you buy and on no account a harsh chemicals regarding example alcohol, Petroleum, Prebnim (a preservative), artificial fragrances another option.
Another issue that faces firms could well with comparison to its maintenance. While it's true that a lot of firms should buy their own servers and have a place prepared, there is the fact that many of parents may not have the skills or personnel that will maintain the device. It can be a lot perform maintaining it. One can hire skilled personnel to do it, nevertheless the salary expenses can burn a huge hole in the pocket. Because of this , it appears reasonable for they then to opt professional hosting companies. Thanks to cloud hosting, firms have more options saved. They can have the processing power of a server, without worrying about maintenance.
A bug is just a device that transmits audio or a plan of video and audio to human being that place device in your home or business. Once upon a time, most bugs for you to be by telephones or walls healthcare priorities . had to get hardwired into the electrical system of your house. With wireless technology, a bug can literally be around your place. The reasons for ahead of time can be many several of options are not authentic. Are you involved in the divorce? Might be your spouse or a curious party towards the soon-to-be-ex is bugging your premises to obtain something in order to or just eavesdrop to find out what your plans are. Sometimes business partners are bugging you, or it become someone (usually an ex that can't let go) with an obsession a person.
India will end up wealthy, business will fly. There will be deemed as a spiritual wind in The indian subcontinent. There will be happiness Buddhism, christianity and Islam will grow and merge with Hinduism the. Prince Sidharta was after all a hindu-prince. India experiened ultimate happiness for men and women under the rule of king Ashoka. It was the golden period of India.
This System works by using a 5 pip trailing stop and a 20 pip take gain. If the signal falls between 5 pips and 20 pips, Forex Ambush 2.0 provides for a trade outlet. If it falls outside of this range, Forex Ambush 2.0 advises you never to trade. These devices does not use hard stop loss technique.
In MLM, there include the in info circle, and people who are not ever. Those not in the winners circle would feel right in justifying that MLM is bad, wrong, and have some of reasons and legitimate past experiences to prove doing it. They will also have a clear account.