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OVERVIEW OF MODELSThere are three types of model: AR (auto regressive) model: a model which depends only on previous outputs of system. MA model( moving average): model which depends only on inputs to system. ARMA(autoregressive moving average): model based on both inputs and outputs . 2
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AUTOREGRESSIVE MODEL &FILTER In an AR model of a time series the current value of the series ,x(n),is expressed as a linear function of previous values plus an error term, e(n),thus:x(n)=-a(1)x(n-1)-a(2)x(n-2)-. . . –a(k)x(n-k)-…- a(p)x(n-p)+e(n){p previous terms & represent a model of order p.}Also written as x(n)=- a(k)x(n-k)+e(n)=- a(k) x(n)+e(n) 3
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POWER SPECTRUM DENSITY OFAR SERIES The power spectrum density, , of the AR series x(n) is required. This is related to power spectrum density of the white noise error signal , ,which is its variance , ,by 6
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YULE-WALKER METHOD The Yule-Walker Method estimates the power spectral density (PSD) of the input using the Yule- Walker AR method. This method, also called the autocorrelation method, fits an autoregressive (AR) model to the windowed input data. An autoregressive model depends on a limited number of parameters, which are estimated from measured noise data. 7
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CALCULATIONS Computation of model parameters- Yule Walker equations8
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CALCULATIONS In an AR model of a time series the current value of the series ,x(n),is expressed as a linear function of previous values plus an error term e(n), thus: x(n) = -a(n)x(n-1)-a(2)x(n-2)- . . . -a(k)x(n-k)- . . . -a(p)x(n-p)+e(n) (1) 9
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CONTD. The optimum model p/ms will be those which minimize the errors , e(n),for each sampled point, x(n), represented by an equation ‘1’.These errors are given by re-ordering equation ‘1’ to e(n) = x(n)+ a((k)x(n-k) (2) A measure of the total error over all samples , N(1 n N ) ,is required . The mean squared error is given by: (3) 10
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CONTD.The optimum value of each p/m is obtained by setting the partialderivative of equation (3) w.r.t. the model p/m to zero, we have: (4) Now, 11
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CONTD. And so equation (4) simplifies to (5) Giving for kth p/m: (6) 12
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CONTD. Writing out the LHS of equation (4) for the e.g. case of k=1,gives 13
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CONTD. Since in the case of autocorrelation functions Rxx(-j) = Rxx(j), the expression may be written as The RHS of equation (6) is equal to – Rxx(1).Equating the left and right sides gives (7) 14
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CONTD. For each value of k,1 ≤k≤p,a similar equation may be written.These equations may be written in matrix form as (8) 15
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CONTD. The model p/ms,a(k), may now be obtained from this set of eqns which are known as Yule Walker (YW) equations. In matrix notation eqn (8) may be writtten (9) Hence ,in principle, (10) Rxx(k-j)is symmetrical → Toeplitz 16
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CONTD. Equation (3) allows calculation of E , but another expression another in terms of autocorrelation functions and the a(k) may be found as follows. Assuming the a(k) are real & expanding equation (3) gives 17
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CONTD. From eqn (5),which is true for all k , it is seen that eqn(11) Hence eqn(11) simplifies to 19
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CONTD. So that finally (12) Equation (12) or (3) and the model p/ms from eqn(10) may now be inserted in eqn of power spectrum density Px(f) to obtain the autoregressive power density spectrum.However , the possible ways of solving eqn(8) for a(k) and the choice of the model order p, must first be described. 20
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SOLUTION OF THE YULE WALKER EQUATIONS The autocorrelation method The covariance method The modified covariance method The Burg method21
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THE AUTOCORRELATIONMETHOD The autocorrelation method is based upon the mean squared error expression in eqn (3) . The Levinson-urbin (kay,1988;Pardey ,Roberts, and Tarassenko.1996) provides a computation efficient way of solving the YW equations of (8) for the model p/ms. This method gives poorer frequency resolution than the other to be described , and is therefore less suitable for shorter data records. 22
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THE COVARIANCE METHOD In this method the limits of summation in eqn (3) are modified to run from n=p to n=N . Also, the average is calculated over N-p terms rather than N.Thus , eqn (3) becomes (13) 23
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CONTD. The equivalent of eqn (8) is (14) where (15) 24
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CONTD. E is given by (16) The p × p matrix Cxx(j,k) is Hermitian and positive semi-definite .Equation (14) may be solved using the Cholensky decomposition method (Lawson & Hanson,1974 ). Only N-p lagged components are summed , so for short data length there could be some end effects. The covariance method results in better spectral resolution than the autocorrelation method. 25
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THE MODIFIED COVARIANCEMETHOD In this method the average of the estimated forward and backward prediction errors is minimized .EQUATION (14) & (16) still apply, but eqn (15) is modified to (17) The method doesn’t guarantee a stable all – pole filter ,but this usually results . It yields statistically stable spectral estimates of high resolution. 26
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THE BURG METHOD This method relies upon aspects beyond the present scope . It produces accurate spectral estimates for AR data. 27
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APPLICATIONS A high-order Yule-Walker method for estimation of the AR parameters of an ARMA model Microwave multi-level band-pass filter using discrete-time Yule-Walker method In radar applications , the number of observations is small (say 63 observations) and asymptotic descriptions do not cover the estimates (better than 1st order Talyer approx.). 28
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