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2016 | OriginalPaper | Buchkapitel

11. Parametric Spectral Methods

verfasst von : Silvia Maria Alessio

Erschienen in: Digital Signal Processing and Spectral Analysis for Scientists

Verlag: Springer International Publishing

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Abstract

In this chapter, parametric methods of spectral estimation are presented. They rely on fitting a proper stochastic model to the data record. The model is supposed to represent the persistence, i.e., autocorrelation, present in the process generating the observed signal. The signal’s spectral characteristics are then derived from the estimated model. This approach requires selecting model type and order (number of parameters), and then estimating the parameters. This can be done in several different ways, and the method of parameter estimation gives its name to the parametric spectral method: we thus have the Yule-Walker method, the covariance and modified covariance methods, Burg’s method and the maximum entropy method. These methods provide better resolution than non-parametric ones, especially when the record is short.

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Vorheriges Kapitel Non-Parametric Spectral Methods

Nächstes Kapitel Singular Spectrum Analysis (SSA)

Since the input is stationary and the filter is stable, the output is also certainly stationary.

We do not need to worry about B(z) and its degree, because the numerator of H(z), representing the polynomial transfer function of an FIR filter, contributes only zeros outside the origin of the z-plane; it has poles in the origin. On the other hand, 1 / A(z) is an IIR rational transfer function that contributes poles outside the origin and zeros in the origin. In summary, outside the origin the poles of H(z) are exclusively due to the zeros of A(z).

A system with rational transfer function is minimum-phase if not only all its poles, but also all its zeros are inside the unit circle, so that both the system and its inverse are causal and stable. A minimum-phase system is called this way because it has an additional useful property: the natural logarithm of the magnitude of the frequency response is related to the phase angle of the frequency response by the Hilbert transform. This implies that for all causal and stable systems that have the same magnitude response, the minimum phase system has its impulse-response energy concentrated near the start of h[n], i.e., it minimizes the delay of the energy in the impulse response. As a result, for all causal and stable systems that have the same magnitude response, the minimum phase system has the minimum group delay.

When the AC absolute values take much longer to decay than the rate associated with the ARMA class of processes discussed in this chapter, the process is often referred to as having long-term memory or long-range persistence. Therefore the memory associated with ARMA processes is usually classified as short-term memory.

This expression is common in stochastic model literature, but does not mean that the IIR filter has no zeros. Actually, as we learned in Sect. 3.2.8 (Table 3.1), a rational transfer function with numerator degree $M=0$ and denominator degree $N=p$ has p poles and no zeros at finite values of z outside the origin of the z-plane, and p zeros in the origin.

The expression for $\gamma _{ex}[l]$ can also be directly derived from the formula for the input-output cross-covariance of an LTI system, reported in Sect. 9.9:

$$\begin{aligned} \gamma _{ex}[l] =\sum _{k=-\infty }^{+\infty }h[k] \gamma _{ee}[l+k] =\sigma _e^2 \sum _{k=-\infty }^{+\infty }h[k] \delta [l+k]=\sigma _e^2 h[-l] \leadsto \gamma _{ex}[l-i] =\sigma _e^2 h[i-l]. \end{aligned}$$

These equations have been named variously in the literature, and are also referred to as normal equations, or Wiener-Hopf equations.

In linear algebra, a square ($n\times n$) symmetric real matrix $\mathbf {M}$ is said to be positive definite if the quadratic form $\mathbf {v}^T \mathbf {M v}$ is positive for every non-zero column vector $\mathbf {v}$ of n real numbers. The symbol $\mathbf {v}^T$ denotes the transpose of $\mathbf {v}$, i.e., the corresponding row vector.

In linear algebra, a Toeplitz matrix (named after Otto Toeplitz), or diagonal-constant matrix, is a matrix in which each descending diagonal from left to right is constant, i.e., all the elements on any line parallel to the main diagonal are identical.

If the roots lie on the unit circle, the AR process will only be stationary in case of noise being identical to zero. In that case a harmonic process will result, consisting of a sum of cosine functions. We might wonder what will happen if the AR process has poles very close to the unit circle. As poles on the unit circle represent a harmonic process, an AR process with poles near the unit circle can be expected to demonstrate some kind of pseudo-periodic behavior (Priestley 1994). In this case the AC can be described as a sum of weakly damped sinusoids. Furthermore, the AR process may exhibit a kind of almost-non-stationary behavior, because the transfer function will be close to instability in the filtering sense. The pole locations will also affect the reliability of the various parameter estimation techniques. It was claimed by Priestley (1994) that YW equations may lead to poor parameter estimates, even for moderately large data samples, if the AR operator has a pole near the unit circle.

This does not mean that the FIR filter has no poles. Actually (see Table 3.1), a rational transfer function with numerator degree $M=q$ and denominator degree $N=0$ has p zeros and no poles at finite values of z outside the origin of the z-plane, and p poles in the origin.

The equation $\sigma _x^2=\sigma _e^2/\left( 1-\alpha ^2\right) $ can be obtained as follows:

$$\begin{aligned} \mathrm {E}\left\{ x[n]^2\right\}= & {} \mathrm {E}\left\{ \left( \alpha x[n-1]+e[n]\right) ^2\right\} = \\= & {} \alpha ^2 \mathrm {E}\left\{ x[n-1]^2\right\} +2\alpha \mathrm {E}\left\{ x[n-1]e[n]\right\} +\mathrm {E}\left\{ e[n]^2\right\} = \alpha ^2 \mathrm {E}\left\{ x[n-1]^2\right\} + \sigma _e^2, \end{aligned}$$

because the white-noise input and the output signal are uncorrelated.

If we substitute the model expression into the last equation, i.e., if we use $x[n-1]=\alpha x[n-2]+e[n-1]$, and then iterate, we get

$$\begin{aligned} \mathrm {E}\left\{ x[n]^2\right\} = \sigma _e^2 + \alpha ^2 \sigma _e^2 + \alpha ^4 \sigma _e^2 +\cdots +\alpha ^{2l} \mathrm {E}\left\{ x[n-l]^2\right\} \end{aligned}$$

that for increasing l becomes, for a centered x[n] for which $\mathrm {E}\left\{ x[n]^2\right\} =\sigma _x^2$,

$$\begin{aligned} \sigma _x^2 = \sigma _e^2 (1+\alpha ^2 + \alpha ^4 +\cdots ) = \frac{\sigma _e^2}{1-\alpha ^2}. \end{aligned}$$

In a similar way, the formula for the AC at lag l could be derived. We can also derive these formulas directly from YW equations:

$$\begin{aligned} \gamma [l]+a_1\gamma [-1]=\sigma ^2_e \delta [l] \end{aligned}$$

provides

$$\begin{aligned} \gamma [0]+a_1\gamma [l-1]=\sigma ^2_e =\gamma [0]+a_1\gamma [1] \end{aligned}$$

and since $\gamma [0]=\sigma ^2_x$,

$$\begin{aligned} \sigma ^2_x=\sigma ^2_e-a_1\gamma [1]=\sigma ^2_e+\alpha \gamma [1]. \end{aligned}$$

Also,

$$\begin{aligned} \gamma [1]+a_1\gamma [0]=0,\qquad \qquad \gamma [1]=-a_1 \sigma ^2_x =\alpha \sigma ^2_x, \end{aligned}$$

hence

$$\begin{aligned} a_1=-\frac{\gamma [1]}{\gamma [0]}=-\rho [1], \qquad \qquad \alpha =\rho [1]. \end{aligned}$$

Proceeding further, we have

$$\begin{aligned} \gamma [2]+a_1\gamma [1]=0,\qquad \qquad \gamma [2]=-a_1 \gamma [1] =\alpha \gamma [1]=\alpha ^2 \sigma ^2_x, \end{aligned}$$

and by iterating, we deduce that in general

$$\begin{aligned} \gamma [l]=\alpha ^l \sigma ^2_x=\rho [1]^l \sigma _x^2, \qquad \qquad \rho [l]=\left( \rho [1]\right) ^l=\alpha ^l. \end{aligned}$$

Moreover we can write

$$\begin{aligned} \sigma ^2_x=\sigma ^2_e+\alpha \gamma [1]=\sigma ^2_e+\alpha ^2\sigma ^2_x, \qquad \qquad \sigma ^2_x=\frac{\sigma ^2_e}{1-\alpha ^2}, \end{aligned}$$

and finally

$$\begin{aligned} \gamma [l]=\alpha ^l \sigma ^2_x=\frac{\sigma ^2_e \alpha ^l}{1-\alpha ^2}. \end{aligned}$$

Recall that, in general, the noise that is not white is termed “colored noise” and has a smooth spectrum with more power at some frequencies with respect to others, in analogy to colored light. A typical colored noise spectrum does not contain narrowband features associated with periodic or quasi-periodic signal components (Sect. 10.2).

The calculation of the AC is straightforward:

$$\begin{aligned} \mathrm {E}\left\{ x[n]x[n+l]\right\}= & {} \mathrm {E}\left\{ \left( e[n]+\alpha e[n-1]\right) \left( e[n+l]+\alpha e[n+l-1]\right) \right\} = \\= & {} \mathrm {E}\left\{ e[n]e[n+l]\right\} \\&+\,\alpha \mathrm {E}\left\{ e[n-1]e[n+l]\right\} +\alpha \mathrm {E}\left\{ e[n]e[n+l-1]\right\} +\alpha ^2\mathrm {E}\left\{ e[n-1]e[n+l-1]\right\} . \end{aligned}$$

The right-hand terms are all zero, except when the indexes of the noise samples involved are equal. Independently of the index, each term of the kind $\mathrm {E}\left\{ \left( e^2[n+l]\right) \right\} $ is equal to $\sigma _e^2$, so we can write

for $l=0$, $\mathrm {E}\left\{ x[n]^2\right\} = \sigma _e^2(1+\alpha ^2)$,
for $l=1$, $\mathrm {E}\left\{ x[n]x[n+1]\right\} = \alpha \sigma _e^2$,
for $l=-1$, $\mathrm {E}\left\{ x[n]x[n-1]\right\} = \alpha \sigma _e^2$,

while for different values of l we find zero.

In statistics and machine learning, overfitting occurs when a statistical model ends up describing random errors or noise instead of the underlying relation. Overfitting generally occurs when a model is excessively complex, such as having too many parameters relative to the number of observations. A model that has been overfit will generally have poor predictive performance, as it can exaggerate minor fluctuations in the data. In other words, in case of overfitting, the estimator is too flexible and captures illusory trends in the data. These illusory trends are often the result of the noise in the observations. The contrary phenomenon, underfitting, occurs when an estimator is not flexible enough to capture the underlying trends in the observed data, usually because of an insufficient number of parameters.

Recall that we are dealing with centered data, which in practice means that the sample mean has been subtracted from each data sample. If the process true mean value were known, we should write $\mathrm{FPE}[p]=[(N+p)/(N-p)]\sigma ^2_{e,p}$.

For example, the 0.975 probability point of the standard normal distribution is 1.96. This means that for a normally-distributed variable, 95 % of the data lies within $1.96 \approx 2$ standard deviations of the mean. The 95 % confidence interval for the AC at lag l is therefore $\pm 1.96/\sqrt{N}$ . For the 99 % confidence interval, the 0.995 probability point of the normal distribution is 2.57, so, 99.5 % of the data lies within $2.57 \approx 3$ standard deviations of the mean. The 99 % confidence interval for the AC is thus $\pm 2.57/\sqrt{N}$. A value outside this confidence interval is evidence that the model residuals are not random at the chosen probability level. If a residual-AC plot for a given series shows that none of the AC samples of the chosen model residuals fall outside the 99 % confidence interval around zero, then the modeling explains the persistence and yields random residuals at that confidence level.

The DOF would be $K-p-q$ in the ARMA case.

Narrowband processes have AC values slowly decreasing with increasing lag, which produce large values of $p_{max}$ in the AR-spectrum estimation procedure. Broadband processes have ACs decreasing faster with lag. For an example of this behavior, compare Fig. 11.11a, which is relative to a broadband AR(2) process (as shown in Fig. 11.13a), with Fig. 11.11a, which is relative to a broadband AR(2) process (as shown in Fig. 11.13b).

This matrix would be Hermitian in the case of complex signals.

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Titel: Parametric Spectral Methods
verfasst von: Silvia Maria Alessio
Verlag: Springer International Publishing
Buch: Digital Signal Processing and Spectral Analysis for Scientists
Print ISBN: 978-3-319-25466-1

Electronic ISBN: 978-3-319-25468-5

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-25468-5_11

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