# Maximum-entropy spectral estimator

auto-regressive spectral estimator

An estimator $f _ {q} ^ { * } ( \lambda )$ for the spectral density $f ( \lambda )$ of a discrete-time stationary stochastic process such that 1) the first $q$ values of the auto-correlations are equal to the sample auto-correlations calculated from the observational data, and 2) the entropy of the Gaussian stochastic process with spectral density $f _ {q} ^ { * } ( \lambda )$ is maximized subject to condition 1). If $N$ sample values $x _ {t}$, $t = 1 \dots N$, are known from observing a realization of a real stationary process $X _ {t}$ having spectral density $f ( \lambda )$, then the maximum-entropy spectral estimator $f _ {q} ^ { * } ( \lambda )$ is defined by the relations

$$\tag{1 } \int\limits _ {- \pi } ^ \pi \cos k \lambda \; f _ {q} ^ { * } ( \lambda ) d \lambda = r _ {k} ^ {*\ } \equiv$$

$$\equiv \ N ^ {-1} \sum _ { j=1 } ^ { N-k } x _ {j} x _ {j+k} ,\ \ k = 0 \dots q ,$$

$$\tag{2 } \int\limits _ {- \pi } ^ \pi \mathop{\rm log} f _ {q} ^ { * } ( \lambda ) d \lambda = \max ,$$

where the sign $\equiv$ denotes "equal by definition" . The maximum-entropy spectral estimator has the form

$$\tag{3 } f _ {q} ^ { * } ( \lambda ) = \frac{\sigma ^ {2} }{2 \pi | 1 + \beta _ {1} \mathop{\rm exp} ( i \lambda ) + \dots + \beta _ {q} \mathop{\rm exp} ( i q \lambda ) | ^ {2} } ,$$

where the coefficients $\beta _ {1} \dots \beta _ {q}$ and $\sigma ^ {2}$ are given by the $q + 1$ equations (1) (see, e.g., [1], [9], [10]). Formula (3) shows that the maximum-entropy spectral estimator coincides with the so-called auto-regressive spectral estimator (introduced in [2], [3]). The positive integer $q$ here plays a role related to that played by the reciprocal width of a spectral window in the case of non-parametric estimation of the spectral density by periodogram smoothing (see Spectral window; Statistical problems in the theory of stochastic processes). There are several methods for estimating the optimal value of $q$ from given observations (see, for example, [1], [4], [5], [8]). The values of the coefficients $\beta _ {1} \dots \beta _ {q} , \sigma ^ {2}$ can be found using a solution of the Yule–Walker equations

$$\tag{4 } r _ {k} ^ {*} + \sum _ { j=1 } ^ { q } \beta _ {j} r _ {| k - j | } ^ {*} = 0 ,\ k = 1 \dots q ,$$

$$\tag{5 } r _ {0} ^ {*} + \sum _ { j=1 } ^ { q } \beta _ {j} r _ {j} ^ {*} = \sigma ^ {2} ;$$

there are also other, numerically more convenient, methods for calculating these coefficients (see, e.g., [1], [4][6], [10]).

In the case of small sample size or spectral densities of complex form, maximum-entropy spectral estimators and parametric spectral estimators (cf. Spectral estimator, parametric), which generalize them, possess definite advantages over non-parametric estimators of $f ( \lambda )$: they usually have a more regular form and possess better resolving power, that is, they permit one to better distinguish close peaks of the graph of the spectral density (see [1], [4][7]). Therefore maximum-entropy spectral estimators are widely used in the applied spectral analysis of a stationary stochastic process.

#### References

 [1] D.G. Childers (ed.) , Modern spectrum analysis , IEEE (1978) [2] E. Parzen, "An approach to empirical time series analysis" Radio Sci. , 68 (1964) pp. 937–951 [3] H. Akaike, "Power spectrum estimation through autoregressive model fitting" Ann. Inst. Stat. Math. , 21 : 3 (1969) pp. 407–419 [4] S.S. Haykin (ed.) , Nonlinear methods of spectral analysis , Springer (1979) [5] S.M. Kay, S.L. Marpl, "Spectrum analysis—a modern perspective" Proc. IEEE , 69 : 11 (1981) pp. 1380–1419 [6] "Spectral estimation" Proc. IEEE , 70 : 9 (1982) ((Special Issue)) [7] V.F. Pisarenko, "Sampling properties of maximum entropy spectral estimation" , Numerical Seismology , Moscow (1977) pp. 118–149 (In Russian) [8] J.G. de Gooyer, B. Abraham, A. Gould, L. Robinson, "Methods for determining the order of an autoregressive-moving average process: A survey" Internat. Stat. Rev. , 55 (1985) pp. 301–329 [9] M.B. Priestley, "Spectral analysis and time series" , 1–2 , Acad. Press (1981) [10] A. Papoulis, "Probability, random variables and stochastic processes" , McGraw-Hill (1984)
How to Cite This Entry:
Maximum-entropy spectral estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximum-entropy_spectral_estimator&oldid=51335
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article