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For some fields such as econometrics (Shore, 1980), oil prospecting (Claerbout, 1976), speech recognition (Levinson and Lieberman, 1981), satellite monitoring (Lavergnat et al., 1980), epilepsy diagnosis (Gersch and Tharp, 1977), and plasma physics (Bloomfield, 1976), there is a need to obtain an estimate of the spectral density (when it exists) in order to gain at least a crude understanding of the frequency content of time series data. An outstanding tutorial on the classical problem of spectral density estimation is given by Kay and Marple (1981). For an excellent collection of fundamental papers dealing with modern spec­ tral density estimation as well as an extensive bibliography on other fields of application, see Childers (1978). To devise a high-performance sample spectral density estimator, one must develop a rational basis for its construction, provide a feasible algorithm, and demonstrate its performance with respect to prescribed criteria. An algorithm is certainly feasible if it can be implemented on a computer, possesses computational efficiency (as measured by compu­ tational complexity analysis), and exhibits numerical stability. An estimator shows high performance if it is insensitive to violations of its underlying assumptions (i.e., robust), consistently shows excellent frequency resolutipn under realistic sample sizes and signal-to-noise power ratios, possesses a demonstrable numerical rate of convergence to the true population spectral density, and/or enjoys demonstrable asymp­ totic statistical properties such as consistency and efficiency.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
For some fields such as econometrics (Shore, 1980), oil prospecting (Claerbout, 1976), speech recognition (Levinson and Lieberman, 1981), satellite monitoring (Lavergnat et al., 1980), epilepsy diagnosis (Gersch and Tharp, 1977), and plasma physics (Bloomfield, 1976), there is a need to obtain an estimate of the spectral density (when it exists) in order to gain at least a crude understanding of the frequency content of time series data. An outstanding tutorial on the classical problem of spectral density estimation is given by Kay and Marple (1981). For an excellent collection of fundamental papers dealing with modern spectral density estimation as well as an extensive bibliography on other fields of application, see Childers (1978).
Wray Britton

2. Imposing the constraints

Abstract
Let Ωn# denote the parental function space for Ωn consisting of all spectral distribution functions F on the torus T° = [-Π, Π] endowed with the Lebesgue σ-algebra T which satisfy the following constraints:
(2.1)
F is real-valued and absolutely continuous with respect to Lebesgue measure (dω) on T°
 
(2.2)
F′ = f, which exists by (2.1), is a strictly positive, even, continuous density on T° which is of bounded variation
 
(2.3)
0 < δ < f(ω) < Γ < ∞ for all ω ε [0, π] (δ and Γ fixed and independent of f and n)
 
(2.4)
f has an absolutely convergent Fourier (cosine) series on [0, π].
 
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3. Selecting the objective functional: conjugate duality

Abstract
Observe that for finite n there are an infinity of spectral densities in which in particular satisfy (2.7). In order to select a unique function from Ωn we need a criterion. Putting aside certain philosophical issue (Jaynes, 1979), consider the negentropy functional:
H:Ωn → R, where:
$$\rm H(F)=\int_{0}^{2 \pi}f(\omega)\log[f(\omega)]d\omega$$
(3.1)
.
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4. Choosing a truncation point for (c*c)k

Abstract
In order to exploit either fast convolution or FFT/WFTA technique (Cooley et al., 1977; Henrici, 1979; Kronsjo, 1979; Lifermann, 1979; Winograd, 1978; Zohar, 1981) in the evaluation of \({\rm Z_n}(\vec{\theta}) = 2\pi {\rm (c*c)_o}\) and \(\rm r_k^{(n)}={(c*c)_k}/(c*c)_o\), we need an a priori truncation point.
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5. Solving for

Abstract
By Theorem 3.16, \(\vec{\theta}^{(1)}: \theta_{1}\) is determined by the following relation:
$$\rm h_1(\theta_1)=\log [Z_1(\theta_1)] - \theta_1r_1=\min !$$
(5.1)
,
where Z11) = 2π(c*c)o and:
$$\rm c_o=1$$
(5.2)
$$\rm 2c_k=k^{-1} \theta_1 c_{k-1} \,\,\, k \ge $$
(5.3)
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6. Solving for

Abstract
By Theorem 3.16, our task is to minimize the (real analytic, transcendentally nonlinear) strictly convex functional \({\rm h_n}(\vec{\theta})\) on the open convex set Θn. For numerical work, we must employ the functional \({\rm h_n}(\theta):=\log [\hat{\rm Z}_{\rm n}(\vec{\theta})] - \rm \sum\limits_{k = 1}^n {\theta _k r_k}\) and {ck, 0 ≤ k ≤ Nt} is determined by recursive relation (3.16.3)-(3.16.4) for some judicious truncation point Nt ≤ ∞. Recall that {rk,1 ≤ k ≤ n} is given for n ≤ N.
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7. Obtaining an initial estimate of

Abstract
Consider the constrained negentropy-rate functional \(\rm H_n^*:\Omega_n\rightarrow R\), where:
$$\rm H_n^*(f)=\int_{0}^{2\pi} \log[f(\omega)]d\omega + \sum \limits_{k=0}^{n}\alpha_k[r_{k}-\int_{0}^{2\pi}\cos (k\omega)f(\omega)d\omega]$$
(7.1)
.
and \(\vec{\alpha} = (\alpha_0, \alpha_1, \ldots, \alpha_n)\prime\) consists of n+1 Lagrange multipliers.
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8. Numerical asymptotics

Abstract
I begin with some classical motivational results:
Proposition 8.1 (F. and M. Riesz; cf. Koosis, 1980, pp. 40–47/100–102) Let F be a spectral distribution which is of bounded variation on To. Let rk denote the kth trigonometric moment of the measure dF, i.e.,
$$\rm r_k = \int_{0}^{2 \pi}\cos (k \omega)dF(\omega)\; \; \; 0 \le k \le \infty$$
(8.1.1)
.
If \(\sum \limits_{k-1}^{\infty} |r_k| < \infty\), then F is absolutely continuous.
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9. Assessing the sample efficiency of and

Abstract
Definition 9.1 Let \({\rm f}(\vec{\rm v}, \omega) \varepsilon \rm C^2 [0,2 \pi]\) be a strictly positive spectral density with vector parameter \(\vec{\rm v} = \rm (v_1, v_2, \ldots, v_n) \varepsilon R^n\). Let \({\rm M_f}(\vec{\rm v})\) and \({\rm I_f}(\vec{\rm v})\) be the n × n real symmetric positive definite matrices defined by
$${\rm M_f}(\vec{\rm v}) = \rm (M_{ij}^{(n)})_{1 \leqslant i, j \leqslant n}$$
(9.1.1)
.
$${\rm I_f}(\vec{\rm v}) = \rm (I_{ij}^{(n)})_{1 \leqslant i, j \leqslant n}$$
(9.1.2)
.
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10. The numerical experiment: preliminaries

Without Abstract
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11. The numerical experiment: results

Abstract
A preliminary determination of f*n and fn revealed that the Areas Index series exhibited a strong frequency bias at ω = 0 as has also been found (Bloomfield, 1976) in the Wölf-Zürich Index over a different time period. To achieve frequency resolution, I subjected the data to the Anscombe-Johnson transformation ym: R → R defined by:
$$\rm y_m=\sin h^{-1}(z_m)$$
(11.1)
,
where \(\rm z_m=(x_m-\bar{x})/s\) with \(\rm \bar{x}\) and s the sample mean and standard deviation of the given time series. Both compositional factors of ym are well known; in particular, the outer factor is the variance stabilizer of the Student t.
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12. Tables and Graphs

Without Abstract
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13. Conclusion

Abstract
From the numerical experiment, we see that it is feasible to reconstruct a sufficiently smooth spectral density on the basis of the minimum negentropy criterion. We achieve a robust spectral density estimator in fn at the expense of increased computational complexity and the possible degradation of the statistical asymptotic properties of \(\vec{\theta}^{(\rm n)}\).
Wray Britton

Backmatter

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