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The Total Variance of a Periodogram-Based Spectral 4556 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 12, DECEMBER 2005 The Total Variance of a Periodogram-Based Spectral Estimate of a Stochastic Process With Spectral Uncertainty and Its Application to Classifier Design Yanwu Zhang, Member, IEEE, Arthur B. Baggeroer, Fellow, IEEE, and James G. Bellingham Abstract—The variance of a spectral estimate of a stochastic with spectral uncertainty. The total variance is derived under process is essential to the formulation and performance of a some assumptions. The result gives a clear and concise expres- spectral classifier. The overall variance of a spectral estimate sion of the contributions of the two variance sources. Thus it originates from two sources: the within-class spectral uncertainty and the variance introduced in the spectral estimation procedure. provides a quantitative guide for devising approaches to reduce In this paper, we derive the total variance of a periodogram-based the total variance. Using the total variance expression, we for- spectral estimate under some assumptions. Using this result, we mulate a linear spectral classifier based on Fisher’s separability formulate a linear spectral classifier based on Fisher’s separa- metric [10]. As an application example, the classifier is used to bility metric. The classifier is used to classify two oceanographic classify two oceanographic processes. processes: ocean convection versus internal waves. Index Terms—Classifier, periodogram, spectral uncertainty. II. TOTAL VARIANCE OF A PERIODOGRAM-BASED SPECTRAL ESTIMATE OF A STOCHASTIC PROCESS WITH I. INTRODUCTION SPECTRAL UNCERTAINTY HE Fourier transform retains all information in the orig- Due to the computational efficiency of fast Fourier transform, T inal time series since the transform basis is complete and the periodogram and its variants [12] are the most commonly orthogonal. By the linear system theory, the Fourier transform used methods for estimating the power spectrum density (PSD) produces spectral outputs that are uncorrelated between disjoint of a stochastic process. Suppose is one realization of a sto- bands. The spectrum over disjoint bands provides a nonredun- chastic process . Its periodogram is defined as [13] dant representation of a signal [1]. Thus we can use spectra to distinguish between different classes of signals. Spectral classi- (1) fication has been applied to classifying speech [2], [3], images where is a window function of duration . stands for [4]–[6], trees [7], and sea beds [8], [9]. the Fourier transform. The variance of a spectral estimate of a stochastic process The periodogram is an asymptotically unbiased es- is essential for a classifier’s formulation and performance [10], timate of the true PSD , but its standard deviation is as [11]. The overall variance of a spectral estimate originates from large as its asymptotic mean [14]. To reduce the PSD estimate’s two sources: the within-class spectral uncertainty and the vari- variance, one can do frequency-domain smoothing (the Daniell ance introduced in the spectral estimation procedure. An analyt- method [12]) with an interval of 1 , and take the smoothed ical derivation of the total variance is not seen in the literature. periodogram as the PSD estimate An insight of the composition of the total variance is needed for formulating a spectral classifier, analyzing its performance, and devising approaches to reduce the total variance and thus im- (2) proving the classifier’s performance. In this paper, we analyze the formation of the total variance where 2 1 is the total number of frequency points included of a periodogram-based spectral estimate of a stochastic process in the smoothing, and is the weighting function. The expectation of formulated in (2) is [14] Manuscript received January 30, 2004. This work was supported in part by the Office of Naval Research under Grants N00014-95-1-1316 and N00014-97-1-0470, in part by the MIT Sea Grant College Program under Grant NA46RG0434, in part by the Ford Professorship of Ocean Engineering, and in part by the David and Lucile Packard Foundation. The associate editor (3) coordinating the review of this manuscript and approving it for publication was Dr. Athina Petropulu. Y. Zhang is with the Monterey Bay Aquarium Research Institute, Moss Under the assumption that is a Gaussian process, the co- Landing, CA 95039 USA (e-mail: [email protected]). A. B. Baggeroer is with the Department of Electrical Engineering and Com- variance of is [14] as shown in (4) at the bottom of the next puter Science, Massachusetts Institute of Technology, Cambridge, MA 02139 page, where is the Fourier transform of the window func- USA (e-mail: [email protected]). tion and is the weight for frequency-domain smoothing. J. G. Bellingham is with the Monterey Bay Aquarium Research Institute, Moss Landing, CA 95039 USA (e-mail: [email protected]). Notation “ ” stresses that the expectation and the co- Digital Object Identifier 10.1109/TSP.2005.859346 variance are conditioned on the true PSD . 1053-587X/$20.00 © 2005 IEEE ZHANG et al.: TOTAL VARIANCE OF PERIODOGRAM-BASED SPECTRAL ESTIMATE OF STOCHASTIC PROCESS 4557 The variance of is obtained directly from (4) as shown , where is the bandwidth of and in (5) at the bottom of the page. When is smooth across the smoothing bandwidth (2 1) , can be pulled out of the integration in (3) Cov when (10) is dependent on the window function and the fre- quency-domain smoothing function . When is a boxcar window, and an -point smoothing with uniform weights is (6) done in the frequency domain, we have . The vari- ance of is thus reduced by a factor of . The cost paid is Under the constraint , the smearing of the PSD estimate within the smoothing band- (6) becomes width, so the total number of frequency points that provide un- correlated PSD estimates is effectively reduced. The duration (7) of determines the bandwidth of . With a longer dura- tion, the window’s bandwidth is smaller, so that the total number Thus as long as is smooth over the smoothing band- of frequency points that provide uncorrelated PSD estimates width, and the window and the weighting functions satisfy the increases. The shape of determines the width of ’s above constraint, is approximately an unbiased estimate mainlobe and the height of its sidelobes. Bartlett, Hanning, and of . Hamming windows have lower sidelobes than that of a boxcar Similarly, under the smoothness assumption, (5) becomes1 window, but their mainlobes are wider. A lower sidelobe means (8), shown at the bottom of the page, where represents less interference from nearby frequencies, but a wider mainlobe the quantity in the curly braces and is called the “effective means a larger bandwidth of , which leads to fewer fre- number of degrees of freedom.” is shown [14] to approx- quency points that provide uncorrelated PSD estimates. imately obey a distribution with 2 degrees of freedom For stochastic processes that belong to the same class, their underlying physical properties can vary in a range. For example, the PSD of ocean internal waves’ vertical flow velocity is depen- (9) dent on the buoyancy frequency [15], which is a function of the vertical profile of water density. The buoyancy frequency typi- Equation (8) shows that the periodogram-based PSD estimate cally varies in a range of several cycles per hour. Corresponding has an inherent variance that is proportional to the square of to different buoyancy frequencies, the PSDs are different. As the true PSD. By (4), we see that the PSD estimates are ap- the buoyancy frequency is uncertain, the PSD of the internal proximately uncorrelated between frequencies farther apart than waves’ vertical flow velocity correspondingly bears an uncer- tainty. Thus the variability of the underlying physical properties 1Zero-frequency is special: Var @HAj @HA % P @HAa# . This spe- cialty is properly treated in computations in this paper, but omitted in expres- causes spectral uncertainty of the stochastic processes within sions herein for the sake of conciseness. the same class. Corresponding to given values of the physical Cov (4) Var (5) Var (8) 4558 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 12, DECEMBER 2005 properties of a stochastic process, its PSD is a determin- istic quantity at any frequency . Over the variation range of the physical properties, however, is actually a random variable at any frequency. We denote the mean of by . Note that the expectation and the covari- ance of derived so far are conditioned on . The successive decomposition of the variance of is illustrated in Fig. 1. Using (7), we have Fig. 1. Relationship between a periodogram-based PSD estimate and the class mean of PSD. (11) Incorporating (14) into (13), we have (15) shown at the Thus, as long as is smooth over the smoothing band- bottom of the page. width, and the window and the weighting functions satisfy the By , integration of the first term equals above-mentioned constraint, is approximately an unbi- ; that of the second term equals Var ; that of the ased estimate of . The variance of at any fre- third term vanishes (note that quency is derived as follows, where argument “ ” is omitted ). Equation (15) then becomes for conciseness, as shown in (12) at the bottom of the page. We integrate the three terms individually and then sum them. Var (16) 1) Integration of the first term in (12) 2) Integration of the second term in (12) (13) where by (7) and (8), and the definition of conditional variance, we get Var (17) Var where we note that and (14) . Var (12) (15) ZHANG et al.: TOTAL VARIANCE OF PERIODOGRAM-BASED SPECTRAL ESTIMATE OF STOCHASTIC PROCESS 4559 3) Integration of the third term in (12) results in (18), shown In this paper, we adopt Fisher’s criterion [10]. It represents at the bottom of the page, noting that clear physical meanings and leads to a linear classifier.
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