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...n possible to obtain the desired unitary factor, U, as any unitary matrix that diagonalizes RHX(r) for some time lag r. As pointed out in [3] it follows from the spectral theorem for normal matrices' =-=[9]-=- that the existence of a unitary matrix V is guaranteed such that for a nonzero time lag T, VHRHx(T)V= diag{di, ...,dk} (2.6) 'A normal matrix M, i.e. MMf=MHM, by the spectral theorem is unitarily dia...

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...rmation, and other non-gaussian features. Some widely used methods for higher-order based source separation include contrast functions, cumulant matching [5][6][7], and independent component analysis =-=[8]-=-. In each the goal is to minimize some cost function measuring the higher-order dependencies between source signals. Non-Stationarity An additional potential characteristic of the source signals that ...

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... here # indicates the Moore-Penrose pseudo-inverse 70 with: P5.2 Maximum Information/Maximum Likelihood One possibility for the cost function, g(o), is the popular infomax cost function introduced in =-=[23]-=-. Here we maximize with respect to A, g(A) = H(h(A#X)) (5.3) where H(y) is the differential entropy for the vector of random processes y, H(y) = - j log(fy(y'))fy (y')dy' (5.4) and h): R"' " -+R"x' is...

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...y')dy' (5.4) and h): R"' " -+R"x' is a finite. An equivalent approach estimate of P given X, i.e. squashing function for restricting the entropy to be [24] is to find P as the maximum likelihood [25] =-=[26]-=- P = max f(XIP) p (5.5) In both cases the distribution of the sources are assumed known in advance and the estimate P of the sources is found as the estimate that most resembles this distribution. 5.3...

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...ces are guaranteed to be distinct so that using Equation (2.11) will not work in practice regardless of the structure of Rx. Information theoretic approaches to this problem have been studied in [11],=-=[12]-=- and other methods in [13],[14]. In each, however, it has been assumed that either GGH is a scaled identity matrix as in (2.10), or that GGH is known a priori. To be robust in this estimation, so that...

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...Regressive (AR) gaussian data show that SOON improves the quality of source separation in comparison to the standard second-order separation algorithms, i.e., Second-Order Blind Identification (SOBI) =-=[3]-=- and SecondOrder Non-Stationary (SONS) blind identification [4]. The sensitivity in performance of SONS and SOON to several algorithmic parameters is also displayed in these experiments. To reduce sen...

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...ndent auto-correlation functions, this second-order source of dependence can be used to unmix the sources. Some popular methods that take advantage of second-order statistics include, factor analysis =-=[1]-=-, Bayesian BSS [2], SOBI [3], and SONS [4]. Additionally a second-order method was developed in [17] that incorporated SOBI for use on low SNR data sets and cases where the signal order is unknown. In...

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...of random processes y, H(y) = - j log(fy(y'))fy (y')dy' (5.4) and h): R"' " -+R"x' is a finite. An equivalent approach estimate of P given X, i.e. squashing function for restricting the entropy to be =-=[24]-=- is to find P as the maximum likelihood [25] [26] P = max f(XIP) p (5.5) In both cases the distribution of the sources are assumed known in advance and the estimate P of the sources is found as the es...

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... Data 4.5.1 The AVIRIS instrument To compare the performance of SOON to that of PCA and NAPCA we have chosen a data set collected with the AVIRIS instrument used at the Jet Propulsion Laboratory [21] =-=[22]-=-. Aviris is 224 channel instrument that records frequencies ranging from 370 to 2500 nm which spans the visible to the mid-infrared bands. The images used here have been collected over Moffet Field, C...

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...his domain, including kurtosis, mutual information, and other non-gaussian features. Some widely used methods for higher-order based source separation include contrast functions, cumulant matching [5]=-=[6]-=-[7], and independent component analysis [8]. In each the goal is to minimize some cost function measuring the higher-order dependencies between source signals. Non-Stationarity An additional potential...

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...The two classic methods most used in the remote sensing community for these purposes are Principal Component Analysis (PCA) and its extension, Noise Adjusted Principal Component Analysis (NAPCA) [19] =-=[20]-=-. 53 4.2 Principal Component Analysis (PCA) 4.2.1 The Method Principal component Analysis is a method for changing the coordinate system of a data set. Formally the process can be described as operati...

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...)fy (y')dy' (5.4) and h): R"' " -+R"x' is a finite. An equivalent approach estimate of P given X, i.e. squashing function for restricting the entropy to be [24] is to find P as the maximum likelihood =-=[25]-=- [26] P = max f(XIP) p (5.5) In both cases the distribution of the sources are assumed known in advance and the estimate P of the sources is found as the estimate that most resembles this distribution...

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...tion functions, this second-order source of dependence can be used to unmix the sources. Some popular methods that take advantage of second-order statistics include, factor analysis [1], Bayesian BSS =-=[2]-=-, SOBI [3], and SONS [4]. Additionally a second-order method was developed in [17] that incorporated SOBI for use on low SNR data sets and cases where the signal order is unknown. In each the cost fun...

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...n this domain, including kurtosis, mutual information, and other non-gaussian features. Some widely used methods for higher-order based source separation include contrast functions, cumulant matching =-=[5]-=-[6][7], and independent component analysis [8]. In each the goal is to minimize some cost function measuring the higher-order dependencies between source signals. Non-Stationarity An additional potent...

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...the data set. The method applicable for this characterization depends on the type of data being analyzed. For time series data many heuristics exist for partitioning the data into stationary segments =-=[18]-=-. An algorithm that computes exact partitioning of the data into stationary segments scales4 as 0(2 -1), where T is the number of 4Consider a time series with T samples, then there exists T-1 position...

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...atrix as in (2.10), or that GGH is known a priori. To be robust in this estimation, so that we may use the less restrictive noise model of this thesis, we adopt a method outlined in the ION algorithm =-=[16]-=- which utilizes a scree plot. The scree plot used here displays the log magnitude-ordered eigenvalues of the sample correlation matrix for the data matrix X as a function of eigenvector number. In Fig...

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... domain, including kurtosis, mutual information, and other non-gaussian features. Some widely used methods for higher-order based source separation include contrast functions, cumulant matching [5][6]=-=[7]-=-, and independent component analysis [8]. In each the goal is to minimize some cost function measuring the higher-order dependencies between source signals. Non-Stationarity An additional potential ch...

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...matrices are guaranteed to be distinct so that using Equation (2.11) will not work in practice regardless of the structure of Rx. Information theoretic approaches to this problem have been studied in =-=[11]-=-,[12] and other methods in [13],[14]. In each, however, it has been assumed that either GGH is a scaled identity matrix as in (2.10), or that GGH is known a priori. To be robust in this estimation, so...

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...istinct so that using Equation (2.11) will not work in practice regardless of the structure of Rx. Information theoretic approaches to this problem have been studied in [11],[12] and other methods in =-=[13]-=-,[14]. In each, however, it has been assumed that either GGH is a scaled identity matrix as in (2.10), or that GGH is known a priori. To be robust in this estimation, so that we may use the less restr...

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...ct so that using Equation (2.11) will not work in practice regardless of the structure of Rx. Information theoretic approaches to this problem have been studied in [11],[12] and other methods in [13],=-=[14]-=-. In each, however, it has been assumed that either GGH is a scaled identity matrix as in (2.10), or that GGH is known a priori. To be robust in this estimation, so that we may use the less restrictiv...

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...ion. The two classic methods most used in the remote sensing community for these purposes are Principal Component Analysis (PCA) and its extension, Noise Adjusted Principal Component Analysis (NAPCA) =-=[19]-=- [20]. 53 4.2 Principal Component Analysis (PCA) 4.2.1 The Method Principal component Analysis is a method for changing the coordinate system of a data set. Formally the process can be described as op...

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...its, merges, and all other cash flows to share holders so that such artifacts would not factor in the resulting pricing behavior. The data was obtained from the Center for Research in Security Prices =-=[27]-=- and provided by Wharton Research Data Services [28]. Here the objective was to obtain the independent components detected by SOON and establish their utility in predicting future pricing behavior and...

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...rs so that such artifacts would not factor in the resulting pricing behavior. The data was obtained from the Center for Research in Security Prices [27] and provided by Wharton Research Data Services =-=[28]-=-. Here the objective was to obtain the independent components detected by SOON and establish their utility in predicting future pricing behavior and explaining market dynamics. A scree plot containing...