Data Mining Practical Machine Learning Tools and Techniques

An overview of the self-organizing map algorithm, on which the papers in this issue are based, is presented in this article.

The Self-Organizing Map

1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra. 4. Parametric Methods for Line Spectra. 5. Filter Bank Methods. 6. Spatial Methods. Appendix A: Linear Algebra and Matrix Analysis Tools. Appendix B: Cramer-Rao Bound Tools. Appendix C: Model Order Selection Tools. Appendix D: Answers to Selected Exercises. Bibliography. References Grouped by Subject. Subject Index.

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Spectral analysis of signals

The performance of the MUSIC and ML methods is studied, and their statistical efficiency is analyzed. The Cramer-Rao bound (CRB) for the estimation problems is derived, and some useful properties of the CRB covariance matrix are established. The relationship between the MUSIC and ML estimators is investigated as well. A numerical study is reported of the statistical efficiency of the MUSIC estimator for the problem of finding the directions of two plane waves using a uniform linear array. An exact description of the results is included. >

MUSIC, maximum likelihood, and Cramer-Rao bound

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Statistical Signal Processing of Complex-Valued Data: Correlation analysis

In this paper we revisit a detector first derived by Reed and Yu [1], generalized by Bliss and Parker [2], and recently studied by Hiltunen, Loubaton, and Chevalier [3], [4]. The problem is to detect a known signal transmitted over an unknown MIMO channel of unknown complex gains and unknown additive noise covariance. The probability distribution of a CFAR detector for this problem was first derived for the SIMO channel in [1]. We generalize this distribution for the case of a MIMO channel, and show that the CFAR detector statistic is distributed as the product of independent scalar beta random variables under the null. Our results, based on the theory of beta distributed random matrices, hold for M symbols transmitted from p transmitters and received at L receivers. The asymptotic results of [3], [4] are based on large random matrix theory, which assumes L and M to be unbounded.

Distribution results for a multi-rank version of the Reed-Yu detector

When discretizing continuous-time systems or signals, one is often interested in preserving a property termed covariance-invariance. In this paper a technique is outlined for synthesizing discrete-time systems and signals which are covariance-invariant with corresponding continuous-time systems and signals. Applications of the technique to process simulation, minimum mean-squared error estimation, and digital filter synthesis are outlined, with example designs presented for covariance-invariant Butterworth and Chebychev digital filters. Based on the frequency response of these designs it is argued that the method of covariance-invariance is superior to the methods of impulse-invariance and bilinear-z as a response matching design technique for the synthesis of digital filters. This superiority is especially apparent at sampling rates that are marginal with respect to filter critical frequencies.

Covariance-invariant signal processing

Non-stationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is non-zero. Since the Karhunen-Loeve expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and an improper internal description. We use the Karhunen-Loeve expansion to solve the problem of detecting non-stationary improper complex random signals in additive white Gaussian noise. Using the deflection criterion we compare the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. The performance gain can be as great as a factor of 2.

The Karhunen-Loeve expansion of improper complex random signals with applications in detection

A feature extraction method for underwater target classification
is developed that exploits the linear dependence (coherence)
between two sonar returns. A canonical coordinate decomposition is
applied to resolve two consecutive acoustic backscattered signals
into their dominant canonical coordinates. The corresponding
canonical correlations are selected as features for classifying
mine-like from non-mine-like objects. Test results are based on a
subset of a wideband data set that has been collected at the
Applied Research Lab (ARL), University of Texas (UT)-Austin. This
subset includes returns from different mine-like and non-mine-like
objects at several aspect angles in a smooth bottom condition. The
test results demonstrate the potential of the canonical
correlation-based feature extraction for underwater target
classification and indicate that canonical correlation features
are indeed robust to variations in aspect angle.

Louis L. Scharf

Papers

Statistical Signal Processing of Complex-Valued Data: Correlation analysis

Distribution results for a multi-rank version of the Reed-Yu detector

Covariance-invariant signal processing

The Karhunen-Loeve expansion of improper complex random signals with applications in detection

Classification of underwater mine-like and non-mine-like objects using canonical correlations