Multivariate & Spectral Signal Analysis

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Abstractions about decomposing signals and multivariate data into structured components, including Fourier-type and elliptic filters, canonical correlation and Karhunen-Loeve analysis, and statistical dependence concepts underlying spectral and dimensionality-reduction methods.

10 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Canonical correlation — In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices.
• Cochran's Theorem — A rank criterion that identifies when a complete quadratic-form partition of centered spherical Gaussian variation yields mutually independent chi-square components.
• Elliptic filter — An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband.
• Fourier Sine Transform — In mathematics, the Fourier sine and cosine transforms are integral equations that decompose arbitrary functions into a sum of sine waves representing the odd component of the function plus cosine waves representing the even component of the function.
• Generalized multidimensional scaling — Generalized multidimensional scaling (GMDS) is an extension of metric multidimensional scaling, in which the target space is non-Euclidean.
• Karhunen–Loève theorem — The importance of the Karhunen–Loève theorem is that it yields the best such basis in the sense that it minimizes the total mean squared error.
• Operational Calculus — A family of methods that represents differentiation, integration, or related operators in an algebraic domain, solves the resulting operator equation, and interprets the inverse representation as a function.
• Primon Gas — In mathematical physics, the primon gas or Riemann gas discovered by Bernard Julia is a model illustrating correspondences between number theory and methods in quantum field theory, statistical mechanics and dynamical systems such as the Lee–Yang theorem.
• Secondary Polynomials — Secondary Polynomials is a recurring orthogonal polynomials, analysis identity in which a difference quotient of an orthogonal polynomial is integrated against its density to generate an associated polynomial sequence.
• Statistical Dependence — Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.