Directed spectral measures quantify, in the frequency domain, directed statistical interactions between time-series variables. Most commonly, the various measures are computed based on a (linear) multi-variate autoregressive (MVAR) model of the data. The various methods quantify in different ways the strength of interaction terms in the Fourier transform of the MVAR model. While these methods a...

The poor practical behavior of (1)-(2) has been known for many years. If the level sets of f resemble long valleys, the sequence {xk} displays a typical zig-zagging trajectory and the speed of convergence is very slow. In the simplest case, in which f is a strictly convex quadratic, the method converges to the solution with a Q-linear rate of convergence whose factor tends to 1 when the conditi...

The purpose of time-series analysis is to detect basic properties of the system that engenders a time series. The hope of predicting the system's future evolution is closely related to the possibility of such detection. The most easily predictable components of a system's evolution are the regular, deterministic ones; hence we look for trends and periodic oscillations. In doing so, it is often ...

1 preprint numerics no. 1/2012 norwegian university of science and technology trondheim, norway This paper discusses numerical solution of boundary value problems using spectral methods combined with nonlinear and adaptive mappings between the reference domain and the physical domain. A brief review of existing methods for adaptive mesh generation is given, and a method for finding close to opt...

For simplicity we will focus on a simple Gaussian Mixture Model. Consider a mixture of k spherical gaussians in R which is the following generative process. Let w ∈ ∆([k]) denote a distribution and let μ1, . . . , μk ∈ R be the mean vectors. Each point xi is generated by first choosing a component hi ∼ w and then xi ∼ N (μhi , I). We are given n samples x1, . . . , xn drawn according to this pr...

This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then p...

Spectral interdependency methods are a means of statistically quantifying the interrelationship between a pair of dynamic processes as a function of frequency or time period of oscillation. The measures of spectral interdependency are derived from the time series recordings of dynamic systems either by using autoregressive modeling (parametric method) or by using direct Fourier or wavelet trans...