348,417 research outputs found
On the Correlation Matrix of the Discrete Fourier Transform and the Fast Solution of Large Toeplitz Systems For Long-Memory Time Series
For long-memory time series, we show that the Toeplitz system §n(f)x = b can be solved in
O(n log5=2 n) operations using a well-known version of the preconditioned conjugate gradient method, where §n(f) is the n£n covariance matrix, f is the spectral density and b is a known vector. Solutions of such systems are needed for optimal linear prediction and interpolation. We establish connections between this preconditioning method and the frequency domain analysis of time series. Indeed, the running time of the algorithm is determined by rate of increase of the condition number of the correlation matrix of the discrete Fourier transform vector, as the sample size tends to 1. We derive an upper bound for this condition number. The bound is of interest in its own right, as it sheds some light on the widely-used but heuristic approximation that the standardized DFT coefficients
are uncorrelated with equal variances. We present applications of the preconditioning methodology to the forecasting and smoothing of volatility in a long memory stochastic volatility model, and to the evaluation of the Gaussian likelihood function of a long-memory model.Statistics Working Papers Serie
On the Correlation Matrix of the Discrete Fourier Transform and the Fast Solution of Large Toeplitz Systems For Long-Memory Time Series
For long-memory time series, we show that the Toeplitz system §n(f)x = b can be solved in
O(n log5=2 n) operations using a well-known version of the preconditioned conjugate gradient method, where §n(f) is the n£n covariance matrix, f is the spectral density and b is a known vector. Solutions of such systems are needed for optimal linear prediction and interpolation. We establish connections between this preconditioning method and the frequency domain analysis of time series. Indeed, the running time of the algorithm is determined by rate of increase of the condition number of the correlation matrix of the discrete Fourier transform vector, as the sample size tends to 1. We derive an upper bound for this condition number. The bound is of interest in its own right, as it sheds some light on the widely-used but heuristic approximation that the standardized DFT coefficients
are uncorrelated with equal variances. We present applications of the preconditioning methodology to the forecasting and smoothing of volatility in a long memory stochastic volatility model, and to the evaluation of the Gaussian likelihood function of a long-memory model.Statistics Working Papers Serie
On the Mathematics of Music: From Chords to Fourier Analysis
Mathematics is a far reaching discipline and its tools appear in many
applications. In this paper we discuss its role in music and signal processing
by revisiting the use of mathematics in algorithms that can extract chord
information from recorded music. We begin with a light introduction to the
theory of music and motivate the use of Fourier analysis in audio processing.
We introduce the discrete and continuous Fourier transforms and investigate
their use in extracting important information from audio data
Novel Fourier Quadrature Transforms and Analytic Signal Representations for Nonlinear and Non-stationary Time Series Analysis
The Hilbert transform (HT) and associated Gabor analytic signal (GAS)
representation are well-known and widely used mathematical formulations for
modeling and analysis of signals in various applications. In this study, like
the HT, to obtain quadrature component of a signal, we propose the novel
discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine
quadrature transforms (FSQTs), designated as Fourier quadrature transforms
(FQTs). Using these FQTs, we propose sixteen Fourier-Singh analytic signal
(FSAS) representations with following properties: (1) real part of eight FSAS
representations is the original signal and imaginary part is the FCQT of the
real part, (2) imaginary part of eight FSAS representations is the original
signal and real part is the FSQT of the real part, (3) like the GAS, Fourier
spectrum of the all FSAS representations has only positive frequencies, however
unlike the GAS, the real and imaginary parts of the proposed FSAS
representations are not orthogonal to each other. The Fourier decomposition
method (FDM) is an adaptive data analysis approach to decompose a signal into a
set of small number of Fourier intrinsic band functions which are AM-FM
components. This study also proposes a new formulation of the FDM using the
discrete cosine transform (DCT) with the GAS and FSAS representations, and
demonstrate its efficacy for improved time-frequency-energy representation and
analysis of nonlinear and non-stationary time series.Comment: 22 pages, 13 figure
- …