Using wavelets for time series forecasting: Does it pay off?

By means of wavelet transform a time series can be decomposed into a time dependent sum of frequency components. As a result we are able to capture seasonalities with time-varying period and intensity, which nourishes the belief that incorporating the wavelet transform in existing forecasting methods can improve their quality. The article aims to verify this by comparing the power of classical and wavelet based techniques on the basis of four time series, each of them having individual characteristics. We find that wavelets do improve the forecasting quality. Depending on the data's characteristics and on the forecasting horizon we either favour a denoising step plus an ARIMA forecast or an multiscale wavelet decomposition plus an ARIMA forecast for each of the frequency components. --Forecasting,Wavelets,ARIMA,Denoising,Multiscale Analysis

Single-trial multiwavelet coherence in application to neurophysiological time series

A method of single-trial coherence analysis is presented, through the application of continuous muldwavelets. Multiwavelets allow the construction of spectra and bivariate statistics such as coherence within single trials. Spectral estimates are made consistent through optimal time-frequency localization and smoothing. The use of multiwavelets is considered along with an alternative single-trial method prevalent in the literature, with the focus being on statistical, interpretive and computational aspects. The multiwavelet approach is shown to possess many desirable properties, including optimal conditioning, statistical descriptions and computational efficiency. The methods. are then applied to bivariate surrogate and neurophysiological data for calibration and comparative study. Neurophysiological data were recorded intracellularly from two spinal motoneurones innervating the posterior,biceps muscle during fictive locomotion in the decerebrated cat

Algorithms for Spectral Analysis of Irregularly Sampled Time Series

In this paper, we present a spectral analysis method based upon least square approximation. Our method deals with nonuniform sampling. It provides meaningful phase information that varies in a predictable way as the samples are shifted in time. We compare least square approximations of real and complex series, analyze their properties for sample count towards infinity as well as estimator behaviour, and show the equivalence to the discrete Fourier transform applied onto uniformly sampled data as a special case. We propose a way to deal with the undesirable side effects of nonuniform sampling in the presence of constant offsets. By using weighted least square approximation, we introduce an analogue to the Morlet wavelet transform for nonuniformly sampled data. Asymptotically fast divide-and-conquer schemes for the computation of the variants of the proposed method are presented. The usefulness is demonstrated in some relevant applications.

Wavelet Analysis and Denoising: New Tools for Economists

This paper surveys the techniques of wavelets analysis and the associated methods of denoising. The Discrete Wavelet Transform and its undecimated version, the Maximum Overlapping Discrete Wavelet Transform, are described. The methods of wavelets analysis can be used to show how the frequency content of the data varies with time. This allows us to pinpoint in time such events as major structural breaks. The sparse nature of the wavelets representation also facilitates the process of noise reduction by nonlinear wavelet shrinkage , which can be used to reveal the underlying trends in economic data. An application of these techniques to the UK real GDP (1873-2001) is described. The purpose of the analysis is to reveal the true structure of the data - including its local irregularities and abrupt changes - and the results are surprising.Wavelets, Denoising, Structural breaks, Trend estimation

A Case Study of Natural Frequency of the Tram Rail Due to Vibration Using Wavelets

Many vibration signals of tram rails due to tram movement are non-stationary and have highly complex time-frequency characteristics. The vibration signal of a rotating wheel involves condition monitoring and fault diagnosis. Many signal analysis methods are able to extract useful information from vibration data. In this paper, we were able to correlate non-linear independent signal acquired using acceleromets at different spots across the city and extract tram rail vibration noise and model the effect of signal noise to identify the frequency characteristics of the rail by characterizing the spectral content of the noise signal using parametric distribution and then by applying non parametric filters to characterize the signal power spectral density using Wavelet Transform (WT) and Parseval’s theorem. The fault can be detected from a given level of resolution. For this purpose, Parseval’s theorem is used as an evaluation criterion to select the optimal level. Associated to envelope analysis, it allows clear visualization of fault frequencies. on the inner rail of the railway line. The time-frequency contour map can easily show the power distribution of signal in time and frequency domain. Moreover, it is a good way to identify the rail track faults involving a breakdown change. The simulative results show that time-frequency contour map have the capabilities to identify the difference of those faults of vibration monitoring. In conclusion, the faults along the rail track can be classified by time-frequency contour map for frequency decomposition. We hereby decompose the high frequency detail of the signal without decomposition after wavelet transform, so as to improve the frequency resolution

A two-factor model for electricity prices with dynamic volatility

The wavelet transform is used to identify a biannual and an annual seasonality in the Phelix Day Peak and to separate the long-term trend from its short-term motion. The short-term/long-term model for commodity prices of Schwartz & Smith (2000) is applied but generalised to account for weekly periodicities and time-varying volatility. Eventually we find a bivariate SARMA-CCC-GARCH model to fit best. Moreover it surpasses the goodness of fit of an univariate GARCH model, which shows that the additional effort of dealing with a two-factor model is worthwile. --Wavelets,Seasonal Filter,Relative Wavelet Energy,Multivariate GARCH,Energy Price Modelling

Complex data processing: fast wavelet analysis on the sphere

In the general context of complex data processing, this paper reviews a recent practical approach to the continuous wavelet formalism on the sphere. This formalism notably yields a correspondence principle which relates wavelets on the plane and on the sphere. Two fast algorithms are also presented for the analysis of signals on the sphere with steerable wavelets.Comment: 20 pages, 5 figures, JFAA style, paper invited to J. Fourier Anal. and Appli