From Fourier to Wavelet Analysis of Time Series

2011

All praises and thanks be to Almighty, the most beneficent, the most merciful, Who bestowed upon me the courage, patience and strength to embark upon this work and carry it out to its successful completion. After Almighty, I must admire to our last and great Prophet, Huzoor (S.A.W.) whose hfe (as a model) provided me the determination and patience to continue my hard work especially in the period of doubt and despair. Dated: OZ-11' ZOli ^g^Khan) ni Chapter 5 is based on analysis of stock market by using wavelets. In this chapter, we have used wavelet transform to analyze data of stock market. This is one of the application of wavelets in time series analysis. The scatter plots of shares of different companies have been plotted and total variation plots of Sensex versus shares of companies are also shown. At the end, an extensive bibliography of the existing literature related to the subject matter of the dissertation is included.

Mathematical Geosciences, 2017

Least-squares spectral analysis, an alternative to the classical Fourier transform, is a method of analyzing unequally spaced and non-stationary time series in their first and second statistical moments. However, when a time series has components with low or high amplitude and frequency variability over time, it is not appropriate to use either the least-squares spectral analysis or Fourier transform. On the other hand, the classical short-time Fourier transform and the continuous wavelet transform do not consider the covariance matrix associated with a time series nor do they consider trends or datum shifts. Moreover, they are not defined for unequally spaced time series. A new method of analyzing time series, namely, the least-squares wavelet analysis is introduced, which is a natural extension of the least-squares spectral analysis. This method decomposes a time series to the time-frequency domain and obtains its spectrogram. In addition, the probability distribution function of the spectrogram is derived that identifies statistically significant peaks. The least-squares wavelet analysis can analyze any non-stationary and unequally spaced time series with components of low or high amplitude and frequency variability, including datum shifts, trends, and constituents of known forms, by taking into account the covariance matrix associated with the time series. The outstanding performance of the proposed method on synthetic time series and a very long baseline interferometry series is demonstrated, and the results are compared with the weighted wavelet Z-transform.

IEEE PES Transmission and Distribution Conference and Exposition, 2006

In continuous-time wavelet analysis, most wavelet presents some kind of symmetry. Based on the Fourier and Hartley transform kernels, a new wavelet multiresolution analysis has been proposed. This approach is based on a pair of orthogonal wavelet functions and can be named as the Fourier-like and Hartley-like wavelet analysis. In this paper we carry a preliminary investigation about potentials applications for the Fourier-like wavelet analysis on signals derived from power systems to illustrate the behavior of such wavelets.

Applied Sciences, 2019

Non-stationary time series (TS) analysis has gained an explosive interest over the recent decades in different applied sciences. In fact, several decomposition methods were developed in order to extract various components (e.g., seasonal, trend and abrupt components) from the non-stationary TS, which allows for an improved interpretation of the temporal variability. The wavelet transform (WT) has been successfully applied over an extraordinary range of fields in order to decompose the non-stationary TS into time-frequency domain. For this reason, the WT method is briefly introduced and reviewed in this paper. In addition, this latter includes different research and applications of the WT to non-stationary TS in seven different applied sciences fields, namely the geo-sciences and geophysics, remote sensing in vegetation analysis, engineering, hydrology, finance, medicine, and other fields, such as ecology, renewable energy, chemistry and history. Finally, five challenges and future w...

Slovak Journal of Civil Engineering, 2015

The paper presents an analysis of changes in the structure of the average annual discharges, average annual air temperature, and average annual precipitation time series in Slovakia. Three time series with lengths of observation from 1961 to 2006 were analyzed. An introduction to spectral analysis with Fourier analysis (FA) is given. This method is used to determine significant periods of a time series. Later in this article a description of a wavelet transform (WT) is reviewed. This method is able to work with non-stationary time series and detect when significant periods are presented. Subsequently, models for the detection of potential changes in the structure of the time series analyzed were created with the aim of capturing changes in the cyclical components and the multiannual variability of the time series selected for Slovakia. Finally, some of the comparisons of the time series analyzed are discussed. The aim of the paper is to show the advantages of time series analysis us...

2013

European Journal of …, 2009

It is shown how a modern extension of Fourier analysis known as wavelet analysis is applied to signals containing multiscale information. First, a continuous wavelet transform is used to analyse the spectrum of a nonstationary signal (one whose form changes in time). The spectral analysis of such a signal gives the strength of the signal in each frequency as a function of time. Next, the theory is specialized to discrete values of time and frequency, and the resulting discrete wavelet transform is shown to be useful for data compression. This paper is addressed to a broad community, from undergraduate to graduate students to general physicists and to specialists in other fields than wavelets.

Journal of the Royal Statistical Society: Series D (The Statistician), 2000

In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this paper gives a relatively accessible introduction to standard wavelet analysis and provides a review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be effective, rather than to researchers who are already familiar with the ®eld. Given that objective, we do not emphasize mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in greater detail and generality if required. The paper ®rst establishes some necessary basic mathematical background and terminology relating to wavelets. It then reviews the more well-established applications of wavelets in statistics including their use in nonparametric regression, density estimation, inverse problems, changepoint problems and in some specialized aspects of time series analysis. Possible extensions to the uses of wavelets in statistics are then considered. The paper concludes with a brief reference to readily available software packages for wavelet analysis.

International Journal of Scientific Research in Computer Science, Engineering and Information Technology

Wavelets are mathematical functions which are used as a basis for writing down other complex functions in an easy way. These cut up data into its frequency components and so that we can study each and every part with more preciseness as it is scaled for our convenience. We may also term wavelets as a tool to decompose signals and trend as a function of time. Wavelets are certainly used in place of the applications of Fourier Analysis as wavelets give more freedom to work on. In this paper, a basic idea of wavelet is provided to a person who is unknown with the idea of function approximation. Apart from pure mathematics areas, wavelets are highly useful tool in analyzing a time series. Wavelets are used for removing noise from a statistical data which is one of the most important job in data analysis. The applications of wavelets not only bars here, but they are also used in quantum physics, artificial intelligence and visual recognition. An important aspect of wavelets, image proces...

2004