# Wavelet Methods For Time Series Analysis Percival Pdf 25

This analysis provides evidence regarding the timely and appropriate measure of correlation changes and the behaviour of sukuk and bond indices globally, which is beneficial to the management of sukuk and bond portfolios.

## wavelet methods for time series analysis percival pdf 25

Bhuiyan, R.A., Rahman, M.P., Saiti, B. and Mat Ghani, G. (2019), "Co-movement dynamics between global sukuk and bond markets: New insights from a wavelet analysis", International Journal of Emerging Markets, Vol. 14 No. 4, pp. 550-581. -12-2017-0521

Wastewater-based epidemiology (WBE) is a novel approach in drug use epidemiology which aims to monitor the extent of use of various drugs in a community. In this study, we investigate functional principal component analysis (FPCA) as a tool for analysing WBE data and compare it to traditional principal component analysis (PCA) and to wavelet principal component analysis (WPCA) which is more flexible temporally.

We analysed temporal wastewater data from 42 European cities collected daily over one week in March 2013. The main temporal features of ecstasy (MDMA) were extracted using FPCA using both Fourier and B-spline basis functions with three different smoothing parameters, along with PCA and WPCA with different mother wavelets and shrinkage rules. The stability of FPCA was explored through bootstrapping and analysis of sensitivity to missing data.

Recently, functional principal component analysis (FPCA) has been explored as a statistical method for analysing wastewater data [18]. The approach was found not only to be well suited for extracting useful information about the different drug loads during the course of a week, but also extracted detailed information that would otherwise be lost when using more traditional statistical methods. It can easily be argued that functional data analysis (FDA) is a reasonable approach to analysing temporal wastewater data [18], but there is a concern that the basis functions of the FDA framework might be too smooth to model the rapid temporal changes in drug load curves that can occur over the course of a week, especially the change between weekdays and weekend. Alternative, more flexible, statistical approaches should also be explored.

Wavelets have a long tradition in time series analysis [19]. Wavelet basis functions are localized in both frequency and time domains simultaneously, allowing for the extraction of features that are less smooth from temporal data [20, 21]. Wavelet-based principal component analysis (WPCA) has recently been applied successfully to analysis of foetal movement monitoring data [22, 23]. The temporally more flexible WPCA could be able to detect rapid temporal changes in wastewater data.

The unit of observation in the analysis is a seven day week starting Wednesday and ending Tuesday. As wavelet analysis generally requires individual time series to have a length of a power of two observations [21], we added the first observation to the end of the time series, generating an eight day time series, for ease of comparison. This additional day is needed only for technical purposes and does not have any impact on the results [21]. Missing data across all the 38 cities was 2.2 %. As standard frequentist functional data analysis (FDA) needs complete data sets for analysis, we performed single imputation [28] using the bootstrapping-based expectation maximization algorithm [29], before proceeding with the analysis on the imputed dataset. Moreover, the wastewater data was heavily skewed, and the data was log-transformed prior to further analysis.

Using traditional PCA, each day of the week is considered a single variable and each PC resulting from the PCA is defined as a linear combination of the original variables. Since in PCA the load of a drug at a given day is assumed to be independent of the drug load at any other day, be it preceding or following days, the correlation between individual days is not taken into account. This assumption is however likely to be violated for wastewater data where consecutive days are naturally correlated in time. PCA on temporal data will yield temporal PCs, but as intra-correlation of individual time series is not modelled, the temporal aspect of the data will be ignored, leading to a lower ability to recover the true underlying signal of interest [30]. So while PCA is a well-known, well-established statistical method for extracting structure in the data, results should be interpreted with care for temporal data.

FDA is a statistical methodology specifically developed for analysing temporal data [25]. The first step of FDA is to fit a mathematical function to the temporal observations, and the statistical analysis is then performed on this mathematical function rather than the raw data. The time series for the 38 European cities were converted into 38 continuous smooth curves using both Fourier and B-spline basis functions. The optimal smoothing was found using the generalized cross validation (GCV) criterion [31]. A single choice of smoothing parameter for all cities is usually recommended [32], but for exploratory purposes we also fitted an optimal individual smoothing parameter for each city. This smoothing removes the random day-to-day variation, e.g. non-systematic error, measurement error and normal fluctuations in the drug load.

Wavelets is a mathematical framework developed for analysing high-dimensional data, such as time series or images [19]. While FDA uses global basis functions, such as trigonometric functions, wavelet basis functions are localized in both time and space, allowing for modelling of less smooth temporal data, even spikes [20, 21]. Wavelet basis functions are generally not expressed explicitly as functions. Instead individual basis functions are specified by recursive difference equations conditioned on a mother wavelet [33]. The mother wavelet Ψ(t) and a corresponding father wavelet φ(t) can be interpreted as a high-pass and low-pass filter of the original data respectively.

Before proceeding with the wavelet principal component analysis (WPCA), for each vector of wavelet coefficients we applied wavelet shrinkage on the coefficients B i to filter out the noise inherited from y i (t) [34]. Numerous thresholding rules exist. In this study, we considered universal thresholding [35] and Bayesian [36] wavelet shrinkage.

Wavelet-based principal component analysis (WPCA) is an application of standard PCA to the wavelet domain [22]. Performing PCA on the smoothed B i coefficients in the wavelet domain using eq. 1, results in a set of new variables which are linear combinations of the smoothed wavelet coefficients B i . PCA in wavelet domain as applied here thus assumes independency of the coefficients B i . Intra-correlation of the individual time series is taken care of during the transformation process from time domain to wavelet domain (eq. 3). Back transforming the PCs obtained in wavelet domain to time domain gives the wavelet principal components (WPCs). The process also provides a score for each individual y i (t) indicating the intensity with which each of the WPC patterns is present in that particular temporal MDMA load curve. While WPCA is temporally more flexible than FPCA, it is also technically less tractable.

Figure 2d-f show the first three wavelet principal components (WPCs) for each of the three mother wavelets. The temporal patterns are qualitatively consistent with those from PCA and functional principal component analysis (FPCA), but the WPC patterns seem to be somewhat more smoothed throughout the week.

In this study, we have explored functional principal component analysis (FPCA) as a tool for analysing temporal wastewater data, comparing it to traditional principal component analysis (PCA) and the temporally more flexible wavelet PCA (WPCA), as well as exploring the robustness of the extracted FPCA patterns and sensitivity to missing data. The results were generally consistent between PCA, FPCA and WPCA. WPCA did not detect any rapid temporal changes in the data. FPCA thus appears not to smooth away essential information in the temporal data, and there is no need to go beyond FPCA to the less tractable WPCA. The analyses establish FPCA using Fourier basis functions and common optimal smoothing as a precise, flexible and stable method for analysing wastewater-based epidemiology (WBE) data.

WBE provides an objective estimate of the use of a specific drug for all people contributing to the wastewater treatment plant in a catchment area over a time period. Recently the advantages of functional data analysis (FDA) over the traditional statistical analyses usually applied to WBE data for information extraction have been demonstrated [18]. FDA is analytically tractable and a well-established mathematical framework for temporal data [25], and a series of R packages for calculations exist [26]. This greatly assists the introduction of more advanced statistical analysis to a novel field within the health sciences, and the initial concern that FDA might over-smooth the underlying temporal process in wastewater data was shown to be non-existent. FDA is indeed sufficiently flexible and stable for the analysis of WBE data.

Even though the patterns extracted by PCA, FPCA and WPCA were qualitatively consistent, the interpretation of the principal components (PCs) and WPCs can be difficult to compare to the FPCs. In PCA, individual days are assumed to be independent variables, that is, the drug load at a given day is independent of the drug load at any other day, so PCA does not take the possible correlation between consecutive days of the temporal data set into account. PCA on WBE is generally not advised, since it treats days as independent entities without controlling for the intra-correlation between them, ignoring the fact that the seven days constitute a single entity, a week. Further, WPCA is an extension of traditional PCA, but less direct than FPCA. In the version of WPCA applied here, PCA is performed on the smoothed wavelet coefficients in wavelet domain, where the wavelet coefficients constitute the independent variables for the subsequent PCA, before back calculating each WPC to time domain. As a result the patterns of the WPCs do not have the same scale as the PCs and FPCs, making direct comparison between the methods difficult.