### 机器学习代写|聚类分析作业代写clustering analysis代考| Wavelet representation of time series

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|聚类分析作业代写clustering analysis代考|Wavelet representation of time series

Time series features such as autocorrelations and partial autocorrelations describe the dynamics of a stationary time series in the time domain, whereas spectral ordinates describe the dynamics of a stationary time series in the frequency domain. When an observed time series is nonstationary in the mean, it first has to be differenced to be made stationary before analyzing its dynamics using autocorrelations, partial autocorrelations and spectral ordinates. When a time series is decomposed into wavelet series, the wavelet coefficients describe the dynamics of a time series in both the time and frequency domains. Furthermore, wavelet analysis is applicable to both stationary and non-stationary time series without the need for differencing a non-stationary time series. While we present just a brief description of relevant aspects of wavelet analysis as applicable to discrete times here, more specific and general details can be found in several books on the topic, one of which is by Percival and Walden (2000). Our descriptions that follow mostly use their notations.

## 机器学习代写|聚类分析作业代写clustering analysis代考|Discrete wavelet transform

The Discrete Wavelet Transform (DWT), which is an orthonormal transform, re-expresses a time series of length $T$ in terms of coefficients that are associated with a particular time and with a particular dyadic scale as well as one or more scaling coefficients. The $j$-th dyadic scale is of the form $2^{j-1}$ where $j=1,2, \ldots, J$, and $J$ is the maximum allowable number of scales.

The number of coefficients at the $j$-th scale is $T / 2^{j}$, provided $T=2^{J}$. In general the wavelet coefficients at scale $2^{j-1}$ are associated with frequencies in the interval $\left[1 / 2^{j+1}, 1 / 2^{j}\right]$. Large time scales give more low frequency information, while small time scales give more high frequency information about the time series. The coefficients are obtained from projecting the time series with translated and dilated versions of a wavelet filter. The DWT is computed using what is known as the pyramid algorithm.

In general, the wavelet coefficients are proportional to the differences of averages of the time series observations at each scale, whereas the scaling coefficients are proportional to the averages of the original series over the largest scale. The scaling coefficients reflect long-term variations, which would exhibit a similar trend to the original series. The DWT re-expresses a time series in terms of coefficients that are associated with a particular time and a particular dyadic scale. These coefficients are fully equivalent to the information contained in the original series in that a time series can be perfectly reconstructed from its DWT coefficients. An important aspect of the DWT is that it de-correlates even highly correlated series; that is, the wavelet coefficients at each scale are approximately uncorrelated.

It is possible to recover the time series $\left{x_{t}, t=1,2, \ldots, T\right}$ from its DWT by synthesis, that is, the multi-resolution analysis (MRA) of a time series which is expressed as
$$x_{t}=\sum_{j=1}^{J} d_{j}+s_{J},$$
where $d_{j}$ is the wavelet detail (series of inverse wavelet coefficients at scale j) and $s_{J}$ is the smooth series which is the inverse of the series of scaling coefficients. Hence a time series and its DWT are actually two representations of the same mathematical entity.

## 机器学习代写|聚类分析作业代写clustering analysis代考|Modified discrete wavelet transform

The maximum overlap discrete wavelet transform (MODWT) is a modification of the DWT. Under the MODWT, the number of wavelet coefficients created will be the same as the number of observations in the original time series. Because the MODWT decomposition retains all possible times at each time scale, the MODWT has the advantage of retaining the time invariant property of the original time series. The MODWT can be used in a similar manner to the DWT in defining a multi-resolution analysis of a given time series. In contrast to the DWT, the MODWT details and smooths are associated with zero phase filters making it easy to line up features in a MRA with the original time series more meaningfully.

Many families of wavelet filters, whose qualities vary according to a number of criteria, are available. Some commonly used filters of width $N$ (where $N$ is an integer) are from the Daubechies family abbreviated as $\mathrm{DB}(N)$. These filters are asymmetric. The Haar filter which is the simplest wavelet filter is a $\mathrm{DB}(2)$ filter. Another family of filters which is a modification of the Daubechies family is the least asymmetric family LA $(N)$ (also referred to as the symmletts family SYM $(N)$ ). These filters are nearly symmetric and have the property of aligning the wavelet coefficients very well with the given time series. The coiflets family of filters, $\mathrm{CF}(N)$ also possess this property and are symmetric filters. Filters from the least symmetric and coiflets families are usually recommended for use with time series because of their good alignment properties.
Example 2.3 Fig. 2.6 shows the total seasonally adjusted retail turnover in Australia from January 2005 to August 2015 (128 months) from the website of the Australian Bureau of Statistics, while Fig. 2.7 shows the MODWT decomposition of this series over 5 scales using the LA(8) filter. $d 1$ to $d 6$ represent the series of wavelet coefficients at five scales and $s 6$ the series of scaling coefficients at the 5 th scale. It can be observed while the series is non-stationary in mean, the wavelet series are stationary in the mean. The $d 1$ series describes the series dynamics over 2-4 months, $d 2$, over 4-8 months, $d 3$, over 8-16 months, $d 4$, over 15-32 months and $d 5$, over 32-64 months. The $s 6$ series describes the variation over the 128-month period.

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## MATLAB代写

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