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

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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|Overview

Time series clustering and classification has relevance in a diverse range of fields which include geology, medicine, environmental science, finance and economics. Clustering is an unsupervised approach to grouping together similar items of interest and was initially applied to cross-sectional data. However, clustering time series data has become a popular research topic over the past three to four decades and a rich literature exists on this topic. A set of time series can be clustered using conventional hierarchical and non-hierarchical methods, fuzzy clustering methods, machine learning methods and modelbased methods.

Actual time series observations can be clustered (e.g., D’Urso, 2000; Coppi and D’Urso, 2001, D’Urso, 2005), or features extracted from the time series can be clustered. Features are extracted in the time, frequency and wavelets domains. Clustering using time domain features such as autocorrelations, partial autocorrelations, and cross-correlations have been proposed by several authors including Goutte et al. (1999), Galeano and Peña (2000), Dose and Cincotti $(2005)$, Singhal and Seborg (2005), Caiado et al. (2006), Basalto et al. (2007), Wang et al. (2007), Takayuki et al. (2006), Ausloos and Lambiotte (2007), Miskiewicz and Ausloos (2008), and D’Urso and Maharaj (2009).

In the frequency domain, features such as the periodogram and spectral and cepstral ordinates are extracted; included in the literature are studies by Kakizawa et al. (1998), Shumway (2003), Caiado et al. (2006), Maharaj and D’Urso $(2010,2011)$.

The features extracted in the wavelets domain are discreet wavelet transforms (DWT), wavelet variances and wavelet correlations and methods have been proposed by authors such as Zhang et al. (2005), Maharaj et al. (2010), D’Urso and Maharaj (2012) and D’Urso et al. (2014). As well, time series

can be modelled and the parameters estimates used as the clustering variables. Studies on the model-based clustering method include those by Piccolo $(1990)$, Tong and Dabas (1990), Maharaj (1996, 2000), Kalpakis et al. (2001), Ramoni et al. ( 2002$)$, Xiong and Yeung (2002), Boets (2005), Singhal and Seborg (2005), Savvides et al. (2008), Otranto (2008), Caiado and Crato (2010), D’Urso et al. (2013), Maharaj et al. (2016) and D’Urso et al. (2016).

Classification is a supervised approach to grouping together items of interest and discriminant analysis and machine learning methods are amongst the approaches that have been used. Initially classification was applied to crosssectional data but a large literature now exists on the classification of time series which includes many very useful applications. These time series classification methods include the use of feature-based, model-based and machine learning techniques. The features are extracted in the time domain (Chandler and Polonok, 2006; Maharaj, 2014), the frequency domain (Kakizawa et al., 1998; Maharaj, 2002; Shumway, 2003) and the wavelets domain (Maharaj, 2005; Maharaj and Alonso, 2007, 2014; Fryzlewicz and Omboa, 2012). Model-based approaches for time series classification include ARIMA models, Gaussian mixture models and Bayesian approaches (Maharaj, 1999, 2000; Sykacek and Roberts, 2002; Liu and Maharaj, 2013; Liu et al., 2014; Kotsifakos and Panagiotis, 2014), while machine learning approaches include classification trees, nearest neighbour methods and support vector machines (DouzalChouakria and Amblard, 2000; Do et al., 2017; Gudmundsson et al., 2008; Zhang et al., 2010).

It should be noted that clustering and classifying data evolving in time is substantially different from classifying static data. Hence, the volume of work on these topics focuses on extracting time series features or considering specific time series models and also understanding the risks of directly extending the common-use metric for static data to time series data.

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

We discuss three examples to illustrate time series clustering and classification before going into detail about these and other approaches in subsequent chapters. The first example illustrates clustering using time domain features, the second is observation-based and the third illustrates classification using wavelet features.

Example 1.1 D’Urso and Maharaj (2009) illustrate through simulated data, crisp clustering (traditional hierarchical and non-hierarchical) and fuzzy clustering of time series using the time domain features of autocorrelations. The aim here is to bring together series generated from the same process in order to understand the classification success. Fig. $1.1$ shows the autocorrelation functions (ACFs) over 10 lags for 12 simulated series, 4 of each generated from an AR(1) process with $\phi=0$ (a white noise process), an AR(1) process with $\phi=0.5$ and an MA(1) process with $\theta=0.9$. The patterns of the $\mathrm{ACFs}$ associated with each process are clearly distinguishable at the early lags. Table $1.1$ show a summary of results of clustering the 12 series, 4 from each process over 1000 simulations. The fuzzy c-means results are subject to specific choices of parameter values. It is clear from the results in Table $1.1$ that the autocorrelations provide good separation features.

## 机器学习代写|聚类分析作业代写clustering analysis代考|Structure of the book

After this chapter, time series concepts essential for what is to follow are discussed in Chapter 2. The rest of the book is divided into three parts. Part 1 consisting of Chapters 3 to 8 is on unsupervised approaches to classifying time series, namely, clustering techniques. Traditional cluster analysis and fuzzy clustering are discussed in Chapters 3 and 4, respectively, and this is followed by observation-based, feature-based, model-based clustering, and other time series clustering approaches in Chapters 5 to 8 .

Part 2 is on supervised classification approaches. This includes featurebased approaches in Chapter 9 and other time series classification approaches in Chapter 10. Throughout the book, many examples of simulated scenarios and real-world applications are provided, and these are mostly drawn from the research of the three authors. Part 3 provides links to software packages, some specific programming scripts used in these applications and simulated scenarios, as well as links to relevant data sets.

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