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

statistics-lab™ 为您的留学生涯保驾护航 在代写聚类分析clustering analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写聚类分析clustering analysis代写方面经验极为丰富，各种代写聚类分析clustering analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|partitioning clustering

In contrast to hierarchical clustering, which yields a successive level of clusters by iterative fusions or divisions, non-hierarchical or partitioning clustering assigns a set of data points into $c$ clusters without any hierarchical structure. This process usually accompanies the optimization of a criterion function, usually the minimization of a objective function representing the within variability of the clusters (Xu and Wunsch, 2009). One of the best-known and most popular non-hierarchical clustering methods is c-means clustering. Another interesting partitioning method is c-medoids clustering. In the following sections, we briefly present these methods and the cluster validity criteria for determining the optimal number of clusters that have to be pre-specified in these methods.

## 机器学习代写|聚类分析作业代写clustering analysis代考|c-Means clustering method

The c-means clustering method (MacQueen, 1967) which is also known as $\mathrm{k}$ means clustering is one of the best-known and most popular clustering methods. It is also commonly known as k-means clustering. The c-means clustering methods seeks an optimal partition of the data by minimizing the sumof-squared-error criterion shown in Eq. (3.1) with an iterative optimization procedure, which belongs to the category of hill-climbing algorithms (Xu and Wunsch, 2009). The basic clustering procedure of c-means clustering is summarized as follows (Everitt et al., 2011; Xu and Wunsch, 2009):

1. Initialize a c-partition randomly or based on some prior knowledge. Calculate the cluster prototypes (centroids or means) (that is, calculate the mean in each cluster considering only the observations belonging to each cluster).
2. Assign each unit in the data set to the nearest cluster by using a suitable distance measure between each pair of units and centroids.
3. Recalculate the cluster prototypes (centroids or means) based on the current partition.
4. Repeat steps 2 and 3 until there is no change for each cluster.

Mathematically, the c-means clustering method is formalized as follows:
$$\begin{array}{r} \min : \sum_{i=1}^{I} \sum_{c=1}^{C} u_{i c} d_{i c}^{2}=\sum_{i=1}^{I} \sum_{c=1}^{C} u_{i c}\left|\mathbf{x}{i}-\mathbf{h}{c}\right|^{2} \ \sum_{c=1}^{C} u_{i c}=1, u_{i c} \geq 0, u_{i c}={0,1} \end{array}$$
where $u_{i c}$ indicates the membership degree of the $i$-th unit to the $c$-th cluster; $u_{i c}={0,1}$, that is, $u_{i c}=1$ when the $i$-th unit belongs to the $c$-th cluster; $u_{i c}=0$ otherwise; $d_{i c}^{2}=\left|\mathbf{x}{i}-\mathbf{h}{c}\right|^{2}$ indicates the squared Euclidean distance between the $i$-th object and the centroid of the $c$-th cluster.

## 机器学习代写|聚类分析作业代写clustering analysis代考|c-Medoids clustering method

By considering the c-medoids clustering method or partitioning around medoids (PAM) method (Kaufman and Rousseeuw, 1987, 1990), units are classified into clusters represented by one of the data points in the cluster (this method is also often referred to as k-medoids). These data points are the prototypes, the so-called medoids. Each medoid synthesizes the cluster information and represents the prototypal features of the clusters and then synthesizes the characteristics of the units belonging to each cluster. Following the c-medoids clustering method, we minimize the objective function represented by the sum (or mathematically equivalent, average) of the dissimilarity of units to their closest representative units. The c-medoids clustering method first computes a set of representative units, the medoids. After finding the set of medoids, each unit of the data set is assigned to the nearest medoid units. The algorithm suggested by Kaufman and Rousseeuw (1990) for the c-medoids clustering method proceeds in two phases:

Phase $1(B U I L D)$ : This phase sequentially selects $c$ “centrally located” units to be used as initial medoids.

Phase $2(S W A P)$ : If the objective function can be reduced by interchanging (swapping) a selected unit with an unselected unit, then the swap is carried out. This is continued until the objective function can no longer be decreased. Then, by considering a set of $I$ units by X (set of the observations) and a subset of $\mathbf{X}$ with $C$ units by $\tilde{\mathbf{X}}$ (set of the medoids) (where $C<<I$ ), we could formalize the model as follows:
$$\begin{array}{r} \min : \sum_{i=1}^{I} \sum_{c=1}^{C} u_{i c} d_{i c}^{2}=\sum_{i=1}^{I} \sum_{c=1}^{C} u_{i c}\left|\mathbf{x}{i}-\tilde{\mathbf{x}}{c}\right|^{2} \ \sum_{c=1}^{C} u_{i c}=1, u_{i c} \geq 0, u_{i c}={0,1} \end{array}$$ where $u_{i c}$ indicates the membership degree of the $i$-th unit to the $c$-th cluster; $u_{i c}={0,1}$, that is, $u_{i c}=1$ when the $i$-th unit belongs to the $c$-th cluster; $u_{i c}=0$ otherwise; $d_{i c}^{2}=\left|\mathbf{x}{i}-\tilde{\mathbf{x}}{c}\right|^{2}$ indicates the squared Euclidean distance between the $i$-th object and the medoid of the $c$-th cluster.

## 机器学习代写|聚类分析作业代写clustering analysis代考|c-Means clustering method

c-means 聚类方法 (MacQueen, 1967)，也称为ķ意味着聚类是最著名和最流行的聚类方法之一。它也通常称为 k-means 聚类。c-means 聚类方法通过最小化公式中所示的平方和误差标准来寻求数据的最佳划分。（3.1）具有迭代优化过程，属于爬山算法的范畴（Xu and Wunsch，2009）。c-means 聚类的基本聚类过程总结如下（Everitt et al., 2011; Xu and Wunsch, 2009）：

1. 随机或基于一些先验知识初始化一个 c 分区。计算集群原型（质心或均值）（即，仅考虑属于每个集群的观测值来计算每个集群中的平均值）。
2. 通过在每对单位和质心之间使用合适的距离度量，将数据集中的每个单位分配给最近的集群。
3. 根据当前分区重新计算集群原型（质心或均值）。
4. 重复步骤 2 和 3，直到每个集群都没有变化。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写聚类分析clustering analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写聚类分析clustering analysis代写方面经验极为丰富，各种代写聚类分析clustering analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Traditional cluster analysis

In many cases, traditional cluster analysis, that is, hierarchical clustering, is used for clustering time series. In this case, first a suitable distance measure inheriting the dynamic features of the time series is defined for comparing time series and, successively, a standard hierarchical (e.g., agglomerative) cluster analysis is applied using the defined distance. For this reason, in this chapter, we briefly describe the traditional clustering methods.

The aim of cluster analysis is to assign units (objects) to clusters so that units within each cluster are similar to one another with respect to observed variables, and the clusters themselves stand apart from one another. In other words, the goal is to divide the units into homogeneous and distinct (well separated) clusters. Generally clustering methods are classified as hierarchical clustering and non-hierarchical clustering (or partitional clustering) methods, based on the properties of the generated clusters (Everitt et al., 2011; Xu and Wunsch, 2009). Hierarchical clustering (see Section 3.3) groups data with a sequence of nested partitions, either from singleton clusters to a cluster including all individuals or vice versa. The former is known as agglomerative clustering, and the latter is called divisive clustering. Both agglomerative and divisive clustering methods organize data into the hierarchical structure based on suitable proximity measures (that is, distance measures (see Section 3.2), dissimilarity measures, similarity indices). In Section 3.3, we focus our attention only on the agglomerative approach. Non-hierarchical clustering (see Section 3.4) directly divides data points into some pre-specified number of clusters without the hierarchical structure. For more details, see Everitt et al. (2011) and Xu and Wunsch (2009).

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

Let $\mathbf{X}=\left{x_{i j}: 1, \ldots, I ; j=1, \ldots J\right}=\left{\mathbf{x}{i}=\left(x{i 1}, \ldots, x_{i j}, \ldots x_{i J}\right)^{\prime}: i=\right.$ $1, \ldots, I}$ be the data matrix where $x_{i j}$ represents the $j$-th variable observed on the $i$-th object and $\mathbf{x}{i}$ represents the vector of the $i$-th observation. The most common class of distance measure used in cluster analysis is the distance class of Minkowski (Everitt et al., 2011): $${ }{r} d_{i l}=\left[\sum_{j=1}^{J}\left|x_{i j}-x_{l j}\right|^{r}\right]^{\frac{1}{r}}, \quad r \geq 1 .$$
For $r=1$, we have the city-block distance (or Manhattan distance):
$${ }{1} d{i l}=\sum_{j=1}^{J}\left|x_{i j}-x_{l j}\right|$$
and for $r=2$, we have the Euclidean distance, probably the most commonly used distance measure in cluster analysis:
$${ }{2} d{i l}=\left[\sum_{j=1}^{J}\left(x_{i j}-x_{l j}\right)^{2}\right]^{\frac{1}{2}} .$$
An interesting weighted version of the previous distance class of Minkowski is (Everitt et al., 2011):
$${ }{r} \tilde{d}{i l}=\left[\sum_{j=1}^{J} w_{j}^{r}\left|x_{i j}-x_{l j}\right|^{r}\right]^{\frac{1}{r}}, \quad r \geq 1$$
and then,
${ }{1} \tilde{d}{i l}=\sum_{j=1}^{J} w_{j}^{1}\left|x_{j l}-x_{l j}\right| \quad(r=1) \quad$ (weighted city-block distance)
${ }{2} \tilde{d}{i l}=\left[\sum_{j=1}^{J} w_{j}^{2}\left(x_{j l}-x_{l j}\right)^{2}\right]^{\frac{1}{2}} \quad(r=2) \quad$ (weighted Euclidean distance)
where $w_{j}(j=1, \ldots, J)$ represents a suitable weight for $j$-th variable.

For using the distance measures in the clustering techniques (that is, in hierarchical clustering) it is useful to collect all the distances for each pair of units in a (squared) matrix form; e.g., the Minkowski distance matrix can be represented as follows:
$${ }{r} \mathbf{D}=\left{{ }{r} d_{i l}=\left[\sum_{j=1}^{J}\left|x_{i j}-x_{l j}\right|^{r}\right]^{\frac{1}{r}}: i, l=1, \ldots, I\right}, r \geq 1 .$$
See Everitt et al. (2011) for more details on the distance measures and their use in cluster analysis.

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

In this section, we focus our attention only on agglomerative methods which are probably the most widely used of the hierarchical methods. They produce a series of partitions of the data: the first consists of $I$ single-member clusters; the last consists of a single cluster containing all $I$ units (Everitt et al., 2011). Agglomerative clustering starts with $I$ clusters, each of which includes exactly one data point. A series of merge operations is then followed that eventually forces all objects into the same group. The general agglomerative clustering can be summarized by the following procedure (Xu and Wunsch, 2009):

1. Start with $I$ singleton clusters. Calculate the proximity matrix, e.g. distance matrix, for the $I$ clusters;
2. In the distance matrix, search the minimal distance $d\left(C_{c}, C_{c^{\prime}}\right)=$ $\min {1 \leq p, q \leq I} d\left(C{p}, C_{q}\right)$, where $d(\cdot, \cdot)$ is the distance function discussed later $p \neq q$
in the following, and combine cluster $C_{c}$ and $C_{c^{\prime}}$ to form a new cluster $C_{c c^{\prime}} ;$
3. Update the distance matrix by computing the distances between the cluster $C_{c c^{\prime}}$ and the other clusters;
4. Repeat steps 2 and 3 until only one cluster remains.
The merging of a pair of clusters or the formation of a new cluster is dependent on the definition of the distance function between two clusters. There exist a large number of distance function definitions between a cluster $C_{q}$ and a new cluster $C_{c c}{ }^{\prime}$ formed by the merge of two clusters $C_{c}$ and $C_{c^{\prime}}$. In the following we show briefly some methods for defining distance functions:
• Single linkage method (nearest neighbor method): the distance between a pair of clusters is determined by the two closest units to the different clusters. Single linkage clustering tends to generate elongated clusters, which causes the chaining effect (Everitt et al., 2011). As a result, two clusters with quite different properties may be connected due to the existence of

noise. However, if the clusters are separated far from each other, the single linkage method works well.

• Complete linkage method: in contrast to single linkage clustering, the complete linkage method uses the farthest distance of a pair of objects to define inter-cluster distance.
• Group average linkage method (unweighted pair group method average, that is, UPGMA): the distance between two clusters is defined as the average of the distances between all pairs of data points, each of which comes from a different cluster.
• Weighted average linkage method (weighted pair group method average, that is, WPGMA): similar to UPGMA, the average linkage is also used to calculate the distance between two clusters. The difference is that the distances between the newly formed cluster and the rest are weighted based on the number of data points in each cluster.
• Centroid linkage method (unweighted pair group method centroid, that is, UPGMC): two clusters are merged based on the distance of their centroids (means).
• Ward’s method (minimum variance method): the aim of Ward’s method is to minimize the increase of the so-called within-class sum of the squared errors.

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

1d一世l=∑j=1Ĵ|X一世j−Xlj|

2d一世l=[∑j=1Ĵ(X一世j−Xlj)2]12.
Minkowski 之前的距离类的一个有趣的加权版本是（Everitt 等人，2011）：
rd~一世l=[∑j=1Ĵ在jr|X一世j−Xlj|r]1r,r≥1

1d~一世l=∑j=1Ĵ在j1|Xjl−Xlj|(r=1)（加权城市街区距离）
2d~一世l=[∑j=1Ĵ在j2(Xjl−Xlj)2]12(r=2)（加权欧几里得距离
）在j(j=1,…,Ĵ)代表一个合适的重量j-th 变量。

{ }{r} \mathbf{D}=\left{{ }{r} d_{i l}=\left[\sum_{j=1}^{J}\left|x_{i j}-x_{l j} \right|^{r}\right]^{\​​frac{1}{r}}: i, l=1, \ldots, I\right}, r \geq 1 。{ }{r} \mathbf{D}=\left{{ }{r} d_{i l}=\left[\sum_{j=1}^{J}\left|x_{i j}-x_{l j} \right|^{r}\right]^{\​​frac{1}{r}}: i, l=1, \ldots, I\right}, r \geq 1 。

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

1. 从…开始一世单例集群。计算邻近矩阵，例如距离矩阵，用于一世集群；
2. 在距离矩阵中，搜索最小距离d(CC,CC′)= 分钟1≤p,q≤一世d(Cp,Cq)， 在哪里d(⋅,⋅)是后面讨论的距离函数p≠q
在下面，并结合集群CC和CC′形成一个新的集群CCC′;
3. 通过计算集群之间的距离来更新距离矩阵CCC′和其他集群；
4. 重复步骤 2 和 3，直到只剩下一个簇。
一对簇的合并或新簇的形成取决于两个簇之间距离函数的定义。一个簇之间存在大量的距离函数定义Cq和一个新的集群CCC′由两个集群合并而成CC和CC′. 下面我们简要介绍一些定义距离函数的方法：
• 单联动法（最近邻法）：一对簇之间的距离由离不同簇最近的两个单元决定。单链接聚类往往会产生拉长的聚类，从而导致链接效应（Everitt et al., 2011）。结果，由于存在

• 完全联动法：与单联动聚类相比，完全联动法使用一对对象的最远距离来定义簇间距离。
• 组平均联动法（unweighted pair group method average，即UPGMA）：两个簇之间的距离定义为所有数据点对之间距离的平均值，每对数据点来自不同的簇。
• 加权平均联动法（weighted pair group method average，即WPGMA）：与UPGMA类似，平均联动也用于计算两个簇之间的距离。不同之处在于，新形成的集群与其余集群之间的距离是根据每个集群中数据点的数量加权的。
• 质心联动法（unweighted pair group method centroid，即UPGMC）：将两个簇根据质心的距离（均值）进行合并。
• Ward 方法（最小方差法）：Ward 方法的目的是使所谓的类内误差平方和的增加最小化。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写聚类分析clustering analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写聚类分析clustering analysis代写方面经验极为丰富，各种代写聚类分析clustering analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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.

X吨=∑j=1Ĵdj+sĴ,

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写聚类分析clustering analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写聚类分析clustering analysis代写方面经验极为丰富，各种代写聚类分析clustering analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Spectral representation of time series

Associated with every stationary stochastic process $\left{X_{t}\right}$ is the spectral density function which is a tool for considering the frequency properties of a stationary time series. The spectral density function, also referred to as the power spectral density function or the spectrum, is the derivative of the spectral distribution function $F(\omega)$, where $\omega$ is the frequency, which is defined as the number of radians per unit time. This is a continuous function that is monotone and bounded in the interval $[0, \pi]$. This derivative is denoted by $f(\omega)$, so that
$$f(\omega)=\frac{d F(\omega)}{d \omega} .$$
When $f(\omega)$ exists, Eq. $2.2$ can be expressed as
$$\gamma_{k}=\int_{0}^{\pi} \cos \omega k f(\omega) d \omega$$
When $k=0$, Eq. $2.3$ becomes
$$\gamma_{0}=\sigma_{X}^{2}=\int_{0}^{\pi} f(\omega) d \omega=F(\pi)$$

The interpretation of the spectrum is that $f(\omega) d \omega$ represents the contribution to variance of components of frequencies in the range $(\omega, \omega+d \omega)$. Eq. 2.4 indicates that the total area under the curve of the spectrum is equal to the variance of the process. A peak in the spectrum indicates an important contribution of variance at frequencies near the values that correspond to the peak.

It should be noted that the autocovariance function and the spectral density function are equivalent ways of describing a stationary stochastic process.
From Eq. 2.3, the corresponding inverse can be obtained, namely
$$f(\omega)=\frac{1}{\pi} \sum_{k=-\infty}^{\infty} \gamma_{k} e^{-i \omega k} .$$
This implies that the spectral density function is the Fourier transform of the autocovariance function. Refer to Chatfield (2004) for details on the Fourier transform. Since $\gamma_{k}$ is an even function of $k$, Eq. $2.5$ can be expressed as
$$f(\omega)=\frac{1}{\pi}\left[\gamma_{0}+2 \sum_{k=1}^{\infty} \gamma_{k} \cos \omega k\right] .$$
The normalized form of the spectral density function is given by
$$f^{}(\omega)=\frac{f(\omega)}{\sigma_{X_{t}}^{2}}=\frac{d F^{}(\omega)}{d \omega} .$$
This is the derivative of the normalized spectral distribution function. Hence, $f^{}(\omega)$ is the Fourier transform of the autocorrelation function, namely, $$f^{}(\omega)=\frac{1}{\pi}\left[1+2 \sum_{k=1}^{\infty} \rho_{k} \cos \omega k\right] .$$
This implies that $f^{*}(\omega) d \omega$ is the proportion of variance in the interval $(\omega, \omega+$ $d \omega$.

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

An estimator of the spectral density function is the periodogram $I(\omega)$ where at an ordinate $p$ it is expressed as:
$$I\left(\omega_{p}\right)=\frac{1}{\pi}\left(c_{0}+2 \sum_{k=1}^{T-1} c_{k} \cos \left(\omega_{p} k\right)\right),$$
where $c_{k}$ is the sample autocovariance coefficient at lag $k, T$ is the length of the observed time series, and $p=1,2, \ldots,(T / 2)-1$. The periodogram is asymptotically unbiased, that is,

$$\lim _{T \rightarrow \infty} E[I(\omega)]=f(\omega)$$
However, $I(\omega)$ is not a consistent estimator of $f(\omega)$. It can be shown that neighbouring periodogram ordinates are asymptotically independent. Refer to Chatfield (2004) for more details on periodogram analysis including the distribution associated with the periodogram ordinates.

Example 2.2 Consider the observed time series of weekly sales of a consumer product from Example 2.1. Fig. 2.4 shows the periodogram of this series. The peak occurs around a normalized frequency between 0 and $0.05$, indicating that most of the largest contribution to the variance of the series is within this frequency range. The frequency range around which a peak occurs gives an indication of the frequency at which that cyclic component may exist.
Note that just as the spectral density function can be normalized, the periodogram can be normalized. In particular, the normalized periodogram is
$$I^{*}(\omega)=\frac{I(\omega)}{\operatorname{Var}\left(x_{t}\right)}$$

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

While the periodogram is useful in assessing whether there are one or more strong cyclic components in a time series, the sampling error associated with its ordinates is quite large and confidence intervals set up around the ordinates would therefore be very wide. Therefore, the periodogram is not a very good estimator of the spectral density function especially when the signal to noise ratio of time series is low, that is, when the time series is very noisy. The periodogram can be smoothed to overcome these problems and there is a vast literature on windows that are used to smooth the periodogram. Refer to Chatfield (2004) for details about smoothing the periodogram.

Fig. $2.5$ shows a smoothed periodogram, from which it is clear that the fluctuations in the periodogram have been smoothed out to better assess which frequency bands account for most of the variance in the time series.

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

F(ω)=dF(ω)dω.

Cķ=∫0圆周率因⁡ωķF(ω)dω

C0=σX2=∫0圆周率F(ω)dω=F(圆周率)

F(ω)=1圆周率∑ķ=−∞∞Cķ和−一世ωķ.

F(ω)=1圆周率[C0+2∑ķ=1∞Cķ因⁡ωķ].

F(ω)=F(ω)σX吨2=dF(ω)dω.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写聚类分析clustering analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写聚类分析clustering analysis代写方面经验极为丰富，各种代写聚类分析clustering analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Stationary models

An autoregressive (AR) model is one for which the current value of the deviation of the process from the mean is expressed as a finite, linear combination of previous values of the process and a shock or error term. Given a stochastic process $\left{X_{t}\right}$, the AR model is expressed as:
$$\phi(B) Z_{t}=\varepsilon_{t}$$
where $Z_{t}=X_{t}-\mu$, is the deviation from the mean, $\varepsilon_{t}$ is a white noise or random process with mean 0 and variance $\sigma_{t}^{2}$,
$$\phi(B)=1-\phi_{1} B-\phi_{2} B^{2}-\ldots-\phi_{p} B^{p}$$
is the autoregressive operator, $p$ is the order of the AR model and $B$ is the backshift operator. In particular, we refer to it as an $\operatorname{AR}(p)$ model. This model is stationary and a necessary requirement for stationarity is that all roots of $\Phi(B)=0$ must lie outside the unit circle.

A moving average (MA) model is one where the current value of the deviation of the process from the mean is expressed as a linear combination of a finite number of previous error terms. The MA model is expressed as:
$$Z_{t}=\theta(B) \varepsilon_{t}$$
where
$$\theta(B)=1-\theta_{1} B-\theta_{2} B^{2}-\ldots-\theta_{q} B^{q},$$
and $q$ is the order of the MA model. In particular, we refer to it as an MA $(q)$ model. This model is also stationary with a similar stationarity condition as that of the AR model applying. While the AR and MA are useful representations of observed time series, it is sometimes useful to include both AR and MA terms in a model, resulting in an autoregressive, moving average (ARMA) model or an ARMA $(p, q)$ model which is expressed as:
$$\phi(B) Z_{t}=\theta(B) \varepsilon_{t}$$
In order for the $\operatorname{AR}(p), \operatorname{MA}(q)$ and $\operatorname{ARMA}(p, q)$ models to be fitted to an observed time series, it is assumed that the series is stationary, that is, it fluctuates about a fixed mean and its variance is constant.

In order to identify a suitable model that may be fitted to an observed stationary time series, we examine the ACF and PACF of this series to determine if it to some extent emulates the theoretical ACF and PACF associated with the model. For an AR(1) model, the ACF shows exponential decay while the PACF is zero beyond lag 1 . Hence, we can infer that an AR(1) model would be

an appropriate fit to an observed time series, when the ACF decays exponentially and when the PACF has a single significant spike at lag 1 . Given that this is the behaviour displayed by the ACF and PACF of the observed series in Example 2.1, we could infer that an AR(1) model is possibly an appropriate model to fit to this series. In general, for an $\operatorname{AR}(p)$ model, with $p \geq 2$, the ACF can show exponential decay or a damped sin wave pattern, whereas the $\mathrm{PACF}$ is zero beyond lag $q$.

For an MA(1) model, the PACF shows exponential decay while the $\mathrm{ACF}$ is zero beyond lag 1 . Hence, if the ACF of an observed stationary time series has a single significant spike at lag 1 , and the PACF decays exponentially, we can infer that an MA(1) model would be an appropriate fit to this series. In general, for an MA $(q)$ model, with $q \geq 2$, the ACF is zero beyond lag $p$, while the PACF can show exponential decay or a damped sin wave pattern. Refer to books such as Chatfield (2004), Makridakis et al. (1998) and Ord and Fildes (2013) for more details about the theoretical behaviour of ARMA models in general.

## 机器学习代写|聚类分析作业代写clustering analysis代考|Non-stationary models

Many time series encountered in various fields exhibit non-stationary behaviour and in particular they do not fluctuate about a fixed level. Although the level about which the series fluctuates may be different at different times, when differences in levels are taken, they may be similar. This is referred to as homogeneous non-stationary behaviour (Box et al., 1994 ) and the series can be represented by a model that requires the $d$-th difference of the process to be stationary. In practice, $d$ is usually no more than 2 . Hence, an ARMA model can be extended to what is known as an autoregressive, integrated moving average (ARIMA) model, or $\operatorname{ARIMA}(p, d, q)$ to represent a homogeneous non-stationary time series. This model is expressed as
$$\phi(B)(1-B)^{d} Z_{t}=\theta(B) \varepsilon_{t} .$$
In practice, time series may also have a seasonal component. Just as the consecutive data points of an observed time series may exhibit AR, MA or ARMA properties, so data separated by a whole season (for example, a year or a quarter) may exhibit similar properties. The ARIMA notation can be extended to incorporate seasonal aspects and in general we have an $\operatorname{ARIMA}(p, d, q)(P, D, Q){s}$ model which can be expressed as $$\phi(B) \Phi(B)(1-B)^{d}\left(1-B^{s}\right)^{D} Z{t}=\theta(B) \Theta(B) \varepsilon_{t}$$
where
\begin{aligned} &\Phi(B)=1-\Phi_{1} B^{s}-\Phi_{2} B^{2 s}-\ldots-\Phi_{P} B^{P_{s}} \ &\Theta(B)=1-\Theta_{1} B^{s}-\Theta_{2} B^{2 s}-\ldots-\Theta_{Q} B^{Q_{s}} \end{aligned} $D$ is the degree of seasonal differencing and $s$ is the number of periods per season. For example, $s=12$ for monthly time series and $s=4$ for quarterly time series. Refer to books such as Makridakis et al. (1994) and Ord and Fildes (2013) for more details about fitting non-stationary models that may be seasonal or not.

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

All these models discussed thus far are linear and are applicable to univariate time series. A popular extension to fitting models to stationary multivariate time series are vector autoregressive moving average models (VARMA). One of the large number of books in which details of these models can be found is Lutkepohl (1991). There are also several classes of non-linear models. A particular class is one that is concerned with modeling changes in variance or the volatility of a time series. These include autoregressive conditionally heteroscedastic (ARCH) and generalized autoregressive conditionally heteroscedastic (GARCH) models. One of the large number of books in which details of these models can be found is Tsay $(2010)$.

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

φ(乙)从吨=e吨

φ(乙)=1−φ1乙−φ2乙2−…−φp乙p

θ(乙)=1−θ1乙−θ2乙2−…−θq乙q,

φ(乙)从吨=θ(乙)e吨

## 机器学习代写|聚类分析作业代写clustering analysis代考|Non-stationary models

φ(乙)(1−乙)d从吨=θ(乙)e吨.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 机器学习代写|聚类分析作业代写clustering analysis代考|Time series features and models

statistics-lab™ 为您的留学生涯保驾护航 在代写聚类分析clustering analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写聚类分析clustering analysis代写方面经验极为丰富，各种代写聚类分析clustering analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Time series features and models

The topic of time series analysis is the subject of a large number of books and journal articles. In this chapter, we highlight fundamental time series concepts, as well as features and models that are relevant to the clustering and classification of time series in subsequent chapters. Much of this material on time series analysis, in much greater detail, is available in books by authors such as Box et al. (1994), Chatfield (2004), Shumway and Stoffer (2016), Percival and Walden (2016) and Ord and Fildes (2013).

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

A stochastic process is defined as a collection of random variables that are ordered in time and defined as a set of points which may be discrete or continuous. We denote the random variable at time $t$ by $X(t)$ if time is continuous or by $X_{t}$ if time is discrete. A continuous stochastic process is described as ${X(t),-\infty<t<\infty}$ while a discrete stochastic process is described as $\left{X_{t}, t=\ldots,-2,-1,0,1,2, \ldots\right}$.

Most statistical problems are concerned with estimating the properties of a population from a sample. The properties of the sample are typically determined by the researcher, including the sample size and whether randomness is incorporated into the selection process. In time series analysis there is a different situation in that the order of observations is determined by time. Although it may be possible to increase the sample size by varying the length of the observed time series, there will be a single outcome of the process and a single observation on the random variable at time $t$. Nevertheless, we may regard the observed time series as just one example of an infinite set of time series that might be observed. The infinite set of time series is called an ensemble. Every member of the ensemble is a possible realization of the stochastic process. The observed time series can be thought of as one possible realization of the stochastic process and is denoted by ${x(t),-\infty<t<\infty}$ if time is continuous or $\left{x_{t}, t=0,1,2, . . T\right}$ if time is discrete. Time series analysis is essentially concerned with evaluating the properties of the underlying probability model from this observed time series. In what follows, we will be working with mainly discrete time series which are realizations of discrete stochastic processes.

Many models for stochastic processes are expressed by means of algebraic expressions relating the random variable at time $t$ to past values of the process, together with values of an unobservable error process. From one such model we may be able to specify the joint distribution of $X_{t_{1}}, X_{t_{2}}, \ldots, X_{t_{k}}$, for any set of times $t_{1}, t_{2}, \ldots, t_{k}$ and any value of $k$. A simple way to describe a stochastic process is to examine the moments of the process, particularly the first and second moments, namely, the mean and autocovariance function.
$$\begin{gathered} \mu_{t}=E\left(X_{t}\right) \ \gamma_{t_{1}, t_{2}}=E\left[\left(X_{t_{1}}-\mu_{t}\right)\left(X_{t_{2}}-\mu_{t}\right)\right] \end{gathered}$$
The variance is a special case of the autocovariance function when $t_{1}=t_{2}$, that is,
$$\sigma_{t}^{2}=E\left[\left(X_{t}-\mu_{t}\right)^{2}\right] .$$
An important class of stochastic processes is that which is stationary. A time series is said to be stationary if the joint distribution of $X_{t_{1}}, X_{t_{2}}, \ldots, X_{t_{k}}$

is the same as that of $X_{t_{1}+\tau}, X_{t_{2}+\tau}, \ldots, X_{t_{k}+\tau}$, for all $t_{1}, t_{2}, \ldots, t_{k}, \tau$. In other words, shifting the time origin by the amount $\tau$ has no effect on the joint distribution which must therefore depend only on the intervals between $t_{1}, t_{2}, \ldots, t_{k}$.
This definition holds for any value of $k$. In particular, if $k=1$, strict stationarity implies that the distribution of $X_{t}$ is the same for all $t$, provided the first two moments are finite and are both constant, that is, $\mu_{t}=\mu$ and $\sigma_{t}^{2}=\sigma^{2}$. If $k=2$, the joint distribution of $X_{t_{1}}$ and $X_{t_{2}}$ depends only on the time difference $t_{1}-t_{2}=\tau$ which is called a lag. Thus the autocovariance function which depends only on $t_{1}-t_{2}$ may be written as
$$\gamma_{\tau}=\operatorname{COV}\left(X_{t}, X_{t+\tau}\right)=E\left[\left(X_{t}-\mu\right)\left(X_{t+\tau}-\mu\right)\right]$$

## 机器学习代写|聚类分析作业代写clustering analysis代考|Autocorrelation and partial autocorrelation functions

Autocovariance and autocorrelation measure the linear relationship between various values of an observed time series that are lagged $k$ periods apart, that is, given an observed time series $\left{x_{t}, t=0,1,2, . T\right}$, we measure the relationship between $x_{t}$ and $x_{t-1}, x_{t}$ and $x_{t-2}, x_{t}$ and $x_{t-3}$, etc.. Thus, the autocorrelation function is an important tool for assessing the degree of dependence in observed time series. It is useful in determining whether or not a time series is stationary. It can suggest possible models that can be fitted to the observed time series and it can detect repeated patterns in a time series such as the presence of a periodic signal which has been buried by noise. The sample autocorrelation function (ACF), $r_{k}, k=1,2, \ldots$ is typically plotted for at least a quarter of the number of lags or thereabouts. The plot is supplemented with $5 \%$ significance limits to enable a graphical check of whether of not dependence is statistically significant at a particular lag.

Partial autocorrelations are used to measure the relationship between $x_{t}$ and $x_{t-k}$, with the effect of the other time lags, $1,2, \ldots, k$-1 removed. It is also useful to plot the partial autocorrelation function (PACF) because it, together with the plot of the ACF, can help inform one on a possible appropriate model that can be fitted to the time series. Refer to any of the references mentioned in Section $2.1$ for more details on the ACF and PACF including their sampling distributions which enable the determination of the significance limits.

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

μ吨=和(X吨) C吨1,吨2=和[(X吨1−μ吨)(X吨2−μ吨)]

σ吨2=和[(X吨−μ吨)2].

Cτ=冠状病毒⁡(X吨,X吨+τ)=和[(X吨−μ)(X吨+τ−μ)]

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写聚类分析clustering analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写聚类分析clustering analysis代写方面经验极为丰富，各种代写聚类分析clustering analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。