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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

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。

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