Matlab Statistical Analysis--Clustering Algorithm Model

Statistical Analysis – Clustering Algorithm Models

distance analysis

data standardization

The Euclidean distance is related to the dimension, so sometimes it is necessary to preprocess the data,
such as standardization.
The command in MATLAB is zscore, the calling format

Z = zscore(X)
输入X表示N行p列的原始观测矩阵，行为个体，列为指标。

输出Z为X的标准化矩阵:
Z = (X–ones(N,1)*mean(X)) ./(ones(N,1)* std(X))，

mean(X)为行向量，表示各个指标的均值估计，
std（X）表示指标的标准差估计。./表示对应元素相除，
ones(N,1)表示元素全为1的行向量，向量的长度为N。

K-means clustering

The algorithm flow of K-means clustering:

1. Specify the K value of the number of clusters to be divided (the number of classes)
2. Randomly select K data objects as initial cluster centers (not necessarily our sample points)
3. Calculate the distance from each of the remaining data objects to the K initial cluster centers, and classify the data objects into the cluster where the center closest to it is located;
4. Adjust the new class and recalculate the center of the new class;
5. Loop steps 3 and 4 to see if the center converges (unchanged). If it converges or reaches the number of iterations, stop the loop;
6. Finish.

K-means clustering features

K-means++ clustering algorithm

SPSS software use

code

%% 
% K-means 算法MATLAB实现
%-------------------------------------------------------------
%{
    
    
利用Matlab软件中的命令: kmeans,可以实现k-means聚类
对于要处理的数据 构造矩阵，矩阵X的每一行为每个个体的实际数据，每一列都是不同的指标
如果提供的数据不是按照规范模式，需要进行矩阵转置
x=y';     %矩阵x的行为个体，列为指标  
[a,b]=kmeans(x,2)　　%分为2类，输出:  a为聚类的结果，b为聚类重心, 每一行表示一个类的重心
使用kmeans进行处理 
%}
%% 数据准备和初始化
clc
clear
load kdata.mat


[a,b]=kmeans(x,3);  %%分为3类输出
x1=x(find(a==1),:)   %提取第1类里的样品
x2=x(find(a==2),:)   %提取第2类里的样品
x3=x(find(a==3),:)   %提取第3类里的样品
sd1=std(x1)
sd2=std(x2)　 
sd3=std(x3)　 % 分别计算第1类和第2类第3类的标准差
plot(x(a==1,1),x(a==1,2),'r.',x(a==2,1),x(a==2,2),'b.',x(a==3,1),x(a==3,2),'g.','MarkerSize',10)  %作出聚类的散点图
title('k-means聚类分析散点图');

Cluster Analysis – Pedigree Analysis

To study the MATLAB implementation of clustering, the implementation steps are roughly as follows:

1. Enter the data matrix, pay attention to the actual meaning of rows and columns;
2. Calculate the distance between samples (row? column?)
   Euclidean distance: d=pdist(A) % Pay attention to calculate the distance between each row in A;
   absolute distance: d= pdist(A,'cityblock');
   Mingshi Distance: d=pdist(A,'minkowski',r); % r must be filled with a specific real number;
   variance weighted distance: d= pdist(A,'seuclid');
   Mahalanobis distance: d= pdist(A,' mahal');
   Note: The output of the above command is a row vector
3. Choose different inter-class distances for clustering
4. Make a pedigree cluster map
5. According to the number of categories, output the clustering results

Try to use the survey data to carry out cluster analysis on 16 regions.

The following table is the summary data of the sample survey of farmers' expenditure in 16 regions in my country in 1982. Each region has investigated six indicators reflecting the average living consumption expenditure per capita.

Pedigree Cluster Diagram

a=load('ho2.txt');%导入数据

d1=pdist(a);% 此时计算出各行之间的欧氏距离，

z1=linkage(d1);

z2=linkage(d1,'complete');

z3=linkage(d1,'average');

z4=linkage(d1,'centroid');

z5=linkage(d1,'ward');

R=[cophenet(z1,d1),cophenet(z2,d1),cophenet(z3,d1),cophenet(z4,d1),cophenet(z5,d1)]

H= dendrogram(z3)

T=cluster(z3,4)  %cluster 创建聚类，并作出谱系图

set(get(gca, 'Title'), 'String', '聚类分析-谱系聚类图');

k-means cluster analysis scatter plot

[a,b]=kmeans(x,4);  %%分为4类输出
x1=x(find(a==1),:)   %提取第1类里的样品
x2=x(find(a==2),:)   %提取第2类里的样品
x3=x(find(a==3),:)   %提取第3类里的样品
x4=x(find(a==4),:)   %提取第3类里的样品
sd1=std(x1)
sd2=std(x2)　 
sd3=std(x3)　 % 分别计算第1类和第2类第3类的标准差
sd4=std(x4)　 % 分别计算第1类和第2类第3类的标准差
plot(x(a==1,1),x(a==1,2),'r.',x(a==2,1),x(a==2,2),'b.',x(a==3,1),x(a==3,2),'g.',x(a==4,1),x(a==4,2),'y.','MarkerSize',15)  %作出聚类的散点图
title('k-means聚类分析散点图');

linkage函数
调用格式：Z=linkage(Y,‘method’)
输入值说明：Y为pdist函数返回的M*(M-1)/2个元素的行向量，用‘method’参数指定的算法计算系统聚类树。
method：可取值如下：
‘single’：最短距离法（默认）；
‘complete’：最长距离法；
‘average’：未加权平均距离法；
‘weighted’： 加权平均法；
‘centroid’：质心距离法；
‘median’：加权质心距离法；
‘ward’：内平方距离法（最小方差算法）
返回值说明：Z为一个包含聚类树信息的（m-1）×3的矩阵，其中前两列为索引标识，表示哪两个序号的样本可以聚为同一类，第三列为这两个样本之间的距离。另外，除了M个样本以外，对于每次新产生的类，依次用M+1、M+2、…来标识