大数据挖掘算法篇之K-Means实例
2013-12-19 12:00
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一、引言
K-Means算法是聚类算法中，应用最为广泛的一种。本文基于欧几里得距离公式：d = sqrt((x1-x2)^+(y1-y2)^)计算二维向量间的距离，作为聚类划分的依据，输入数据为二维数据两列数据，输出结果为聚类中心和元素划分结果。输入数据格式如下：
二、欧几里得距离：
欧几里得距离定义： 欧几里得距离（ Euclidean distance）也称欧氏距离，在n维空间内，最短的线的长度即为其欧氏距离。它是一个通常采用的距离定义，它是在m维空间中两个点之间的真实距离。
在二维和三维空间中的欧式距离的就是两点之间的距离，二维的公式是
d = sqrt((x1-x2)^+(y1-y2)^)
三维的公式是
d=sqrt((x1-x2)^+(y1-y2)^+(z1-z2)^)
推广到n维空间，欧式距离的公式是
d=sqrt( ∑(xi1-xi2)^ ) 这里i=1,2..n
xi1表示第一个点的第i维坐标,xi2表示第二个点的第i维坐标
n维欧氏空间是一个点集,它的每个点可以表示为(x(1),x(2),...x(n)),其中x(i)(i=1,2...n)是实数,称为x的第i个坐标,两个点x和y=(y(1),y(2)...y(n))之间的距离d(x,y)定义为上面的公式.
欧氏距离看作信号的相似程度。 距离越近就越相似，就越容易相互干扰，误码率就越高。
三、代码示例
四、主调程序
五、输出结果
K-Means算法是聚类算法中，应用最为广泛的一种。本文基于欧几里得距离公式：d = sqrt((x1-x2)^+(y1-y2)^)计算二维向量间的距离，作为聚类划分的依据，输入数据为二维数据两列数据，输出结果为聚类中心和元素划分结果。输入数据格式如下：
1 18 2 2 3 2 4 0.0 0.0 5 1.0 0.0 6 0.0 1.0 7 2.0 1.0 8 1.0 2.0 9 2.0 2.0 10 2.0 0.0 11 0.0 2.0 12 7.0 6.0 13 7.0 7.0 14 7.0 8.0 15 8.0 6.0 16 8.0 7.0 17 8.0 8.0 18 8.0 9.0 19 9.0 7.0 20 9.0 8.0 21 9.0 9.0 22
二、欧几里得距离：
欧几里得距离定义： 欧几里得距离（ Euclidean distance）也称欧氏距离，在n维空间内，最短的线的长度即为其欧氏距离。它是一个通常采用的距离定义，它是在m维空间中两个点之间的真实距离。
在二维和三维空间中的欧式距离的就是两点之间的距离，二维的公式是
d = sqrt((x1-x2)^+(y1-y2)^)
三维的公式是
d=sqrt((x1-x2)^+(y1-y2)^+(z1-z2)^)
推广到n维空间，欧式距离的公式是
d=sqrt( ∑(xi1-xi2)^ ) 这里i=1,2..n
xi1表示第一个点的第i维坐标,xi2表示第二个点的第i维坐标
n维欧氏空间是一个点集,它的每个点可以表示为(x(1),x(2),...x(n)),其中x(i)(i=1,2...n)是实数,称为x的第i个坐标,两个点x和y=(y(1),y(2)...y(n))之间的距离d(x,y)定义为上面的公式.
欧氏距离看作信号的相似程度。 距离越近就越相似，就越容易相互干扰，误码率就越高。
三、代码示例
1 /****************************************************************************
2 * *
3 * KMEANS *
4 * *
5 *****************************************************************************/
6
7 #include <stdio.h>
8 #include <stdlib.h>
9 #include <string.h>
10 #include <conio.h>
11 #include <math.h>
12
13 // FUNCTION PROTOTYPES
14
15
16 // DEFINES
17 #define SUCCESS 1
18 #define FAILURE 0
19 #define TRUE 1
20 #define FALSE 0
21 #define MAXVECTDIM 20
22 #define MAXPATTERN 20
23 #define MAXCLUSTER 10
24
25
26
27
28
29 char *f2a(double x, int width){
30 char cbuf[255];
31 char *cp;
32 int i,k;
33 int d,s;
34 cp=fcvt(x,width,&d,&s);
35 if (s) {
36 strcpy(cbuf,"-");
37 }
38 else {
39 strcpy(cbuf," ");
40 } /* endif */
41 if (d>0) {
42 for (i=0; i<d; i++) {
43 cbuf[i+1]=cp[i];
44 } /* endfor */
45 cbuf[d+1]=0;
46 cp+=d;
47 strcat(cbuf,".");
48 strcat(cbuf,cp);
49 } else {
50 if (d==0) {
51 strcat(cbuf,".");
52 strcat(cbuf,cp);
53 }
54 else {
55 k=-d;
56 strcat(cbuf,".");
57 for (i=0; i<k; i++) {
58 strcat(cbuf,"0");
59 } /* endfor */
60 strcat(cbuf,cp);
61 } /* endif */
62 } /* endif */
63 cp=&cbuf[0];
64 return cp;
65 }
66
67
68
69
70 // ***** Defined structures & classes *****
71 struct aCluster {
72 double Center[MAXVECTDIM];
73 int Member[MAXPATTERN]; //Index of Vectors belonging to this cluster
74 int NumMembers;
75 };
76
77 struct aVector {
78 double Center[MAXVECTDIM];
79 int Size;
80 };
81
82 class System {
83 private:
84 double Pattern[MAXPATTERN][MAXVECTDIM+1];
85 aCluster Cluster[MAXCLUSTER];
86 int NumPatterns; // Number of patterns
87 int SizeVector; // Number of dimensions in vector
88 int NumClusters; // Number of clusters
89 void DistributeSamples(); // Step 2 of K-means algorithm
90 int CalcNewClustCenters();// Step 3 of K-means algorithm
91 double EucNorm(int, int); // Calc Euclidean norm vector
92 int FindClosestCluster(int); //ret indx of clust closest to pattern
93 //whose index is arg
94 public:
95 void system();
96 int LoadPatterns(char *fname); // Get pattern data to be clustered
97 void InitClusters(); // Step 1 of K-means algorithm
98 void RunKMeans(); // Overall control K-means process
99 void ShowClusters(); // Show results on screen
100 void SaveClusters(char *fname); // Save results to file
101 void ShowCenters();
102 };
103 //输出聚类中心
104 void System::ShowCenters(){
105 int i,j;
106 printf("Cluster centers:\n");
107 for (i=0; i<NumClusters; i++) {
108 Cluster[i].Member[0]=i;
109 printf("ClusterCenter[%d]=(%f,%f)\n",i,Cluster[i].Center[0],Cluster[i].Center[1]);
110 } /* endfor */
111 printf("\n");
112 getchar();
113 }
114
115 //读取文件
116 int System::LoadPatterns(char *fname)
117 {
118 FILE *InFilePtr;
119 int i,j;
120 double x;
121 if((InFilePtr = fopen(fname, "r")) == NULL)
122 return FAILURE;
123 fscanf(InFilePtr, "%d", &NumPatterns); // Read # of patterns 18数据量
124 fscanf(InFilePtr, "%d", &SizeVector); // Read dimension of vector 2维度
125 fscanf(InFilePtr, "%d", &NumClusters); // Read # of clusters for K-Means 2簇
126 for (i=0; i<NumPatterns; i++) { // For each vector
127 for (j=0; j<SizeVector; j++) { // create a pattern
128 fscanf(InFilePtr,"%lg",&x); // consisting of all elements
129 Pattern[i][j]=x;
130 } /* endfor */
131 } /* endfor */
132 //输出所有数据元素
133 printf("Input patterns:\n");
134 for (i=0; i<NumPatterns; i++) {
135 printf("Pattern[%d]=(%2.3f,%2.3f)\n",i,Pattern[i][0],Pattern[i][1]);
136 } /* endfor */
137 printf("\n--------------------\n");
138 getchar();
139 return SUCCESS;
140 }
141 //***************************************************************************
142 // InitClusters *
143 // Arbitrarily assign a vector to each of the K clusters *
144 // We choose the first K vectors to do this *
145 //***************************************************************************
146 //初始化聚类中心
147 void System::InitClusters(){
148 int i,j;
149 printf("Initial cluster centers:\n");
150 for (i=0; i<NumClusters; i++) {
151 Cluster[i].Member[0]=i;
152 for (j=0; j<SizeVector; j++) {
153 Cluster[i].Center[j]=Pattern[i][j];
154 } /* endfor */
155 } /* endfor */
156 for (i=0; i<NumClusters; i++) {
157 printf("ClusterCenter[%d]=(%f,%f)\n",i,Cluster[i].Center[0],Cluster[i].Center[1]); //untransplant
158 } /* endfor */
159 printf("\n");
160 getchar();
161 }
162 //运行KMeans
163 void System::RunKMeans(){
164 int converged;
165 int pass;
166 pass=1;
167 converged=FALSE;
168 //第N次聚类
169 while (converged==FALSE) {
170 printf("PASS=%d\n",pass++);
171 DistributeSamples();
172 converged=CalcNewClustCenters();
173 ShowCenters();
174 getchar();
175 } /* endwhile */
176 }
177 //在二维和三维空间中的欧式距离的就是两点之间的距离，二维的公式是
178 //d = sqrt((x1-x2)^+(y1-y2)^)
179 //通过这种运算，就可以把所有列的属性都纳入进来
180 double System::EucNorm(int p, int c){ // Calc Euclidean norm of vector difference
181 double dist,x; // between pattern vector, p, and cluster
182 int i; // center, c.
183 char zout[128];
184 char znum[40];
185 char *pnum;
186 //
187 pnum=&znum[0];
188 strcpy(zout,"d=sqrt(");
189 printf("The distance from pattern %d to cluster %d is calculated as:\n",p,c);
190 dist=0;
191 for (i=0; i<SizeVector ;i++){
192 //拼写字符串
193 x=(Cluster[c].Center[i]-Pattern[p][i])*(Cluster[c].Center[i]-Pattern[p][i]);
194 strcat(zout,f2a(x,4));
195 if (i==0)
196 strcat(zout,"+");
197 //计算距离
198 dist += (Cluster[c].Center[i]-Pattern[p][i])*(Cluster[c].Center[i]-Pattern[p][i]);
199 } /* endfor */
200 printf("%s)\n",zout);
201 return dist;
202 }
203 //查找最近的群集
204 int System::FindClosestCluster(int pat){
205 int i, ClustID;
206 double MinDist, d;
207 MinDist =9.9e+99;
208 ClustID=-1;
209 for (i=0; i<NumClusters; i++) {
210 d=EucNorm(pat,i);
211 printf("Distance from pattern %d to cluster %d is %f\n\n",pat,i,sqrt(d));
212 if (d<MinDist) {
213 MinDist=d;
214 ClustID=i;
215 } /* endif */
216 } /* endfor */
217 if (ClustID<0) {
218 printf("Aaargh");
219 exit(0);
220 } /* endif */
221 return ClustID;
222 }
223 //
224 void System::DistributeSamples(){
225 int i,pat,Clustid,MemberIndex;
226 //Clear membership list for all current clusters
227 for (i=0; i<NumClusters;i++){
228 Cluster[i].NumMembers=0;
229 }
230 for (pat=0; pat<NumPatterns; pat++) {
231 //Find cluster center to which the pattern is closest
232 Clustid= FindClosestCluster(pat);//查找最近的聚类中心
233 printf("patern %d assigned to cluster %d\n\n",pat,Clustid);
234 //post this pattern to the cluster
235 MemberIndex=Cluster[Clustid].NumMembers;
236 Cluster[Clustid].Member[MemberIndex]=pat;
237 Cluster[Clustid].NumMembers++;
238 } /* endfor */
239 }
240 //计算新的群集中心
241 int System::CalcNewClustCenters(){
242 int ConvFlag,VectID,i,j,k;
243 double tmp[MAXVECTDIM];
244 char xs[255];
245 char ys[255];
246 char nc1[20];
247 char nc2[20];
248 char *pnc1;
249 char *pnc2;
250 char *fpv;
251
252 pnc1=&nc1[0];
253 pnc2=&nc2[0];
254 ConvFlag=TRUE;
255 printf("The new cluster centers are now calculated as:\n");
256 for (i=0; i<NumClusters; i++) { //for each cluster
257 pnc1=itoa(Cluster[i].NumMembers,nc1,10);
258 pnc2=itoa(i,nc2,10);
259 strcpy(xs,"Cluster Center");
260 strcat(xs,nc2);
261 strcat(xs,"(1/");
262 strcpy(ys,"(1/");
263 strcat(xs,nc1);
264 strcat(ys,nc1);
265 strcat(xs,")(");
266 strcat(ys,")(");
267 for (j=0; j<SizeVector; j++) { // clear workspace
268 tmp[j]=0.0;
269 } /* endfor */
270 for (j=0; j<Cluster[i].NumMembers; j++) { //traverse member vectors
271 VectID=Cluster[i].Member[j];
272 for (k=0; k<SizeVector; k++) { //traverse elements of vector
273 tmp[k] += Pattern[VectID][k]; // add (member) pattern elmnt into temp
274 if (k==0) {
275 strcat(xs,f2a(Pattern[VectID][k],3));
276 } else {
277 strcat(ys,f2a(Pattern[VectID][k],3));
278 } /* endif */
279 } /* endfor */
280 if(j<Cluster[i].NumMembers-1){
281 strcat(xs,"+");
282 strcat(ys,"+");
283 }
284 else {
285 strcat(xs,")");
286 strcat(ys,")");
287 }
288 } /* endfor */
289 for (k=0; k<SizeVector; k++) { //traverse elements of vector
290 tmp[k]=tmp[k]/Cluster[i].NumMembers;
291 if (tmp[k] != Cluster[i].Center[k])
292 ConvFlag=FALSE;
293 Cluster[i].Center[k]=tmp[k];
294 } /* endfor */
295 printf("%s,\n",xs);
296 printf("%s\n",ys);
297 } /* endfor */
298 return ConvFlag;
299 }
300 //输出聚类
301 void System::ShowClusters(){
302 int cl;
303 for (cl=0; cl<NumClusters; cl++) {
304 printf("\nCLUSTER %d ==>[%f,%f]\n", cl,Cluster[cl].Center[0],Cluster[cl].Center[1]);
305 } /* endfor */
306 }
307
308 void System::SaveClusters(char *fname){
309 }四、主调程序
1 void main(int argc, char *argv[])
2 {
3
4 System kmeans;
5 /*
6 if (argc<2) {
7 printf("USAGE: KMEANS PATTERN_FILE\n");
8 exit(0);
9 }*/
10 if (kmeans.LoadPatterns("KM2.DAT")==FAILURE ){
11 printf("UNABLE TO READ PATTERN_FILE:%s\n",argv[1]);
12 exit(0);
13 }
14
15 kmeans.InitClusters();
16 kmeans.RunKMeans();
17 kmeans.ShowClusters();
18 }五、输出结果
1 Input patterns: 2 Pattern[0]=(0.000,0.000) 3 Pattern[1]=(1.000,0.000) 4 Pattern[2]=(0.000,1.000) 5 Pattern[3]=(2.000,1.000) 6 Pattern[4]=(1.000,2.000) 7 Pattern[5]=(2.000,2.000) 8 Pattern[6]=(2.000,0.000) 9 Pattern[7]=(0.000,2.000) 10 Pattern[8]=(7.000,6.000) 11 Pattern[9]=(7.000,7.000) 12 Pattern[10]=(7.000,8.000) 13 Pattern[11]=(8.000,6.000) 14 Pattern[12]=(8.000,7.000) 15 Pattern[13]=(8.000,8.000) 16 Pattern[14]=(8.000,9.000) 17 Pattern[15]=(9.000,7.000) 18 Pattern[16]=(9.000,8.000) 19 Pattern[17]=(9.000,9.000) 20 21 -------------------- 22 23 Initial cluster centers: 24 ClusterCenter[0]=(0.000000,0.000000) 25 ClusterCenter[1]=(1.000000,0.000000) 26 27 28 PASS=1 29 The distance from pattern 0 to cluster 0 is calculated as: 30 d=sqrt( .0000+ .0000) 31 Distance from pattern 0 to cluster 0 is 0.000000 32 33 The distance from pattern 0 to cluster 1 is calculated as: 34 d=sqrt( 1.0000+ .0000) 35 Distance from pattern 0 to cluster 1 is 1.000000 36 37 patern 0 assigned to cluster 0 38 39 The distance from pattern 1 to cluster 0 is calculated as: 40 d=sqrt( 1.0000+ .0000) 41 Distance from pattern 1 to cluster 0 is 1.000000 42 43 The distance from pattern 1 to cluster 1 is calculated as: 44 d=sqrt( .0000+ .0000) 45 Distance from pattern 1 to cluster 1 is 0.000000 46 47 patern 1 assigned to cluster 1 48 49 The distance from pattern 2 to cluster 0 is calculated as: 50 d=sqrt( .0000+ 1.0000) 51 Distance from pattern 2 to cluster 0 is 1.000000 52 53 The distance from pattern 2 to cluster 1 is calculated as: 54 d=sqrt( 1.0000+ 1.0000) 55 Distance from pattern 2 to cluster 1 is 1.414214 56 57 patern 2 assigned to cluster 0 58 59 The distance from pattern 3 to cluster 0 is calculated as: 60 d=sqrt( 4.0000+ 1.0000) 61 Distance from pattern 3 to cluster 0 is 2.236068 62 63 The distance from pattern 3 to cluster 1 is calculated as: 64 d=sqrt( 1.0000+ 1.0000) 65 Distance from pattern 3 to cluster 1 is 1.414214 66 67 patern 3 assigned to cluster 1 68 69 The distance from pattern 4 to cluster 0 is calculated as: 70 d=sqrt( 1.0000+ 4.0000) 71 Distance from pattern 4 to cluster 0 is 2.236068 72 73 The distance from pattern 4 to cluster 1 is calculated as: 74 d=sqrt( .0000+ 4.0000) 75 Distance from pattern 4 to cluster 1 is 2.000000 76 77 patern 4 assigned to cluster 1 78 79 The distance from pattern 5 to cluster 0 is calculated as: 80 d=sqrt( 4.0000+ 4.0000) 81 Distance from pattern 5 to cluster 0 is 2.828427 82 83 The distance from pattern 5 to cluster 1 is calculated as: 84 d=sqrt( 1.0000+ 4.0000) 85 Distance from pattern 5 to cluster 1 is 2.236068 86 87 patern 5 assigned to cluster 1 88 89 The distance from pattern 6 to cluster 0 is calculated as: 90 d=sqrt( 4.0000+ .0000) 91 Distance from pattern 6 to cluster 0 is 2.000000 92 93 The distance from pattern 6 to cluster 1 is calculated as: 94 d=sqrt( 1.0000+ .0000) 95 Distance from pattern 6 to cluster 1 is 1.000000 96 97 patern 6 assigned to cluster 1 98 99 The distance from pattern 7 to cluster 0 is calculated as: 100 d=sqrt( .0000+ 4.0000) 101 Distance from pattern 7 to cluster 0 is 2.000000 102 103 The distance from pattern 7 to cluster 1 is calculated as: 104 d=sqrt( 1.0000+ 4.0000) 105 Distance from pattern 7 to cluster 1 is 2.236068 106 107 patern 7 assigned to cluster 0 108 109 The distance from pattern 8 to cluster 0 is calculated as: 110 d=sqrt( 49.0000+ 36.0000) 111 Distance from pattern 8 to cluster 0 is 9.219544 112 113 The distance from pattern 8 to cluster 1 is calculated as: 114 d=sqrt( 36.0000+ 36.0000) 115 Distance from pattern 8 to cluster 1 is 8.485281 116 117 patern 8 assigned to cluster 1 118 119 The distance from pattern 9 to cluster 0 is calculated as: 120 d=sqrt( 49.0000+ 49.0000) 121 Distance from pattern 9 to cluster 0 is 9.899495 122 123 The distance from pattern 9 to cluster 1 is calculated as: 124 d=sqrt( 36.0000+ 49.0000) 125 Distance from pattern 9 to cluster 1 is 9.219544 126 127 patern 9 assigned to cluster 1 128 129 The distance from pattern 10 to cluster 0 is calculated as: 130 d=sqrt( 49.0000+ 64.0000) 131 Distance from pattern 10 to cluster 0 is 10.630146 132 133 The distance from pattern 10 to cluster 1 is calculated as: 134 d=sqrt( 36.0000+ 64.0000) 135 Distance from pattern 10 to cluster 1 is 10.000000 136 137 patern 10 assigned to cluster 1 138 139 The distance from pattern 11 to cluster 0 is calculated as: 140 d=sqrt( 64.0000+ 36.0000) 141 Distance from pattern 11 to cluster 0 is 10.000000 142 143 The distance from pattern 11 to cluster 1 is calculated as: 144 d=sqrt( 49.0000+ 36.0000) 145 Distance from pattern 11 to cluster 1 is 9.219544 146 147 patern 11 assigned to cluster 1 148 149 The distance from pattern 12 to cluster 0 is calculated as: 150 d=sqrt( 64.0000+ 49.0000) 151 Distance from pattern 12 to cluster 0 is 10.630146 152 153 The distance from pattern 12 to cluster 1 is calculated as: 154 d=sqrt( 49.0000+ 49.0000) 155 Distance from pattern 12 to cluster 1 is 9.899495 156 157 patern 12 assigned to cluster 1 158 159 The distance from pattern 13 to cluster 0 is calculated as: 160 d=sqrt( 64.0000+ 64.0000) 161 Distance from pattern 13 to cluster 0 is 11.313708 162 163 The distance from pattern 13 to cluster 1 is calculated as: 164 d=sqrt( 49.0000+ 64.0000) 165 Distance from pattern 13 to cluster 1 is 10.630146 166 167 patern 13 assigned to cluster 1 168 169 The distance from pattern 14 to cluster 0 is calculated as: 170 d=sqrt( 64.0000+ 81.0000) 171 Distance from pattern 14 to cluster 0 is 12.041595 172 173 The distance from pattern 14 to cluster 1 is calculated as: 174 d=sqrt( 49.0000+ 81.0000) 175 Distance from pattern 14 to cluster 1 is 11.401754 176 177 patern 14 assigned to cluster 1 178 179 The distance from pattern 15 to cluster 0 is calculated as: 180 d=sqrt( 81.0000+ 49.0000) 181 Distance from pattern 15 to cluster 0 is 11.401754 182 183 The distance from pattern 15 to cluster 1 is calculated as: 184 d=sqrt( 64.0000+ 49.0000) 185 Distance from pattern 15 to cluster 1 is 10.630146 186 187 patern 15 assigned to cluster 1 188 189 The distance from pattern 16 to cluster 0 is calculated as: 190 d=sqrt( 81.0000+ 64.0000) 191 Distance from pattern 16 to cluster 0 is 12.041595 192 193 The distance from pattern 16 to cluster 1 is calculated as: 194 d=sqrt( 64.0000+ 64.0000) 195 Distance from pattern 16 to cluster 1 is 11.313708 196 197 patern 16 assigned to cluster 1 198 199 The distance from pattern 17 to cluster 0 is calculated as: 200 d=sqrt( 81.0000+ 81.0000) 201 Distance from pattern 17 to cluster 0 is 12.727922 202 203 The distance from pattern 17 to cluster 1 is calculated as: 204 d=sqrt( 64.0000+ 81.0000) 205 Distance from pattern 17 to cluster 1 is 12.041595 206 207 patern 17 assigned to cluster 1 208 209 The new cluster centers are now calculated as: 210 Cluster Center0(1/3)( .000+ .000+ .000), 211 (1/3)( .000+ 1.000+ 2.000) 212 Cluster Center1(1/15)( 1.000+ 2.000+ 1.000+ 2.000+ 2.000+ 7.000+ 7.000+ 7.000+ 8 213 .000+ 8.000+ 8.000+ 8.000+ 9.000+ 9.000+ 9.000), 214 (1/15)( .000+ 1.000+ 2.000+ 2.000+ .000+ 6.000+ 7.000+ 8.000+ 6.000+ 7.000+ 8.00 215 0+ 9.000+ 7.000+ 8.000+ 9.000) 216 Cluster centers: 217 ClusterCenter[0]=(0.000000,1.000000) 218 ClusterCenter[1]=(5.866667,5.333333)
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