KNN(k-Nearest Neighbour):K-近邻算法,主要思想可以归结为一个成语:物以类聚
1.1 工作原理给定一个训练数据集,对新的输入实例,在训练数据集中找到与该实例最邻近的 k (k <= 20)个实例,这 k 个实例的多数属于某个类,
就把该输入实例分为这个类。
https://www.cnblogs.com/ybjourney/p/4702562.html给出的例子很形象,这里借用一下。
如下图,绿色圆要被决定赋予哪个类,是红色三角形还是蓝色四方形?如果K=3,由于红色三角形所占比例为2/3,绿色圆将被赋予红色三角形那个类,
如果K=5,由于蓝色四方形比例为3/5,因此绿色圆被赋予蓝色四方形类。
由此也说明了KNN算法的结果很大程度取决于K的选择。
1.2 欧氏距离公式计算两个向量点xA和xB之间的距离
1.3 分类决策规则(如多数表为指示函数,即当时为 1,否则为0。
1.4 算法流程对未知类别属性的数据集中的每个点依次执行以下操作:
1. 计算已知类别数据集中的点与当前点之间的距离;
2. 按照距离递增次序排序;
3. 选取与当前点距离最小的 k 个点;
4. 确定前 k 个点所在类别的出现频率;
5. 返回前 k 个点出现频率最高的类别作为当前点的预测分类;
2. 代码实现python3.6
每个方法的作用,以及每行代码的作用,同样我都做了详细的注解。
希望大家最好自己能实现一下,特别是在运算时 list,array,matrix之间的关系以及运用场景,
只有在你自己实现时,才能理清这三者的作用以及关系。
2.1 输入数据datingTestSet2.txt :约会网站数据(三种类型:不喜欢的人,魅力一般的人,极具魅力的人)
1 40920 8.326976 0.953952 3
2 14488 7.153469 1.673904 2
3 26052 1.441871 0.805124 1
4 75136 13.147394 0.428964 1
5 38344 1.669788 0.134296 1
6 72993 10.141740 1.032955 1
7 35948 6.830792 1.213192 3
8 42666 13.276369 0.543880 3
9 67497 8.631577 0.749278 1
10 35483 12.273169 1.508053 3
11 50242 3.723498 0.831917 1
12 63275 8.385879 1.669485 1
13 5569 4.875435 0.728658 2
14 51052 4.680098 0.625224 1
15 77372 15.299570 0.331351 1
16 43673 1.889461 0.191283 1
17 61364 7.516754 1.269164 1
18 69673 14.239195 0.261333 1
19 15669 0.000000 1.250185 2
20 28488 10.528555 1.304844 3
21 6487 3.540265 0.822483 2
22 37708 2.991551 0.833920 1
23 22620 5.297865 0.638306 2
24 28782 6.593803 0.187108 3
25 19739 2.816760 1.686209 2
26 36788 12.458258 0.649617 3
27 5741 0.000000 1.656418 2
28 28567 9.968648 0.731232 3
29 6808 1.364838 0.640103 2
30 41611 0.230453 1.151996 1
31 36661 11.865402 0.882810 3
32 43605 0.120460 1.352013 1
33 15360 8.545204 1.340429 3
34 63796 5.856649 0.160006 1
35 10743 9.665618 0.778626 2
36 70808 9.778763 1.084103 1
37 72011 4.932976 0.632026 1
38 5914 2.216246 0.587095 2
39 14851 14.305636 0.632317 3
40 33553 12.591889 0.686581 3
41 44952 3.424649 1.004504 1
42 17934 0.000000 0.147573 2
43 27738 8.533823 0.205324 3
44 29290 9.829528 0.238620 3
45 42330 11.492186 0.263499 3
46 36429 3.570968 0.832254 1
47 39623 1.771228 0.207612 1
48 32404 3.513921 0.991854 1
49 27268 4.398172 0.975024 1
50 5477 4.276823 1.174874 2
51 14254 5.946014 1.614244 2
52 68613 13.798970 0.724375 1
53 41539 10.393591 1.663724 3
54 7917 3.007577 0.297302 2
55 21331 1.031938 0.486174 2
56 8338 4.751212 0.064693 2
57 5176 3.692269 1.655113 2
58 18983 10.448091 0.267652 3
59 68837 10.585786 0.329557 1
60 13438 1.604501 0.069064 2
61 48849 3.679497 0.961466 1
62 12285 3.795146 0.696694 2
63 7826 2.531885 1.659173 2
64 5565 9.733340 0.977746 2
65 10346 6.093067 1.413798 2
66 1823 7.712960 1.054927 2
67 9744 11.470364 0.760461 3
68 16857 2.886529 0.934416 2
69 39336 10.054373 1.138351 3
70 65230 9.972470 0.881876 1
71 2463 2.335785 1.366145 2
72 27353 11.375155 1.528626 3
73 16191 0.000000 0.605619 2
74 12258 4.126787 0.357501 2
75 42377 6.319522 1.058602 1
76 25607 8.680527 0.086955 3
77 77450 14.856391 1.129823 1
78 58732 2.454285 0.222380 1
79 46426 7.292202 0.548607 3
80 32688 8.745137 0.857348 3
81 64890 8.579001 0.683048 1
82 8554 2.507302 0.869177 2
83 28861 11.415476 1.505466 3
84 42050 4.838540 1.680892 1
85 32193 10.339507 0.583646 3
86 64895 6.573742 1.151433 1
87 2355 6.539397 0.462065 2
88 0 2.209159 0.723567 2
89 70406 11.196378 0.836326 1
90 57399 4.229595 0.128253 1
91 41732 9.505944 0.005273 3
92 11429 8.652725 1.348934 3
93 75270 17.101108 0.490712 1
94 5459 7.871839 0.717662 2
95 73520 8.262131 1.361646 1
96 40279 9.015635 1.658555 3
97 21540 9.215351 0.806762 3
98 17694 6.375007 0.033678 2
99 22329 2.262014 1.022169 1
100 46570 5.677110 0.709469 1
101 42403 11.293017 0.207976 3
102 33654 6.590043 1.353117 1
103 9171 4.711960 0.194167 2
104 28122 8.768099 1.108041 3
105 34095 11.502519 0.545097 3
106 1774 4.682812 0.578112 2
107 40131 12.446578 0.300754 3
108 13994 12.908384 1.657722 3
109 77064 12.601108 0.974527 1
110 11210 3.929456 0.025466 2
111 6122 9.751503 1.182050 3
112 15341 3.043767 0.888168 2
113 44373 4.391522 0.807100 1
114 28454 11.695276 0.679015 3
115 63771 7.879742 0.154263 1
116 9217 5.613163 0.933632 2
117 69076 9.140172 0.851300 1
118 24489 4.258644 0.206892 1
119 16871 6.799831 1.221171 2
120 39776 8.752758 0.484418 3
121 5901 1.123033 1.180352 2
122 40987 10.833248 1.585426 3
123 7479 3.051618 0.026781 2
124 38768 5.308409 0.030683 3
125 4933 1.841792 0.028099 2
126 32311 2.261978 1.605603 1
127 26501 11.573696 1.061347 3
128 37433 8.038764 1.083910 3
129 23503 10.734007 0.103715 3
130 68607 9.661909 0.350772 1
131 27742 9.005850 0.548737 3
132 11303 0.000000 0.539131 2
133 0 5.757140 1.062373 2
134 32729 9.164656 1.624565 3
135 24619 1.318340 1.436243 1
136 42414 14.075597 0.695934 3
137 20210 10.107550 1.308398 3
138 33225 7.960293 1.219760 3
139 54483 6.317292 0.018209 1
140 18475 12.664194 0.595653 3
141 33926 2.906644 0.581657 1
142 43865 2.388241 0.913938 1
143 26547 6.024471 0.486215 3
144 44404 7.226764 1.255329 3
145 16674 4.183997 1.275290 2
146 8123 11.850211 1.096981 3
147 42747 11.661797 1.167935 3
148 56054 3.574967 0.494666 1
149 10933 0.000000 0.107475 2
150 18121 7.937657 0.904799 3
151 11272 3.365027 1.014085 2
152 16297 0.000000 0.367491 2
153 28168 13.860672 1.293270 3
154 40963 10.306714 1.211594 3
155 31685 7.228002 0.670670 3
156 55164 4.508740 1.036192 1
157 17595 0.366328 0.163652 2
158 1862 3.299444 0.575152 2
159 57087 0.573287 0.607915 1
160 63082 9.183738 0.012280 1
161 51213 7.842646 1.060636 3
162 6487 4.750964 0.558240 2
163 4805 11.438702 1.556334 3
164 30302 8.243063 1.122768 3
165 68680 7.949017 0.271865 1
166 17591 7.875477 0.227085 2
167 74391 9.569087 0.364856 1
168 37217 7.750103 0.869094 3
169 42814 0.000000 1.515293 1
170 14738 3.396030 0.633977 2
171 19896 11.916091 0.025294 3
172 14673 0.460758 0.689586 2
173 32011 13.087566 0.476002 3
174 58736 4.589016 1.672600 1
175 54744 8.397217 1.534103 1
176 29482 5.562772 1.689388 1
177 27698 10.905159 0.619091 3
178 11443 1.311441 1.169887 2
179 56117 10.647170 0.980141 3
180 39514 0.000000 0.481918 1
181 26627 8.503025 0.830861 3
182 16525 0.436880 1.395314 2
183 24368 6.127867 1.102179 1
184 22160 12.112492 0.359680 3
185 6030 1.264968 1.141582 2
186 6468 6.067568 1.327047 2
187 22945 8.010964 1.681648 3
188 18520 3.791084 0.304072 2
189 34914 11.773195 1.262621 3
190 6121 8.339588 1.443357 2
191 38063 2.563092 1.464013 1
192 23410 5.954216 0.953782 1
193 35073 9.288374 0.767318 3
194 52914 3.976796 1.043109 1
195 16801 8.585227 1.455708 3
196 9533 1.271946 0.796506 2
197 16721 0.000000 0.242778 2
198 5832 0.000000 0.089749 2
199 44591 11.521298 0.300860 3
200 10143 1.139447 0.415373 2
201 21609 5.699090 1.391892 2
202 23817 2.449378 1.322560 1
203 15640 0.000000 1.228380 2
204 8847 3.168365 0.053993 2
205 50939 10.428610 1.126257 3
206 28521 2.943070 1.446816 1
207 32901 10.441348 0.975283 3
208 42850 12.478764 1.628726 3
209 13499 5.856902 0.363883 2
210 40345 2.476420 0.096075 1
211 43547 1.826637 0.811457 1
212 70758 4.324451 0.328235 1
213 19780 1.376085 1.178359 2
214 44484 5.342462 0.394527 1
215 54462 11.835521 0.693301 3
216 20085 12.423687 1.424264 3
217 42291 12.161273 0.071131 3
218 47550 8.148360 1.649194 3
219 11938 1.531067 1.549756 2
220 40699 3.200912 0.309679 1
221 70908 8.862691 0.530506 1
222 73989 6.370551 0.369350 1
223 11872 2.468841 0.145060 2
224 48463 11.054212 0.141508 3
225 15987 2.037080 0.715243 2
226 70036 13.364030 0.549972 1
227 32967 10.249135 0.192735 3
228 63249 10.464252 1.669767 1
229 42795 9.424574 0.013725 3
230 14459 4.458902 0.268444 2
231 19973 0.000000 0.575976 2
232 5494 9.686082 1.029808 3
233 67902 13.649402 1.052618 1
234 25621 13.181148 0.273014 3
235 27545 3.877472 0.401600 1
236 58656 1.413952 0.451380 1
237 7327 4.248986 1.430249 2
238 64555 8.779183 0.845947 1
239 8998 4.156252 0.097109 2
240 11752 5.580018 0.158401 2
241 76319 15.040440 1.366898 1
242 27665 12.793870 1.307323 3
243 67417 3.254877 0.669546 1
244 21808 10.725607 0.588588 3
245 15326 8.256473 0.765891 2
246 20057 8.033892 1.618562 3
247 79341 10.702532 0.204792 1
248 15636 5.062996 1.132555 2
249 35602 10.772286 0.668721 3
250 28544 1.892354 0.837028 1
251 57663 1.019966 0.372320 1
252 78727 15.546043 0.729742 1
253 68255 11.638205 0.409125 1
254 14964 3.427886 0.975616 2
255 21835 11.246174 1.475586 3
256 7487 0.000000 0.645045 2
257 8700 0.000000 1.424017 2
258 26226 8.242553 0.279069 3
259 65899 8.700060 0.101807 1
260 6543 0.812344 0.260334 2
261 46556 2.448235 1.176829 1
262 71038 13.230078 0.616147 1
263 47657 0.236133 0.340840 1
264 19600 11.155826 0.335131 3
265 37422 11.029636 0.505769 3
266 1363 2.901181 1.646633 2
267 26535 3.924594 1.143120 1
268 47707 2.524806 1.292848 1
269 38055 3.527474 1.449158 1
270 6286 3.384281 0.889268 2
271 10747 0.000000 1.107592 2
272 44883 11.898890 0.406441 3
273 56823 3.529892 1.375844 1
274 68086 11.442677 0.696919 1
275 70242 10.308145 0.422722 1
276 11409 8.540529 0.727373 2
277 67671 7.156949 1.691682 1
278 61238 0.720675 0.847574 1
279 17774 0.229405 1.038603 2
280 53376 3.399331 0.077501 1
281 30930 6.157239 0.580133 1
282 28987 1.239698 0.719989 1
283 13655 6.036854 0.016548 2
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285 40409 12.393001 1.571281 3
286 13605 9.627613 0.935842 2
287 26400 11.130453 0.597610 3
288 13491 8.842595 0.349768 3
289 30232 10.690010 1.456595 3
290 43253 5.714718 1.674780 3
291 55536 3.052505 1.335804 1
292 8807 0.000000 0.059025 2
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448 8579 1.933803 1.374388 2
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451 4364 2.661254 0.946117 2
452 4985 0.988432 1.305027 2
453 37068 2.063741 1.125946 1
454 41137 2.220590 0.690754 1
455 67759 6.424849 0.806641 1
456 11831 1.156153 1.613674 2
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478 171 8.132449 0.039062 2
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484 82008 8.362736 1.285939 1
485 25054 9.084726 1.606312 3
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2.2 KNN算法实现
myKNN.py
1 # -*- coding: utf-8 -*-
2 """
3 Created on Mon Sep 17 15:58:58 2018
4 KNN(K-Nearest Neighbor) K-近邻算法
5 @author: weixw
6 """
7
8 import numpy as np
9 import operator
10 #输入:行测试数据集,训练数据集,标签数据集,用于选择最近邻居的数目
11 #功能:根据欧氏距离公式,找到与未知类别的测试数据距离最小的 k 个点,
12 # 以这 k 个点出现频率最高的类别座位测试数据的预测分类。
13 # 欧氏距离公式:测试数据与训练数据对应位置作差,平方和,然后开方
14 #输出:测试数据预测分类结果
15 def classify(testDataSet, trainingDataSet, labelList, k):
16 #训练数据集行数
17 trainingDataSetSize = trainingDataSet.shape[0]
18 #np.tile(testDataSet, (trainingDataSetSize,1)沿X轴复制1倍(相当于没有复制),再沿Y轴复制trainingDataSetSize倍,维数:1000*3
19 #欧氏距离公式实现
20 #1 测试数据 - 训练数据
21 diffMat = np.mat(np.tile(testDataSet, (trainingDataSetSize, 1)) - trainingDataSet)
22 #2 差平方(需要将matrix转化为数组,否则报错)
23 sqDiffMat = diffMat.A**2
24 #3 按行求和 axis = 0(默认按列) axis = 1(按行)
25 sqDistances = sqDiffMat.sum(axis = 1)
26 #4 开方
27 distances = sqDistances**0.5
28 #agrsort():从小到大排序,返回欧氏距离最小值对应的索引列表
29 sortedDistIndicies = distances.argsort()
30 #预测分类计数
31 predictClassCount = {}
32 #多数表决方式,选择 k 个欧氏距离最小值
33 for i in range(k):
34 #找到索引对应的标签值
35 voteLabel = labelList[sortedDistIndicies[i]]
36 #预测标签值字典,存储索引标签值预测次数
37 predictClassCount[voteLabel] = predictClassCount.get(voteLabel, 0) 1
38 #对象按值逆向(由大到小)排序
39 # sorted(iterable[, cmp[, key[, reverse]]])
40 # itemgetter(1) 取第一项结果
41 sortedPredictClassCount = sorted(predictClassCount.items(), key = operator.itemgetter(1), reverse = True)
42 return sortedPredictClassCount[0][0]
43
44
45
46 #输入:数据文件
47 #功能:加载文件,文件最后一列是标签数据,分离特征数据集与标签数据集
48 # 自动检测多少列特征数据并分离
49 #输出:特征数据集矩阵,标签数据集矩阵
50 def loadDataSet(fileName):
51 #特征数据列长度
52 numberFeat = len(open(fileName).readline().split(' ')) - 1
53 dataSet = []; labelSet = []
54 fr = open(fileName)
55 for line in fr.readlines():
56 lineArr = []
57 #去除收尾空格,然后分割每一列
58 curLine = line.strip().split(' ')
59 #保存每一列特征数据
60 for i in range(numberFeat):
61 lineArr.append(float(curLine[i]))
62 dataSet.append(lineArr)
63 labelSet.append(float(curLine[-1]))
64 return np.mat(dataSet), labelSet
65
66 #输入:原始特征数据集
67 #功能:数据归一化,使每类数据都在同一范围内 (0, 1) 变化
68 # 归一化公式:newValue = (oldValue - min)/(max - min)
69 #输出:归一化后特征数据集,范围数组大小(分母),列最小值数组
70 def autoNorm(dataMat):
71 #min(axis) 无参数:所有值中最小值;axis = 0:每列最小值;axis = 1:每行最小值
72 #求出每列最小值
73 minValsMat = dataMat.min(0)
74 #求出每列最大值
75 maxValsMat = dataMat.max(0)
76 #计算差值(对应位置相减)
77 rangesMat = maxValsMat - minValsMat
78 #归一化特征数据集初始化,维数:1000*3
79 normDataMat = np.zeros(np.shape(dataMat))
80 #原始数据集行数目
81 m = dataMat.shape[0]
82 #归一化公式分子实现
83 #np.tile(minVals, (m,1)沿X轴复制1倍(相当于没有复制),再沿Y轴复制m倍,维数:1000*3
84 normDataMat = dataMat - np.tile(minValsMat, (m, 1))
85 #归一化公式实现,求得归一化结果
86 normDataMat = normDataMat/np.tile(rangesMat, (m, 1))
87 return normDataMat, rangesMat, minValsMat
88
89 #输入:特征数据集矩阵,标签数据集列表,测试数据与训练数据比例,用于选择最近邻居的数目
90 #功能:求出测试特征数据集预测分类结果
91 # 1.解析文件
92 # 2.通过ratio确定测试数据集
93 # 3.归一化
94 # 4.对每一行测试数据运用欧氏距离公式以及多数表决方式预测分类结果
95 # 5.求出整个测试数据集的预测分类结果
96 #输出:测试数据预测分类结果
97 def dataClassify(dataMat, labelList, ratio, k):
98
99 #特征数据集归一化
100 normDataMat, rangesMat, minValsMat = autoNorm(dataMat)
101 #归一化特征数据集行数目
102 m = normDataMat.shape[0]
103 #测试数据集行数目(也就知道训练数据集行数)
104 testDataNum = int(m*ratio)
105 #预测分类错误计数
106 errorCount = 0.0
107 for i in range(testDataNum):
108 #求出测试数据集每行预测分类
109 classifierResult = classify(normDataMat[i, :], normDataMat[testDataNum:m, :], labelList[testDataNum:m], k)
110 print ("the classifier result is: %d, the real answer is: %d"% (classifierResult, labelList[i]))
111 #统计错误预测分类
112 if(classifierResult != labelList[i]):
113 errorCount = 1.0
114 print ("the total error count is %d"% errorCount)
115 print ("the total error rate is: %f"%(errorCount/float(testDataNum)))
116
117
118 #绘制散点图
119 def drawScatter(filename):
120 import matplotlib.pyplot as plt
121 #加载文件,分离特征数据集和标签数据集
122 dataMat, labelList = loadDataSet(filename)
123 #矩阵转化为数组
124 dataArr = dataMat.A
125 #创建一副图画
126 plt.figure()
127 #保存标签类型相同的索引值(观察标签数据集,有3种不同类型)
128 label_idx1 = []; label_idx2 = []; label_idx3 = []
129 #遍历标签数组,索引,值
130 for index, value in enumerate(labelList):
131 if(value == 1):
132 label_idx1.append(index)
133 elif(value == 2):
134 label_idx2.append(index)
135 else:
136 label_idx3.append(index)
137 #scatter(x,y,s,maker,color,label)
138 #x,y必须是数组类型,s表示形状大小,maker:形状
139 plt.scatter(dataArr[label_idx1, 1], dataArr[label_idx1, 2], marker = 'x', color = 'm', label = 'no like', s = 30)
140 plt.scatter(dataArr[label_idx2, 1], dataArr[label_idx2, 2], marker = ' ', color = 'c', label = 'like', s = 50)
141 plt.scatter(dataArr[label_idx3, 1], dataArr[label_idx3, 2], marker = 'o', color = 'r', label = 'very like', s = 15)
142 plt.legend(loc = 'upper right')
143
144
2.3 测试代码
1 # -*- coding: utf-8 -*-
2 """
3 Created on Tue Sep 18 14:07:14 2018
4 测试KNN算法
5 @author: weixw
6 """
7 import myKNN as mk
8 #前50%是测试数据,后50%作为训练数据
9 ratio = 0.5
10 #选择邻居数目
11 #errCount:31 errRate:6.2%
12 k = 4
13 #errCount:30 errRate:6.0%
14 #k = 8
15 #errCount:30 errRate:6.0%
16 #k = 12
17 #errCount:33 errRate:6.6%
18 #k = 16
19 #errCount:32 errRate:6.4%
20 #k = 20
21
22
23
24 fileName = 'datingTestSet2.txt'
25 #绘制数据散点图
26 mk.drawScatter(fileName)
27 #加载文件,分离特征数据集和标签数据集
28 dataMat, labelList = mk.loadDataSet(fileName)
29 #预测测试数据结果
30 mk.dataClassify(dataMat, labelList, ratio, k)
2.4 运行结果
输入数据的散点图:
k = 4 ,ratio = 0.5(一半测试数据,一半训练数据)时分类结果:
在 k为不同值时运行结果:
可以看出,并不是 k越大,正确率越高,会产生过拟合。
3. 优缺点优点:1. 简单,易于理解,易于实现,无需训练;
2. 精度高,对异常值不敏感;
缺点:计算复杂度高,空间复杂度高。
,