应用多元统计分析（7）第五章 判别分析

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
import numpy as np
import pandas as pd
#### ---------------------------------------------
#### (一) 距离判别
#### ---------------------------------------------

#### ---------------------------------------------
#### 两个总体距离判别问题
#### 1.例 5.2.3 
#### ---------------------------------------------
dat = pd.read_excel("F:\基础数学课\应用多元统计分析\examp5.2.3.xlsx",header = 1)

G1 = dat.iloc[:21,:-1]
G2 = dat.iloc[21:,:-1]

x1_bar = G1.mean()
x2_bar = G2.mean()
u_bar = (x1_bar + x2_bar)/2

S1 = G1.cov()
S2 = G2.cov()

n1 = G1.shape[0]
n2 = G2.shape[0]

Sp = (n1-1)/(n1+n2-2)*S1 + (n2-1)/(n1+n2-2)*S2

alpha = np.linalg.inv(Sp).dot(x1_bar-x2_bar)

#### 判别函数 W(X) = alpha.T.dot(X-u_bar)
def W(X):
    if alpha.T.dot(X-u_bar) > 0:
        return [1 , "G1:破产"]
    else:
        return [2 , "G2：不破产"]

Xnew = np.array([0.2,0.05,1.5,0.5])

W(Xnew)

## 将 46 家公司全部判断一遍
pred = dat.iloc[:,:-1].agg(W,axis=1) # agg() 函数

## 计算正确率 
1-sum(np.abs(pd.DataFrame(list(pred.values)).iloc[:,0] - dat.iloc[:,-1]))/dat.shape[0]

## 交叉验证法
## 线性判别 cross validation
def linear_dis2(data): 
    G1 = data[data.iloc[:,-1] == 1].iloc[:,:-1]
    G2 = data[data.iloc[:,-1] == 2].iloc[:,:-1]

    G1 = dat.iloc[:21,:-1] ;  G2 = dat.iloc[21:,:-1]

    x1_bar = G1.mean()  ;  x2_bar = G2.mean()
    u_bar = (x1_bar + x2_bar)/2

    S1 = G1.cov() ; S2 = G2.cov()

    n1 = G1.shape[0] ;  n2 = G2.shape[0]

    Sp = (n1-1)/(n1+n2-2)*S1 + (n2-1)/(n1+n2-2)*S2

    alpha = np.linalg.inv(Sp).dot(x1_bar-x2_bar)

    #### 判别函数 W(X) = alpha.T.dot(X-u_bar)
    def W(X):
        if alpha.T.dot(X-u_bar) > 0:
            return [1 , "G1:破产"]
        else:
            return [2 , "G2：不破产"]
    
    return W

f2 = linear_dis2(dat)

f2(Xnew)

## 交叉验证法
res = []

for i in range(46):
    data = np.delete(dat.values,i,0)
    data = pd.DataFrame(data)
    f_temp = linear_dis2(data)
    res.append(f_temp(dat.iloc[i,:-1].values))


## 刀切法
train_N = int(np.ceil(dat.shape[0]*0.8))
np.random.seed(123)
# np.random.randint(0,46,train_N)
# np.random.shuffle()
train_sample = np.random.choice(46,train_N,replace = False)

train_set = dat.iloc[train_sample,:]

test_sample =list(set(np.arange(46))-set(train_sample))
test_set = dat.iloc[train_sample,:]

f_train = linear_dis2(train_set)
f_train(test_set.iloc[0,:-1])

test_set.iloc[:,:-1].agg(f_train,axis=1)
test_set.iloc[:,-1]

#### ---------------------------------------------
#### 两个总体距离判别问题
#### 2.iris
#### ---------------------------------------------

# 距离判别
## 两个总体
iris = pd.read_csv('F:\基础数学课\应用多元统计分析\iris.csv')

u1_hat,u2_hat = iris.groupby('species').agg(np.mean).to_numpy()

S1 = iris.groupby('species').cov().to_numpy()[:4,]
S2 = iris.groupby('species').cov().to_numpy()[4:,]

n1,n2 = iris.groupby('species').count().iloc[:,0]
### n1,n2 = np.array(iris.groupby('Species').count()['Sepal.Length'])

u_bar = 1/2*(u1_hat+u2_hat)
S = (n1-1)/(n1+n2-2)*S1 + (n2-1)/(n1+n2-2)*S2

alpha = np.linalg.inv(S).dot(u1_hat-u2_hat)

def f1(X):
    return alpha.T.dot(X-u_bar)

result = iris.iloc[:,:-1].apply(f1,axis=1)>0
pd.DataFrame(result)
result.value_counts()

#### ---------------------------------------------
#### 多个总体距离判别问题
#### 1.iris
#### ---------------------------------------------
iris = pd.read_csv("F:\基础数学课\应用多元统计分析\iris.csv")
iris.groupby('species').count()

train_data1 = iris.iloc[:40,:]
train_data2 = iris.iloc[50:90,:]
train_data3 = iris.iloc[100:140,:]

u1 = train_data1.mean()
u2 = train_data2.mean()
u3 = train_data3.mean()

S1 = train_data1.cov()
S2 = train_data2.cov()
S3 = train_data3.cov()
Sp = 1/3*(S1+S2+S3)

I1 = np.linalg.inv(Sp).dot(u1)
C1 = -0.5*u1.T.dot(np.linalg.inv(Sp)).dot(u1)
def W1(X):
    return I1.T.dot(X)+C1

I2 = np.linalg.inv(Sp).dot(u2)
C2 = -0.5*u2.T.dot(np.linalg.inv(Sp)).dot(u2)
def W2(X):
    return I2.T.dot(X)+C2

I3 = np.linalg.inv(Sp).dot(u3)
C3 = -0.5*u3.T.dot(np.linalg.inv(Sp)).dot(u3)
def W3(X):
    return I3.T.dot(X)+C3

data41 = iris.iloc[40,:-1]
### iris.iloc[40,:-1]

W1(data41)
W2(data41)
W3(data41)

np.where([2,9,4]==np.amax([2,9,4]))

def W(X):
    v=np.array([W1(X),W2(X),W3(X)])
    return np.where(v==np.amax(v))[0].tolist()[0]

W(data41)

test_data = pd.concat([iris.iloc[40:50],iris.iloc[90:100],iris.iloc[140:150]]).iloc[:,:-1]

test_data.agg(W,axis=1)

#### ---------------------------------------------
#### (二) Bayes判别
#### ---------------------------------------------

iris = pd.read_csv("F:\基础数学课\应用多元统计分析\iris.csv")
q = np.array([0.1,0.3,0.6])
C = np.array([[0,3,9],[12,0,2],[20,40,0]])

###假设数据集服从多元正态分布

train_data1 = iris.iloc[:40,:]
train_data2 = iris.iloc[50:90,:]
train_data3 = iris.iloc[100:140,:]

u1 = train_data1.mean()
u2 = train_data2.mean()
u3 = train_data3.mean()

S1 = train_data1.cov()
S2 = train_data2.cov()
S3 = train_data3.cov()
Sp = 1/3*(S1+S2+S3)

def fx(x,u,S):
    return np.exp(-0.5*(x-u).T.dot(np.linalg.inv(S)).dot(x-u))

def f1x(x):
    return fx(x,u=u1,S=Sp)

def f2x(x):
    return fx(x,u=u2,S=Sp)

def f3x(x):
    return fx(x,u=u3,S=Sp)

def h(x,j):
    return np.sum(q*C[:,j-1]*np.array([f1x(x),f2x(x),f3x(x)]))

xnew = np.array([5,3,2,1])

h(xnew,1)

#### ---------------------------------------------
####  (三) Fisher判别
#### ---------------------------------------------
##协差阵不等
iris = pd.read_csv("F:\基础数学课\应用多元统计分析\iris.csv")
iris.groupby('species').count()

train_data1 = iris.iloc[:40,:]
train_data2 = iris.iloc[50:90,:]
train_data3 = iris.iloc[100:140,:]

u1 = train_data1.mean()
u2 = train_data2.mean()
u3 = train_data3.mean()

S1 = train_data1.cov()
S2 = train_data2.cov()
S3 = train_data3.cov()
Sp = 1/3*(S1+S2+S3)

k = 3
p = 4
E = S1+S2+S3
M = pd.concat([pd.DataFrame(u1),pd.DataFrame(u2),pd.DataFrame(u3)],axis=1).T.values
B = M.T.dot(np.eye(3)-1/3*np.ones((3,3))).dot(M) 

lamda, T = np.linalg.eig(np.linalg.inv(E).dot(B))

lamda_max = lamda.max()
# t1 = T[0] #特征根为列向量
t1 = T[:,0]

# 代入测试集
test_data1 = iris[40:50]
test_data2 = iris[90:100]
test_data3 = iris[140:150]
test_data  = pd.concat([test_data1,test_data2,test_data3])

## 有问题？？
def U(x):
    return t1.T.dot(x.T)

for i in range(10):
    X = iris[i:i+1]
    print(U(X))

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
#### ---------------------------------------------
# 利用iris数据集，每个类前40条作为训练集，后10条作为测试集
# 1. 进行Fisher判别，计算判别准确率
# 2. 进行贝叶斯判别，，计算判别准确率, 设3个总体都服从正态分布，先验概率为 0.2，0.3, 0.5, 误判概率矩阵为： [[0, 24, 40],[ 12, 0, 8][14, 27,0]]
#### ---------------------------------------------
import numpy as np
import pandas as pd

iris = pd.read_csv("F:\基础数学课\应用多元统计分析\iris.csv")
iris.groupby('species').count()
train_data1 = iris[:40].drop(columns='species')
train_data2 = iris[50:90].drop(columns='species')
train_data3 = iris[100:140].drop(columns='species')

test_data1 = iris[40:50].drop(columns='species')
test_data2 = iris[90:100].drop(columns='species')
test_data3 = iris[140:150].drop(columns='species')
test_data  = pd.concat([test_data1,test_data2,test_data3])

iris = iris.drop(columns='species')


u1 = train_data1.mean().to_list()
u2 = train_data2.mean().to_list()
u3 = train_data3.mean().to_list()

S1 = train_data1.cov()
S2 = train_data2.cov()
S3 = train_data3.cov()
Sp = 1/3*(S1+S2+S3)

# 假设三个总体都服从四元正态分布
k = 3
p = 4

#### ---------------------------------------------
#### 1. 进行Fisher判别，计算判别准确率
#### ---------------------------------------------

E = S1+S2+S3
M = pd.concat([pd.DataFrame(u1),pd.DataFrame(u2),pd.DataFrame(u3)],axis=1).T.values
B = M.T.dot(np.eye(3)-1/3*np.ones((3,3))).dot(M) 

lamda, T = np.linalg.eig(np.linalg.inv(E).dot(B))
lamda_max = lamda.max()
t1 = T[:,0]

# 利用课本的Fisher第一判别函数
def fisher_y1(X):
    dis1 = abs(t1.dot(X.T)-t1.dot(u1))
    dis2 = abs(t1.dot(X.T)-t1.dot(u2))
    dis3 = abs(t1.dot(X.T)-t1.dot(u3))
    dis = np.array([dis1,dis2,dis3])

    return np.where(dis == np.amin(dis))[0][0]+1

# 代入测试集
Species_fisher = []
# 判断 'species'
for i in range(30):
    X = np.array(test_data[i:i+1])[0]
    kind_fisher = fisher_y1(X)
    Species_fisher.append(kind_fisher)

# 计算判别准确率
real_species = np.hstack(([1]*10,[2]*10,[3]*10)).tolist()
real_species == Species_fisher

pd.DataFrame(np.column_stack((real_species,Species_fisher)),columns=('real_species','Species_fisher'))

#### ---------------------------------------------
#### 2. Bayes判别
#### ---------------------------------------------

# 先验概率
q = np.array([0.2,0.3,0.5])
# 误判代价矩阵
C = np.array([[0,24,40],[12,0,8],[14,27,0]])

# 概率密度函数
def f(X,u,S):
    return ((2*np.pi)**(-p/2)*np.linalg.det(S)**(-0.5))*np.exp(-0.5*(X-u).T.dot(np.linalg.inv(S)).dot(X-u))

def f1(X):
    return f(X,u1,S1)
def f2(X):
    return f(X,u2,S2)
def f3(X):
    return f(X,u3,S3)

def F(X):
    return np.array((f1(X),f2(X),f3(X)))

# 判别函数
def bayes_func(X):
    ECM1 = np.sum(q*F(X)*C[:,0])
    ECM2 = np.sum(q*F(X)*C[:,1])
    ECM3 = np.sum(q*F(X)*C[:,2])

    return np.where([ECM1,ECM2,ECM3]==min([ECM1,ECM2,ECM3]))[0][0]+1

# 代入测试集
Species_bayes = []
for i in range(30):
    X = np.array(test_data[i:i+1])[0]
    kind_bayes = bayes_func(X)
    Species_bayes.append(kind_bayes)

# 计算判别准确率
real_species = np.hstack(([1]*10,[2]*10,[3]*10)).tolist()
real_species == Species_bayes
np.array(real_species) == np.array(Species_bayes)

pd.DataFrame(np.column_stack((real_species,Species_bayes)),columns=('real_species','Species_bayes'))