## Fisher's Linear Discriminant

The first example shows the implementation of Fisher's Linear Classifier for 2-class problem and this algorithm is precisely described in book "Pattern Recognition and Machine Learning" by Christopher M Bishop (p 186, Section 4.1). The main idea of this algorithm is that we try to reduce the dimensionality of input vector X and project it onto 1D space using the equation y=W.T X where W.T - row vector of weights, and we adjust the weight vector W and choose the projection that maximizes the class separation. The following program use the famouse data set Iris with 150 number of instances and 4 attributes (4D space), target vector which contains labels: "Iris-setosa", "Iris-virginica", "Iris-versicolor", therefore, we have 3 classes, but, in this case, we may assume that we have class 1 with labels "Iris-setosa" and class 2 with other instances. Iris data set is available here: http://archive.ics.uci.edu/ml/datasets/Iris/ or here (comma separated format) - bezdekIris.data.txt

```   1 from __future__ import division
2 import numpy as np
3 import matplotlib.pyplot as plt
4
6   f=open("Iris.txt", 'r')
7   lines=[line.strip() for line in f.readlines()]
8   f.close()
9
10   lines=[line.split(",") for line in lines if line]
11
12   class1=np.array([line[:4] for line in lines if line[-1]=="Iris-setosa"], dtype=np.float)
13
14   class2=np.array([line[:4] for line in lines if line[-1]!="Iris-setosa"], dtype=np.float)
15
16   return class1, class2
17
18
19 def main():
20
22
23   mean1=np.mean(class1, axis=0)
24   mean2=np.mean(class2, axis=0)
25
26   #calculate variance within class
27   Sw=np.dot((class1-mean1).T, (class1-mean1))+np.dot((class2-mean2).T, (class2-mean2))
28
29   #calculate weights which maximize linear separation
30   w=np.dot(np.linalg.inv(Sw), (mean2-mean1))
31
32   print "vector of max weights", w
33   #projection of classes on 1D space
34   plt.plot(np.dot(class1, w), [0]*class1.shape[0], "bo", label="Iris-setosa")
35   plt.plot(np.dot(class2, w), [0]*class2.shape[0], "go", label="Iris-versicolor and Iris-virginica")
36   plt.legend()
37
38   plt.show()
39
40 main()
```

## Probabilistic Generative Model

This program is the implementation of Probabilistic Generative Model for K-class problem which is also described in book "Pattern Recognition and Machine Learning" by Christopher M Bishop (p 196, Section 4.2). We try to learn the class-conditional densities (likelihood) p(x|Ck) for each class K, and prior probability density p(Ck), then we can compute posterior probability p(Ck|x) by using Bayes rule. Here we assume that p(x|Ck) are 4D Gaussians with parameters uk - mean vector of class K, Sk - covariance matrix of class K, also p(Ck) for all k is 1/3. Then we compute so called quantities ak (variables pc's in the program) and if ak>>aj for all k!=j then assign p(Ck|x)=1 and p(Cj|x)=0.

```   1 from __future__ import division
2 import numpy as np
3 import matplotlib.pyplot as plt
4 import math
5
7   f=open("Iris.txt", 'r')
8   lines=[line.strip() for line in f.readlines()]
9   f.close()
10
11   lines=[line.split(",") for line in lines if line]
12
13   data=np.array([line[:4] for line in lines if line], dtype=np.float)
14
15   class1=np.array([line[:4] for line in lines if line[-1]=="Iris-setosa"], dtype=np.float)
16
17   class2=np.array([line[:4] for line in lines if line[-1]=="Iris-virginica"], dtype=np.float)
18
19   class3=np.array([line[:4] for line in lines if line[-1]=="Iris-versicolor"], dtype=np.float)
20
21  #list of class labels
22   labels=[]
23   for line in lines:
24    strt=line.pop()
25    labels.append(strt)
26   #create array of labels
27   labels=[line.split(",") for line in labels if line]
28   t=np.zeros(shape=(150, 3))
29   #create target vector encoded according to 1-of-K scheme
30   for i in xrange(len(data)):
31    if labels[i]==["Iris-setosa"]: t[i][0]=1
32    elif labels[i]==["Iris-versicolor"]: t[i][1]=1
33    elif labels[i]==["Iris-virginica"]: t[i][2]=1
34
35   return class1, class2, class3, data, t
36
37 def gaussian(x, mean, cov):
38   xm=np.reshape((x-mean), (-1, 1))
39   px=1/(math.pow(2.0*math.pi, 2))*1/math.sqrt(np.linalg.det(cov))*math.exp(-(np.dot(np.dot(xm.T, np.linalg.inv(cov)), xm))/2)
40   return px
41
42 def main():
43  class1, class2, class3, data, t=read_data()
44
45  count=np.zeros(shape=(150,1))
46  t_assigned=np.zeros(shape=(150, 3))
47  cov=np.zeros(shape=(3, 4, 4))
48  mean=np.zeros(shape=(3, 4))
49
50   #compute means for each class
51  mean1=class1.mean(axis=0)
52  mean2=class2.mean(axis=0)
53  mean3=class3.mean(axis=0)
54   #compute covariance matrices, such that the columns are variables and rows are observations of variables
55  cov1=np.cov(class1, rowvar=0)
56  cov2=np.cov(class2, rowvar=0)
57  cov3=np.cov(class3, rowvar=0)
58
59 #compute gaussian likelihood functions p(x|Ck) for each class
60  for i in xrange(len(data)):
61      px1=(1/3.0)*gaussian(data[i], mean1, cov1)
62      px2=(1/3.0)*gaussian(data[i], mean2, cov2)
63      px3=(1/3.0)*gaussian(data[i], mean3, cov3)
64      m=np.max([px1, px2, px3])
65  #compute posterior probability p(Ck|x) assuming that p(x|Ck) is gaussian and the entire expression is wrapped by sigmoid function
66      pc1=((math.exp(px1)*math.exp(-m))*math.exp(m))/((math.exp(px2)*math.exp(-m)+math.exp(px3)*math.exp(-m))*math.exp(m))
67      pc2=((math.exp(px2)*math.exp(-m))*math.exp(m))/((math.exp(px1)*math.exp(-m)+math.exp(px3)*math.exp(-m))*math.exp(m))
68      pc3=((math.exp(px3)*math.exp(-m))*math.exp(m))/((math.exp(px1)*math.exp(-m)+math.exp(px2)*math.exp(-m))*math.exp(m))
69  #assign p(Ck|x)=1 if p(Ck|x)>>p(Cj|x) for all j!=k
70      if pc1>pc2 and pc1>pc3: t_assigned[i][0]=1
71      elif pc3>pc1 and pc3>pc2: t_assigned[i][1]=1
72      elif pc2>pc1 and pc2>pc3: t_assigned[i][2]=1
73  #count the number of misclassifications
74      for j in xrange(3):
75       if t[i][j]-t_assigned[i][j]!=0: count[i]=1
76
77  cov=[cov1, cov2, cov3]
78  mean=[mean1, mean2, mean3]
79
80  t1=np.zeros(shape=(len(class1), 1))
81  t2=np.zeros(shape=(len(class2), 1))
82  t3=np.zeros(shape=(len(class3), 1))
83  for i in xrange(len(data)):
84   for j in xrange(len(class1)):
85    if t_assigned[i][0]==1: t1[j]=1
86    elif t_assigned[i][1]==1: t2[j]=2
87    elif t_assigned[i][2]==1: t3[j]=3
88
89  plt.plot(t1, "bo", label="Iris-setosa")
90  plt.plot(t2, "go", label="Iris-versicolor")
91  plt.plot(t3, "ro", label="Iris-virginica")
92  plt.legend()
93  plt.show()
94
95  print "number of misclassifications", sum(count), "assigned labels to data points", t_assigned, "target data", t
96
97 main()
```

This program resulted in the number of misclassifications = 3 out of all 150 instances

Cookbook/LinearClassification (last edited 2011-05-29 04:05:46 by DmitriyRybalkin)