FDA 정리 및 코드 구현

Fisher Discriminant Analysis (FDA)

import numpy as np

import matplotlib.pyplot as plt

%matplotlib inline

# generating data set  

n0 = 200  
n1 = 200  

sigma = \[\[0.9, -0.4\],  
         \[-0.4, 0.3\]\]

np.random.seed(0)

x0 = np.random.multivariate\_normal(\[2.5,2.5\], sigma, n0) # data in class 0

x1 = np.random.multivariate\_normal(\[1,1\], sigma, n1) # data in class 1

print(x0.shape)

print(x1.shape)

(200, 2) (200, 2)

mu0 = np.mean(x0, axis = 0)

mu1 = np.mean(x1, axis = 0)

S0 = np.dot((x0-mu0).T, x0-mu0)/(n0-1)

S1 = np.dot((x1-mu1).T, x1-mu1)/(n1-1)

w = np.linalg.solve(n0/(n0+n1)\*S0 + n1/(n0+n1)\*S1, mu0 - mu1)

print(w)

[ 9.64218334 18.00307461]

dmu = (mu0 - mu1)\[:,np.newaxis\]

Sb = np.dot(dmu, dmu.T)

#print(Sb.shape)

Sw = n0/(n0+n1)\*S0 + n1/(n0+n1)\*S1

L = np.linalg.cholesky(Sw)

#print(L)

print(np.allclose(Sw, np.dot(L,L.T)))

L\_inv = np.linalg.inv(L)

Sb\_new = np.dot(np.dot(L\_inv, Sb), L\_inv.T)

eigenvalues, eigenvectors = np.linalg.eigh(Sb\_new)

print(eigenvalues)

print(eigenvectors)

w2 = np.dot(L\_inv.T, eigenvectors\[:,-1:\])

print(np.dot(np.dot(w2.T, Sb), w2) / np.dot(np.dot(w2.T, Sw), w2))

print(w/w2\[:,0\])

True
[ 0. 42.37927574]
[[-0.97005478 0.24288622]
[ 0.24288622 0.97005478]]
[[42.37927574]]
[6.50993669 6.50993669]

plt.figure(figsize = (10, 8))

plt.plot(x0\[:,0\], x0\[:,1\], 'r.')

plt.plot(x1\[:,0\], x1\[:,1\], 'b.')

xp = np.arange(-4, 6, 0.1)

yp = w\[1\]/w\[0\]\*xp

plt.plot(xp, yp, 'k')

plt.axis('equal')

plt.ylim(\[-2, 6\])

plt.show()

thres = np.sum(w \* 1/2 \* (mu1+mu0))

plt.figure(figsize = (10, 8))

plt.plot(x0\[:,0\], x0\[:,1\], 'r.')

plt.plot(x1\[:,0\], x1\[:,1\], 'b.')

xp = np.arange(-4, 6, 0.1)

yp = (thres - w\[0\] \* xp)/w\[1\]

plt.plot(xp, yp, 'k')

plt.axis('equal')

plt.ylim(\[-2, 6\])

plt.show()

y1 = np.dot(x0,w)

y2 = np.dot(x1,w)

plt.figure(figsize = (10, 8))

plt.hist(y1, 21, color = 'r', rwidth = 0.5)

plt.hist(y2, 21, color = 'b', rwidth = 0.5)

plt.show()

Scikit-learn

\# reshape data

X = np.vstack(\[x0, x1\])

y = np.vstack(\[np.ones(\[n0, 1\]), np.zeros(\[n1, 1\])\])

from sklearn import discriminant\_analysis

clf = discriminant\_analysis.LinearDiscriminantAnalysis()

clf.fit(X, np.ravel(y))

\# projection

X\_LDA = clf.transform(X)

plt.figure(figsize = (10, 8))

plt.hist(X\_LDA\[0:200\], 21, color = 'r', rwidth = 0.5)

plt.hist(X\_LDA\[200:400\], 21, color = 'b', rwidth = 0.5)

plt.show()

clf.coef\_, clf.intercept\_

(array([[ 9.64218334, 18.00307461]]), array([-48.06306799]))

clf.coef\_/w, thres

(array([[1., 1.]]), 48.06306799151288)