 # Fisher's linear discriminant

Mar 1st, 2021
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1. import numpy as np
2. from scipy.stats import norm
3.
4. # Separates two classes and calculates
5. # rejection criteria and probability.
6. # Input: classes a and b, fraction of class a
7. # fa, data point x = (x1, x2)
8. # Output: a rejection criterion a0,
9. # b selection efficiency be,
10. # probabilities for classes pa and pb.
11. def fdisc(a, b, fa, x1, x2):
12.     acm = np.cov(a[:,0], a[:,1]) # covariances
13.     bcm = np.cov(b[:,0], b[:,1])
14.
15.     apb = acm + bcm # sum of covariances
16.     apbi = np.linalg.inv(apb) # invert
17.
18.     # means for classes a and b
19.     am = np.array((np.mean(a[:,0]), np.mean(a[:,1])))
20.     bm = np.array((np.mean(b[:,0]), np.mean(b[:,1])))
21.
22.     c = apbi @ (am - bm).T # projection vector
23.
24.     anew = c @ a.T # x3 data
25.     bnew = c @ b.T
26.
27.     anew = np.sort(anew)
28.     bnew = np.sort(bnew)
29.     a0 = anew[int((1. - fa) * len(anew))] # pick value based on fraction
30.     be = len(bnew[bnew < a0]) / len(bnew) # efficiency
31.
32.     xnew = c @ (x1, x2) # point (x1, x2)
33.     pa = norm.cdf(xnew, c @ am, np.std(anew)) # calculate probabilities
34.     pb = 1.-norm.cdf(xnew, c @ bm, np.std(bnew))
35.
36.     return a0, be, pa, pb
37.
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