Class UAC

Univariate Additive Classifier

A classifier of the form:
    Classify x as class y = sgn( s(x) + c sgn(x-b) )
where s(x) is a score function, (c,b) are unknown parameters.
Suppose there are N training samples (x_n, y_n, s(x_n)).
Let:
    + B(b) = (sgn(x_1-b), sgn(x_2-b), ..., sgn(x_N-b))
    + F(c,b) = (sgn(s(x_1) + c sgn(x_1-b)), ..., sgn(s(x_N) + c sgn(x_N-b)))
When training with N samples (x_n, y_n, F(x_n)), it is provable that
    + R can be partitioned into at most N+1 intervals such that in any
      interval, B(b) is fixed for different values of b
    + Given b is known, R can be partitioned into at most N+1 intervals such
      that in any interval, F(c,b) is fixed for different values of c
    + Given b is fixed but unknown, R can be partitioned into at most 2N+1
      intervals such that in any interval, F(c,b) is fixed for different
      values of c
s(x) must not be 0 for all x_n

Find b, c such that:
If crit = 0: to minimize thelambda*FRR(b,c)+FAR(b,c),
    where thelambda = param1
If crit = 1: to minimize FAR(b,c) subject to FRR(b,c) <= 1-minDR,
    where minDR = param1
If crit = 2: to minimize FRR(b,c) subject to FAR(b,c) <= maxFAR,
    where maxFAR = param1
Return:
    bparam: (b,c)
    bsol: (FRR,FAR)