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I have several data points:

X = np.array([0.01,0.05,0.2,1,10,100,400,2000])
Y = np.array([623,3245.5,710564,332456,540124.5,1851657,5679457,10096844])

And I want to fit these data points to the sigmoid function :

S=D+(A-D)/(1+(X/C)**B)

A,B,C,D are four parameters need to be determined by giving the initial value p(A0,B0,C0,D0).

In addition, each point should add a weight factor to the residual , thus making the sum of square reach global minimum :

weight = ([100,90,80,70,60,50,40,30])

I am a novice in algorithm as well as coding, can anyone give some ideas on how to fit these data? Million thanks!

Here is my Levenberg-Maquardt code:

from numpy import *

def function(p):

    r = array([10*(p[1]-p[0]**2),(1-p[0])])
    fp = dot(transpose(r),r) #= 100*(p[1]-p[0]**2)**2 + (1-p[0])**2
    J = (array([[-20*p[0],10],[-1,0]]))
    grad = dot(transpose(J),transpose(r))
    return fp,r,grad,J

def lm(p0,tol=10**(-5),maxits=100):

    nvars=shape(p0)[0]
    nu=0.01
    p = p0
    fp,r,grad,J = function(p)
    e = sum(dot(transpose(r),r))
    nits = 0
    while nits<maxits and linalg.norm(grad)>tol:
        nits += 1
        fp,r,grad,J = function(p)
        H=dot(transpose(J),J) + nu*eye(nvars)

    pnew = zeros(shape(p))
    nits2 = 0
    while (p!=pnew).all() and nits2<maxits:
        nits2 += 1
        dp,resid,rank,s = linalg.lstsq(H,grad)
        pnew = p-dp
        fpnew,rnew,gradnew,Jnew = function(pnew)
        enew = sum(dot(transpose(rnew),rnew))
        rho = linalg.norm(dot(transpose(r),r)-dot(transpose(rnew),rnew))
        rho /= linalg.norm(dot(transpose(grad),pnew-p))

        if rho>0:
            update = 1
            p = pnew
            e = enew
        if rho>0.25:
            nu=nu/10
        else: 
            nu=nu*10
            update = 0
        print fp, p, e, linalg.norm(grad), nu

p0 = array([-1.92,2])

lm(p0)
ali_m
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canfee
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  • Here's a good example of implementing gradient descent with numpy: http://stackoverflow.com/a/17796231/866915 – ErikR Nov 14 '15 at 14:53
  • Is there a reason why you're not using [`scipy.optimize.curve_fit`](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html#scipy.optimize.curve_fit) or [`scipy.optimize.minimize`](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html#scipy.optimize.minimize) for this? – ali_m Nov 14 '15 at 15:58
  • thank you @ErikR！ I know there have some useful tools like scipy.optimize.leastsq and scipy.optimize.curve_fit, but I always can't get a satisfying result. So I want to rewrite these codes.@ali_m – canfee Nov 15 '15 at 14:30

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