Implementation of Gauss-Newton method from Wikipedia example
I am relatively new to Python and am trying to implement the Gauss-Newton method, specifically an example on the Wikipedia page for it ( Gauss-Newton algorithm , 3 examples). Here's what I've done so far:
import scipy
import numpy as np
import math
import scipy.misc
from matplotlib import pyplot as plt, cm, colors
S = [0.038,0.194,.425,.626,1.253,2.500,3.740]
rate = [0.050,0.127,0.094,0.2122,0.2729,0.2665,0.3317]
iterations = 5
rows = 7
cols = 2
B = np.matrix([[.9],[.2]]) # original guess for B
Jf = np.zeros((rows,cols)) # Jacobian matrix from r
r = np.zeros((rows,1)) #r equations
def model(Vmax, Km, Sval):
return ((vmax * Sval) / (Km + Sval))
def partialDerB1(B2,xi):
return round(-(xi/(B2+xi)),10)
def partialDerB2(B1,B2,xi):
return round(((B1*xi)/((B2+xi)*(B2+xi))),10)
def residual(x,y,B1,B2):
return (y - ((B1*x)/(B2+x)))
for i in range(0,iterations):
sumOfResid=0
#calculate Jr and r for this iteration.
for j in range(0,rows):
r[j,0] = residual(S[j],rate[j],B[0],B[1])
sumOfResid = sumOfResid + (r[j,0] * r[j,0])
Jf[j,0] = partialDerB1(B[1],S[j])
Jf[j,1] = partialDerB2(B[0],B[1],S[j])
Jft = np.transpose(Jf)
B = B + np.dot((np.dot(Jft,Jf)**-1),(np.dot(Jft,r)))
print B
The sum of the squares of the residuals is increasing rather than going to zero with each iteration, and my resulting vector is B
increasing.
I am having a hard time figuring out where my problem is and any help would be appreciated.
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1 answer
You are wrong about the beta update code: it should be
B = B - np.dot(np.dot( inv(np.dot(Jft, Jf)), Jft), r)
instead of **-1
on the matrix to calculate the inverse matrix
import scipy
import numpy as np
from numpy.linalg import inv
import math
import scipy.misc
#from matplotlib import pyplot as plt, cm, colors
S = [0.038,0.194,.425,.626,1.253,2.500,3.740]
rate = [0.050,0.127,0.094,0.2122,0.2729,0.2665,0.3317]
iterations = 5
rows = 7
cols = 2
B = np.matrix([[.9],[.2]]) # original guess for B
print(B)
Jf = np.zeros((rows,cols)) # Jacobian matrix from r
r = np.zeros((rows,1)) #r equations
def model(Vmax, Km, Sval):
return ((Vmax * Sval) / (Km + Sval))
def partialDerB1(B2,xi):
return round(-(xi/(B2+xi)),10)
def partialDerB2(B1,B2,xi):
return round(((B1*xi)/((B2+xi)*(B2+xi))),10)
def residual(x,y,B1,B2):
return (y - ((B1*x)/(B2+x)))
#
for _ in xrange(iterations):
sumOfResid=0
#calculate Jr and r for this iteration.
for j in xrange(rows):
r[j,0] = residual(S[j],rate[j],B[0],B[1])
sumOfResid += (r[j,0] * r[j,0])
Jf[j,0] = partialDerB1(B[1],S[j])
Jf[j,1] = partialDerB2(B[0],B[1],S[j])
Jft = Jf.T
B -= np.dot(np.dot( inv(np.dot(Jft,Jf)),Jft),r)
print B
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