SciPy comes with a least squares Levenberg-Marquardt implementation. This allows you to minimize functions. By defining your function as the difference between some measurements and your model function, you can fit a model to those measurements.

Sometimes your model contains multiple functions. You can also minimize for all functions using this approach:

- Define your functions that you like to minimize A(p0), B(P1), …

their cumulative paramaters will be a tuple (p0, p1, …). - Define your function to be minimized as f(x0), where x0 is expanded to the parameter tuple.
- The function f returns a vector of differences between discrete measured sample and the individual functions A, B etc.
- Let SciPy minimize this function, starting with a reasonably selected initial parameter vector.

This is an example implementation:

import math

import scipy.optimize

measured = {

1: [ 0, 0.02735, 0.47265 ],

6: [ 0.0041, 0.09335, 0.40255 ],

10: [ 0.0133, 0.14555, 0.34115 ],

20: [ 0.0361, 0.205, 0.2589 ],

30: [ 0.06345, 0.23425, 0.20225 ],

60: [ 0.132, 0.25395, 0.114 ],

90: [ 0.2046, 0.23445, 0.06095 ],

120: [ 0.2429, 0.20815, 0.04895 ],

180: [ 0.31755, 0.1618, 0.02065 ],

240: [ 0.3648, 0.121, 0.0142 ],

315: [ 0.3992, 0.0989, 0.00195 ]

}

def A( x, a, k ):

return a * math.exp( -x * k )

def B( x, a, k, l ):

return k * a / ( l - k ) * ( math.exp( -k * x ) - math.exp( -l * x ) )

def C( x, a, k, l ):

return a * ( 1 - l / ( l - k ) * math.exp( -x * k ) + k / ( l - k ) * math.exp( -x * l ) )

def f( x0 ):

a, k, l = x0

error = []

for x in measured:

error += [ C( x, a, k, l ) - measured[ x ][ 0 ],

B( x, a, k, l ) - measured[ x ][ 1 ],

A( x, a, k ) - measured[ x ][ 2 ]

]

return error

def main():

x0 = ( 0.46, 0.01, 0.001 ) # initial parameters for a, k and l

x, cov, infodict, mesg, ier = scipy.optimize.leastsq( f, x0, full_output = True, epsfcn = 1.0e-2 )

print x

if __name__ == "__main__":

main()

SciPy returns a lot more information, not only the final parameters. See their documentation for details. You also may want to tweak epsfcn for a better fit. This depends on your functions shape and properties.