/usr/lib/python2.7/dist-packages/Scientific/Functions/LeastSquares.py is in python-scientific 2.9.4-1.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | # This module contains functions to do general non-linear
# least squares fits.
#
# Written by Konrad Hinsen <hinsen@cnrs-orleans.fr>
# last revision: 2008-8-18
#
"""
Non-linear least squares fitting
Usage example::
from Scientific.N import exp
def f(param, t):
return param[0]*exp(-param[1]/t)
data_quantum = [(100, 3.445e+6),(200, 2.744e+7),
(300, 2.592e+8),(400, 1.600e+9)]
data_classical = [(100, 4.999e-8),(200, 5.307e+2),
(300, 1.289e+6),(400, 6.559e+7)]
print leastSquaresFit(f, (1e13,4700), data_classical)
def f2(param, t):
return 1e13*exp(-param[0]/t)
print leastSquaresFit(f2, (3000.,), data_quantum)
"""
from Scientific import N, LA
from FirstDerivatives import DerivVar
from Scientific import IterationCountExceededError
def _chiSquare(model, parameters, data):
n_param = len(parameters)
chi_sq = 0.
alpha = N.zeros((n_param, n_param))
for point in data:
sigma = 1
if len(point) == 3:
sigma = point[2]
f = model(parameters, point[0])
chi_sq = chi_sq + ((f-point[1])/sigma)**2
d = N.array(f[1])/sigma
alpha = alpha + d[:,N.NewAxis]*d
return chi_sq, alpha
def leastSquaresFit(model, parameters, data, max_iterations=None,
stopping_limit = 0.005,
validator = None):
"""General non-linear least-squares fit using the
X{Levenberg-Marquardt} algorithm and X{automatic differentiation}.
@param model: the function to be fitted. It will be called
with two parameters: the first is a tuple containing all fit
parameters, and the second is the first element of a data point (see
below). The return value must be a number. Since automatic
differentiation is used to obtain the derivatives with respect to the
parameters, the function may only use the mathematical functions known
to the module FirstDerivatives.
@type model: callable
@param parameters: a tuple of initial values for the
fit parameters
@type parameters: C{tuple} of numbers
@param data: a list of data points to which the model
is to be fitted. Each data point is a tuple of length two or
three. Its first element specifies the independent variables
of the model. It is passed to the model function as its first
parameter, but not used in any other way. The second element
of each data point tuple is the number that the return value
of the model function is supposed to match as well as possible.
The third element (which defaults to 1.) is the statistical
variance of the data point, i.e. the inverse of its statistical
weight in the fitting procedure.
@type data: C{list}
@returns: a list containing the optimal parameter values
and the chi-squared value describing the quality of the fit
@rtype: C{(list, float)}
"""
n_param = len(parameters)
p = ()
i = 0
for param in parameters:
p = p + (DerivVar(param, i),)
i = i + 1
id = N.identity(n_param)
l = 0.001
chi_sq, alpha = _chiSquare(model, p, data)
niter = 0
while 1:
delta = LA.solve_linear_equations(alpha+l*N.diagonal(alpha)*id,
-0.5*N.array(chi_sq[1]))
next_p = map(lambda a,b: a+b, p, delta)
if validator is not None:
while not validator(*next_p):
delta *= 0.8
next_p = map(lambda a,b: a+b, p, delta)
next_chi_sq, next_alpha = _chiSquare(model, next_p, data)
if next_chi_sq > chi_sq:
l = 10.*l
else:
l = 0.1*l
if chi_sq[0] - next_chi_sq[0] < stopping_limit: break
p = next_p
chi_sq = next_chi_sq
alpha = next_alpha
niter = niter + 1
if max_iterations is not None and niter == max_iterations:
raise IterationCountExceededError
return [p[0] for p in next_p], next_chi_sq[0]
#
# The special case of n-th order polynomial fits
# was contributed by David Ascher. Note: this could also be
# done with linear least squares, e.g. from LinearAlgebra.
#
def _polynomialModel(params, t):
r = 0.0
for i in range(len(params)):
r = r + params[i]*N.power(t, i)
return r
def polynomialLeastSquaresFit(parameters, data):
"""
Least-squares fit to a polynomial whose order is defined by
the number of parameter values.
@note: This could also be done with a linear least squares fit
from L{Scientific.LA}
@param parameters: a tuple of initial values for the polynomial
coefficients
@type parameters: C{tuple}
@param data: the data points, as for L{leastSquaresFit}
@type data: C{list}
"""
return leastSquaresFit(_polynomialModel, parameters, data)
# Test code
if __name__ == '__main__':
from Scientific.N import exp
def f(param, t):
return param[0]*exp(-param[1]/t)
data_quantum = [(100, 3.445e+6),(200, 2.744e+7),
(300, 2.592e+8),(400, 1.600e+9)]
data_classical = [(100, 4.999e-8),(200, 5.307e+2),
(300, 1.289e+6),(400, 6.559e+7)]
print leastSquaresFit(f, (1e13,4700), data_classical)
def f2(param, t):
return 1e13*exp(-param[0]/t)
print leastSquaresFit(f2, (3000.,), data_quantum)
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