Newer
Older
.. algorithm::
.. summary::
.. alias::
.. properties::
Description
-----------
Calculates a measure of the goodness of a fit in four ways.
The ChiSquared property returns the sum of squares of differences between the calculated and measured values:
:math:`\chi_{1}^{2} = \sum_{i} (y_i - f_i)^2`
where :math:`y_i` and :math:`f_i` are the measured and calculated values at i-th point.
The ChiSquaredDividedByDOF is ChiSquared divided by the number of degrees of freedom (DOF):
:math:`\chi_{2}^{2} = \frac{1}{DOF}\sum_{i} (y_i - f_i)^2`
:math:`DOF = N_d - N_p` where :math:`N_d` is the number of data points used in fitting and :math:`N_p`
is the number of free (not fixed or tied) parameters of the function.
The ChiSquaredWeighted property sums the squares of the differences divided by the data errors:
:math:`\chi_{3}^{2} = \sum_{i} \left(\frac{y_i - f_i}{\sigma_i}\right)^2`
Finally, ChiSquaredWeightedDividedByDOF is
:math:`\chi_{4}^{2} = \chi_{3}^{2} / DOF`
Parameter errors
================
Setting the Output property to a non-empty string makes the algorithm explore the surface of the :math:`\chi^{2}`
around its minimum and estimate the standard deviations for the parameters. The value of the property is a base name
for two output table workspaces: '<Output>_errors' and '<Output>_pdf'. The former workspace contains parameter error
estimates and the latter shows :math:`\chi^{2}`'s 1d slices along each parameter (keeping all other fixed).
The error table has the following columns:
=============== ===========
Column Description
=============== ===========
Parameter Parameter name
Value Parameter value passed with the Function property
Value at Min The minimum point of the 1d slice of the :math:`\chi^{2}`. If the Function is at the minimum then
Value at Min should be equal to Value.
Left Error The negative deviation from the minimum point equivalent to :math:`1\sigma`. Estimated from analisys
of the surface.
Right Error The positive deviation from the minimum point equivalent to :math:`1\sigma`. Estimated from analisys
of the surface.
Quadratic Error :math:`1\sigma` standard deviation in the quadratic approximation of the surface.
Chi2 Min The value of :math:`\chi^{2}` at the minimum relative to the test point.
=============== ===========
The pdf table contains slices of the :math:`\chi^{2}` along each parameter. It has 3 column per parameter. The first column of the 3
is the parameter values, the second has the :math:`\chi^{2}` and the third is the probability density function normalised to
have 1 at the maximum.
.. testcode:: CalculateChiSquaredExample
import numpy as np
# Create a data set
x = np.linspace(0,1,10)
y = 1.0 + 2.0 * x
e = np.sqrt(y)
ws = CreateWorkspace(DataX=x, DataY=y, DataE=e)
# Define a function
func = 'name=LinearBackground,A0=1.1,A1=1.9'
# Calculate the chi squared
chi2,chi2dof,chi2W,chi2Wdof = CalculateChiSquared(func,ws)
print 'Chi squared is %s' % chi2
print 'Chi squared / DOF is %s' % chi2dof
print 'Chi squared weighted is %s' % chi2W
print 'Chi squared weighted / DOF is %s' % chi2Wdof
print
# Define a function that models the data exactly
func = 'name=LinearBackground,A0=1.0,A1=2.0'
# Calculate the chi squared
chi2,chi2dof,chi2W,chi2Wdof = CalculateChiSquared(func,ws)
print 'Chi squared is %s' % chi2
print 'Chi squared / DOF is %s' % chi2dof
print 'Chi squared weighted is %s' % chi2W
print 'Chi squared weighted / DOF is %s' % chi2Wdof
Output:
.. testoutput:: CalculateChiSquaredExample
Chi squared is 0.0351851851852
Chi squared / DOF is 0.00439814814815
Chi squared weighted is 0.0266028783977
Chi squared weighted / DOF is 0.00332535979971
Chi squared is 0.0
Chi squared / DOF is 0.0
Chi squared weighted is 0.0
Chi squared weighted / DOF is 0.0
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
**Example 2**
.. testcode::
import numpy as np
# Create a workspace and fill it with some gaussian data and some noise
n = 100
x = np.linspace(-10,10,n)
y = np.exp(-x*x/2) + np.random.normal(0.0, 0.01, n)
e = [1] * n
ws = CreateWorkspace(x,y,e)
# Gefine a Gaussian with exactly the same parameters that were used to
# generate the data
fun_t = 'name=Gaussian,Height=%s,PeakCentre=%s,Sigma=%s'
fun = fun_t % (1, 0, 1)
# Test the chi squared.
CalculateChiSquared(fun,ws,Output='Test0')
# Check the Test0_errors table and see that the parameters are not at minimum
# Fit the function
res = Fit(fun,ws,Output='out')
# res[3] is a table with the fitted parameters
nParams = res[3].rowCount() - 1
params = [res[3].cell(i,1) for i in range(nParams)]
# Build a new function and populate it with the fitted parameters
fun = fun_t % tuple(params)
# Test the chi squared.
CalculateChiSquared(fun,ws,Output='Test1')
# Check the Test1_errors table and see that the parameters are at minimum now
.. sourcelink::