3.12

CalculateChiSquared v1

../_images/CalculateChiSquared-v1_dlg.png

CalculateChiSquared dialog.

Summary

Calculate chi squared for a function and a data set in a workspace.

Properties

Name Direction Type Default Description
Function InOut Function Mandatory Parameters defining the fitting function and its initial values
InputWorkspace Input Workspace Mandatory Name of the input Workspace
IgnoreInvalidData Input boolean False Flag to ignore infinities, NaNs and data with zero errors.
DomainType Input string Simple The type of function domain to use: Simple, Sequential, or Parallel. Allowed values: [‘Simple’, ‘Sequential’, ‘Parallel’]
EvaluationType Input string CentrePoint The way the function is evaluated on histogram data sets. If value is “CentrePoint” then function is evaluated at centre of each bin. If it is “Histogram” then function is integrated within the bin and the integrals returned. Allowed values: [‘CentrePoint’, ‘Histogram’]
PeakRadius Input number 0 A value of the peak radius the peak functions should use. A peak radius defines an interval on the x axis around the centre of the peak where its values are calculated. Values outside the interval are not calculated and assumed zeros.Numerically the radius is a whole number of peak widths (FWHM) that fit into the interval on each side from the centre. The default value of 0 means the whole x axis.
ChiSquared Output number   Output value of chi squared.
ChiSquaredDividedByDOF Output number   Output value of chi squared divided by the number of degrees of freedom (NofData - nOfParams).
ChiSquaredDividedByNData Output number   Output value of chi squared divided by the number of data points).
ChiSquaredWeighted Output number   Output value of weighted chi squared.
ChiSquaredWeightedDividedByDOF Output number   Output value of weighted chi squared divided by the number of degrees of freedom (NofData - nOfParams).
ChiSquaredWeightedDividedByNData Output number   Output value of weighted chi squared divided by the number of data points).
Output Input string   A base name for output workspaces.
Weighted Input boolean False Option to use the weighted chi squared in error estimation. Default is false.

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:

\chi_{1}^{2} = \sum_{i} (y_i - f_i)^2

where y_i and 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):

\chi_{2}^{2} = \frac{1}{DOF}\sum_{i} (y_i - f_i)^2

DOF = N_d - N_p where N_d is the number of data points used in fitting and 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:

\chi_{3}^{2} = \sum_{i} \left(\frac{y_i - f_i}{\sigma_i}\right)^2

Finally, ChiSquaredWeightedDividedByDOF is

\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 \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 \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 \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 1\sigma. Estimated from analisys of the surface.
Right Error The positive deviation from the minimum point equivalent to 1\sigma. Estimated from analisys of the surface.
Quadratic Error 1\sigma standard deviation in the quadratic approximation of the surface.
Chi2 Min The value of \chi^{2} at the minimum relative to the test point.

The pdf table contains slices of the \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 \chi^{2} and the third is the probability density function normalised to have 1 at the maximum.

Usage

Example 1

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,chi2ndata,chi2W,chi2Wdof,chi2Wndata = CalculateChiSquared(func,ws)

print('Chi squared is {:.13f}'.format(chi2))
print('Chi squared / DOF is {:.14f}'.format(chi2dof))
print('Chi squared / NDATA is {:.14f}'.format(chi2ndata))
print('Chi squared weighted is {:.11f}'.format(chi2W))
print('Chi squared weighted / DOF is {:.14f}'.format(chi2Wdof))
print('Chi squared weighted / NDATA is {:.12f}'.format(chi2Wndata))

# Define a function that models the data exactly
func = 'name=LinearBackground,A0=1.0,A1=2.0'

# Calculate the chi squared
chi2,chi2dof,chi2ndata,chi2W,chi2Wdof,chi2Wndata = CalculateChiSquared(func,ws)

print('Chi squared is {:.1f}'.format(chi2))
print('Chi squared / DOF is {:.1f}'.format(chi2dof))
print('Chi squared / NDATA is {:.1f}'.format(chi2ndata))
print('Chi squared weighted is {:.1f}'.format(chi2W))
print('Chi squared weighted / DOF is {:.1f}'.format(chi2Wdof))
print('Chi squared weighted / NDATA is {:.1f}'.format(chi2Wndata))

Output:

Chi squared is 0.0351851851852
Chi squared / DOF is 0.00439814814815
Chi squared / NDATA is 0.00351851851852
Chi squared weighted is 0.02660287840
Chi squared weighted / DOF is 0.00332535979971
Chi squared weighted / NDATA is 0.002660287840
Chi squared is 0.0
Chi squared / DOF is 0.0
Chi squared / NDATA is 0.0
Chi squared weighted is 0.0
Chi squared weighted / DOF is 0.0
Chi squared weighted / NDATA is 0.0

Example 2

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

Categories: Algorithms | Optimization

Source

C++ source: CalculateChiSquared.cpp (last modified: 2017-03-01)

C++ header: CalculateChiSquared.h (last modified: 2016-02-12)