Table of Contents
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. |
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:
where and
are the measured and calculated values at i-th point.
The ChiSquaredDividedByDOF is ChiSquared divided by the number of degrees of freedom (DOF):
where
is the number of data points used in fitting and
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:
Finally, ChiSquaredWeightedDividedByDOF is
Setting the Output property to a non-empty string makes the algorithm explore the surface of the
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
‘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 ![]() |
Left Error | The negative deviation from the minimum point equivalent to ![]() |
Right Error | The positive deviation from the minimum point equivalent to ![]() |
Quadratic Error | ![]() |
Chi2 Min | The value of ![]() |
The pdf table contains slices of the along each parameter. It has 3 column per parameter. The first column of the 3
is the parameter values, the second has the
and the third is the probability density function normalised to
have 1 at the maximum.
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: Algorithm Index | Optimization
C++ source: CalculateChiSquared.cpp (last modified: 2018-10-05)
C++ header: CalculateChiSquared.h (last modified: 2018-10-05)