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# CalculateChiSquared v1¶

## 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’]

StepSizeMethod

Input

string

Default

The way the step size is calculated for numerical derivatives. See the section about step sizes in the Fit algorithm documentation to understand the difference between “Default” and “Sqrt epsilon”. Allowed values: [‘Default’, ‘Sqrt epsilon’]

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).

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

$$\chi_{4}^{2} = \chi_{3}^{2} / DOF$$

## 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

Categories: AlgorithmIndex | Optimization