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# FitGaussianPeaks v1¶

## Summary¶

Fits a list of gaussian peaks returning the parameters and the value of a cost function.

## Properties¶

Name

Direction

Type

Default

Description

InputWorkspace

Input

Workspace

Mandatory

Workspace with peaks to be identified

PeakGuessTable

Input

TableWorkspace

Mandatory

Table containing the guess for the peak position

CentreTolerance

Input

number

1

Tolerance value used in looking for peak centre

EstimatedPeakSigma

Input

number

3

Estimate of the peak half width

MinPeakSigma

Input

number

0.1

Minimum value for the standard deviation of a peak

MaxPeakSigma

Input

number

30

Maximum value for the standard deviation of a peak

EstimateFitWindow

Input

boolean

True

If checked, algorithm attempts to calculate number of data points to use from EstimatePeakSigma, if unchecked algorithm will use FitWindowSize argument

FitWindowSize

Input

number

5

Number of data point used to fit peaks, minimum allowed value is 5, value must be an odd number

GeneralFitTolerance

Input

number

0.1

Tolerance for the constraint in the general fit

RefitTolerance

Input

number

0.001

Tolerance for the constraint in the refitting

PeakProperties

Output

TableWorkspace

peak_table

Table containing the properties of the peaks

RefitPeakProperties

Output

TableWorkspace

refit_peak_table

Table containing the properties of the peaks that had to be fitted twice as the firsttime the error was unreasonably large

FitCost

Output

TableWorkspace

fit_cost

Table containing the value of both chi2 and poisson cost functions for the fit

## Description¶

The algorithm takes:

• a table with a single column where each row contains the centre of the peak

• a workspace containing the data to be fitted in the first spectra and a background for the data. This is important to produce sensible results from the Poisson weight.

It then performs the following steps:

1. Perform a preliminary fit of each peak separately to obtain an estimate for the peak parameters.

2. Fit all the peaks together using a gaussian for each peak.

3. Create a table where every row contains the parameters and error for a peak.

4. For some peaks the fit might produce unreasonably large errors (above $$10^7$$). These peaks will not be included in the table. Instead they will be refitted with tighter constraints that will return sensible values for the parameters most of the times. These parameters will be inserted in a second table structured as the first.

5. The fit will be evaluated using unweighted $$\chi^2$$ and a poisson cost function (see below). For the Poisson fit, the background is added. The result of the two are included in a third table.

## Cost functions¶

• $$\chi^2$$:

The result of the fit is compared with the data using the equation: $${1 \over N} \sum_{i=1}^{N} {(m_i - d_i) \over \sigma_i}^2$$

Where $$m_i$$ is the i-th data point of the result of the fit, $$d_i$$ is the i-th data point of the data to be fitted and $$\sigma_i$$ is the error on the i-th data point.

• Poisson:

The result of the fit and input data are filtered to remove zeros in the fitted data. They are then compared using the equation: $$\sum_{i=1}^{N} (-m_i + d_i \ln(m_i))$$

Where $$m_i$$ is the i-th data point of the result of the fit and $$d_i$$ is the i-th data point of the data to be fitted. This is the natural logarithm of the cost, calculated as: $$\prod_{i=1}^{N} \exp(-m_i + d_i \ln(m_i))$$

Example - Finding two simple gaussian peaks.

# Function for a gaussian peak
def gaussian(xvals, centre, height, sigma):
exp_val = (xvals - centre) / (np.sqrt(2) * sigma)

return height * np.exp(-exp_val * exp_val)

# Creating two peaks
x_values = np.linspace(0, 100, 1001)
centre = [25, 75]
height = [35, 20]
width = [10, 5]
y_values = gaussian(x_values, centre, height, width)
y_values += gaussian(x_values, centre, height, width)
background = 10 * np.ones(len(x_values))

# Generating a table with a guess of the position of the centre of the peaks
peak_table = CreateEmptyTableWorkspace()

# Generating a workspace with the data and a flat background
data_ws = CreateWorkspace(DataX=np.concatenate((x_values, x_values)),
DataY=np.concatenate((y_values, background)),
DataE=np.sqrt(np.concatenate((y_values, background))),
NSpec=2)

# Fitting the data
parameters, refitted_parameters, cost = FitGaussianPeaks(
InputWorkspace=data_ws,
PeakGuessTable=peak_table,
CentreTolerance=3.0,
EstimatedPeakSigma=5,
MinPeakSigma=0.0,
MaxPeakSigma=30.0,
GeneralFitTolerance=0.1,
RefitTolerance=0.001
)

peak1 = parameters.row(0)
peak2 = parameters.row(1)
print('Peak 1: centre={:.2f}+/-{:.2f}, height={:.2f}+/-{:.2f}, sigma={:.2f}+/-{:.1f}'
.format(peak1['centre'], peak1['error centre'],
peak1['height'], peak1['error height'],
peak1['sigma'], peak1['error sigma']))
print('Peak 2: centre={:.2f}+/-{:.2f}, height={:.2f}+/-{:.2f}, sigma={:.2f}+/-{:.1f}'
.format(peak2['centre'], peak2['error centre'],
peak2['height'], peak2['error height'],
peak2['sigma'], peak2['error sigma']))
print('Chi2 cost: {:.3f}'.format(cost.column(0)))
print('Poisson cost: {:.3f}'.format(cost.column(1)))


Output (the number on your machine may differ slightly from these:

Peak 1: centre=25.00+/-0.11, height=35.00+/-0.47, sigma=10.00+/-0.1
Peak 2: centre=75.00+/-0.10, height=20.00+/-0.49, sigma=5.00+/-0.1
Chi2 cost: 0.000
Poisson cost: 46444.723


Categories: AlgorithmIndex | Optimization\PeakFinding

## Source¶

Python: FitGaussianPeaks.py