Fit Constraint¶

How constraints on parameters work¶

Consider the scenario where the aim is to fit a lorenzian function to a 1D dataset but a constraint applied on the peak centre parameter. Assume the 1D dataset consists of \(N\) data points \((x_1,y_1^{obs}), (x_2,y_2^{obs}), ... (x_N,y_N^{obs})\), where \(x_i\) is the ith x-value and \(y_i^{obs}\) is the ith observed value for that x-value. Write the lorentzian function as:

where he lorentzian fitting parameters here are

• \(h\) - height of peak (at maximum)
• \(x0\) - centre of peak
• \(w\) - half-width at half-maximum

\(x_i\) is the x-value of the ith data point and \(y_i^{cal}\) is the lorentzian calculated value at that data point.

We want to apply a constraint on the x0 parameter, i.e. the centre of the peak. For example, apply the constraint that \(x0\) should be in between \(x0_{min}\) and \(x0_{max}\). If this is not satisfied we then add the following penalty function to \(y_i^{cal}\) if \(x0 < x0_{min}\):

where \(C\) is a constant (default 1000). The penalty function when \(x0 > x0_{max}\) takes the form:

.

If more than one constraint is defined, then for each violated constraint a penalty of the type defined above is added to the calculated fitting function.

If the penalty C is not the default value of 1000, then the constraint penalty value will be included whenever the function is converted to a string. For example:

from mantid.simpleapi import *
myFunction = Gaussian(Height=1.0, PeakCentre=3.0, Sigma=1.0)
myFunction.constrain("PeakCentre < 6")
print(myFunction)
myFunction.setConstraintPenaltyFactor("PeakCentre", 10.0)
print(myFunction)
myFunction.constrain('Sigma > 0')
print(myFunction)

will output:

name=Gaussian,Height=1,PeakCentre=3,Sigma=1,constraints=(PeakCentre<6)
name=Gaussian,Height=1,PeakCentre=3,Sigma=1,constraints=(PeakCentre<6,penalty=10)
name=Gaussian,Height=1,PeakCentre=3,Sigma=1,constraints=(PeakCentre<6,penalty=10,0<Sigma)`

Category: Concepts