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:

$y_i^{cal}(h, x0, w) = h \left( \frac{w^2}{(x_i-x0)^2+w^2} \right)$

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 $x0 < x0_{min}</$ :

$p_i = C(x0_{min}-x0)*R(x_i)$

where $C$ is a constant (default 1000) and $R(x_i)$ a spiky function which takes the value 1 for the first and last data point and for every 10th data point from the 1st data point, but is otherwise zero. The penalty function when $x0 > x0_{min}</$ takes the form:

$p_i = C(x0-x0_{max})*R(x_i)$

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.

Category: Concepts