How to apply a constraint to a power law model so that it passes through a specific data point (Python)

Question

I would like to develop a power law model using lmfit (or another library). Is there any way I can apply a constraint or bound to the model so that it has to pass through a specific data point (exp. the point (1, 2))

plm = models.PowerLawModel()

params = plm.guess(y_data, x=x_data)
result = plm.fit(y_data, params, x=x_data, )
const = round(result.best_values['amplitude'], 3)
exp = round(result.best_values['exponent'], 3)
best = result.best_fit

Answer 1

For a Power Law model, which is just y = Amplitude * x**Exponent, it would be pretty easy to write a constraint to force the model to always go through a point (x0, y0). You want Amplitude to be such that

 y0 = Amplitude*x0**Exponent

for any value of Exponent. That is

 Amplitude = y0*x0**(-Exponent)

You can specify that in lmfit with something like

plm = models.PowerLawModel()
params = plm.guess(y_data, x=x_data)
params['amplitude'].expr = '{0:g}*{1:g}**(-exponent)'.format(y0, x0)

This will make amplitude not vary independently from exponent in the fit, and always take values that depend on the value of exponent and this expression.