ex_piecewise.rst

Example: Piecewise continuous function

Piecewise continuus functions can be tricky to fit. However, using the symfit interface this process is made a lot easier. Suppose we want to fit to the following model:

$$\begin{aligned} f(x) = \begin{cases} x^2 - a x & \text{if}\quad x \leq x_0 \\\ a x + b & \text{otherwise} \end{cases} \end{aligned}$$

The script below gives an example of how to fit such a model:

../../examples/piecewise2.py

This code prints:

Parameter Value        Standard Deviation
a         -4.780338e-02 None
b         1.205443e+00 None
x0        1.051163e+00 None
Fitting status message: Optimization terminated successfully.
Number of iterations:   18
Regression Coefficient: 0.9849188499599985

Judging from this graph, another possible solution would have been to also demand a continuous derivative at the point x0. This can be achieved by setting the following constraints instead:

constraints = [
    Eq(y1.diff(x).subs({x: x0}), y2.diff(x).subs({x: x0})),
    Eq(y1.subs({x: x0}), y2.subs({x: x0}))
]

This gives the following plot:

and the following fit report:

Parameter Value        Standard Deviation
a         8.000000e-01 None
b         -6.400000e-01 None
x0        8.000000e-01 None
Fitting status message: Optimization terminated successfully.
Number of iterations:   3
Regression Coefficient: 0.8558226069368662

The first fit is therefore the prevered one, but it does show you how easy it is to set these constraints using symfit.