python - Fitting a vector function with curve_fit in Scipy

Question

I want to fit a function with vector output using Scipy's curve_fit (or something more appropriate if available). For example, consider the following function:

import numpy as np
def fmodel(x, a, b):
    return np.vstack([a*np.sin(b*x), a*x**2 - b*x, a*np.exp(b/x)])

Each component is a different function but they share the parameters I wish to fit. Ideally, I would do something like this:

x = np.linspace(1, 20, 50)
a = 0.1
b = 0.5
y = fmodel(x, a, b)
y_noisy = y + 0.2 * np.random.normal(size=y.shape)

from scipy.optimize import curve_fit
popt, pcov = curve_fit(f=fmodel, xdata=x, ydata=y_noisy, p0=[0.3, 0.1])

But curve_fit does not work with functions with vector output, and an error Result from function call is not a proper array of floats. is thrown. What I did instead is to flatten out the output like this:

def fmodel_flat(x, a, b):
    return fmodel(x[0:len(x)/3], a, b).flatten()

popt, pcov = curve_fit(f=fmodel_flat, xdata=np.tile(x, 3),
                       ydata=y_noisy.flatten(), p0=[0.3, 0.1])

and this works. If instead of a vector function I am actually fitting several functions with different inputs as well but which share model parameters, I can concatenate both input and output.

Is there a more appropriate way to fit vector function with Scipy or perhaps some additional module? A main consideration for me is efficiency - the actual functions to fit are much more complex and fitting can take some time, so if this use of curve_fit is mangled and is leading to excessive runtimes I would like to know what I should do instead.

Answer 1

If I can be so blunt as to recommend my own package symfit, I think it does precisely what you need. An example on fitting with shared parameters can be found in the docs.

Your specific problem stated above would become:

from symfit import variables, parameters, Model, Fit, sin, exp

x, y_1, y_2, y_3 = variables('x, y_1, y_2, y_3')
a, b = parameters('a, b')
a.value = 0.3
b.value = 0.1

model = Model({
    y_1: a * sin(b * x), 
    y_2: a * x**2 - b * x, 
    y_3: a * exp(b / x),
})

xdata = np.linspace(1, 20, 50)
ydata = model(x=xdata, a=0.1, b=0.5)
y_noisy = ydata + 0.2 * np.random.normal(size=(len(model), len(xdata)))

fit = Fit(model, x=xdata, y_1=y_noisy[0], y_2=y_noisy[1], y_3=y_noisy[2])
fit_result = fit.execute()

Check out the docs for more!