# Fitting Types¶

## Fit (Least Squares)¶

The default fitting object does least-squares fitting:

from symfit import parameters, variables, Fit
import numpy as np

# Define a model to fit to.
a, b = parameters('a, b')
x = variables('x')
model = a * x + b

# Generate some data
xdata = np.linspace(0, 100, 100)  # From 0 to 100 in 100 steps
a_vec = np.random.normal(15.0, scale=2.0, size=(100,))
b_vec = np.random.normal(100.0, scale=2.0, size=(100,))
# Point scattered around the line 5 * x + 105
ydata = a_vec * xdata + b_vec

fit = Fit(model, xdata, ydata)
fit_result = fit.execute()

The Fit object also supports standard deviations. In order to provide these, it’s nicer to use a named model:

a, b = parameters('a, b')
x, y = variables('x, y')
model = {y: a * x + b}

fit = Fit(model, x=xdata, y=ydata, sigma_y=sigma)

Warning

symfit assumes these sigma to be from measurement errors by default, and not just as a relative weight. This means the standard deviations on parameters are calculated assuming the absolute size of sigma is significant. This is the case for measurement errors and therefore for most use cases symfit was designed for. If you only want to use the sigma for relative weights, then you can use absolute_sigma=False as a keyword argument.

Please note that this is the opposite of the convention used by scipy’s curve_fit(). Looking through their mailing list this seems to have been implemented the opposite way for historical reasons, and was understandably never changed so as not to lose backwards compatibility. Since this is a new project, we don’t have that problem.

## Constrained Least Squares Fit¶

The Fit takes a constraints keyword; a list of relationships between the parameters that has to be respected. As an example of fitting with constraints, we could imagine fitting the angles of a triangle:

a, b, c = parameters('a, b, c')
a_i, b_i, c_i = variables('a_i, b_i, c_i')

model = {a_i: a, b_i: b, c_i: c}

data = np.array([
[10.1, 9., 10.5, 11.2, 9.5, 9.6, 10.],
[102.1, 101., 100.4, 100.8, 99.2, 100., 100.8],
[71.6, 73.2, 69.5, 70.2, 70.8, 70.6, 70.1],
])

fit = Fit(
model=model,
a_i=data[0],
b_i=data[1],
c_i=data[2],
constraints=[Equality(a + b + c, 180)]
)
fit_result = fit.execute()

The line constraints=[Equality(a + b + c, 180)] ensures the our basic knowledge of geometry is respected despite my sloppy measurements.

Note

Under the hood, a different Minimizer is used to perform a constrained fit. Fit tries to select the right Minimizer based on the problem you present it with. See Fit for more.

## Likelihood¶

Given a dataset and a model, what values should the model’s parameters have to make the observed data most likely? This is the principle of maximum likelihood and the question the Likelihood object can answer for you.

Example:

from symfit import Parameter, Variable, exp
from symfit.core.objectives import LogLikelihood
import numpy as np

# Define the model for an exponential distribution (numpy style)
beta = Parameter('beta')
x = Variable('x')
model = (1 / beta) * exp(-x / beta)

# Draw 100 samples from an exponential distribution with beta=5.5
data = np.random.exponential(5.5, 100)

# Do the fitting!
fit = Fit(model, data, objective=LogLikelihood)
fit_result = fit.execute()

fit_result is a normal FitResults object. As always, bounds on parameters and even constraints are supported.

Multiple data sets can be likelihood fitted simultaneously by merging this example with that of global fitting, see Example: Global Likelihood fitting in the example section.

## Minimize/Maximize¶

Minimize or Maximize a model subject to bounds and/or constraints. As an example, I present an example from the scipy docs.

Suppose we want to maximize the following function:

$f(x,y) = 2xy + 2x - x^2 - 2y^2$

Subject to the following constraints:

$x^3 - y = 0$
$y - 1 >= 0$

In SciPy code the following lines are needed:

def func(x, sign=1.0):
""" Objective function """
return sign*(2*x[0]*x[1] + 2*x[0] - x[0]**2 - 2*x[1]**2)

def func_deriv(x, sign=1.0):
""" Derivative of objective function """
dfdx0 = sign*(-2*x[0] + 2*x[1] + 2)
dfdx1 = sign*(2*x[0] - 4*x[1])
return np.array([ dfdx0, dfdx1 ])

cons = ({'type': 'eq',
'fun' : lambda x: np.array([x[0]**3 - x[1]]),
'jac' : lambda x: np.array([3.0*(x[0]**2.0), -1.0])},
{'type': 'ineq',
'fun' : lambda x: np.array([x[1] - 1]),
'jac' : lambda x: np.array([0.0, 1.0])})

res = minimize(func, [-1.0,1.0], args=(-1.0,), jac=func_deriv,
constraints=cons, method='SLSQP', options={'disp': True})

Takes a couple of read-throughs to make sense, doesn’t it? Let’s do the same problem in symfit:

from symfit import parameters, Maximize, Eq, Ge

x, y = parameters('x, y')
model = 2*x*y + 2*x - x**2 -2*y**2
constraints = [
Eq(x**3 - y, 0),
Ge(y - 1, 0),
]

fit = Fit(- model, constraints=constraints)
fit_result = fit.execute()

Done! symfit will determine all derivatives automatically, no need for you to think about it. Notice the minus sign in the call to Fit. This is because Fit will always minimize, so in order to achieve maximization we should minimize - model.

Warning

You might have noticed that x and y are Parameter’s in the above problem, which may strike you as weird. However, it makes perfect sense because in this problem they are parameters to be optimised, not independent variables. Furthermore, this way of defining it is consistent with the treatment of Variable’s and Parameter’s in symfit. Be aware of this when minimizing such problems, as the whole process won’t work otherwise.

## ODE Fitting¶

Fitting to a system of ordinary differential equations (ODEs) is also remarkedly simple with symfit. Let’s do a simple example from reaction kinetics. Suppose we have a reaction A + A -> B with rate constant $$k$$. We then need the following system of rate equations:

\begin{align}\begin{aligned}\frac{dA}{dt} = -k A^2\\\frac{dB}{dt} = k A^2\end{aligned}\end{align}

In symfit, this becomes:

model_dict = {
D(a, t): - k * a**2,
D(b, t): k * a**2,
}

We see that the symfit code is already very readable. Let’s do a fit to this:

tdata = np.array([10, 26, 44, 70, 120])
adata = 10e-4 * np.array([44, 34, 27, 20, 14])
a, b, t = variables('a, b, t')
k = Parameter('k', 0.1)
a0 = 54 * 10e-4

model_dict = {
D(a, t): - k * a**2,
D(b, t): k * a**2,
}

ode_model = ODEModel(model_dict, initial={t: 0.0, a: a0, b: 0.0})

fit = Fit(ode_model, t=tdata, a=adata, b=None)
fit_result = fit.execute()

That’s it! An ODEModel behaves just like any other model object, so Fit knows how to deal with it! Note that since we don’t know the concentration of B, we explicitly set b=None when calling Fit so it will be ignored.

Warning

Fitting to ODEs is extremely difficult from an algorithmic point of view since these systems are usually very sensitive to the parameters. Using (very) good initial guesses for the parameters and initial values is critical.

Upon every iteration of performing the fit, the ODEModel is integrated again from the initial point using the new guesses for the parameters.

We can plot it just like always:

# Generate some data
tvec = np.linspace(0, 500, 1000)

A, B = ode_model(t=tvec, **fit_result.params)
plt.plot(tvec, A, label='[A]')
plt.plot(tvec, B, label='[B]')
plt.legend()
plt.show()

As an example of the power of symfit’s ODE syntax, let’s have a look at a system with 2 equilibria: compound AA + B <-> AAB and AAB + B <-> BAAB.

In symfit these can be implemented as:

AA, B, AAB, BAAB, t = variables('AA, B, AAB, BAAB, t')
k, p, l, m = parameters('k, p, l, m')

AA_0 = 10  # Some made up initial amound of [AA]
B = AA_0 - BAAB + AA  # [B] is not independent.

model_dict = {
D(BAAB, t): l * AAB * B - m * BAAB,
D(AAB, t): k * A * B - p * AAB - l * AAB * B + m * BAAB,
D(A, t): - k * A * B + p * AAB,
}

The result is as readable as one can reasonably expect from a multicomponent system (and while using chemical notation). Let’s plot the model for some kinetics constants:

model = ODEModel(model_dict, initial={t: 0.0, AA: AA_0, AAB: 0.0, BAAB: 0.0})

# Generate some data
tdata = np.linspace(0, 3, 1000)
# Eval the normal way.
AA, AAB, BAAB = model(t=tdata, k=0.1, l=0.2, m=0.3, p=0.3)

plt.plot(tdata, AA, color='red', label='[AA]')
plt.plot(tdata, AAB, color='blue', label='[AAB]')
plt.plot(tdata, BAAB, color='green', label='[BAAB]')
plt.plot(tdata, B(BAAB=BAAB, AA=AA), color='pink', label='[B]')
# plt.plot(tdata, AA + AAB + BAAB, color='black', label='total')
plt.legend()
plt.show()

More common examples, such as dampened harmonic oscillators also work as expected:

# Oscillator strength
k = Parameter('k')
# Mass, just there for the physics
m = 1
# Dampening factor
gamma = Parameter('gamma')

x, v, t = symfit.variables('x, v, t')

# Define the force based on Hooke's law, and dampening
a = (-k * x - gamma * v)/m
model_dict = {
D(x, t): v,
D(v, t): a,
}
ode_model = ODEModel(model_dict, initial={t: 0, v: 0, x: 1})

# Let's create some data...
times = np.linspace(0, 15, 150)
data = ode_model(times, k=11, gamma=0.9, m=m.value).x
# ... and add some noise to it.
noise = np.random.normal(1, 0.1, data.shape)  # 10% error
data *= noise

fit = Fit(ode_model, t=times, x=data)
fit_result = fit.execute()

Note

Evaluating the model above will produce a named tuple with values for both x and v. Since we are only interested in the values for x, we immediately select it with .x.

## Fitting multiple datasets¶

A common fitting problem is to fit to multiple datasets. This is sometimes referred to as global fitting. In such fits parameters might be shared between the fits to the different datasets. The same syntax used for ODE fitting makes this problem very easy to solve in symfit.

As a simple example, suppose we have two datasets measuring exponential decay, with the same background, but the different amplitude and decay rate.

$f(x) = y_0 + a * e^{- b * x}$

In order to fit to this, we define the following model:

x_1, x_2, y_1, y_2 = variables('x_1, x_2, y_1, y_2')
y0, a_1, a_2, b_1, b_2 = parameters('y0, a_1, a_2, b_1, b_2')

model = Model({
y_1: y0 + a_1 * exp(- b_1 * x_1),
y_2: y0 + a_2 * exp(- b_2 * x_2),
})

Note that y0 is shared between the components. Fitting is then done in the normal way:

fit = Fit(model, x_1=xdata1, x_2=xdata2, y_1=ydata1, y_2=ydata2)
fit_result = fit.execute()

Any Model that comes to mind is fair game. Behind the scenes symfit will build a least squares function where the residues of all the components are added squared, ready to be minimized. Unlike in the above example, the x-axis does not even have to be shared between the components.

Warning

The regression coefficient is not properly defined for vector-valued models, but it is still listed! Until this is fixed, please recalculate it on your own for every component using the bestfit parameters.

Do not cite the overall $$R^2$$ given by symfit.

## Fitting multidimensional datasets¶

So far we have only considered only considered problems with a single independent variable, but in the real world it is quite common to have problems with multiple independent variables. For example, a specific property over a grid, like the temperature of a surface. In that case, you have a three-dimensional dataset consisting of (x-, y-coordinates, temperature), and our model is a function $$T(x, y; \vec{p})$$, where $$\vec{p}$$ indicates the collection of parameters to be determined during the fit.

Let’s work this out with the following mathematical model. We have a polynomial function with two coefficients, representing two terms of mixed order in x and y:

$$T(x, y) = z = c_2 x^4 y^5 + c_1 x y^2$$

Secondly, we have to implement our model:

x, y, z = variables('x, y, z')
c1, c2 = parameters('c1, c2')
model_dict = {z: Poly( {(1, 2): c1, (4, 5): c2}, x ,y)}
model = Model(model_dict)
# prints z(x, y; c1, c2) = Poly(c2*x**4*y**5 + c1*x*y**2, x, y, domain='ZZ[c1,c2]')

Now we can fit this polynomial model to some mock data. We have to be careful that xdata, ydata and zdata are two-dimensional:

x = np.linspace(0, 100, 100)
y = np.linspace(0, 100, 100)
xdata, ydata = np.meshgrid(x, y)
zdata = 42 * xdata**4 * ydata**5 + 3.14 * xdata * ydata**2

fit = Fit(model, x=xdata, y=ydata, z=zdata)
fit_result = fit.execute()

In conclusion, we made a mathematical model for a multidimensional function with two fit parameters, implemented this model, and fed it data to get a result.

## Global Minimization¶

Very often, there are multiple solutions to a fitting (or minimisation) problem. These are local minima of the objective function. The best solution, of course, is the global minimum, but most minimization algorithms will only find a local minimum, and thus the answer you get will depend on the initial values of your parameters. This can be incredibly annoying if you have no further knowledge about your system.

Luckily, global minimizers exist which are not influenced by the initial guesses for your parameters. In symfit, two such algorithms from scipy have been wrapped for this purpose. Firstly, the differential_evolution() algorithm from scipy is wrapped as DifferentialEvolution. Secondly, the basinhopping() algorithm is available as BasinHopping. To use these minimizers, just tell Fit:

from symfit import Parameter, Variable, Model, Fit
from symfit.core.minimizers import DifferentialEvolution

x = Parameter('x')
x.min, x.max = -100, 100
x.value = -2.5
y = Variable('y')

model = Model({y: x**4 - 10 * x**2 - x})  # Skewed Mexican hat
fit = Fit(model, minimizer=DifferentialEvolution)
fit_result = fit.execute()

However, due to how this algorithm works, it’s not great at finding the exact minimum (but it will find it if given enough time). You can work around this by “chaining” minimizers: first, run a global minimization to (hopefully) get close to your answer, and then polish it off using a local minimizer:

fit = Fit(model, minimizer=[DifferentialEvolution, BFGS])

Note

Global minimizers such as differential evolution and basin-hopping are rather sensitive to their hyperparameters. You might need to play with those to get appropriate results, e.g.:

fit.execute(DifferentialEvolution={'popsize': 20, 'recombination': 0.9})

Note

There is no way to guarantee that the minimum found is actually the global minimum. Unfortunately, there is no way around this. Therefore, you should always critically inspect the results.

## Constrained Basin-Hopping¶

Worthy of special mention is the ease with which constraints or bounds can be added to symfit.core.minimizers.BasinHopping when used through the symfit.core.fit.Fit interface. As a very simple example, we shall compare to an example from the scipy docs:

import numpy as np
from scipy.optimize import basinhopping

def func2d(x):
f = np.cos(14.5 * x[0] - 0.3) + (x[1] + 0.2) * x[1] + (x[0] + 0.2) * x[0]
df = np.zeros(2)
df[0] = -14.5 * np.sin(14.5 * x[0] - 0.3) + 2. * x[0] + 0.2
df[1] = 2. * x[1] + 0.2
return f, df

minimizer_kwargs = {"method":"L-BFGS-B", "jac":True}
x0 = [1.0, 1.0]
ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs, niter=200)

Let’s compare to the same functionality in symfit:

import numpy as np
from symfit.core.minimizers import BasinHopping
from symfit import parameters, Fit, cos

x0 = [1.0, 1.0]
x1, x2 = parameters('x1, x2', value=x0)

model = cos(14.5 * x1 - 0.3) + (x2 + 0.2) * x2 + (x1 + 0.2) * x1

fit = Fit(model, minimizer=BasinHopping)
fit_result = fit.execute(niter=200)

No minimizer_kwargs have to be provided, as symfit will automatically compute and provide the jacobian and select a minimizer. In this case, symfit will choose BFGS. When bounds are provided, symfit will switch to using L-BFGS-B instead. Setting bounds is as simple as:

x1.min = 0.0
x1.max = 100.0

However, the real strength of the symfit syntax lies in providing constraints:

constraints = [Eq(x1, x2)]
fit = Fit(model, minimizer=BasinHopping, constraints=constraints)

This artificial example will make sure x1 == x2 after fitting. If you have read the Minimize/Maximize section, you will know how much work this would be in pure scipy.

In general, the separate components of the model can be whatever you need them to be. You can mix and match which variables and parameters should be coupled and decoupled ad lib. Some examples are given below.

Same parameters and same function, different (in)dependent variables:

datasets = [data_1, data_2, data_3, data_4, data_5, data_6]

xs = variables('x_1, x_2, x_3, x_4, x_5, x_6')
ys = variables('y_1, y_2, y_3, y_4, y_5, y_6')
zs = variables(', '.join('z_{}'.format(i) for i in range(1, 7)))
a, b = parameters('a, b')

model_dict = {
z: a/(y * b) *  exp(- a * x)
for x, y, z in zip(xs, ys, zs)
}

## What if the model is unnamed?¶

Then you’ll have to use the ordering. Variables throughout symfit’s objects are internally ordered in the following way: first independent variables, then dependent variables, then sigma variables, and lastly parameters when applicable. Within each group, alphabetical ordering applies.

It is therefore always possible to assign data to variables in an unambiguous way using this ordering. For example:

fit = Fit(model, x_data, y_data, sigma_y_data)