Fitting Types¶
Fit (LeastSquares)¶
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,))
ydata = a_vec * xdata + b_vec # Point scattered around the line 5 * x + 105
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)
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 ‘wrong’ way
for historical reasons, and was understandably never changed so as not to loose backwards compatibility.
Since this is a new project, we don’t have that problem.
Fit
currently simply wraps NumericalLeastSquares
, but might become more intelligent in the future.
(Non)LinearLeastSquares¶
The LinearLeastSquares
implements the analytical solution to Least Squares fitting.
When your model is linear in it’s parameters, consider using this rather than the default
NumericalLeastSquares
since this gives the exact solution in one step, no iteration and
no guesses needed.
NonLinearLeastSquares
is the generalization to non-linear models. It works by approximating
the model by a linear one around the value of your guesses and repeating that process iteratively.
This process is therefore very sensitive to getting good initial guesses.
Note’s on these objects:
- Use
NonLinearLeastSquares
instead ofLinearLeastSquares
unless you have a reason not to.NonLinearLeastSquares
will behave exactly the same asLinearLeastSquares
when the model is linear. - Bounds are currently ignored by both. This is because for linear models there can only be one solution. For non-linear models it simply hasn’t been considered yet.
- When performance matters, use
NumericalLeastSquares
instead ofNonLinearLeastSquares
. These analytical objects are implemented in pure python and are therefore massively outgunned byNumericalLeastSquares
which is ultimately a wrapper to MINPACK.
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, Likelihood, exp
import numpy as np
# Define the model for an exponential distribution (numpy style)
beta = Parameter()
x = Variable()
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 = Likelihood(model, data)
fit_result = fit.execute()
Off-course fit_result
is a normal FitResults
object. Because scipy.optimize.minimize
is used to do the actual work, bounds on parameters, and even constraints are supported. For more information on this subject, check out symfit
‘s Minimize
.
Minimize/Maximize¶
Minimize or Maximize a model subject to bounds and/or constraints. It is a wrapper to scipy.optimize.minimize
. As an
example I present an example from the scipy
docs.
Suppose we want to maximize the following function:
Subject to the following constraints:
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 = Maximize(model, constraints=constraints)
fit_result = fit.execute()
Done! symfit
will determine all derivatives automatically, no need for you to think about it.
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 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 using Minimize
, as the whole process won’t work otherwise.
ODE Fitting¶
Fitting to a system of 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:
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(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.
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.scatter(tdata, adata)
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 <-> d.
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()
Global FItting¶
In a global fitting problem, we fit to multiple datasets where one or more
parameters might be shared. 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 different amplitude and decay rate.
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()
Warning
The regression coeeficient 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
.
Advanced usage¶
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(6)))
a, b = parameters('a, b')
model_dict = {
z: a/(y * b) * exp(- a * x)
for x, y, z in zip(xs, ys, zs)
}
How Does Fit
Work?¶
How does Fit
get from a (named) model and some data to a fit? Consider the following example:
from symfit import parameters, variables, Fit
a, b = parameters('a, b')
x, y = variables('x, y')
model = {y: a * x + b}
fit = Fit(model, x=x_data, y=y_data, sigma_y=sigma_data)
fit_result = fit.execute()
The first thing symfit
does is build \(\chi^2\) for your model:
chi_squared = sum((y - f)**2/sigmas[y]**2 for y, f in model.items())
In this line sigmas
is a dict which contains all vars that where given a value, or returns 1 otherwise.
This \(\chi^2\) is then transformed into a python function which can then be used to do the numerical calculations:
vars, params = seperate_symbols(chi_squared)
py_chi_squared = lambdify(vars + params, chi_squared)
We are now almost there. Just two steps left. The first is to wrap all the data into the py_chi_squared
function using partial
into the function to be optimized:
from functools import partial
error = partial(py_chi_squared, **data_per_var)
where data_per_var
is a dict containing variable names: value pairs.
Now all that is left is to call leastsqbound
and have it find the best fit parameters:
best_fit_parameters, covariance_matrix = leastsqbound(
error,
self.guesses,
self.eval_jacobian,
self.bounds,
)
That’s it! Finally there are some steps to generate a FitResult object, but these are not important for our current discussion.
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 unambiguis way using this ordering. In the above example:
fit = Fit(model, x_data, y_data, sigma_data)