I am trying to write a curve fitting function which returns the optimal parameters a, b and c, here is a simplified example:
import numpy
import scipy
from scipy.optimize import curve_fit
def f(x, a, b, c):
return x * 2*a + 4*b - 5*c
xdata = numpy.array([1,3,6,8,10])
ydata = numpy.array([ 0.91589774, 4.91589774, 10.91589774, 14.91589774, 18.91589774])
popt, pcov = scipy.optimize.curve_fit(f, xdata, ydata)
This works fine, but I want to give the user a chance to supply some (or none) of the parameters a, b or c, in which case they should be treated as constants and not estimated. How can I write f so that it fits only the parameters not supplied by the user?
Basically, I need to define f dynamically with the correct arguments. For instance if a was known by the user, f becomes:
def f(x, b, c):
a = global_version_of_a
return x * 2*a + 4*b - 5*c
Taking a page from the collections.namedtuple playbook, you can use exec to "dynamically" define func:
import numpy as np
import scipy.optimize as optimize
import textwrap
funcstr=textwrap.dedent('''\
def func(x, {p}):
return x * 2*a + 4*b - 5*c
''')
def make_model(**kwargs):
params=set(('a','b','c')).difference(kwargs.keys())
exec funcstr.format(p=','.join(params)) in kwargs
return kwargs['func']
func=make_model(a=3, b=1)
xdata = np.array([1,3,6,8,10])
ydata = np.array([ 0.91589774, 4.91589774, 10.91589774, 14.91589774, 18.91589774])
popt, pcov = optimize.curve_fit(func, xdata, ydata)
print(popt)
# [ 5.49682045]
Note the line
func=make_model(a=3, b=1)
You can pass whatever parameters you like to make_model. The parameters you pass to make_model become fixed constants in func. Whatever parameters remain become free parameters that optimize.curve_fit will try to fit.
For example, above, a=3 and b=1 become fixed constants in func. Actually, the exec statement places them in func's global namespace. func is thus defined as a function of x and the single parameter c. Note the return value for popt is an array of length 1 corresponding to the remaining free parameter c.
Regarding textwrap.dedent: In the above example, the call to textwrap.dedent is unnecessary. But in a "real-life" script, where funcstr is defined inside a function or at a deeper indentation level, textwrap.dedent allows you to write
def foo():
funcstr=textwrap.dedent('''\
def func(x, {p}):
return x * 2*a + 4*b - 5*c
''')
instead of the visually unappealing
def foo():
funcstr='''\
def func(x, {p}):
return x * 2*a + 4*b - 5*c
'''
Some people prefer
def foo():
funcstr=(
'def func(x, {p}):\n'
' return x * 2*a + 4*b - 5*c'
)
but I find quoting each line separately and adding explicit EOL characters a bit onerous. It does save you a function call however.
I usually use a lambda for this purpose.
user_b, user_c = get_user_vals()
opt_fun = lambda x, a: f(x, a, user_b, user_c)
popt, pcov = scipy.optimize.curve_fit(opt_fun, xdata, ydata)
If you want a simple solution based on curve_fit, I'd suggest that you wrap your function in a class. Minimal example:
import numpy
from scipy.optimize import curve_fit
class FitModel(object):
def f(self, x, a, b, c):
return x * 2*a + 4*b - 5*c
def f_a(self, x, b, c):
return self.f(x, self.a, b, c)
# user supplies a = 1.0
fitModel = FitModel()
fitModel.a = 1.0
xdata = numpy.array([1,3,6,8,10])
ydata = numpy.array([ 0.91589774, 4.91589774, 10.91589774, 14.91589774, 18.91589774])
initial = (1.0,2.0)
popt, pconv = curve_fit(fitModel.f_a, xdata, ydata, initial)
There is already a package that does this:
https://lmfit.github.io/lmfit-py/index.html
From the README:
"LMfit-py provides a Least-Squares Minimization routine and class
with a simple, flexible approach to parameterizing a model for
fitting to data. Named Parameters can be held fixed or freely
adjusted in the fit, or held between lower and upper bounds. In
addition, parameters can be constrained as a simple mathematical
expression of other Parameters."
def f(x, a = 10, b = 15, c = 25):
return x * 2*a + 4*b - 5*c
If the user doesn't supply an argument for the parameter in question, whatever you specified on the right-hand side of the = sign will be used:
e.g.:
f(5, b = 20) will evaluate to return 5 * 2*10 + 4*20 - 5*25 and
f(7) will evaluate to return 7 * 2*10 + 4*15 - 5*25
Related
I've got a little project for my college and I need to write a method which fits an array to some function, here's it's part:
def Linear(self,x,a,b):
return a*x+b
def Quadratic(self, x, a,b,c):
return a*(x**2)+b*x+c
def Sinusoid(self,t, a, gam, omega, phi, offset):
return np.e ** (gam * t) * a * np.sin((2 * np.pi * omega * t) + phi) + offset
def Fit(self, name):
func= getattr(App, name)
self.fit_params, self.covariance_matrix = curve_fit(func, self.t, self.a, maxfev= 100000)
But it returns absolutely wrong values and also doesn't even work for Sinusoid function (ptimizeWarning: Covariance of the parameters could not be estimated warnings.warn('Covariance of the parameters could not be estimated'). I've already checked if it's not an issue with getattr function but it works correctly.
I'm running out of ideas where the issue is.
There are several problems here.
Linear, Quadratic and Sinusoid do not need self, you can define those as staticmethod:
#This is the decorator use to define staticmethod
#staticmethod
def Linear(x,a,b):
return a*x+b
The same applies to other methods (except Fit).
It will help when they are called in the future. When you call them with curve_fit(func,...), for example, if func is Sinusoid, what you are doing is
curve_fit(Sinusoid, ...) and not curve_fit(self.Sinusoid,...). When curve_fit use Sinusoid, the first argument may be understood as the self, therefore, you get and error.
If you define
#staticmethod
def Sinusoid(t, a, gam, omega, phi, offset):
return np.e ** (gam * t) * a * np.sin((2 * np.pi * omega * t) + phi) + offset
a, gam, omega, phi and offset are constants to fit. Therefore, when you call curve_fit, you need to pass this constants as params:
scipy.optimize.curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, check_finite=True, bounds=- inf, inf, method=None, jac=None, **kwargs)
To clarify: The first argument is the func, the second is your experimental data for x, and the third, is the data for y. And after that all the others kwargs, like maxfev. I don't know if self.a is the data for y, or if you want to set an initial value for a. I'm going to guess you want to set the initial value of a. You need to do it like this: p0=[x1,x2,x3,x4,..., xn] where xi is the initial value of the i-th constant. If you don't pass p0 as an argument to curve_fit, the default value for each constant is going to be 1. But here it comes the last problem:
The diferent functions use a diferent number arguments, so for the first one, Linear, you need to do p0 = [a,1]. For the second, p0 = [a, 1, 1]. And for the last one, the Sinusoid one, p0 = [a, 1, 1, 1, 1]
I don't really know if you can pass p0 = [a] to all of the functions and curve_fit will magically understand. But try so and if curve_fit doesn't let you, use condictionals:
def Fit(self, name):
if func = Linear:
p0 = [a,1]
elif func = Quadratic:
p0 = [a,1,1]
elif func = Sinusoid:
p0 = [a,1,1,1,1]
self.fit_params, self.covariance_matrix = curve_fit(func, self.t, self.y_values, p0=p0, maxfev= 100000)
Where self.y_values is the experimental data as an array-like element.
If self.a is the data for y you can cut all of this mambojambo and let Fit define as it currently is. But don't forguet the #staticmethod in the other functions.
Hope this solves your problem.
I would like to perform some tests on my optimisation routine using scipy.optimize.minimize, in particular graphing the convergence (or the rather objective function) each iteration, over multiple tests.
Suppose I have the following linearly constrained quadratic optimisation problem:
minimise: x_i Q_ij x_j + a|x_i|
subject to: sum(x_i) = 1
I can code this as:
def _fun(x, Q, a):
c = np.einsum('i,ij,j->', x, Q, x)
p = np.sum(a * np.abs(x))
return c + p
def _constr(x):
return np.sum(x) - 1
And I will implement the optimisation in scipy as:
x_0 = # some initial vector
x_soln = scipy.optimise.minimize(_fun, x_0, args=(Q, a), method='SLSQP',
constraints={'type': 'eq', 'fun': _constr})
I see that there is a callback argument but which only accepts a single argument of the parameter values at each iteration. How can I utilise this in a more esoteric case where I might have other arguments that need to be supplied to my callback function?
The way I solved this was use a generic callback cache object referenced each time from my callback function. Let's say you want to do 20 tests and plot the objective function after each iteration in the same chart. You will need an outer loop to run 20 tests, but we'll create that later.
First lets create a class that will store all iteration objective function values for us, and a couple extra bits and pieces:
class OpObj(object):
def __init__(self, Q, a):
self.Q, self.a = Q, a
rv = np.random.rand()
self.x_0 = np.array([rv, (1-rv)/2, (1-rv)/2])
self.f = np.full(shape=(500,), fill_value=np.NaN)
self.count = 0
def _fun(self, x):
return _fun(x, self.Q, self.a)
Also lets add a callback function that manipulates that class obj. Don't worry that it has more than one argument for now since we'll fix this later. Just make sure the first parameter is the solution variables.
def cb(xk, obj=None):
obj.f[obj.count] = obj._fun(xk)
obj.count += 1
All this does is use the object's functions and values to update itself, counting the number of iterations each time. This function will be called after each iteration.
Putting this all together all we need is two more things: 1) some matplotlib-ing to do the plot, and fixing the callback to have only one argument. We can do that with a decorator which is exactly what functools partial does. It returns a function with less arguments than the original. So the final code looks like this:
import matplotlib.pyplot as plt
import scipy.optimize as op
import numpy as np
from functools import partial
Q = np.array([[1.0, 0.75, 0.45], [0.75, 1.0, 0.60], [0.45, 0.60, 1.0]])
a = 1.0
def _fun(x, Q, a):
c = np.einsum('i,ij,j->', x, Q, x)
p = np.sum(a * np.abs(x))
return c + p
def _constr(x):
return np.sum(x) - 1
class OpObj(object):
def __init__(self, Q, a):
self.Q, self.a = Q, a
rv = np.random.rand()
self.x_0 = np.array([rv, (1-rv)/2, (1-rv)/2])
self.f = np.full(shape=(500,), fill_value=np.NaN)
self.count = 0
def _fun(self, x):
return _fun(x, self.Q, self.a)
def cb(xk, obj=None):
obj.f[obj.count] = obj._fun(xk)
obj.count += 1
fig, ax = plt.subplots(1,1)
x = np.linspace(1,500,500)
for test in range(20):
op_obj = OpObj(Q, a)
x_soln = op.minimize(_fun, op_obj.x_0, args=(Q, a), method='SLSQP',
constraints={'type': 'eq', 'fun': _constr},
callback=partial(cb, obj=op_obj))
ax.plot(x, op_obj.f)
ax.set_ylim((1.71,1.76))
plt.show()
I have a system of two first order ODEs, which are nonlinear, and hence difficult to solve analytically in a closed form. I want to fit the numerical solution to this system of ODEs to a data set. My data set is for only one of the two variables that are part of the ODE system. How do I go about this?
This didn't help because there's only one variable there.
My code which is currently leading to an error is:
import numpy as np
from scipy.integrate import odeint
from scipy.optimize import curve_fit
def f(y, t, a, b, g):
S, I = y # S, I are supposed to be my variables
Sdot = -a * S * I
Idot = (a - b) * S * I + (b - g - b * I) * I
dydt = [Sdot, Idot]
return dydt
def y(t, a, b, g, y0):
y = odeint(f, y0, t, args=(a, b, g))
return y.ravel()
I_data =[] # I have data only for I, not for S
file = open('./ratings_showdown.csv')
for e_raw in file.read().split('\r\n'):
try:
e=float(e_raw); I_data.append(e)
except ValueError:
continue
data_t = range(len(I_data))
popt, cov = curve_fit(y, data_t, I_data, [.05, 0.02, 0.01, [0.99,0.01]])
#want to fit I part of solution to data for variable I
#ERROR here, ValueError: setting an array element with a sequence
a_opt, b_opt, g_opt, y0_opt = popt
print("a = %g" % a_opt)
print("b = %g" % b_opt)
print("g = %g" % g_opt)
print("y0 = %g" % y0_opt)
import matplotlib.pyplot as plt
t = np.linspace(0, len(data_y), 2000)
plt.plot(data_t, data_y, '.',
t, y(t, a_opt, b_opt, g_opt, y0_opt), '-')
plt.gcf().set_size_inches(6, 4)
#plt.savefig('out.png', dpi=96) #to save the fit result
plt.show()
This type of ODE fitting becomes a lot easier in symfit, which I wrote specifically as a user friendly wrapper to scipy. I think it will be very useful for your situation because the decreased amount of boiler-plate code simplifies things a lot.
From the docs and applied roughly to your problem:
from symfit import variables, parameters, Fit, D, ODEModel
S, I, t = variables('S, I, t')
a, b, g = parameters('a, b, g')
model_dict = {
D(S, t): -a * S * I,
D(I, t): (a - b) * S * I + (b - g - b * I) * I
}
ode_model = ODEModel(model_dict, initial={t: 0.0, S: 0.99, I: 0.01})
fit = Fit(ode_model, t=tdata, I=I_data, S=None)
fit_result = fit.execute()
Check out the docs for more :)
So I figured out the problem.
The curve_fit() function apparently returns a list as it's second return value. So, instead of passing the initial conditions as a list [0.99,0.01], I passed them separately as 0.99 and 0.01.
I have the option to add bounds to sio.curve_fit. Is there a way to expand upon this bounds feature that involves a function of the parameters? In other words, say I have an arbitrary function with two or more unknown constants. And then let's also say that I know the sum of all of these constants is less than 10. Is there a way I can implement this last constraint?
import numpy as np
import scipy.optimize as sio
def f(x, a, b, c):
return a*x**2 + b*x + c
x = np.linspace(0, 100, 101)
y = 2*x**2 + 3*x + 4
popt, pcov = sio.curve_fit(f, x, y, \
bounds = [(0, 0, 0), (10 - b - c, 10 - a - c, 10 - a - b)]) # a + b + c < 10
Now, this would obviously error, but I think it helps to get the point across. Is there a way I can incorporate a constraint function involving the parameters to a curve fit?
Thanks!
With lmfit, you would define 4 parameters (a, b, c, and delta). a and b can vary freely. delta is allowed to vary, but has a maximum value of 10 to represent the inequality. c would be constrained to be delta-a-b (so, there are still 3 variables: c will vary, but not independently from the others). If desired, you could also put bounds on the values for a, b, and c. Without testing, your code would be approximately::
import numpy as np
from lmfit import Model, Parameters
def f(x, a, b, c):
return a*x**2 + b*x + c
x = np.linspace(0, 100.0, 101)
y = 2*x**2 + 3*x + 4.0
fmodel = Model(f)
params = Parameters()
params.add('a', value=1, vary=True)
params.add('b', value=4, vary=True)
params.add('delta', value=5, vary=True, max=10)
params.add('c', expr = 'delta - a - b')
result = fmodel.fit(y, params, x=x)
print(result.fit_report())
Note that if you actually get to a situation where the constraint expression or bounds dictate the values for the parameters, uncertainties may not be estimated.
curve_fit and least_squares only accept box constraints. In scipy.optimize, SLSQP can deal with more complicated constraints.
For curve fitting specifically, you can have a look at lmfit package.
I am trying to fit a function which takes as input 2 independent variables x,y and 3 parameters to be found a,b,c. This is my test code:
import numpy as np
from scipy.optimize import curve_fit
def func(x,y, a, b, c):
return a*np.exp(-b*(x+y)) + c
y= x = np.linspace(0,4,50)
z = func(x,y, 2.5, 1.3, 0.5) #works ok
#generate data to be fitted
zn = z + 0.2*np.random.normal(size=len(x))
popt, pcov = curve_fit(func, x,y, zn) #<--------Problem here!!!!!
But i am getting the error: "func() takes exactly 5 arguments (51 given)". How can pass my arguments x,y correctly?
A look at the documentation of scipy.optimize.curve_fit() is all it takes. The prototype is
scipy.optimize.curve_fit(f, xdata, ydata, p0=None, sigma=None, **kw)
The documentation states curve_fit() is called with the target function as the first argument, the independent variable(s) as the second argument, the dependent variable as the third argument ans the start values for the parameters as the forth argument. You tried to call the function in a completely different way, so it's not surprising it does not work. Specifically, you passed zn as the p0 parameter – this is why the function was called with so many parameters.
The documentation also describes how the target function is called:
f: callable
The model function, f(x, ...). It must take the independent variable as the first argument and the parameters to fit as separate remaining arguments.
xdata : An N-length sequence or an (k,N)-shaped array
for functions with k predictors. The independent variable where the data is measured.
You try to uses to separate arguments for the dependent variables, while it should be a single array of arguments. Here's the code fixed:
def func(x, a, b, c):
return a * np.exp(-b * (x[0] + x[1])) + c
N = 50
x = np.linspace(0,4,50)
x = numpy.array([x, x]) # Combine your `x` and `y` to a single
# (2, N)-array
z = func(x, 2.5, 1.3, 0.5)
zn = z + 0.2 * np.random.normal(size=x.shape[1])
popt, pcov = curve_fit(func, x, zn)
Try to pass the first two array parameters to func as a tuple and modify func to accept a tuple of parameters
Normally it is expected the curvefit would accept an x and y parameter func(x) as an input to fit the curve. Strangely in your example as your x parameter is not a single value but two values (not sure why), you have to modify your function so that it accept the x as a single parameter and expands it within.
Generally speaking, three dimensional curve fitting should be handled in a different manner from what you are trying to achieve. You can take a look into the following SO post which tried to fit a three dimensional scatter with a line.
>>> def func((x,y), a, b, c):
return a*np.exp(-b*(x+y)) + c
>>> y= x = np.linspace(0,4,50)
>>> z = func((x,y), 2.5, 1.3, 0.5) #works ok
>>> zn = z + 0.2*np.random.normal(size=len(x))
>>> popt, pcov = curve_fit(func, (x,y), zn)