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Fit Functions
Fit functions are a set of callable classes designed to aid in fitting analytical functions to data. A fit function class combines the following functionality:
An analytical function that is callable with given parameters or fitted parameters.
Curve fitting functionality (usually SciPy’s curve_fit() or linregress()), which stores the fit statistics and parameters into the class. This makes the function easily callable with the fitted parameters.
Error propagation calculations.
A root solver that returns either the known analytical solutions or uses SciPy’s fsolve() to calculate the roots.
[1]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from plasmapy.analysis import fit_functions as ffuncs
plt.rcParams["figure.figsize"] = [10.5, 0.56 * 10.5]
Contents:
Fit function basics
There is an ever expanding collection of fit functions, but this notebook will use ExponentialPlusLinear as an example.
A fit function class has no required arguments at time of instantiation.
[2]:
# basic instantiation
explin = ffuncs.ExponentialPlusLinear()
# fit parameters are not set yet
(explin.params, explin.param_errors)
[2]:
(None, None)
Each fit parameter is given a name.
[3]:
[3]:
('a', 'alpha', 'm', 'b')
These names are used throughout the fit function’s documentation, as well as in its __repr__
, __str__
, and latex_str
methods.
[4]:
(explin, explin.__str__(), explin.latex_str)
[4]:
(f(x) = a exp(alpha x) + m x + b <class 'plasmapy.analysis.fit_functions.ExponentialPlusLinear'>,
'f(x) = a exp(alpha x) + m x + b',
'a \\, \\exp(\\alpha x) + m x + b')
Fitting to data
Fit functions provide the curve_fit() method to fit the analytical function to a set of \((x, y)\) data. This is typically done with SciPy’s curve_fit() function, but fitting is done with SciPy’s linregress() for the Linear fit function.
Let’s generate some noisy data to fit to…
[5]:
params = (5.0, 0.1, -0.5, -8.0) # (a, alpha, m, b)
xdata = np.linspace(-20, 15, num=100)
ydata = explin.func(xdata, *params) + np.random.normal(0.0, 0.6, xdata.size)
plt.plot(xdata, ydata)
plt.xlabel("X", fontsize=14)
plt.ylabel("Y", fontsize=14)
[5]:
Text(0, 0.5, 'Y')
The fit function curve_fit()
shares the same signature as SciPy’s curve_fit(), so any **kwargs
will be passed on. By default, only the \((x, y)\) values are needed.
[6]:
explin.curve_fit(xdata, ydata)
Getting fit results
After fitting, the fitted parameters, uncertainties, and coefficient of determination, or \(r^2\), values can be retrieved through their respective properties, params
, parame_errors
, and rsq
.
[7]:
(explin.params, explin.params.a, explin.params.alpha)
[7]:
(FitParamTuple(a=5.059183439679306, alpha=0.09978375300092507, m=-0.496878180813518, b=-8.011825037449219),
5.059183439679306,
0.09978375300092507)
[8]:
(explin.param_errors, explin.param_errors.a, explin.param_errors.alpha)
[8]:
(FitParamTuple(a=1.1373736632679126, alpha=0.009906559729235313, m=0.05097776786330378, b=1.1636984023536545),
1.1373736632679126,
0.009906559729235313)
[9]:
[9]:
0.9398278915373817
Fit function is callable
Now that parameters are set, the fit function is callable.
[10]:
explin(0)
[10]:
-2.9526415977699125
Associated errors can also be generated.
[11]:
y, y_err = explin(np.linspace(-1, 1, num=10), reterr=True)
(y, y_err)
[11]:
(array([-2.93621835, -2.9439722 , -2.94942414, -2.95252256, -2.9532147 ,
-2.9514466 , -2.94716309, -2.94030777, -2.93082299, -2.91864977]),
array([1.55513147, 1.56993604, 1.58541766, 1.60159539, 1.6184888 ,
1.63611794, 1.65450339, 1.67366623, 1.69362812, 1.71441123]))
Known uncertainties in \(x\) can be specified too.
[12]:
y, y_err = explin(np.linspace(-1, 1, num=10), reterr=True, x_err=0.1)
(y, y_err)
[12]:
(array([-2.93621835, -2.9439722 , -2.94942414, -2.95252256, -2.9532147 ,
-2.9514466 , -2.94716309, -2.94030777, -2.93082299, -2.91864977]),
array([1.55513661, 1.56993886, 1.58541883, 1.60159562, 1.61848882,
1.63611851, 1.65450528, 1.67367026, 1.69363512, 1.71442206]))
Plotting results
[13]:
# plot original data
plt.plot(xdata, ydata, marker="o", linestyle=" ", label="Data")
ax = plt.gca()
ax.set_xlabel("X", fontsize=14)
ax.set_ylabel("Y", fontsize=14)
ax.axhline(0.0, color="r", linestyle="--")
# plot fitted curve + error
yfit, yfit_err = explin(xdata, reterr=True)
ax.plot(xdata, yfit, color="orange", label="Fit")
ax.fill_between(
xdata,
yfit + yfit_err,
yfit - yfit_err,
color="orange",
alpha=0.12,
zorder=0,
label="Fit Error",
)
# plot annotations
plt.legend(fontsize=14, loc="upper left")
txt = f"$f(x) = {explin.latex_str}$\n$r^2 = {explin.rsq:.3f}$\n"
for name, param, err in zip(explin.param_names, explin.params, explin.param_errors, strict=False):
txt += f"{name} = {param:.3f} $\\pm$ {err:.3f}\n"
txt_loc = [-13.0, ax.get_ylim()[1]]
txt_loc = ax.transAxes.inverted().transform(ax.transData.transform(txt_loc))
txt_loc[0] -= 0.02
txt_loc[1] -= 0.05
ax.text(
txt_loc[0],
txt_loc[1],
txt,
fontsize="large",
transform=ax.transAxes,
va="top",
linespacing=1.5,
)
[13]:
Text(0.20727272727272733, 0.95, '$f(x) = a \\, \\exp(\\alpha x) + m x + b$\n$r^2 = 0.940$\na = 5.059 $\\pm$ 1.137\nalpha = 0.100 $\\pm$ 0.010\nm = -0.497 $\\pm$ 0.051\nb = -8.012 $\\pm$ 1.164\n')
Root solving
An exponential plus a linear offset has no analytical solutions for its roots, except for a few specific cases. To get around this, ExponentialPlusLinear().root_solve() uses SciPy’s fsolve() to calculate it’s roots. If a fit function has analytical solutions to its roots (e.g. Linear().root_solve()), then the method is overridden with the known solution.
[14]:
root, err = explin.root_solve(-15.0)
(root, err)
[14]:
(-13.468778765709326, nan)
Let’s use Linear().root_solve() as an example for a known solution.
[15]:
lin = ffuncs.Linear(params=(1.0, -5.0), param_errors=(0.1, 0.1))
root, err = lin.root_solve()
(root, err)
[15]:
(5.0, 0.5099019513592785)