Least squares fitting of models to data¶
This is a quick introduction to statsmodels for physical scientists (e.g. physicists, astronomers) or engineers.
Why is this needed?
Because most of statsmodels was written by statisticians and they use a different terminology and sometimes methods, making it hard to know which classes and functions are relevant and what their inputs and outputs mean.
[1]:
import numpy as np
import pandas as pd
import statsmodels.api as sm
Linear models¶
Assume you have data points with measurements y at positions x as well as measurement errors y_err.
How can you use statsmodels to fit a straight line model to this data?
For an extensive discussion see Hogg et al. (2010), “Data analysis recipes: Fitting a model to data” … we’ll use the example data given by them in Table 1.
So the model is f(x) = a * x + b and on Figure 1 they print the result we want to reproduce … the best-fit parameter and the parameter errors for a “standard weighted least-squares fit” for this data are: * a = 2.24 +- 0.11 * b = 34 +- 18
[2]:
data = """
x y y_err
201 592 61
244 401 25
47 583 38
287 402 15
203 495 21
58 173 15
210 479 27
202 504 14
198 510 30
158 416 16
165 393 14
201 442 25
157 317 52
131 311 16
166 400 34
160 337 31
186 423 42
125 334 26
218 533 16
146 344 22
"""
try:
from StringIO import StringIO
except ImportError:
from io import StringIO
data = pd.read_csv(StringIO(data), delim_whitespace=True).astype(float)
# Note: for the results we compare with the paper here, they drop the first four points
data.head()
[2]:
| x | y | y_err | |
|---|---|---|---|
| 0 | 201.0 | 592.0 | 61.0 |
| 1 | 244.0 | 401.0 | 25.0 |
| 2 | 47.0 | 583.0 | 38.0 |
| 3 | 287.0 | 402.0 | 15.0 |
| 4 | 203.0 | 495.0 | 21.0 |
To fit a straight line use the weighted least squares class WLS … the parameters are called: * exog = sm.add_constant(x) * endog = y * weights = 1 / sqrt(y_err)
Note that exog must be a 2-dimensional array with x as a column and an extra column of ones. Adding this column of ones means you want to fit the model y = a * x + b, leaving it off means you want to fit the model y = a * x.
And you have to use the option cov_type='fixed scale' to tell statsmodels that you really have measurement errors with an absolute scale. If you do not, statsmodels will treat the weights as relative weights between the data points and internally re-scale them so that the best-fit model will have chi**2 / ndf = 1.
[3]:
exog = sm.add_constant(data["x"])
endog = data["y"]
weights = 1.0 / (data["y_err"] ** 2)
wls = sm.WLS(endog, exog, weights)
results = wls.fit(cov_type="fixed scale")
print(results.summary())
WLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.400
Model: WLS Adj. R-squared: 0.367
Method: Least Squares F-statistic: 193.5
Date: Wed, 02 Nov 2022 Prob (F-statistic): 4.52e-11
Time: 17:02:51 Log-Likelihood: -119.06
No. Observations: 20 AIC: 242.1
Df Residuals: 18 BIC: 244.1
Df Model: 1
Covariance Type: fixed scale
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
const 213.2735 14.394 14.817 0.000 185.062 241.485
x 1.0767 0.077 13.910 0.000 0.925 1.228
==============================================================================
Omnibus: 0.943 Durbin-Watson: 2.901
Prob(Omnibus): 0.624 Jarque-Bera (JB): 0.181
Skew: -0.205 Prob(JB): 0.914
Kurtosis: 3.220 Cond. No. 575.
==============================================================================
Notes:
[1] Standard Errors are based on fixed scale
Check against scipy.optimize.curve_fit¶
[4]:
# You can use `scipy.optimize.curve_fit` to get the best-fit parameters and parameter errors.
from scipy.optimize import curve_fit
def f(x, a, b):
return a * x + b
xdata = data["x"]
ydata = data["y"]
p0 = [0, 0] # initial parameter estimate
sigma = data["y_err"]
popt, pcov = curve_fit(f, xdata, ydata, p0, sigma, absolute_sigma=True)
perr = np.sqrt(np.diag(pcov))
print("a = {0:10.3f} +- {1:10.3f}".format(popt[0], perr[0]))
print("b = {0:10.3f} +- {1:10.3f}".format(popt[1], perr[1]))
a = 1.077 +- 0.077
b = 213.273 +- 14.394
Check against self-written cost function¶
[5]:
# You can also use `scipy.optimize.minimize` and write your own cost function.
# This does not give you the parameter errors though ... you'd have
# to estimate the HESSE matrix separately ...
from scipy.optimize import minimize
def chi2(pars):
"""Cost function."""
y_model = pars[0] * data["x"] + pars[1]
chi = (data["y"] - y_model) / data["y_err"]
return np.sum(chi ** 2)
result = minimize(fun=chi2, x0=[0, 0])
popt = result.x
print("a = {0:10.3f}".format(popt[0]))
print("b = {0:10.3f}".format(popt[1]))
a = 1.077
b = 213.274
Non-linear models¶
[6]:
# TODO: we could use the examples from here:
# http://probfit.readthedocs.org/en/latest/api.html#probfit.costfunc.Chi2Regression