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:                Tue, 03 Aug 2021   Prob (F-statistic):           4.52e-11
Time:                        09:19:16   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