8. Supervised Learning: Regression¶
8.1. Lets use simulated data for the examples below¶
First we will model the distance of 100 supernovas (for a particular cosmology as a function of redshift.
We rely on that astroML has a common API with scikit-learn, extending the functionality of the latter.
import numpy as np from astropy.cosmology import LambdaCDM from astroML.datasets import generate_mu_z z_sample, mu_sample, dmu = generate_mu_z(100, random_state=0) cosmo = LambdaCDM(H0=70, Om0=0.30, Ode0=0.70, Tcmb0=0) z = np.linspace(0.01, 2, 1000) mu_true = cosmo.distmod(z)
8.2. Simple linear regression¶
Regression defined as the relation between a dependent variable, , and a set of independent variables, , that describes the expectation value of y given x: .
We will start with the most familiar linear regression, a straight-line fit to data. A straight-line fit is a model of the form where is commonly known as the slope, and is commonly known as the intercept.
We can use Scikit-Learn’s LinearRegression estimator to fit this data and construct the best-fit line:
from sklearn.linear_model import LinearRegression as LinearRegression_sk linear_sk = LinearRegression_sk() linear_sk.fit(z_sample[:,None], mu_sample) mu_fit_sk = linear_sk.predict(z[:, None])
8.3. Measurement Errors in Linear Regression¶
Modifications to LinearRegression in astroML take measurement errors into account on the dependent variable. The API is the same as for the Scikit-Learn version above:
from astroML.linear_model import LinearRegression linear = LinearRegression() linear.fit(z_sample[:,None], mu_sample, dmu) mu_fit = linear.predict(z[:, None])
8.4. Measurement errors in both dependent and independent variables¶
Use simulation data from Kelly 2007 where there is measurement error on the observed values and as well as intrinsic scatter in the regression relationship:
from astroML.datasets import simulation_kelly simulated_values = simulation_kelly(size=100, scalex=0.2, scaley=0.2, alpha=2, beta=1) ksi, eta, xi, yi, xi_error, yi_error, alpha_in, beta_in = simulated_values
Now we take into account errors both on the dependent and independent
variables. The functionality is provided in the new class,
from astroML.linear_model import LinearRegressionwithErrors linreg_xy_err = LinearRegressionwithErrors() linreg_xy_err.fit(xi, yi, yi_error, xi_error)
Now plot the regression: