# Measurement Errors in Linear Regression¶

## Lets use simulated data here.¶

First we will model the distance of 100 supernovae (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.

%matplotlib inline

import numpy as np
from matplotlib import pyplot as plt

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)

plt.errorbar(z_sample, mu_sample, dmu, fmt='.')


## Simple linear regression¶

Regression defined as the relation between a dependent variable, $$y$$, and a set of independent variables, $$x$$, that describes the expectation value of y given x: $$E[y|x]$$

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 $$$y = ax + b$$$$where$$a$$is commonly known as the *slope*, and$$b$ 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])

#------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(8, 6))

ax.plot(z, mu_fit_sk, '-k')
ax.plot(z, mu_true, '--', c='gray')
ax.errorbar(z_sample, mu_sample, dmu, fmt='.k', ecolor='gray', lw=1)

ax.set_xlim(0.01, 1.8)
ax.set_ylim(36.01, 48)

ax.set_ylabel(r'$\mu$')
ax.set_xlabel(r'$z$')

plt.show()


## Measurement errors¶

Modifications to LinearRegression in astroML take measurement errors into account on the dependent variable.

from astroML.linear_model import LinearRegression

linear = LinearRegression()
linear.fit(z_sample[:,None], mu_sample, dmu)

mu_fit = linear.predict(z[:, None])

#------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(8, 6))

ax.plot(z, mu_fit_sk, '-k')
ax.plot(z, mu_fit, '-k', color='red')

ax.plot(z, mu_true, '--', c='gray')
ax.errorbar(z_sample, mu_sample, dmu, fmt='.k', ecolor='gray', lw=1)

ax.set_xlim(0.01, 1.8)
ax.set_ylim(36.01, 48)

ax.set_ylabel(r'$\mu$')
ax.set_xlabel(r'$z$')

plt.show()


## Basis function regression¶

If we consider a function in terms of the sum of bases (this can be polynomials, Gaussians, quadratics, cubics) then we can solve for the coefficients using regression.

### Polynomial basis functions¶

polynomial regression:

$y=a_0+a_1x+a_2x^2+a_3x^3+...$

Notice that this is still a linear model—the linearity refers to the fact that the coefficients $$a_n$$ never multiply or divide each other. What we have effectively done here is to take our one-dimensional $$x$$ values and projected them into a higher dimension, so that a linear fit can fit more complicated relationships between $$x$$ and $$y$$.

from astroML.linear_model import PolynomialRegression

# 2nd degree polynomial regression
polynomial = PolynomialRegression(degree=2)
polynomial.fit(z_sample[:,None], mu_sample, dmu)

mu_fit_poly = polynomial.predict(z[:, None])

#------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(8, 6))

ax.plot(z, mu_fit_poly, '-k', color='red')

ax.plot(z, mu_true, '--', c='gray')
ax.errorbar(z_sample, mu_sample, dmu, fmt='.k', ecolor='gray', lw=1)

ax.set_xlim(0.01, 1.8)
ax.set_ylim(36.01, 48)

ax.set_ylabel(r'$\mu$')
ax.set_xlabel(r'$z$')

plt.show()


### Gaussian basis functions¶

Of course, other basis functions are possible. For example, one useful pattern is to fit a model that is not a sum of polynomial bases, but a sum of Gaussian bases. E.g. we could substitute $$x^2$$ for Gaussians (where we fix $$\sigma$$ and $$\mu$$ and fit for the amplitude) as long as the attribute we are fitting for is linear. This is called basis function regression.

from astroML.linear_model import BasisFunctionRegression

#------------------------------------------------------------
# Define our number of Gaussians
nGaussians = 10
basis_mu = np.linspace(0, 2, nGaussians)[:, None]
basis_sigma = 3 * (basis_mu[1] - basis_mu[0])

gauss_basis = BasisFunctionRegression('gaussian', mu=basis_mu, sigma=basis_sigma)
gauss_basis.fit(z_sample[:,None], mu_sample, dmu)

mu_fit_gauss = gauss_basis.predict(z[:, None])

#------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(8, 6))

ax.plot(z, mu_fit_gauss, '-k', color='red')

ax.plot(z, mu_true, '--', c='gray')
ax.errorbar(z_sample, mu_sample, dmu, fmt='.k', ecolor='gray', lw=1)

ax.set_xlim(0.01, 1.8)
ax.set_ylim(36.01, 48)

ax.set_ylabel(r'$\mu$')
ax.set_xlabel(r'$z$')

plt.show()