Linear Regression, also called Ordinary LeastSquares (OLS) Regression, is probably the most commonly used technique in Statistical Learning. It is also the oldest, dating back to the eighteenth century and the work of Carl Friedrich Gauss and AdrienMarie Legendre. It is also one of the easier and more intuitive techniques to understand, and it provides a good basis for learning more advanced concepts and techniques. This posting explains how to perform linear regression using the statsmodels Python package, we will discuss the single variable case and defer multiple regression to a future post.
This is part of a series of blog posts to show how to do common statistical learning techniques in Python. We provide only a small amount of background on the concepts and techniques we cover, so if you'd like a more thorough explanation check out Introduction to Statistical Learning or sign up for the free online course by the authors here. If you are just here to learn how to do it in Python skip directly to the examples below.
Statsmodels
Statsmodel is a Python library designed for more statisticallyoriented approaches to data analysis, with an emphasis on econometric analyses. It integrates well with the pandas and numpy libraries we covered in a previous post. It also has built in support for many of the statistical tests to check the quality of the fit and a dedicated set of plotting functions to visualize and diagnose the fit. Scikitlearn also has support for linear regression, including many forms of regularized regression lacking in statsmodels, but it lacks the rich set of statistical tests and diagnostics that have been developed for linear models.
Linear Regression and Ordinary Least Squares
Linear regression is one of the simplest and most commonly used modeling techniques. It makes very strong assumptions about the relationship between the predictor variables (the X) and the response (the Y). It assumes that this relationship takes the form:
\(y = \beta_0 + \beta_1 * x\)
Ordinary Least Squares is the simplest and most common estimator in which the two \(\beta\)s are chosen to minimize the square of the distance between the predicted values and the actual values. Even though this model is quite rigid and often does not reflect the true relationship, this still remains a popular approach for several reasons. For one, it is computationally cheap to calculate the coefficients. It is also easier to interpret than more sophisticated models, and in situations where the goal is understanding a simple model in detail, rather than estimating the response well, they can provide insight into what the model captures. Finally, in situations where there is a lot of noise, it may be hard to find the true functional form, so a constrained model can perform quite well compared to a complex model which is more affected by noise.
The resulting model is represented as follows:
\(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 * x\)
Here the hats on the variables represent the fact that they are estimated from the data we have available. The \(\beta\)s are termed the parameters of the model or the coefficients. \(\beta_0\) is called the constant term or the intercept.
Ordinary Least Squares Using Statsmodels
The statsmodels package provides several different classes that provide different options for linear regression. Getting started with linear regression is quite straightforward with the OLS module.
To start with we load the Longley dataset of US macroeconomic data from the Rdatasets website.
# load numpy and pandas for data manipulation
import numpy as np
import pandas as pd
# load statsmodels as alias ``sm``
import statsmodels.api as sm
# load the longley dataset into a pandas data frame  first column (year) used as row labels
df = pd.read_csv('http://vincentarelbundock.github.io/Rdatasets/csv/datasets/longley.csv', index_col=0)
df.head()
GNP.deflator  GNP  Unemployed  Armed.Forces  Population  Year  Employed  

1947  83.0  234.289  235.6  159.0  107.608  1947  60.323 
1948  88.5  259.426  232.5  145.6  108.632  1948  61.122 
1949  88.2  258.054  368.2  161.6  109.773  1949  60.171 
1950  89.5  284.599  335.1  165.0  110.929  1950  61.187 
1951  96.2  328.975  209.9  309.9  112.075  1951  63.221 
We will use the variable Total Derived Employment ('Employed'
) as our response y
and Gross National Product ('GNP'
) as our predictor X
.
We take the single response variable and store it separately. We also add a constant term so that we fit the intercept of our linear model.
y = df.Employed # response
X = df.GNP # predictor
X = sm.add_constant(X) # Adds a constant term to the predictor
X.head()
const  GNP  

1947  1  234.289 
1948  1  259.426 
1949  1  258.054 
1950  1  284.599 
1951  1  328.975 
Now we perform the regression of the predictor on the response, using the sm.OLS
class and and its initialization OLS(y, X)
method. This method takes as an input two arraylike objects: X
and y
. In general, X
will either be a numpy array or a pandas data frame with shape (n, p)
where n
is the number of data points and p
is the number of predictors. y
is either a onedimensional numpy array or a pandas series of length n
.
<span class="n">est</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">OLS</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
We then need to fit the model by calling the OLS object's fit()
method. Ignore the warning about the kurtosis test if it appears, we have only 16 examples in our dataset and the test of the kurtosis is valid only if there are more than 20 examples.
est = est.fit()
est.summary()
/usr/local/lib/python2.7/distpackages/scipy/stats/stats.py:1276: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=16
int(n))
Dep. Variable:  Employed  Rsquared:  0.967 

Model:  OLS  Adj. Rsquared:  0.965 
Method:  Least Squares  Fstatistic:  415.1 
Date:  Sat, 08 Feb 2014  Prob (Fstatistic):  8.36e12 
Time:  01:28:29  LogLikelihood:  14.904 
No. Observations:  16  AIC:  33.81 
Df Residuals:  14  BIC:  35.35 
Df Model:  1 
coef  std err  t  P>t  [95.0% Conf. Int.]  

const  51.8436  0.681  76.087  0.000  50.382 53.305 
GNP  0.0348  0.002  20.374  0.000  0.031 0.038 
Omnibus:  1.925  DurbinWatson:  1.619 

Prob(Omnibus):  0.382  JarqueBera (JB):  1.215 
Skew:  0.664  Prob(JB):  0.545 
Kurtosis:  2.759  Cond. No.  1.66e+03 
After visualizing the relationship we will explain the summary. First, we need the coefficients of the fit.
<span class="n">est</span><span class="o">.</span><span class="n">params</span>
const 51.843590
GNP 0.034752
dtype: float64
# Make sure that graphics appear inline in the iPython notebook
%pylab inline
# We pick 100 hundred points equally spaced from the min to the max
X_prime = np.linspace(X.GNP.min(), X.GNP.max(), 100)[:, np.newaxis]
X_prime = sm.add_constant(X_prime) # add constant as we did before
# Now we calculate the predicted values
y_hat = est.predict(X_prime)
plt.scatter(X.GNP, y, alpha=0.3) # Plot the raw data
plt.xlabel("Gross National Product")
plt.ylabel("Total Employment")
plt.plot(X_prime[:, 1], y_hat, 'r', alpha=0.9) # Add the regression line, colored in red
Populating the interactive namespace from numpy and matplotlib
[<matplotlib.lines.Line2D at 0x4444350>]
Statsmodels also provides a formulaic interface that will be familiar to users of R. Note that this requires the use of a different api to statsmodels, and the class is now called ols
rather than OLS
. The argument formula
allows you to specify the response and the predictors using the column names of the input data frame data
.
# import formula api as alias smf
import statsmodels.formula.api as smf
# formula: response ~ predictors
est = smf.ols(formula='Employed ~ GNP', data=df).fit()
est.summary()
Dep. Variable:  Employed  Rsquared:  0.967 

Model:  OLS  Adj. Rsquared:  0.965 
Method:  Least Squares  Fstatistic:  415.1 
Date:  Sat, 08 Feb 2014  Prob (Fstatistic):  8.36e12 
Time:  01:28:29  LogLikelihood:  14.904 
No. Observations:  16  AIC:  33.81 
Df Residuals:  14  BIC:  35.35 
Df Model:  1 
coef  std err  t  P>t  [95.0% Conf. Int.]  

Intercept  51.8436  0.681  76.087  0.000  50.382 53.305 
GNP  0.0348  0.002  20.374  0.000  0.031 0.038 
Omnibus:  1.925  DurbinWatson:  1.619 

Prob(Omnibus):  0.382  JarqueBera (JB):  1.215 
Skew:  0.664  Prob(JB):  0.545 
Kurtosis:  2.759  Cond. No.  1.66e+03 
This summary provides quite a lot of information about the fit. The parts of the table we think are the most important are bolded in the description below.
The left part of the first table provides basic information about the model fit:
Element  Description 

Dep. Variable  Which variable is the response in the model 
Model  What model you are using in the fit 
Method  How the parameters of the model were calculated 
No. Observations  The number of observations (examples) 
DF Residuals  Degrees of freedom of the residuals. Number of observations  number of parameters 
DF Model  Number of parameters in the model (not including the constant term if present) 
The right part of the first table shows the goodness of fit
Element  Description 

Rsquared  The coefficient of determination. A statistical measure of how well the regression line approximates the real data points 
Adj. Rsquared  The above value adjusted based on the number of observations and the degreesoffreedom of the residuals 
Fstatistic  A measure how significant the fit is. The mean squared error of the model divided by the mean squared error of the residuals 
Prob (Fstatistic)  The probability that you would get the above statistic, given the null hypothesis that they are unrelated 
Loglikelihood  The log of the likelihood function. 
AIC  The Akaike Information Criterion. Adjusts the loglikelihood based on the number of observations and the complexity of the model. 
BIC  The Bayesian Information Criterion. Similar to the AIC, but has a higher penalty for models with more parameters. 
The second table reports for each of the coefficients
Description  

The name of the term in the model  
coef  The estimated value of the coefficient 
std err  The basic standard error of the estimate of the coefficient. More sophisticated errors are also available. 
t  The tstatistic value. This is a measure of how statistically significant the coefficient is. 
P > t  Pvalue that the nullhypothesis that the coefficient = 0 is true. If it is less than the confidence level, often 0.05, it indicates that there is a statistically significant relationship between the term and the response. 
[95.0% Conf. Interval]  The lower and upper values of the 95% confidence interval

Finally, there are several statistical tests to assess the distribution of the residuals
Element  Description 

Skewness  A measure of the symmetry of the data about the mean. Normallydistributed errors should be symmetrically distributed about the mean (equal amounts above and below the line). 
Kurtosis  A measure of the shape of the distribution. Compares the amount of data close to the mean with those far away from the mean (in the tails). 
Omnibus  D'Angostino's test. It provides a combined statistical test for the presence of skewness and kurtosis. 
Prob(Omnibus)  The above statistic turned into a probability 
JarqueBera  A different test of the skewness and kurtosis 
Prob (JB)  The above statistic turned into a probability 
DurbinWatson  A test for the presence of autocorrelation (that the errors are not independent.) Often important in timeseries analysis 
Cond. No  A test for multicollinearity (if in a fit with multiple parameters, the parameters are related with each other). 
As a final note, if you don't want to include a constant term in your model, you can exclude it using the minus operator.
# Fit the nointercept model
est_no_int = smf.ols(formula='Employed ~ GNP  1', data=df).fit()
# We pick 100 hundred points equally spaced from the min to the max
X_prime_1 = pd.DataFrame({'GNP': np.linspace(X.GNP.min(), X.GNP.max(), 100)})
X_prime_1 = sm.add_constant(X_prime_1) # add constant as we did before
y_hat_int = est.predict(X_prime_1)
y_hat_no_int = est_no_int.predict(X_prime_1)
fig = plt.figure(figsize=(8,4))
splt = plt.subplot(121)
splt.scatter(X.GNP, y, alpha=0.3) # Plot the raw data
plt.ylim(30, 100) # Set the yaxis to be the same
plt.xlabel("Gross National Product")
plt.ylabel("Total Employment")
plt.title("With intercept")
splt.plot(X_prime[:, 1], y_hat_int, 'r', alpha=0.9) # Add the regression line, colored in red
splt = plt.subplot(122)
splt.scatter(X.GNP, y, alpha=0.3) # Plot the raw data
plt.xlabel("Gross National Product")
plt.title("Without intercept")
splt.plot(X_prime[:, 1], y_hat_no_int, 'r', alpha=0.9) # Add the regression line, colored in red
[<matplotlib.lines.Line2D at 0x47eab50>]
But notice that this may not be the best idea... :)
Correlation and Causation
Clearly there is a relationship or correlation between GNP and total employment. So does that mean a change in GNP cause a change in total employment? Or does a change in total employment cause a change in GNP? This is a subject we will explore in the next post.
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