Regression in Python


This is a very quick run-through of some basic statistical concepts, adapted from Lab 4 in Harvard's CS109 course. Please feel free to try the original lab if you're feeling ambitious :-) The CS109 git repository also has the solutions if you're stuck.

  • Linear Regression Models
  • Prediction using linear regression
  • Some re-sampling methods
    • Train-Test splits
    • Cross Validation

Linear regression is used to model and predict continuous outcomes while logistic regression is used to model binary outcomes. We'll see some examples of linear regression as well as Train-test splits.

The packages we'll cover are: statsmodels, seaborn, and scikit-learn. While we don't explicitly teach statsmodels and seaborn in the Springboard workshop, those are great libraries to know.




In [1]:
# special IPython command to prepare the notebook for matplotlib and other libraries
%pylab inline 

import numpy as np
import pandas as pd
import scipy.stats as stats
import matplotlib.pyplot as plt
import sklearn

import seaborn as sns

# special matplotlib argument for improved plots
from matplotlib import rcParams
sns.set_style("whitegrid")
sns.set_context("poster")


Populating the interactive namespace from numpy and matplotlib

Part 1: Linear Regression

Purpose of linear regression


Given a dataset $X$ and $Y$, linear regression can be used to:

  • Build a predictive model to predict future values of $X_i$ without a $Y$ value.
  • Model the strength of the relationship between each dependent variable $X_i$ and $Y$
    • Sometimes not all $X_i$ will have a relationship with $Y$
    • Need to figure out which $X_i$ contributes most information to determine $Y$
  • Linear regression is used in so many applications that I won't warrant this with examples. It is in many cases, the first pass prediction algorithm for continuous outcomes.

A brief recap (feel free to skip if you don't care about the math)


Linear Regression is a method to model the relationship between a set of independent variables $X$ (also knowns as explanatory variables, features, predictors) and a dependent variable $Y$. This method assumes the relationship between each predictor $X$ is linearly related to the dependent variable $Y$.

$$ Y = \beta_0 + \beta_1 X + \epsilon$$

where $\epsilon$ is considered as an unobservable random variable that adds noise to the linear relationship. This is the simplest form of linear regression (one variable), we'll call this the simple model.

  • $\beta_0$ is the intercept of the linear model

  • Multiple linear regression is when you have more than one independent variable

    • $X_1$, $X_2$, $X_3$, $\ldots$
$$ Y = \beta_0 + \beta_1 X_1 + \ldots + \beta_p X_p + \epsilon$$

  • Back to the simple model. The model in linear regression is the conditional mean of $Y$ given the values in $X$ is expressed a linear function.
$$ y = f(x) = E(Y | X = x)$$

http://www.learner.org/courses/againstallodds/about/glossary.html

  • The goal is to estimate the coefficients (e.g. $\beta_0$ and $\beta_1$). We represent the estimates of the coefficients with a "hat" on top of the letter.
$$ \hat{\beta}_0, \hat{\beta}_1 $$
  • Once you estimate the coefficients $\hat{\beta}_0$ and $\hat{\beta}_1$, you can use these to predict new values of $Y$
$$\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x_1$$
  • How do you estimate the coefficients?
    • There are many ways to fit a linear regression model
    • The method called least squares is one of the most common methods
    • We will discuss least squares today

Estimating $\hat\beta$: Least squares


Least squares is a method that can estimate the coefficients of a linear model by minimizing the difference between the following:

$$ S = \sum_{i=1}^N r_i = \sum_{i=1}^N (y_i - (\beta_0 + \beta_1 x_i))^2 $$

where $N$ is the number of observations.

  • We will not go into the mathematical details, but the least squares estimates $\hat{\beta}_0$ and $\hat{\beta}_1$ minimize the sum of the squared residuals $r_i = y_i - (\beta_0 + \beta_1 x_i)$ in the model (i.e. makes the difference between the observed $y_i$ and linear model $\beta_0 + \beta_1 x_i$ as small as possible).

The solution can be written in compact matrix notation as

$$\hat\beta = (X^T X)^{-1}X^T Y$$

We wanted to show you this in case you remember linear algebra, in order for this solution to exist we need $X^T X$ to be invertible. Of course this requires a few extra assumptions, $X$ must be full rank so that $X^T X$ is invertible, etc. This is important for us because this means that having redundant features in our regression models will lead to poorly fitting (and unstable) models. We'll see an implementation of this in the extra linear regression example.

Note: The "hat" means it is an estimate of the coefficient.


Part 2: Boston Housing Data Set

The Boston Housing data set contains information about the housing values in suburbs of Boston. This dataset was originally taken from the StatLib library which is maintained at Carnegie Mellon University and is now available on the UCI Machine Learning Repository.

Load the Boston Housing data set from sklearn


This data set is available in the sklearn python module which is how we will access it today.


In [2]:
from sklearn.datasets import load_boston
boston = load_boston()

In [3]:
boston.keys()


Out[3]:
dict_keys(['DESCR', 'target', 'feature_names', 'data'])

In [4]:
boston.data.shape


Out[4]:
(506, 13)

In [6]:
# Print column names
print(boston.feature_names)


['CRIM' 'ZN' 'INDUS' 'CHAS' 'NOX' 'RM' 'AGE' 'DIS' 'RAD' 'TAX' 'PTRATIO'
 'B' 'LSTAT']

In [7]:
# Print description of Boston housing data set
print(boston.DESCR)


Boston House Prices dataset
===========================

Notes
------
Data Set Characteristics:  

    :Number of Instances: 506 

    :Number of Attributes: 13 numeric/categorical predictive
    
    :Median Value (attribute 14) is usually the target

    :Attribute Information (in order):
        - CRIM     per capita crime rate by town
        - ZN       proportion of residential land zoned for lots over 25,000 sq.ft.
        - INDUS    proportion of non-retail business acres per town
        - CHAS     Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
        - NOX      nitric oxides concentration (parts per 10 million)
        - RM       average number of rooms per dwelling
        - AGE      proportion of owner-occupied units built prior to 1940
        - DIS      weighted distances to five Boston employment centres
        - RAD      index of accessibility to radial highways
        - TAX      full-value property-tax rate per $10,000
        - PTRATIO  pupil-teacher ratio by town
        - B        1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town
        - LSTAT    % lower status of the population
        - MEDV     Median value of owner-occupied homes in $1000's

    :Missing Attribute Values: None

    :Creator: Harrison, D. and Rubinfeld, D.L.

This is a copy of UCI ML housing dataset.
http://archive.ics.uci.edu/ml/datasets/Housing


This dataset was taken from the StatLib library which is maintained at Carnegie Mellon University.

The Boston house-price data of Harrison, D. and Rubinfeld, D.L. 'Hedonic
prices and the demand for clean air', J. Environ. Economics & Management,
vol.5, 81-102, 1978.   Used in Belsley, Kuh & Welsch, 'Regression diagnostics
...', Wiley, 1980.   N.B. Various transformations are used in the table on
pages 244-261 of the latter.

The Boston house-price data has been used in many machine learning papers that address regression
problems.   
     
**References**

   - Belsley, Kuh & Welsch, 'Regression diagnostics: Identifying Influential Data and Sources of Collinearity', Wiley, 1980. 244-261.
   - Quinlan,R. (1993). Combining Instance-Based and Model-Based Learning. In Proceedings on the Tenth International Conference of Machine Learning, 236-243, University of Massachusetts, Amherst. Morgan Kaufmann.
   - many more! (see http://archive.ics.uci.edu/ml/datasets/Housing)

Now let's explore the data set itself.


In [8]:
bos = pd.DataFrame(boston.data)
bos.head()


Out[8]:
0 1 2 3 4 5 6 7 8 9 10 11 12
0 0.00632 18.0 2.31 0.0 0.538 6.575 65.2 4.0900 1.0 296.0 15.3 396.90 4.98
1 0.02731 0.0 7.07 0.0 0.469 6.421 78.9 4.9671 2.0 242.0 17.8 396.90 9.14
2 0.02729 0.0 7.07 0.0 0.469 7.185 61.1 4.9671 2.0 242.0 17.8 392.83 4.03
3 0.03237 0.0 2.18 0.0 0.458 6.998 45.8 6.0622 3.0 222.0 18.7 394.63 2.94
4 0.06905 0.0 2.18 0.0 0.458 7.147 54.2 6.0622 3.0 222.0 18.7 396.90 5.33

There are no column names in the DataFrame. Let's add those.


In [9]:
bos.columns = boston.feature_names
bos.head()


Out[9]:
CRIM ZN INDUS CHAS NOX RM AGE DIS RAD TAX PTRATIO B LSTAT
0 0.00632 18.0 2.31 0.0 0.538 6.575 65.2 4.0900 1.0 296.0 15.3 396.90 4.98
1 0.02731 0.0 7.07 0.0 0.469 6.421 78.9 4.9671 2.0 242.0 17.8 396.90 9.14
2 0.02729 0.0 7.07 0.0 0.469 7.185 61.1 4.9671 2.0 242.0 17.8 392.83 4.03
3 0.03237 0.0 2.18 0.0 0.458 6.998 45.8 6.0622 3.0 222.0 18.7 394.63 2.94
4 0.06905 0.0 2.18 0.0 0.458 7.147 54.2 6.0622 3.0 222.0 18.7 396.90 5.33

Now we have a pandas DataFrame called bos containing all the data we want to use to predict Boston Housing prices. Let's create a variable called PRICE which will contain the prices. This information is contained in the target data.


In [11]:
print(boston.target.shape)


(506,)

In [12]:
bos['PRICE'] = boston.target
bos.head()


Out[12]:
CRIM ZN INDUS CHAS NOX RM AGE DIS RAD TAX PTRATIO B LSTAT PRICE
0 0.00632 18.0 2.31 0.0 0.538 6.575 65.2 4.0900 1.0 296.0 15.3 396.90 4.98 24.0
1 0.02731 0.0 7.07 0.0 0.469 6.421 78.9 4.9671 2.0 242.0 17.8 396.90 9.14 21.6
2 0.02729 0.0 7.07 0.0 0.469 7.185 61.1 4.9671 2.0 242.0 17.8 392.83 4.03 34.7
3 0.03237 0.0 2.18 0.0 0.458 6.998 45.8 6.0622 3.0 222.0 18.7 394.63 2.94 33.4
4 0.06905 0.0 2.18 0.0 0.458 7.147 54.2 6.0622 3.0 222.0 18.7 396.90 5.33 36.2

EDA and Summary Statistics


Let's explore this data set. First we use describe() to get basic summary statistics for each of the columns.


In [13]:
bos.describe()


Out[13]:
CRIM ZN INDUS CHAS NOX RM AGE DIS RAD TAX PTRATIO B LSTAT PRICE
count 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000 506.000000
mean 3.593761 11.363636 11.136779 0.069170 0.554695 6.284634 68.574901 3.795043 9.549407 408.237154 18.455534 356.674032 12.653063 22.532806
std 8.596783 23.322453 6.860353 0.253994 0.115878 0.702617 28.148861 2.105710 8.707259 168.537116 2.164946 91.294864 7.141062 9.197104
min 0.006320 0.000000 0.460000 0.000000 0.385000 3.561000 2.900000 1.129600 1.000000 187.000000 12.600000 0.320000 1.730000 5.000000
25% 0.082045 0.000000 5.190000 0.000000 0.449000 5.885500 45.025000 2.100175 4.000000 279.000000 17.400000 375.377500 6.950000 17.025000
50% 0.256510 0.000000 9.690000 0.000000 0.538000 6.208500 77.500000 3.207450 5.000000 330.000000 19.050000 391.440000 11.360000 21.200000
75% 3.647423 12.500000 18.100000 0.000000 0.624000 6.623500 94.075000 5.188425 24.000000 666.000000 20.200000 396.225000 16.955000 25.000000
max 88.976200 100.000000 27.740000 1.000000 0.871000 8.780000 100.000000 12.126500 24.000000 711.000000 22.000000 396.900000 37.970000 50.000000

Scatter plots


Let's look at some scatter plots for three variables: 'CRIM', 'RM' and 'PTRATIO'.

What kind of relationship do you see? e.g. positive, negative? linear? non-linear?


In [14]:
plt.scatter(bos.CRIM, bos.PRICE)
plt.xlabel("Per capita crime rate by town (CRIM)")
plt.ylabel("Housing Price")
plt.title("Relationship between CRIM and Price")


Out[14]:
<matplotlib.text.Text at 0x1139d5438>

Your turn: Create scatter plots between RM and PRICE, and PTRATIO and PRICE. What do you notice?


In [15]:
#your turn: scatter plot between *RM* and *PRICE*
plt.scatter(bos.RM, bos.PRICE)
plt.xlabel("average number of rooms per dwelling (RM)")
plt.ylabel("Housing Price")
plt.title("Relationship between CRIM and Price")


Out[15]:
<matplotlib.text.Text at 0x114510748>

In [16]:
#your turn: scatter plot between *PTRATIO* and *PRICE*
plt.scatter(bos.PTRATIO, bos.PRICE)
plt.xlabel("pupil-teacher ratio by town (PTRATIO)")
plt.ylabel("Housing Price")
plt.title("Relationship between CRIM and Price")


Out[16]:
<matplotlib.text.Text at 0x1148330f0>

Your turn: What are some other numeric variables of interest? Plot scatter plots with these variables and PRICE.


In [19]:
#your turn: create some other scatter plots
plt.scatter(bos.LSTAT, bos.PRICE)
plt.xlabel("pupil-teacher ratio by town (PTRATIO)")
plt.ylabel("Housing Price")
plt.title("Relationship between CRIM and Price")


Out[19]:
<matplotlib.text.Text at 0x114ef50b8>

Scatter Plots using Seaborn


Seaborn is a cool Python plotting library built on top of matplotlib. It provides convenient syntax and shortcuts for many common types of plots, along with better-looking defaults.

We can also use seaborn regplot for the scatterplot above. This provides automatic linear regression fits (useful for data exploration later on). Here's one example below.


In [20]:
sns.regplot(y="PRICE", x="RM", data=bos, fit_reg = True)


Out[20]:
<matplotlib.axes._subplots.AxesSubplot at 0x114eedd30>

Histograms



In [21]:
plt.hist(bos.CRIM)
plt.title("CRIM")
plt.xlabel("Crime rate per capita")
plt.ylabel("Frequencey")
plt.show()


Your turn:

  • Plot histograms for RM and PTRATIO, along with the two variables you picked in the previous section.

In [27]:
#your turn
plt.hist(bos.RM)
plt.title("RM")
plt.xlabel("average number of rooms per dwelling (RM)")
plt.ylabel("Frequencey")
plt.show()



In [26]:
#your turn
plt.hist(bos.PTRATIO)
plt.title("PTRATIO")
plt.xlabel("pupil-teacher ratio by town (PTRATIO)")
plt.ylabel("Frequencey")
plt.show()


Linear regression with Boston housing data example


Here,

$Y$ = boston housing prices (also called "target" data in python)

and

$X$ = all the other features (or independent variables)

which we will use to fit a linear regression model and predict Boston housing prices. We will use the least squares method as the way to estimate the coefficients.

We'll use two ways of fitting a linear regression. We recommend the first but the second is also powerful in its features.

Fitting Linear Regression using statsmodels


Statsmodels is a great Python library for a lot of basic and inferential statistics. It also provides basic regression functions using an R-like syntax, so it's commonly used by statisticians. While we don't cover statsmodels officially in the Data Science Intensive, it's a good library to have in your toolbox. Here's a quick example of what you could do with it.


In [28]:
# Import regression modules
# ols - stands for Ordinary least squares, we'll use this
import statsmodels.api as sm
from statsmodels.formula.api import ols

In [30]:
# statsmodels works nicely with pandas dataframes
# The thing inside the "quotes" is called a formula, a bit on that below
m = ols('PRICE ~ RM',bos).fit()
print(m.summary())


                            OLS Regression Results                            
==============================================================================
Dep. Variable:                  PRICE   R-squared:                       0.484
Model:                            OLS   Adj. R-squared:                  0.483
Method:                 Least Squares   F-statistic:                     471.8
Date:                Thu, 03 Nov 2016   Prob (F-statistic):           2.49e-74
Time:                        13:36:35   Log-Likelihood:                -1673.1
No. Observations:                 506   AIC:                             3350.
Df Residuals:                     504   BIC:                             3359.
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
Intercept    -34.6706      2.650    -13.084      0.000       -39.877   -29.465
RM             9.1021      0.419     21.722      0.000         8.279     9.925
==============================================================================
Omnibus:                      102.585   Durbin-Watson:                   0.684
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              612.449
Skew:                           0.726   Prob(JB):                    1.02e-133
Kurtosis:                       8.190   Cond. No.                         58.4
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Interpreting coefficients

There is a ton of information in this output. But we'll concentrate on the coefficient table (middle table). We can interpret the RM coefficient (9.1021) by first noticing that the p-value (under P>|t|) is so small, basically zero. We can interpret the coefficient as, if we compare two groups of towns, one where the average number of rooms is say $5$ and the other group is the same except that they all have $6$ rooms. For these two groups the average difference in house prices is about $9.1$ (in thousands) so about $\$9,100$ difference. The confidence interval fives us a range of plausible values for this difference, about ($\$8,279, \$9,925$), deffinitely not chump change.

statsmodels formulas


This formula notation will seem familiar to R users, but will take some getting used to for people coming from other languages or are new to statistics.

The formula gives instruction for a general structure for a regression call. For statsmodels (ols or logit) calls you need to have a Pandas dataframe with column names that you will add to your formula. In the below example you need a pandas data frame that includes the columns named (Outcome, X1,X2, ...), bbut you don't need to build a new dataframe for every regression. Use the same dataframe with all these things in it. The structure is very simple:

Outcome ~ X1

But of course we want to to be able to handle more complex models, for example multiple regression is doone like this:

Outcome ~ X1 + X2 + X3

This is the very basic structure but it should be enough to get you through the homework. Things can get much more complex, for a quick run-down of further uses see the statsmodels help page.

Let's see how our model actually fit our data. We can see below that there is a ceiling effect, we should probably look into that. Also, for large values of $Y$ we get underpredictions, most predictions are below the 45-degree gridlines.

Your turn: Create a scatterpot between the predicted prices, available in m.fittedvalues and the original prices. How does the plot look?


In [35]:
# your turn
sns.regplot(y=bos.PRICE, x=m.fittedvalues, fit_reg = True)
plt.xlabel('predicted price')


Out[35]:
<matplotlib.text.Text at 0x117179f28>

Fitting Linear Regression using sklearn


In [47]:
from sklearn.linear_model import LinearRegression
X = bos.drop('PRICE', axis = 1)

# This creates a LinearRegression object
lm = LinearRegression()
lm


Out[47]:
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

What can you do with a LinearRegression object?


Check out the scikit-learn docs here. We have listed the main functions here.

Main functions Description
lm.fit() Fit a linear model
lm.predit() Predict Y using the linear model with estimated coefficients
lm.score() Returns the coefficient of determination (R^2). A measure of how well observed outcomes are replicated by the model, as the proportion of total variation of outcomes explained by the model

What output can you get?


In [42]:
# Look inside lm object
# lm.<tab>
Output Description
lm.coef_ Estimated coefficients
lm.intercept_ Estimated intercept

Fit a linear model


The lm.fit() function estimates the coefficients the linear regression using least squares.


In [48]:
# Use all 13 predictors to fit linear regression model
lm.fit(X, bos.PRICE)


Out[48]:
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

Your turn: How would you change the model to not fit an intercept term? Would you recommend not having an intercept?


In [45]:
lm = LinearRegression(fit_intercept=False)
lm.fit(X, bos.PRICE)


Out[45]:
LinearRegression(copy_X=True, fit_intercept=False, n_jobs=1, normalize=False)

Estimated intercept and coefficients

Let's look at the estimated coefficients from the linear model using 1m.intercept_ and lm.coef_.

After we have fit our linear regression model using the least squares method, we want to see what are the estimates of our coefficients $\beta_0$, $\beta_1$, ..., $\beta_{13}$:

$$ \hat{\beta}_0, \hat{\beta}_1, \ldots, \hat{\beta}_{13} $$

In [49]:
print('Estimated intercept coefficient:', lm.intercept_)


Estimated intercept coefficient: 36.4911032804

In [50]:
print('Number of coefficients:', len(lm.coef_))


Number of coefficients: 13

In [69]:
# The coefficients
pd.DataFrame(list(zip(X.columns, lm.coef_)), columns = ['features', 'estimatedCoefficients'])


Out[69]:
features estimatedCoefficients
0 CRIM -0.107171
1 ZN 0.046395
2 INDUS 0.020860
3 CHAS 2.688561
4 NOX -17.795759
5 RM 3.804752
6 AGE 0.000751
7 DIS -1.475759
8 RAD 0.305655
9 TAX -0.012329
10 PTRATIO -0.953464
11 B 0.009393
12 LSTAT -0.525467

Predict Prices

We can calculate the predicted prices ($\hat{Y}_i$) using lm.predict.

$$ \hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1 X_1 + \ldots \hat{\beta}_{13} X_{13} $$

In [70]:
# first five predicted prices
lm.predict(X)[0:5]


Out[70]:
array([ 30.00821269,  25.0298606 ,  30.5702317 ,  28.60814055,  27.94288232])

Your turn:

  • Histogram: Plot a histogram of all the predicted prices
  • Scatter Plot: Let's plot the true prices compared to the predicted prices to see they disagree (we did this with statsmodels before).

In [72]:
# your turn
sns.distplot(lm.predict(X), kde=False)


Out[72]:
<matplotlib.axes._subplots.AxesSubplot at 0x11829b908>

In [74]:
sns.regplot(lm.predict(X), bos.PRICE)
plt.xlabel('predicted prices')


Out[74]:
<matplotlib.text.Text at 0x118359a90>

Residual sum of squares

Let's calculate the residual sum of squares

$$ S = \sum_{i=1}^N r_i = \sum_{i=1}^N (y_i - (\beta_0 + \beta_1 x_i))^2 $$

In [76]:
print(np.sum((bos.PRICE - lm.predict(X)) ** 2))


11080.2762841

Mean squared error


This is simple the mean of the residual sum of squares.

Your turn: Calculate the mean squared error and print it.


In [79]:
#your turn
print(np.mean((bos.PRICE - lm.predict(X)) ** 2))


21.8977792177

Relationship between PTRATIO and housing price


Try fitting a linear regression model using only the 'PTRATIO' (pupil-teacher ratio by town)

Calculate the mean squared error.


In [80]:
lm = LinearRegression()
lm.fit(X[['PTRATIO']], bos.PRICE)


Out[80]:
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

In [82]:
msePTRATIO = np.mean((bos.PRICE - lm.predict(X[['PTRATIO']])) ** 2)
print(msePTRATIO)


62.6522000138

We can also plot the fitted linear regression line.


In [83]:
plt.scatter(bos.PTRATIO, bos.PRICE)
plt.xlabel("Pupil-to-Teacher Ratio (PTRATIO)")
plt.ylabel("Housing Price")
plt.title("Relationship between PTRATIO and Price")

plt.plot(bos.PTRATIO, lm.predict(X[['PTRATIO']]), color='blue', linewidth=3)
plt.show()


Your turn


Try fitting a linear regression model using three independent variables

  1. 'CRIM' (per capita crime rate by town)
  2. 'RM' (average number of rooms per dwelling)
  3. 'PTRATIO' (pupil-teacher ratio by town)

Calculate the mean squared error.


In [88]:
# your turn
lm = LinearRegression()
lm.fit(X[['PTRATIO', 'RM', 'CRIM']], bos.PRICE)


Out[88]:
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

In [89]:
msePTRATIO = np.mean((bos.PRICE - lm.predict(X[['PTRATIO', 'RM', 'CRIM']])) ** 2)
print(msePTRATIO)


34.3237965647

Other important things to think about when fitting a linear regression model


  • **Linearity**. The dependent variable $Y$ is a linear combination of the regression coefficients and the independent variables $X$.
  • **Constant standard deviation**. The SD of the dependent variable $Y$ should be constant for different values of X.
    • e.g. PTRATIO
  • **Normal distribution for errors**. The $\epsilon$ term we discussed at the beginning are assumed to be normally distributed. $$ \epsilon_i \sim N(0, \sigma^2)$$ Sometimes the distributions of responses $Y$ may not be normally distributed at any given value of $X$. e.g. skewed positively or negatively.
  • **Independent errors**. The observations are assumed to be obtained independently.
    • e.g. Observations across time may be correlated

Part 3: Training and Test Data sets

Purpose of splitting data into Training/testing sets


Let's stick to the linear regression example:

  • We built our model with the requirement that the model fit the data well.
  • As a side-effect, the model will fit THIS dataset well. What about new data?
    • We wanted the model for predictions, right?
  • One simple solution, leave out some data (for testing) and train the model on the rest
  • This also leads directly to the idea of cross-validation, next section.

One way of doing this is you can create training and testing data sets manually.


In [91]:
X_train = X[:-50]
X_test = X[-50:]
Y_train = bos.PRICE[:-50]
Y_test = bos.PRICE[-50:]
print(X_train.shape)
print(X_test.shape)
print(Y_train.shape)
print(Y_test.shape)


(456, 13)
(50, 13)
(456,)
(50,)

Another way, is to split the data into random train and test subsets using the function train_test_split in sklearn.cross_validation. Here's the documentation.


In [95]:
from sklearn.model_selection import train_test_split

X_train, X_test, Y_train, Y_test = train_test_split(
    X, bos.PRICE, test_size=0.33, random_state = 5)
print(X_train.shape)
print(X_test.shape)
print(Y_train.shape)
print(Y_test.shape)


(339, 13)
(167, 13)
(339,)
(167,)

Your turn: Let's build a linear regression model using our new training data sets.

  • Fit a linear regression model to the training set
  • Predict the output on the test set

In [103]:
# your turn
lm = LinearRegression()
lm.fit(X_train, Y_train)


Out[103]:
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

In [104]:
lm.predict(X_test)


Out[104]:
array([ 37.46723562,  31.39154701,  27.1201962 ,   6.46843347,
        33.62966737,   5.67067989,  27.03946671,  29.92704748,
        26.35661334,  22.45246021,  32.20504441,  21.78641653,
        23.41138441,  33.60894362,  28.28619511,  15.13859055,
         0.30087325,  18.71850376,  14.4706712 ,  11.10823598,
         2.69494197,  19.21693734,  38.41159345,  24.36936442,
        31.61493439,  11.42210397,  24.92862188,  23.31178043,
        22.7764079 ,  20.65081211,  16.035198  ,   7.07978633,
        17.65509209,  22.81470561,  29.21943405,  18.61354566,
        28.37701843,   8.80516873,  41.65140459,  34.02910176,
        20.1868926 ,   3.95600857,  29.69124564,  12.18081256,
        27.19403498,  30.63699231,  -6.24952457,  19.9462404 ,
        21.55123979,  13.36478173,  20.39068171,  19.87353324,
        23.57656877,  13.40141285,  17.66457201,  24.77424747,
        35.31476509,  15.48318159,  28.50764575,  21.72575404,
        20.58142839,  26.08460856,  14.51816968,  32.37494056,
        20.80917392,  12.18932524,  19.45551285,  25.23390429,
        21.77302317,  21.30227044,  20.58222113,  26.74635016,
        17.53006166,  18.7191946 ,  19.03026793,  25.76553031,
        21.8757557 ,  15.70891861,  35.12411848,  18.04488652,
        22.43612549,  39.4000555 ,  22.30677551,  14.9738331 ,
        25.29300631,  17.3200635 ,  18.58435124,  10.01693133,
        19.62408198,  17.24471407,  36.26263664,  17.55591517,
        21.10848471,  19.08435242,  24.72519887,  28.0878012 ,
        12.25474746,  22.40592558,  21.00483315,  13.51073355,
        23.09169468,  21.48906423,  14.14959117,  42.75677509,
         2.01088993,  21.9914102 ,  18.32505073,  22.59335404,
        28.93052931,  18.49024451,  27.61537531,  24.65547955,
        20.32508475,  32.66905896,  19.72975821,  12.8254    ,
        22.68957624,  18.2350211 ,  19.40432885,  16.19144346,
        21.77804736,  35.50387944,  22.24038654,  20.20025029,
        24.54270446,  25.29795497,  20.50220669,  23.0150761 ,
        23.38446711,  40.91809141,  37.84906745,  27.54024335,
        12.53470565,  15.90588084,  18.25352202,  21.62847325,
        15.77967465,   5.62636735,  24.00046271,  30.37118947,
        23.01126707,  18.29104509,  16.194709  ,  21.60846672,
        34.71665914,  23.40506116,  30.13747943,  18.0951727 ,
        22.16844264,  29.0922559 ,  13.36146671,  31.8608905 ,
        13.1643678 ,  13.91761543,  26.52314446,  31.39481197,
        10.62913801,  24.6869924 ,  28.95650935,  32.31758322,
        15.87113569,  29.94335724,   9.71836876,  34.70520017,
        25.70410195,  20.15430904,  15.3946584 ])

Your turn:

Calculate the mean squared error

  • using just the test data
  • using just the training data

Are they pretty similar or very different? What does that mean?


In [106]:
# your turn
mse_train = np.mean((Y_train - lm.predict(X_train)) ** 2)
mse_test = np.mean((Y_test - lm.predict(X_test)) ** 2)
print(mse_train)
print(mse_test)

# train is better, so overfitted on the traindata somewhat


19.5467584735
28.5413672756

Residual plots


In [107]:
plt.scatter(lm.predict(X_train), lm.predict(X_train) - Y_train, c='b', s=40, alpha=0.5)
plt.scatter(lm.predict(X_test), lm.predict(X_test) - Y_test, c='g', s=40)
plt.hlines(y = 0, xmin=0, xmax = 50)
plt.title('Residual Plot using training (blue) and test (green) data')
plt.ylabel('Residuals')


Out[107]:
<matplotlib.text.Text at 0x1185fb438>

Your turn: Do you think this linear regression model generalizes well on the test data?


In [108]:
# Yes, I think so, since the datapoints of the test and traindata are pretty similar

K-fold Cross-validation as an extension of this idea


A simple extension of the Test/train split is called K-fold cross-validation.

Here's the procedure:

  • randomly assign your $n$ samples to one of $K$ groups. They'll each have about $n/k$ samples
  • For each group $k$:
    • Fit the model (e.g. run regression) on all data excluding the $k^{th}$ group
    • Use the model to predict the outcomes in group $k$
    • Calculate your prediction error for each observation in $k^{th}$ group (e.g. $(Y_i - \hat{Y}_i)^2$ for regression, $\mathbb{1}(Y_i = \hat{Y}_i)$ for logistic regression).
  • Calculate the average prediction error across all samples $Err_{CV} = \frac{1}{n}\sum_{i=1}^n (Y_i - \hat{Y}_i)^2$

Luckily you don't have to do this entire process all by hand (for loops, etc.) every single time, sci-kit learn has a very nice implementation of this, have a look at the documentation.

Your turn (extra credit): Implement K-Fold cross-validation using the procedure above and Boston Housing data set using $K=4$. How does the average prediction error compare to the train-test split above?


In [119]:
from sklearn.model_selection import cross_val_score
from sklearn.linear_model import LinearRegression
clf = LinearRegression()
scores = cross_val_score(clf, bos.drop('PRICE', axis = 1), bos.PRICE, cv=4, scoring='neg_mean_squared_error')
scores*-1  
# train/test split was 28. The variation is pretty big in the different folds, so it is in between the highest and lowest
# but the one train/test split did not show this variation, which is important to know


Out[119]:
array([ 11.69977729,  39.0720141 ,  57.59307099,  61.59301573])

In [ ]: