# 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 [39]:

# 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.
$$y = f(x) = E(Y | X = x)$$

• 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 [40]:




In [41]:

boston.keys()




Out[41]:

dict_keys(['target', 'data', 'feature_names', 'DESCR'])




In [42]:

boston.data.shape




Out[42]:

(506, 13)




In [43]:

# Print column names
print (boston.feature_names)




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




In [44]:

# 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
- 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 [45]:

bos = pd.DataFrame(boston.data)




Out[45]:

text-align: right;
}

text-align: left;
}

.dataframe tbody tr th {
vertical-align: top;
}

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 [46]:

bos.columns = boston.feature_names




Out[46]:

text-align: right;
}

text-align: left;
}

.dataframe tbody tr th {
vertical-align: top;
}

CRIM
ZN
INDUS
CHAS
NOX
RM
AGE
DIS
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 [47]:

print (boston.target.shape)




(506,)




In [48]:

bos['PRICE'] = boston.target




Out[48]:

text-align: right;
}

text-align: left;
}

.dataframe tbody tr th {
vertical-align: top;
}

CRIM
ZN
INDUS
CHAS
NOX
RM
AGE
DIS
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 [49]:

bos.describe()




Out[49]:

text-align: right;
}

text-align: left;
}

.dataframe tbody tr th {
vertical-align: top;
}

CRIM
ZN
INDUS
CHAS
NOX
RM
AGE
DIS
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 [50]:

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[50]:

<matplotlib.text.Text at 0x960b410>



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



In [51]:

#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 RM and Price")




Out[51]:

<matplotlib.text.Text at 0x9655b50>




In [52]:

#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 PTRATIO and Price")




Out[52]:

<matplotlib.text.Text at 0x83c2830>



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



In [53]:

#your turn: create some other scatter plots
plt.scatter(bos.AGE, bos.PRICE)
plt.xlabel("proportion of owner-occupied units built prior to 1940 (AGE)")
plt.ylabel("Housing Price")
plt.title("Relationship between House Ages and Price")




Out[53]:

<matplotlib.text.Text at 0xaabfcf0>



### 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 [54]:

sns.regplot(y="PRICE", x="RM", data=bos, fit_reg = True)




Out[54]:

<matplotlib.axes._subplots.AxesSubplot at 0x8e208f0>



### Histograms

Histograms are a useful way to visually summarize the statistical properties of numeric variables. They can give you an idea of the mean and the spread of the variables as well as outliers.



In [55]:

plt.hist(bos.CRIM)
plt.title("CRIM")
plt.xlabel("Crime rate per capita")
plt.ylabel("Frequency")
plt.show()






Your turn: Plot separate histograms and one for RM, one for PTRATIO. Any interesting observations?



In [56]:

plt.hist(bos.RM)
plt.title("RM")
plt.xlabel("average number of rooms per dwelling")
plt.ylabel("Frequency")
plt.show()







In [57]:

# Histogram for pupil-teacher ratio by town
plt.hist(bos.PTRATIO)
plt.title("PTRATIO")
plt.xlabel("pupil-teacher ratio by town")
plt.ylabel("Frequency")
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 [58]:

# 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 [59]:

# 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
Method:                 Least Squares   F-statistic:                     471.8
Date:                Sat, 17 Jun 2017   Prob (F-statistic):           2.49e-74
Time:                        14:39:37   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|      [0.025      0.975]
------------------------------------------------------------------------------
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.



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 [73]: # your turn plt.scatter(bos.PRICE, m.fittedvalues) plt.xlabel("Housing Price") plt.ylabel("Predicted Housing Price") plt.title("Relationship between Predicted and Actual Price")   Out[73]: <matplotlib.text.Text at 0xb7b9cb0>  ### Fitting Linear Regression using sklearn  In [61]: from sklearn.linear_model import LinearRegression X = bos.drop('PRICE', axis = 1) # This creates a LinearRegression object lm = LinearRegression() lm   Out[61]: 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 [62]: # 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 [63]: # Use all 13 predictors to fit linear regression model lm.fit(X, bos.PRICE)   Out[63]: 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? ### 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 [64]: print ('Estimated intercept coefficient:', lm.intercept_)   Estimated intercept coefficient: 36.4911032804   In [65]: print ('Number of coefficients:', len(lm.coef_))   Number of coefficients: 13   In [67]: # The coefficients pd.DataFrame(list(zip(X.columns, lm.coef_)), columns = ['features', 'estimatedCoefficients'])   Out[67]: .dataframe thead tr:only-child th { text-align: right; } .dataframe thead th { text-align: left; } .dataframe tbody tr th { vertical-align: top; } 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 [68]: # first five predicted prices lm.predict(X)[0:5]   Out[68]: 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 [74]: # your turn # Plot a histogram of all the predicted prices plt.hist(lm.predict(X)) plt.title("Predicted Prices") plt.xlabel("Predicted Prices") plt.ylabel("Frequency") plt.show() # Let's plot the true prices compared to the predicted prices to see they disagree plt.scatter(bos.PRICE, lm.predict(X)) plt.xlabel("Housing Price") plt.ylabel("Predicted Housing Price") plt.title("Relationship between Predicted and Actual Price")   Out[74]: <matplotlib.text.Text at 0xbf98a10>  ### 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 [75]: 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 [80]: #your turn print ('Mean squared error: ', np.mean((bos.PRICE - lm.predict(X)) ** 2))   Mean squared error: 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 [81]: lm = LinearRegression() lm.fit(X[['PTRATIO']], bos.PRICE)   Out[81]: 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 [101]: # your turn lm.fit(X[['CRIM']], bos.PRICE) print ('(MSE) Per capita crime rate by town: ', np.mean((bos.PRICE - lm.predict(X[['CRIM']])) ** 2)) lm.fit(X[['RM']], bos.PRICE) print ('(MSE) Average number of rooms per dwelling: ', np.mean((bos.PRICE - lm.predict(X[['RM']])) ** 2)) lm.fit(X[['PTRATIO']], bos.PRICE) print ('(MSE) Pupil-teacher ratio by town: ', np.mean((bos.PRICE - lm.predict(X[['PTRATIO']])) ** 2))   (MSE) Per capita crime rate by town: 71.8523466653 (MSE) Average number of rooms per dwelling: 43.6005517712 (MSE) Pupil-teacher ratio by town: 62.6522000138  ## 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  In [123]: sns.set(font_scale=.8) sns.heatmap(X.corr(), vmax=.8, square=True, annot=True)   Out[123]: <matplotlib.axes._subplots.AxesSubplot at 0xcf56310>  # 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 [85]: 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 [86]: X_train, X_test, Y_train, Y_test = sklearn.cross_validation.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 [105]: # your turn # Fit a linear regression model to the training set lm.fit(X_train, Y_train) lm.predict(X_test)   Out[105]: 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 [107]: # your turn # Calculate MSE using just the test data print ('(MSE) using just the test data: ', np.mean((Y_test - lm.predict(X_test)) ** 2)) # Calculate MSE using just the training data print ('(MSE) using just the training data: ', np.mean((Y_train - lm.predict(X_train)) ** 2))   (MSE) using just the test data: 28.5413672756 (MSE) using just the training data: 19.5467584735  Are they pretty similar or very different? What does that mean? -> They are very different because the model us based on training data so it will be accurate compared to the test data. The model is not exposed to test data so it will give a greater mean square error. It means there are data in test data which are different with the training data. #### Residual plots  In [108]: 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[108]: <matplotlib.text.Text at 0xb2cf070>  Your turn: Do you think this linear regression model generalizes well on the test data? -> No, the scatter points are not close to zero so the model needs improvements. Check the features to see highly correlated predictors and remove one of them or check the parameters of the model and do fine-tuning. ### 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 [127]:

from sklearn import cross_validation, linear_model

# If the estimator is a classifier and y is either binary or multiclass, StratifiedKFold is used.
# In all other cases, KFold is used
scores = cross_validation.cross_val_score(lm, X, bos.PRICE, scoring='mean_squared_error', cv=4)

# This will print metric for evaluation
print ('(MSE) Using k-fold: ', np.mean(scores))
print ('The K-fold cross-validation is not performaing well compared to the previous train-test split above')




(MSE) Using k-fold:  -42.4894695275
The K-fold cross-validation is not performaing well compared to the previous train-test split above