Simple linear regression



In [1]:
%matplotlib inline
import numpy as np
import pandas as pd
import scipy.stats as stats
import matplotlib.pyplot as plt
import sklearn


/Users/user/anaconda/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')

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

In [4]:
#keys of the dictionary
boston.keys()


Out[4]:
['data', 'feature_names', 'DESCR', 'target']

In [6]:
#display shape of the dataset
boston.data.shape


Out[6]:
(506, 13)

In [9]:
#Feature names
boston.feature_names


Out[9]:
array(['CRIM', 'ZN', 'INDUS', 'CHAS', 'NOX', 'RM', 'AGE', 'DIS', 'RAD',
       'TAX', 'PTRATIO', 'B', 'LSTAT'], 
      dtype='|S7')

In [11]:
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)


In [12]:
#lets convert it into a panda dataframe
bos_housing = pd.DataFrame(boston.data)
bos_housing.head()


Out[12]:
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

In [65]:
#Adding the target to the data
bos_housing.head()
bos_housing['final_prize'] = boston.target
bos_housing.head()


Out[65]:
0 1 2 3 4 5 6 7 8 9 10 11 12 final_prize
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

In [66]:
#now onto linear regression
from sklearn.linear_model import LinearRegression
X = bos_housing.drop('final_prize',axis=1)

In [67]:
X


Out[67]:
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
5 0.02985 0.0 2.18 0.0 0.458 6.430 58.7 6.0622 3.0 222.0 18.7 394.12 5.21
6 0.08829 12.5 7.87 0.0 0.524 6.012 66.6 5.5605 5.0 311.0 15.2 395.60 12.43
7 0.14455 12.5 7.87 0.0 0.524 6.172 96.1 5.9505 5.0 311.0 15.2 396.90 19.15
8 0.21124 12.5 7.87 0.0 0.524 5.631 100.0 6.0821 5.0 311.0 15.2 386.63 29.93
9 0.17004 12.5 7.87 0.0 0.524 6.004 85.9 6.5921 5.0 311.0 15.2 386.71 17.10
10 0.22489 12.5 7.87 0.0 0.524 6.377 94.3 6.3467 5.0 311.0 15.2 392.52 20.45
11 0.11747 12.5 7.87 0.0 0.524 6.009 82.9 6.2267 5.0 311.0 15.2 396.90 13.27
12 0.09378 12.5 7.87 0.0 0.524 5.889 39.0 5.4509 5.0 311.0 15.2 390.50 15.71
13 0.62976 0.0 8.14 0.0 0.538 5.949 61.8 4.7075 4.0 307.0 21.0 396.90 8.26
14 0.63796 0.0 8.14 0.0 0.538 6.096 84.5 4.4619 4.0 307.0 21.0 380.02 10.26
15 0.62739 0.0 8.14 0.0 0.538 5.834 56.5 4.4986 4.0 307.0 21.0 395.62 8.47
16 1.05393 0.0 8.14 0.0 0.538 5.935 29.3 4.4986 4.0 307.0 21.0 386.85 6.58
17 0.78420 0.0 8.14 0.0 0.538 5.990 81.7 4.2579 4.0 307.0 21.0 386.75 14.67
18 0.80271 0.0 8.14 0.0 0.538 5.456 36.6 3.7965 4.0 307.0 21.0 288.99 11.69
19 0.72580 0.0 8.14 0.0 0.538 5.727 69.5 3.7965 4.0 307.0 21.0 390.95 11.28
20 1.25179 0.0 8.14 0.0 0.538 5.570 98.1 3.7979 4.0 307.0 21.0 376.57 21.02
21 0.85204 0.0 8.14 0.0 0.538 5.965 89.2 4.0123 4.0 307.0 21.0 392.53 13.83
22 1.23247 0.0 8.14 0.0 0.538 6.142 91.7 3.9769 4.0 307.0 21.0 396.90 18.72
23 0.98843 0.0 8.14 0.0 0.538 5.813 100.0 4.0952 4.0 307.0 21.0 394.54 19.88
24 0.75026 0.0 8.14 0.0 0.538 5.924 94.1 4.3996 4.0 307.0 21.0 394.33 16.30
25 0.84054 0.0 8.14 0.0 0.538 5.599 85.7 4.4546 4.0 307.0 21.0 303.42 16.51
26 0.67191 0.0 8.14 0.0 0.538 5.813 90.3 4.6820 4.0 307.0 21.0 376.88 14.81
27 0.95577 0.0 8.14 0.0 0.538 6.047 88.8 4.4534 4.0 307.0 21.0 306.38 17.28
28 0.77299 0.0 8.14 0.0 0.538 6.495 94.4 4.4547 4.0 307.0 21.0 387.94 12.80
29 1.00245 0.0 8.14 0.0 0.538 6.674 87.3 4.2390 4.0 307.0 21.0 380.23 11.98
... ... ... ... ... ... ... ... ... ... ... ... ... ...
476 4.87141 0.0 18.10 0.0 0.614 6.484 93.6 2.3053 24.0 666.0 20.2 396.21 18.68
477 15.02340 0.0 18.10 0.0 0.614 5.304 97.3 2.1007 24.0 666.0 20.2 349.48 24.91
478 10.23300 0.0 18.10 0.0 0.614 6.185 96.7 2.1705 24.0 666.0 20.2 379.70 18.03
479 14.33370 0.0 18.10 0.0 0.614 6.229 88.0 1.9512 24.0 666.0 20.2 383.32 13.11
480 5.82401 0.0 18.10 0.0 0.532 6.242 64.7 3.4242 24.0 666.0 20.2 396.90 10.74
481 5.70818 0.0 18.10 0.0 0.532 6.750 74.9 3.3317 24.0 666.0 20.2 393.07 7.74
482 5.73116 0.0 18.10 0.0 0.532 7.061 77.0 3.4106 24.0 666.0 20.2 395.28 7.01
483 2.81838 0.0 18.10 0.0 0.532 5.762 40.3 4.0983 24.0 666.0 20.2 392.92 10.42
484 2.37857 0.0 18.10 0.0 0.583 5.871 41.9 3.7240 24.0 666.0 20.2 370.73 13.34
485 3.67367 0.0 18.10 0.0 0.583 6.312 51.9 3.9917 24.0 666.0 20.2 388.62 10.58
486 5.69175 0.0 18.10 0.0 0.583 6.114 79.8 3.5459 24.0 666.0 20.2 392.68 14.98
487 4.83567 0.0 18.10 0.0 0.583 5.905 53.2 3.1523 24.0 666.0 20.2 388.22 11.45
488 0.15086 0.0 27.74 0.0 0.609 5.454 92.7 1.8209 4.0 711.0 20.1 395.09 18.06
489 0.18337 0.0 27.74 0.0 0.609 5.414 98.3 1.7554 4.0 711.0 20.1 344.05 23.97
490 0.20746 0.0 27.74 0.0 0.609 5.093 98.0 1.8226 4.0 711.0 20.1 318.43 29.68
491 0.10574 0.0 27.74 0.0 0.609 5.983 98.8 1.8681 4.0 711.0 20.1 390.11 18.07
492 0.11132 0.0 27.74 0.0 0.609 5.983 83.5 2.1099 4.0 711.0 20.1 396.90 13.35
493 0.17331 0.0 9.69 0.0 0.585 5.707 54.0 2.3817 6.0 391.0 19.2 396.90 12.01
494 0.27957 0.0 9.69 0.0 0.585 5.926 42.6 2.3817 6.0 391.0 19.2 396.90 13.59
495 0.17899 0.0 9.69 0.0 0.585 5.670 28.8 2.7986 6.0 391.0 19.2 393.29 17.60
496 0.28960 0.0 9.69 0.0 0.585 5.390 72.9 2.7986 6.0 391.0 19.2 396.90 21.14
497 0.26838 0.0 9.69 0.0 0.585 5.794 70.6 2.8927 6.0 391.0 19.2 396.90 14.10
498 0.23912 0.0 9.69 0.0 0.585 6.019 65.3 2.4091 6.0 391.0 19.2 396.90 12.92
499 0.17783 0.0 9.69 0.0 0.585 5.569 73.5 2.3999 6.0 391.0 19.2 395.77 15.10
500 0.22438 0.0 9.69 0.0 0.585 6.027 79.7 2.4982 6.0 391.0 19.2 396.90 14.33
501 0.06263 0.0 11.93 0.0 0.573 6.593 69.1 2.4786 1.0 273.0 21.0 391.99 9.67
502 0.04527 0.0 11.93 0.0 0.573 6.120 76.7 2.2875 1.0 273.0 21.0 396.90 9.08
503 0.06076 0.0 11.93 0.0 0.573 6.976 91.0 2.1675 1.0 273.0 21.0 396.90 5.64
504 0.10959 0.0 11.93 0.0 0.573 6.794 89.3 2.3889 1.0 273.0 21.0 393.45 6.48
505 0.04741 0.0 11.93 0.0 0.573 6.030 80.8 2.5050 1.0 273.0 21.0 396.90 7.88

506 rows × 13 columns


In [51]:
lr = LinearRegression()
lr


Out[51]:
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

In [52]:
lr.fit(X,bos_housing.final_prize)


Out[52]:
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

In [53]:
print "Intercept Coefficient", lr.intercept_


Intercept Coefficient 36.4911032804

In [54]:
len(lr.coef_)
lr.coef_


Out[54]:
array([ -1.07170557e-01,   4.63952195e-02,   2.08602395e-02,
         2.68856140e+00,  -1.77957587e+01,   3.80475246e+00,
         7.51061703e-04,  -1.47575880e+00,   3.05655038e-01,
        -1.23293463e-02,  -9.53463555e-01,   9.39251272e-03,
        -5.25466633e-01])

In [55]:
pd.DataFrame(zip(X.columns,lr.coef_), columns = ['Features','Estimated'] )


Out[55]:
Features Estimated
0 0 -0.107171
1 1 0.046395
2 2 0.020860
3 3 2.688561
4 4 -17.795759
5 5 3.804752
6 6 0.000751
7 7 -1.475759
8 8 0.305655
9 9 -0.012329
10 10 -0.953464
11 11 0.009393
12 12 -0.525467

In [70]:
#Now lets see how well are predictions are. lets pass the first 5 values (Ideally we should be using unseen data)
lr.predict(X[0:5])


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

In [72]:
plt.scatter(bos_housing.final_prize, lr.predict(X))
plt.xlabel("Prices:")
plt.ylabel("Predicted Prices:")


Out[72]:
<matplotlib.text.Text at 0x11850e650>

In [73]:
# From the visualization it is clear that as the prices go up , our prediction quality degrades
# lets try mean squared error
mseFull = np.mean((bos_housing.final_prize - lr.predict(X))**2)

In [74]:
mseFull


Out[74]:
21.897779217687486

In [ ]:
# Again ideally training on the full data set was a mistake. What we should have done is divieded the data..
# ..into training and test datasets. We will follow that in the upcoming examples