Multiple Linear Regression¶

In [1]:
import pandas as pd
import numpy as np
from matplotlib.pyplot import subplots

house = pd.read_csv('house.csv')
house.head(3)
Out[1]:
Order PID MS SubClass MS Zoning Lot Frontage Lot Area Street Alley Lot Shape Land Contour ... Pool Area Pool QC Fence Misc Feature Misc Val Mo Sold Yr Sold Sale Type Sale Condition SalePrice
0 1 526301100 20 RL 141.0 31770 Pave NaN IR1 Lvl ... 0 NaN NaN NaN 0 5 2010 WD Normal 215000
1 2 526350040 20 RH 80.0 11622 Pave NaN Reg Lvl ... 0 NaN MnPrv NaN 0 6 2010 WD Normal 105000
2 3 526351010 20 RL 81.0 14267 Pave NaN IR1 Lvl ... 0 NaN NaN Gar2 12500 6 2010 WD Normal 172000

3 rows × 82 columns

In [2]:
house = house[house['Bldg Type'] == '1Fam']
house = house[house['Sale Condition'] == 'Normal']
sales = house[['SalePrice', '1st Flr SF', '2nd Flr SF', 'Total Bsmt SF', 
               'Garage Area', 'Wood Deck SF', 'Open Porch SF', 'Lot Area', 
               'Year Built', 'Yr Sold']]
sales.head(3)
Out[2]:
SalePrice 1st Flr SF 2nd Flr SF Total Bsmt SF Garage Area Wood Deck SF Open Porch SF Lot Area Year Built Yr Sold
0 215000 1656 0 1080.0 528.0 210 62 31770 1960 2010
1 105000 896 0 882.0 730.0 140 0 11622 1961 2010
2 172000 1329 0 1329.0 312.0 393 36 14267 1958 2010
In [3]:
sales['SalePrice'].corr(sales['1st Flr SF'])
Out[3]:
np.float64(0.6424662541030226)
In [4]:
sales['SalePrice'].corr(sales['2nd Flr SF'])
Out[4]:
np.float64(0.3575218942800825)
In [5]:
sales['SalePrice'].corr((sales['2nd Flr SF'] + sales['1st Flr SF']))
Out[5]:
np.float64(0.7821920556134877)
In [7]:
n = sales.shape[0] # Number of observations
X = np.ones((n, 1))
X = np.insert(X, 1, sales['1st Flr SF'], axis=1)
X = np.insert(X, 2, sales['2nd Flr SF'], axis=1)
y = sales['SalePrice'].reset_index(drop=True)
X.shape
Out[7]:
(2002, 3)
In [9]:
Z = np.matmul(X.T, X)
t = np.linalg.inv(Z)@X.T@y
MSE = (1/n) * np.linalg.norm(y - X@t)**2
print('MSE = ', MSE)
print('Theta = ', t)

MSE =  1705349005.3197255
Theta =  [-29985.3612312     155.19406373     91.35680257]
In [17]:
#Using Gradient Descent
learning_rate = 0.00000001
t = np.zeros(3).T
MSE = (1/n) * np.linalg.norm(y - X@t)**2
print('MSE, Theta:', MSE, t)
for i in range(11):
    t = t + (2*learning_rate/n) * (X.T @ (y - X@t))
    MSE = (1/n) * np.linalg.norm(y - X@t)**2
    print('MSE, Theta:', MSE, t)

MSE = (1/n) * np.linalg.norm(y - X@t)**2
print('MSE, Theta:', MSE, t)

MSE, Theta: 37483374720.76574 [0. 0. 0.]
MSE, Theta: 35329134615.48689 [3.58369254e-03 4.44420993e+00 1.45915604e+00]
MSE, Theta: 33305851706.1281 [7.05555785e-03 8.74978866e+00 2.87760804e+00]
MSE, Theta: 31405556990.79167 [1.04190511e-02 1.29210261e+01 4.25659660e+00]
MSE, Theta: 29620766460.99461 [1.36775206e-02 1.69620795e+01 5.59732406e+00]
MSE, Theta: 27944451584.46335 [1.68342113e-02 2.08769776e+01 6.90095566e+00]
MSE, Theta: 26370011584.380478 [1.98922677e-02 2.46696247e+01 8.16862069e+00]
MSE, Theta: 24891247405.749187 [2.28547373e-02 2.83438043e+01 9.40141360e+00]
MSE, Theta: 23502337266.195404 [2.57245732e-02 3.19031829e+01 1.06003951e+01]
MSE, Theta: 22197813694.776936 [2.85046374e-02 3.53513139e+01 1.17665931e+01]
MSE, Theta: 20972541968.237617 [3.11977034e-02 3.86916405e+01 1.29010040e+01]
MSE, Theta: 19821699859.656364 [3.38064589e-02 4.19274997e+01 1.40045933e+01]
MSE, Theta: 19821699859.656364 [3.38064589e-02 4.19274997e+01 1.40045933e+01]
In [ ]: