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Linear Regression
What does linear regression on a single variable do? How do we evaluate the accuracy of the model? What is a residual?
Best Line
We find the line that best fits the data. It is one of the most popular tools in statistics.\(
h(x) = ax + b \)
\(
h_\theta = \theta_0 + \theta_1 x_1
\)
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
# Training Dataset
X = np.array([30, 46, 60, 65, 77, 95]).reshape(6,1)
Y = np.array([31, 30, 80, 49, 70, 118])
# Learn a prediction function
r = LinearRegression().fit(X, Y)
a = r.coef_[0].round(1)
b = r.intercept_.round(1)
# Predict unknown
x1 = 80
y1 = a*x1 + b
print(f'h(x) = {a}x + {b}')
print(f'h({x1}) = {y1}')
# Draw graphics
fig, ax = plt.subplots()
plt.ylim(0, 140)
plt.xlim(0, 140)
ax.plot(X, Y, 'x', color='g', label='training data') # Dataset points
ax.plot(x1, y1, 'o', color='r', label=f'h({x1}) = {y1}') # Unknown point
ax.plot(X, a*X + b, label=f'h(x) = {b} + {a}x') # Function line
plt.legend()
plt.show()
"""
h(x) = 1.3x + -18.0
h(80) = 86.0
"""
Two variables
Lets predict the CO2 emission of a car based on the size of the engine. With multiple regression we can throw in more variables, like the weight of the car. This will make the prediction more accurate.\(
h(x) = ax + by + c
\)
\(
h_\theta = \theta_0 + \theta_1 x_1 + \theta_2 x_2
\)
from sklearn.linear_model import LinearRegression
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import pathlib
DIR = pathlib.Path(__file__).resolve().parent
# Training Dataset
with open(DIR / 'data/cars.csv') as file:
df = pd.read_csv(file)
X = df[[
'Weight',
'Volume',
]].values
y = df['CO2'].values
# Learn a prediction function
r = LinearRegression().fit(X, y)
# Draw surface
fig = plt.figure()
Ax, Ay = np.meshgrid(
np.linspace(df.Weight.min(), df.Weight.max(), 100),
np.linspace(df.Volume.min(), df.Volume.max(), 100)
)
onlyX = pd.DataFrame({'Weight': Ax.ravel(), 'Volume': Ay.ravel()})
fittedY = r.predict(onlyX)
fittedY = np.array(fittedY)
ax = fig.add_subplot(111, projection='3d')
ax.scatter(df['Weight'], df['Volume'], df['CO2'], c='g', marker='x', alpha=0.5)
ax.plot_surface(Ax, Ay, fittedY.reshape(Ax.shape), color='b', alpha=0.3)
ax.set_xlabel('Weight')
ax.set_ylabel('Volume')
ax.set_zlabel('CO2')
# Predictions
X1 = [1600, 1252] # Honda Civic, 1600, 1252 / CO2: 94
y1 = r.predict([X1]) # CO2: 101.5
X2 = [1200, 780] # Unknown car
y2 = r.predict([X2]) # CO2: 94.8
print(df, "\n")
print("Honda Civic, 1600, 1252 / CO2:", y1.round(1).item())
print("Unknow car, 1200, 780 / CO2:", y2.round(1).item())
ax.plot(X1[0], X1[1], y1[0], 'o', color='r')
ax.plot(X2[0], X2[1], y2[0], 's', color='g')
plt.show()
"""
Car Model Volume Weight CO2
0 Toyoty Aygo 1000 790 99
1 Mitsubishi Space Star 1200 1160 95
2 Skoda Citigo 1000 929 95
3 Fiat 500 900 865 90
4 Mini Cooper 1500 1140 105
5 VW Up! 1000 929 105
6 Skoda Fabia 1400 1109 90
7 Mercedes A-Class 1500 1365 92
8 Ford Fiesta 1500 1112 98
9 Audi A1 1600 1150 99
10 Hyundai I20 1100 980 99
11 Suzuki Swift 1300 990 101
12 Ford Fiesta 1000 1112 99
13 Honda Civic 1600 1252 94
14 Hundai I30 1600 1326 97
15 Opel Astra 1600 1330 97
16 BMW 1 1600 1365 99
17 Mazda 3 2200 1280 104
18 Skoda Rapid 1600 1119 104
19 Ford Focus 2000 1328 105
20 Ford Mondeo 1600 1584 94
21 Opel Insignia 2000 1428 99
22 Mercedes C-Class 2100 1365 99
23 Skoda Octavia 1600 1415 99
24 Volvo S60 2000 1415 99
25 Mercedes CLA 1500 1465 102
26 Audi A4 2000 1490 104
27 Audi A6 2000 1725 114
28 Volvo V70 1600 1523 109
29 BMW 5 2000 1705 114
30 Mercedes E-Class 2100 1605 115
31 Volvo XC70 2000 1746 117
32 Ford B-Max 1600 1235 104
33 BMW 216 1600 1390 108
34 Opel Zafira 1600 1405 109
35 Mercedes SLK 2500 1395 120
Honda Civic, 1600, 1252 / CO2: 101.5
Unknow car, 1200, 780 / CO2: 94.8
"""
Multiple variables
The multiple variables form of hypothesis function.\(
h_\theta = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \cdots + \theta_n x_n
\)
from sklearn.linear_model import LinearRegression
import matplotlib.pyplot as plt
import numpy as np, sys
import pandas as pd
import pathlib
DIR = pathlib.Path(__file__).resolve().parent
with open(DIR / 'data/real_estate.csv') as file:
df = pd.read_csv(file)
# Features
X = df[[
'X1 transaction date',
'X2 house age',
'X3 distance to the nearest MRT station',
'X4 number of convenience stores',
'X5 latitude',
'X6 longitude',
]].values
# Label
y = df['Y house price of unit area'].values
# Train the model
r = LinearRegression().fit(X, y)
# Predictions
X1 = [2013.17, 13, 732.85, 0, 24.98, 121.53] # price: 39 (train data)
X2 = [2013.58, 16.6, 323.69, 6, 24.98, 121.54] # price: 51 (train data)
X3 = [2013.17, 33, 732.85, 0, 24.98, 121.53] # ?
print(df.head())
print('Predict training item1, price =', r.predict([X1]).round(1).item())
print('Predict training item2, price =', r.predict([X2]).round(1).item())
print('Predict unknow item, price =', r.predict([X3]).round(1).item())
"""
No X1 transaction date X2 house age ...
0 1 2012.917 32.0 ...
1 2 2012.917 19.5 ...
2 3 2013.583 13.3 ...
3 4 2013.500 13.3 ...
4 5 2012.833 5.0 ...
[5 rows x 8 columns]
Predict training item1, price = 38.8
Predict training item2, price = 48.5
Predict unknow item, price = 33.4
"""
Residuals
Evaluate the model fit using SSR sum of squared residuals.
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
# Training Dataset
X = np.array([30, 46, 60, 65, 77, 95]).reshape(6,1)
Y = np.array([31, 30, 80, 49, 70, 118])
# Learn a prediction function
r = LinearRegression().fit(X, Y)
a = r.coef_[0].round(1)
b = r.intercept_.round(1)
# Evaluate the model
P = [] # Predictions (on training dataset)
R = [] # Residuals
SSR = 0 # Sum of squared residuals
for i in range(len(X)):
P = np.append(P, -18 + 1.3*X[i])
R = np.append(R, Y[i] - P[i])
SSR += R[i] ** 2
print(f'Prediction function: f(x) = {a}x + {b}')
print('Residuals:', R)
print(f'SSR = {SSR.round(2).item()}')
# Draw graphics
fig, ax = plt.subplots()
plt.ylim(0, 140)
plt.xlim(0, 140)
ax.plot(X, Y, 'x', color='g', label='training data') # Dataset points
ax.plot(X, a*X + b, label=f'h(x) = {b} + {a}x') # Function line
for i in range(len(X)): # Residuals
ax.plot([X[i], X[i]], [P[i], Y[i]], '-', color='c')
plt.legend()
plt.show()
"""
Prediction function: f(x) = 1.3x + -18.0
Residuals: [10. -11.8 20. -17.5 -12.1 12.5]
SSR = 1248.15
"""
Last update: 223 days ago
