Regression on Python (part 1)

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

Here are the data used:

path = 'https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/DA0101EN/automobileEDA.csv'
df = pd.read_csv(path)
df.head()

SIMPLE LINEAR REGRESSION

from sklearn.linear_model import LinearRegression

#create the linear regression object
lm = LinearRegression()

#highway-mpg for predict price
X = df[['highway-mpg']]
Y = df[['price']]
lm.fit(X,Y)

#predict for first 5 raw of the table
lm.predict(X[0:5])

#value of intercept and coefficient
lm.intercept_
lm.coef_

MULTIPLE LINEAR REGRESSION

Z = df[['horsepower', 'curb-weight', 'engine-size', 'highway-mpg']]

lm.fit(Z, df['price'])

MODEL EVALUATION USING VISUALIZATION

import seaborn as sns

Simple linear regression

width = 12
height = 10
plt.figure(figsize=(width, height))
sns.regplot(x="highway-mpg", y="price", data=df)
plt.ylim(0,)     #y axis starts from 0

image.png

Is more correlated with price "peak-rpm" or "highway-mpg"?

df[["peak-rpm","highway-mpg","price"]].corr()

highway-mpg is more correlated: image.png

Residual Plot it's a graph that shows the residuals on the vertical y-axis and the independent variable on the horizontal x-axis.
We can see from this residual plot that the residuals are not randomly spread around the x-axis, which leads us to believe that maybe a non-linear model is more appropriate for this data.

width = 12
height = 10
plt.figure(figsize=(width, height))
sns.residplot(df['highway-mpg'], df['price'])
plt.show()

image.png

Multiple linear regression

#distribution plot: fitted values vs actual values
Y_hat = lm.predict(Z)    #make the prediction

plt.figure(figsize=(width, height))
ax1 = sns.distplot(df['price'], hist=False, color="r", label="Actual Value")
sns.distplot(Y_hat, hist=False, color="b", label="Fitted Values" , ax=ax1)
plt.title('Actual vs Fitted Values for Price')
plt.xlabel('Price (in dollars)')
plt.ylabel('Proportion of Cars')
plt.show()
plt.close()

image.png

Fitted reasonable close to actual, but there is space for improvement

POLYNOMIAL REGRESSION

Function for plotting the data:

def PlotPolly(model, independent_variable, dependent_variabble, Name):
    x_new = np.linspace(15, 55, 100)
    y_new = model(x_new)
    plt.plot(independent_variable, dependent_variabble, '.', x_new, y_new, '-')
    plt.title('Polynomial Fit with Matplotlib for Price ~ Length')
    ax = plt.gca()
    ax.set_facecolor((0.898, 0.898, 0.898))
    fig = plt.gcf()
    plt.xlabel(Name)
    plt.ylabel('Price of Cars')
    plt.show()
    plt.close()

#variables
x = df['highway-mpg']
y = df['price']

Here we use a polynomial of the 3rd order (cubic):

f = np.polyfit(x, y, 3)
p = np.poly1d(f)
print(p)
PlotPolly(p, x, y, 'highway-mpg')

image.png

#coefficients
np.polyfit(x, y, 3)    # y = a + b1X + b2X² + b3

Multivariate Polynomial function: with four X variables we will have 15 coefficient

from sklearn.preprocessing import PolynomialFeatures
pr=PolynomialFeatures(degree=2)
pr
Z_pr=pr.fit_transform(Z)
Z.shape     #The original data is of 201 samples and 4 features
Z_pr.shape     #after the transformation, there 201 samples and 15 features

MEASURES FOR IN-SAMPLE EVALUATION

Simple linear regression:

lm.fit(X, Y)     #highway_mpg_fit
lm.score(X, Y)

R^2: 0.4965911884339176

Yhat=lm.predict(X)     #predictions
from sklearn.metrics import mean_squared_error
mean_squared_error(df['price'], Yhat)

MSE: 31635042.944639888

Multiple linear regression:

lm.fit(Z, df['price'])     #fit the model
lm.score(Z, df['price'])

R^2: 0.8093562806577457

Y_predict_multifit = lm.predict(Z)     #predictions
mean_squared_error(df['price'], Y_predict_multifit)

MSE: 11980366.87072649

Polynomial linear regression:

from sklearn.metrics import r2_score     #we use a different function
r2_score(y, p(x))

R^2: 0.674194666390652

mean_squared_error(df['price'], p(x))

MSE: 20474146.426361218

Comparing these three models, we conclude that the MLR model is the best model to be able to predict price from our dataset.