import matplotlib.pyplot as plt
import pandas as pd
import pylab as pl
import numpy as np
%matplotlib inline
!wget -O FuelConsumption.csv https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/ML0101ENv3/labs/FuelConsumptionCo2.csv
Understanding the Data
FuelConsumption.csv:
We have downloaded a fuel consumption dataset, FuelConsumption.csv, which contains model-specific fuel consumption ratings and estimated carbon dioxide emissions for new light-duty vehicles for retail sale in Canada. Dataset source
MODELYEAR e.g. 2014
MAKE e.g. Acura
MODEL e.g. ILX
VEHICLE CLASS e.g. SUV
ENGINE SIZE e.g. 4.7
CYLINDERS e.g 6
TRANSMISSION e.g. A6
FUEL CONSUMPTION in CITY(L/100 km) e.g. 9.9
FUEL CONSUMPTION in HWY (L/100 km) e.g. 8.9
FUEL CONSUMPTION COMB (L/100 km) e.g. 9.2
CO2 EMISSIONS (g/km) e.g. 182 --> low --> 0
Reading the data in
df = pd.read_csv("FuelConsumption.csv")
# take a look at the dataset
df.head()
MODELYEAR MAKE MODEL VEHICLECLASS ENGINESIZE CYLINDERS TRANSMISSION FUELTYPE FUELCONSUMPTION_CITY FUELCONSUMPTION_HWY FUELCONSUMPTION_COMB FUELCONSUMPTION_COMB_MPG CO2EMISSIONS
0 2014 ACURA ILX COMPACT 2.0 4 AS5 Z 9.9 6.7 8.5 33 196
1 2014 ACURA ILX COMPACT 2.4 4 M6 Z 11.2 7.7 9.6 29 221
2 2014 ACURA ILX HYBRID COMPACT 1.5 4 AV7 Z 6.0 5.8 5.9 48 136
3 2014 ACURA MDX 4WD SUV - SMALL 3.5 6 AS6 Z 12.7 9.1 11.1 25 255
4 2014 ACURA RDX AWD SUV - SMALL 3.5 6 AS6 Z 12.1 8.7 10.6 27 244
Lets select some features that we want to use for regression.
cdf = df[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB','CO2EMISSIONS']]
cdf.head(9)
cdf = df[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB','CO2EMISSIONS']]
cdf.head(9)
ENGINESIZE CYLINDERS FUELCONSUMPTION_COMB CO2EMISSIONS
0 2.0 4 8.5 196
1 2.4 4 9.6 221
2 1.5 4 5.9 136
3 3.5 6 11.1 255
4 3.5 6 10.6 244
5 3.5 6 10.0 230
6 3.5 6 10.1 232
7 3.7 6 11.1 255
8 3.7 6 11.6 267
Lets plot Emission values with respect to Engine size:
plt.scatter(cdf.ENGINESIZE, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
Creating train and test dataset
Train/Test Split involves splitting the dataset into training and testing sets respectively, which are mutually exclusive. After which, you train with the training set and test with the testing set.
msk = np.random.rand(len(df)) < 0.8
train = cdf[msk]
test = cdf[~msk]
Polynomial regression¶
Sometimes, the trend of data is not really linear, and looks curvy. In this case we can use Polynomial regression methods. In fact, many different regressions exist that can be used to fit whatever the dataset looks like, such as quadratic, cubic, and so on, and it can go on and on to infinite degrees.
In essence, we can call all of these, polynomial regression, where the relationship between the independent variable x and the dependent variable y is modeled as an nth degree polynomial in x. Lets say you want to have a polynomial regression (let's make 2 degree polynomial):
𝑦=𝑏+𝜃1𝑥+𝜃2𝑥2
Now, the question is: how we can fit our data on this equation while we have only x values, such as Engine Size? Well, we can create a few additional features: 1, 𝑥 , and 𝑥2 .
PloynomialFeatures() function in Scikit-learn library, drives a new feature sets from the original feature set. That is, a matrix will be generated consisting of all polynomial combinations of the features with degree less than or equal to the specified degree. For example, lets say the original feature set has only one feature, ENGINESIZE. Now, if we select the degree of the polynomial to be 2, then it generates 3 features, degree=0, degree=1 and degree=2:
from sklearn.preprocessing import PolynomialFeatures
from sklearn import linear_model
train_x = np.asanyarray(train[['ENGINESIZE']])
train_y = np.asanyarray(train[['CO2EMISSIONS']])
test_x = np.asanyarray(test[['ENGINESIZE']])
test_y = np.asanyarray(test[['CO2EMISSIONS']])
poly = PolynomialFeatures(degree=2)
train_x_poly = poly.fit_transform(train_x)
train_x_poly
array([[ 1. , 2.4 , 5.76],
[ 1. , 1.5 , 2.25],
[ 1. , 3.5 , 12.25],
...,
[ 1. , 3.2 , 10.24],
[ 1. , 3. , 9. ],
[ 1. , 3.2 , 10.24]])
fit_transform takes our x values, and output a list of our data raised from power of 0 to power of 2 (since we set the degree of our polynomial to 2).
𝑣1𝑣2⋮𝑣𝑛 ⟶ [1[1⋮[1𝑣1𝑣2⋮𝑣𝑛𝑣21]𝑣22]⋮𝑣2𝑛]
in our example
2.2.41.5⋮ ⟶ [1[1[1⋮2.2.41.5⋮4.]5.76]2.25]⋮
It looks like feature sets for multiple linear regression analysis, right? Yes. It Does. Indeed, Polynomial regression is a special case of linear regression, with the main idea of how do you select your features. Just consider replacing the 𝑥 with 𝑥1 , 𝑥21 with 𝑥2 , and so on. Then the degree 2 equation would be turn into:
𝑦=𝑏+𝜃1𝑥1+𝜃2𝑥2
Now, we can deal with it as 'linear regression' problem. Therefore, this polynomial regression is considered to be a special case of traditional multiple linear regression. So, you can use the same mechanism as linear regression to solve such a problems.
so we can use LinearRegression() function to solve it:
clf = linear_model.LinearRegression()
train_y_ = clf.fit(train_x_poly, train_y)
# The coefficients
print ('Coefficients: ', clf.coef_)
print ('Intercept: ',clf.intercept_)
Coefficients: [[ 0. 51.86678532 -1.70694689]]
Intercept: [105.78768144]
As mentioned before, Coefficient and Intercept , are the parameters of the fit curvy line. Given that it is a typical multiple linear regression, with 3 parameters, and knowing that the parameters are the intercept and coefficients of hyperplane, sklearn has estimated them from our new set of feature sets. Lets plot it:
plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS, color='blue')
XX = np.arange(0.0, 10.0, 0.1)
yy = clf.intercept_[0]+ clf.coef_[0][1]*XX+ clf.coef_[0][2]*np.power(XX, 2)
plt.plot(XX, yy, '-r' )
plt.xlabel("Engine size")
plt.ylabel("Emission")
Text(0, 0.5, 'Emission')
Evaluation
from sklearn.metrics import r2_score
test_x_poly = poly.fit_transform(test_x)
test_y_ = clf.predict(test_x_poly)
print("Mean absolute error: %.2f" % np.mean(np.absolute(test_y_ - test_y)))
print("Residual sum of squares (MSE): %.2f" % np.mean((test_y_ - test_y) ** 2))
print("R2-score: %.2f" % r2_score(test_y_ , test_y) )
Mean absolute error: 24.77
Residual sum of squares (MSE): 1073.01
R2-score: 0.63
Practice
Try to use a polynomial regression with the dataset but this time with degree three (cubic). Does it result in better accuracy?
# write your code here
poly3 = PolynomialFeatures(degree=3)
train_x_poly3 = poly3.fit_transform(train_x)
clf3 = linear_model.LinearRegression()
train_y3_ = clf3.fit(train_x_poly3, train_y)
# The coefficients
print ('Coefficients: ', clf3.coef_)
print ('Intercept: ',clf3.intercept_)
plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS, color='blue')
XX = np.arange(0.0, 10.0, 0.1)
yy = clf3.intercept_[0]+ clf3.coef_[0][1]*XX + clf3.coef_[0][2]*np.power(XX, 2) + clf3.coef_[0][3]*np.power(XX, 3)
plt.plot(XX, yy, '-r' )
plt.xlabel("Engine size")
plt.ylabel("Emission")
test_x_poly3 = poly3.fit_transform(test_x)
test_y3_ = clf3.predict(test_x_poly3)
print("Mean absolute error: %.2f" % np.mean(np.absolute(test_y3_ - test_y)))
print("Residual sum of squares (MSE): %.2f" % np.mean((test_y3_ - test_y) ** 2))
print("R2-score: %.2f" % r2_score(test_y3_ , test_y) )
Coefficients: [[ 0. 32.67645817 3.58375078 -0.43783753]]
Intercept: [126.05296936]
Mean absolute error: 24.71
Residual sum of squares (MSE): 1068.14
R2-score: 0.64
