Data Analysis with Python - Model Evaluation and Refinement
import pandas as pd
import numpy as np
# Import clean data
path = 'https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/DA0101EN/module_5_auto.csv'
df = pd.read_csv(path)
df.to_csv('module_5_auto.csv')
First lets only use numeric data
df=df._get_numeric_data()
df.head()
Libraries for plotting
%%capture
! pip install ipywidgets
from IPython.display import display
from IPython.html import widgets
from IPython.display import display
from ipywidgets import interact, interactive, fixed, interact_manual
Functions for plotting
def DistributionPlot(RedFunction, BlueFunction, RedName, BlueName, Title):
width = 12
height = 10
plt.figure(figsize=(width, height))
ax1 = sns.distplot(RedFunction, hist=False, color="r", label=RedName)
ax2 = sns.distplot(BlueFunction, hist=False, color="b", label=BlueName, ax=ax1)
plt.title(Title)
plt.xlabel('Price (in dollars)')
plt.ylabel('Proportion of Cars')
plt.show()
plt.close()
def PollyPlot(xtrain, xtest, y_train, y_test, lr,poly_transform):
width = 12
height = 10
plt.figure(figsize=(width, height))
#training data
#testing data
# lr: linear regression object
#poly_transform: polynomial transformation object
xmax=max([xtrain.values.max(), xtest.values.max()])
xmin=min([xtrain.values.min(), xtest.values.min()])
x=np.arange(xmin, xmax, 0.1)
plt.plot(xtrain, y_train, 'ro', label='Training Data')
plt.plot(xtest, y_test, 'go', label='Test Data')
plt.plot(x, lr.predict(poly_transform.fit_transform(x.reshape(-1, 1))), label='Predicted Function')
plt.ylim([-10000, 60000])
plt.ylabel('Price')
plt.legend()
Part 1: Training and Testing
y_data = df['price']
x_data=df.drop('price',axis=1)
from sklearn.model_selection import train_test_split
x_train, x_test, y_train, y_test = train_test_split(x_data, y_data, test_size=0.15, random_state=1)
print("number of test samples :", x_test.shape[0])
print("number of training samples:",x_train.shape[0])
number of test samples : 31
number of training samples: 170
x_train1, x_test1, y_train1, y_test1 = train_test_split(x_data, y_data, test_size=0.4, random_state=0)
print("number of test samples :", x_test1.shape[0])
print("number of training samples:",x_train1.shape[0])
number of test samples : 81
number of training samples: 120
Let's import LinearRegression from the module linear_model.
from sklearn.linear_model import LinearRegression
We create a Linear Regression object:
lre=LinearRegression()
we fit the model using the feature horsepower
lre.fit(x_train[['horsepower']], y_train)
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None,
normalize=False)
Let's Calculate the R^2 on the test data:
lre.score(x_test[['horsepower']], y_test)
0.707688374146705
we can see the R^2 is much smaller using the test data.
lre.score(x_train[['horsepower']], y_train)
Find the R^2 on the test data using 90% of the data for training data
x_train1, x_test1, y_train1, y_test1 = train_test_split(x_data, y_data, test_size=0.1, random_state=0)
lre.fit(x_train1[['horsepower']],y_train1)
lre.score(x_test1[['horsepower']],y_test1)
0.7340722810055448
Wednesday, October 16, 2019
Thursday, October 3, 2019
Polynomial Regression
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
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
Tuesday, October 1, 2019
Multiple Linear Regression
Importing Needed packages
import matplotlib.pyplot as plt
import pandas as pd
import pylab as pl
import numpy as np
%matplotlib inline
Downloading Data
!wget -O FuelConsumption.csv https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/ML0101ENv3/labs/FuelConsumptionCo2.csv
Understanding the Data
Reading the data in
df = pd.read_csv("FuelConsumption.csv")
# take a look at the dataset
df.head()
Lets plot Emission values with respect to Engine size:
cdf = df[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_CITY','FUELCONSUMPTION_HWY','FUELCONSUMPTION_COMB','CO2EMISSIONS']]
cdf.head(9)
Creating train and test dataset
plt.scatter(cdf.ENGINESIZE, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
msk = np.random.rand(len(df)) < 0.8
train = cdf[msk]
test = cdf[~msk]
Train data distribution
plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
Multiple Regression Model
from sklearn import linear_model
regr = linear_model.LinearRegression()
x = np.asanyarray(train[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB']])
y = np.asanyarray(train[['CO2EMISSIONS']])
regr.fit (x, y)
# The coefficients
print ('Coefficients: ', regr.coef_)
Ordinary Least Squares (OLS)
OLS is a method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by minimizing the sum of the squares of the differences between the target dependent variable and those predicted by the linear function. In other words, it tries to minimizes the sum of squared errors (SSE) or mean squared error (MSE) between the target variable (y) and our predicted output ( 𝑦̂ ) over all samples in the dataset.
Prediction
y_hat= regr.predict(test[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB']])
x = np.asanyarray(test[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB']])
y = np.asanyarray(test[['CO2EMISSIONS']])
print("Residual sum of squares: %.2f"
% np.mean((y_hat - y) ** 2))
# Explained variance score: 1 is perfect prediction
print('Variance score: %.2f' % regr.score(x, y))
Residual sum of squares: 485.04
Variance score: 0.88
explained variance regression score:
If y_hat is the estimated target output, y the corresponding (correct) target output, and Var is Variance, the square of the standard deviation, then the explained variance is estimated as follow:
𝚎𝚡𝚙𝚕𝚊𝚒𝚗𝚎𝚍𝚅𝚊𝚛𝚒𝚊𝚗𝚌𝚎(y𝑦,y_hat)=1−𝑉𝑎𝑟{𝑦−𝑦̂ }𝑉𝑎𝑟{𝑦}
The best possible score is 1.0, lower values are worse
import matplotlib.pyplot as plt
import pandas as pd
import pylab as pl
import numpy as np
%matplotlib inline
Downloading Data
!wget -O FuelConsumption.csv https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/ML0101ENv3/labs/FuelConsumptionCo2.csv
Understanding the Data
Reading the data in
df = pd.read_csv("FuelConsumption.csv")
# take a look at the dataset
df.head()
Lets plot Emission values with respect to Engine size:
cdf = df[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_CITY','FUELCONSUMPTION_HWY','FUELCONSUMPTION_COMB','CO2EMISSIONS']]
cdf.head(9)
Creating train and test dataset
plt.scatter(cdf.ENGINESIZE, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
msk = np.random.rand(len(df)) < 0.8
train = cdf[msk]
test = cdf[~msk]
Train data distribution
plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
Multiple Regression Model
from sklearn import linear_model
regr = linear_model.LinearRegression()
x = np.asanyarray(train[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB']])
y = np.asanyarray(train[['CO2EMISSIONS']])
regr.fit (x, y)
# The coefficients
print ('Coefficients: ', regr.coef_)
Ordinary Least Squares (OLS)
OLS is a method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by minimizing the sum of the squares of the differences between the target dependent variable and those predicted by the linear function. In other words, it tries to minimizes the sum of squared errors (SSE) or mean squared error (MSE) between the target variable (y) and our predicted output ( 𝑦̂ ) over all samples in the dataset.
Prediction
y_hat= regr.predict(test[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB']])
x = np.asanyarray(test[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB']])
y = np.asanyarray(test[['CO2EMISSIONS']])
print("Residual sum of squares: %.2f"
% np.mean((y_hat - y) ** 2))
# Explained variance score: 1 is perfect prediction
print('Variance score: %.2f' % regr.score(x, y))
Residual sum of squares: 485.04
Variance score: 0.88
explained variance regression score:
If y_hat is the estimated target output, y the corresponding (correct) target output, and Var is Variance, the square of the standard deviation, then the explained variance is estimated as follow:
𝚎𝚡𝚙𝚕𝚊𝚒𝚗𝚎𝚍𝚅𝚊𝚛𝚒𝚊𝚗𝚌𝚎(y𝑦,y_hat)=1−𝑉𝑎𝑟{𝑦−𝑦̂ }𝑉𝑎𝑟{𝑦}
The best possible score is 1.0, lower values are worse
Simple Linear Regression
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
df = pd.read_csv("FuelConsumption.csv")
# take a look at the dataset
df.head()
# summarize the data
df.describe()
cdf = df[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB','CO2EMISSIONS']]
cdf.head(9)
viz = cdf[['CYLINDERS','ENGINESIZE','CO2EMISSIONS','FUELCONSUMPTION_COMB']]
viz.hist()
plt.show()
plt.scatter(cdf.FUELCONSUMPTION_COMB, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("FUELCONSUMPTION_COMB")
plt.ylabel("Emission")
plt.show()
plt.scatter(cdf.ENGINESIZE, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
plt.scatter(cdf.CYLINDERS, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("Cylinders")
plt.ylabel("Emission")
plt.show()
msk = np.random.rand(len(df)) < 0.8
train = cdf[msk]
test = cdf[~msk]
plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
from sklearn import linear_model
regr = linear_model.LinearRegression()
train_x = np.asanyarray(train[['ENGINESIZE']])
train_y = np.asanyarray(train[['CO2EMISSIONS']])
regr.fit (train_x, train_y)
# The coefficients
print ('Coefficients: ', regr.coef_)
print ('Intercept: ',regr.intercept_)
Coefficients: [[39.30843025]]
Intercept: [125.07479893]
plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS, color='blue')
plt.plot(train_x, regr.coef_[0][0]*train_x + regr.intercept_[0], '-r')
plt.xlabel("Engine size")
plt.ylabel("Emission")
Text(0, 0.5, 'Emission')
from sklearn.metrics import r2_score
test_x = np.asanyarray(test[['ENGINESIZE']])
test_y = np.asanyarray(test[['CO2EMISSIONS']])
test_y_hat = regr.predict(test_x)
print("Mean absolute error: %.2f" % np.mean(np.absolute(test_y_hat - test_y)))
print("Residual sum of squares (MSE): %.2f" % np.mean((test_y_hat - test_y) ** 2))
print("R2-score: %.2f" % r2_score(test_y_hat , test_y) )
Mean absolute error: 23.61
Residual sum of squares (MSE): 959.87
R2-score: 0.70
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
df = pd.read_csv("FuelConsumption.csv")
# take a look at the dataset
df.head()
# summarize the data
df.describe()
cdf = df[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB','CO2EMISSIONS']]
cdf.head(9)
viz = cdf[['CYLINDERS','ENGINESIZE','CO2EMISSIONS','FUELCONSUMPTION_COMB']]
viz.hist()
plt.show()
plt.scatter(cdf.FUELCONSUMPTION_COMB, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("FUELCONSUMPTION_COMB")
plt.ylabel("Emission")
plt.show()
plt.scatter(cdf.ENGINESIZE, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
plt.scatter(cdf.CYLINDERS, cdf.CO2EMISSIONS, color='blue')
plt.xlabel("Cylinders")
plt.ylabel("Emission")
plt.show()
msk = np.random.rand(len(df)) < 0.8
train = cdf[msk]
test = cdf[~msk]
plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS, color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
from sklearn import linear_model
regr = linear_model.LinearRegression()
train_x = np.asanyarray(train[['ENGINESIZE']])
train_y = np.asanyarray(train[['CO2EMISSIONS']])
regr.fit (train_x, train_y)
# The coefficients
print ('Coefficients: ', regr.coef_)
print ('Intercept: ',regr.intercept_)
Coefficients: [[39.30843025]]
Intercept: [125.07479893]
plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS, color='blue')
plt.plot(train_x, regr.coef_[0][0]*train_x + regr.intercept_[0], '-r')
plt.xlabel("Engine size")
plt.ylabel("Emission")
Text(0, 0.5, 'Emission')
from sklearn.metrics import r2_score
test_x = np.asanyarray(test[['ENGINESIZE']])
test_y = np.asanyarray(test[['CO2EMISSIONS']])
test_y_hat = regr.predict(test_x)
print("Mean absolute error: %.2f" % np.mean(np.absolute(test_y_hat - test_y)))
print("Residual sum of squares (MSE): %.2f" % np.mean((test_y_hat - test_y) ** 2))
print("R2-score: %.2f" % r2_score(test_y_hat , test_y) )
Mean absolute error: 23.61
Residual sum of squares (MSE): 959.87
R2-score: 0.70
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