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
--2019-04-15 11:01:48-- https://s3-api.us-geo.objectstorage.softlayer.net/cf-courses-data/CognitiveClass/ML0101ENv3/labs/FuelConsumptionCo2.csv Resolving s3-api.us-geo.objectstorage.softlayer.net (s3-api.us-geo.objectstorage.softlayer.net)... 18.104.22.168 Connecting to s3-api.us-geo.objectstorage.softlayer.net (s3-api.us-geo.objectstorage.softlayer.net)|22.214.171.124|:443... connected. HTTP request sent, awaiting response... 200 OK Length: 72629 (71K) [text/csv] Saving to: ‘FuelConsumption.csv’ FuelConsumption.csv 100%[=====================>] 70.93K --.-KB/s in 0.04s 2019-04-15 11:01:48 (1.63 MB/s) - ‘FuelConsumption.csv’ saved [72629/72629]
Did you know? When it comes to Machine Learning, you will likely be working with large datasets. As a business, where can you host your data? IBM is offering a unique opportunity for businesses, with 10 Tb of IBM Cloud Object Storage: Sign up now for free
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
df = pd.read_csv("FuelConsumption.csv") # take a look at the dataset df.head()
|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_CITY','FUELCONSUMPTION_HWY','FUELCONSUMPTION_COMB','CO2EMISSIONS']] cdf.head(9)
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()
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. This will provide a more accurate evaluation on out-of-sample accuracy because the testing dataset is not part of the dataset that have been used to train the data. It is more realistic for real world problems.
This means that we know the outcome of each data point in this dataset, making it great to test with! And since this data has not been used to train the model, the model has no knowledge of the outcome of these data points. So, in essence, it’s truly an out-of-sample testing.
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()
In reality, there are multiple variables that predict the Co2emission. When more than one independent variable is present, the process is called multiple linear regression. For example, predicting co2emission using FUELCONSUMPTION_COMB, EngineSize and Cylinders of cars. The good thing here is that Multiple linear regression is the extension of simple linear 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_)
Coefficients: [[10.55398564 7.83589194 9.49083236]]
As mentioned before, Coefficient and Intercept , are the parameters of the fit line. Given that it is a multiple linear regression, with 3 parameters, and knowing that the parameters are the intercept and coefficients of hyperplane, sklearn can estimate them from our data. Scikit-learn uses plain Ordinary Least Squares method to solve this problem.
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 () over all samples in the dataset.
OLS can find the best parameters using of the following methods:
- Solving the model parameters analytically using closed-form equations - Using an optimization algorithm (Gradient Descent, Stochastic Gradient Descent, Newton’s Method, etc.)
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: 715.98 Variance score: 0.84
explained variance regression score:
If 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:
The best possible score is 1.0, lower values are worse.
# write your code here regr = linear_model.LinearRegression() x = np.asanyarray(train[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_CITY','FUELCONSUMPTION_HWY']]) y = np.asanyarray(train[['CO2EMISSIONS']]) regr.fit (x, y) print ('Coefficients: ', regr.coef_) y_= regr.predict(test[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_CITY','FUELCONSUMPTION_HWY']]) x = np.asanyarray(test[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_CITY','FUELCONSUMPTION_HWY']]) y = np.asanyarray(test[['CO2EMISSIONS']]) print("Residual sum of squares: %.2f"% np.mean((y_ - y) ** 2)) print('Variance score: %.2f' % regr.score(x, y))
Coefficients: [[10.66448127 7.38345472 6.31291384 2.82232368]] Residual sum of squares: 717.06 Variance score: 0.84
Double-click here for the solution.
IBM SPSS Modeler is a comprehensive analytics platform that has many machine learning algorithms. It has been designed to bring predictive intelligence to decisions made by individuals, by groups, by systems – by your enterprise as a whole. A free trial is available through this course, available here: SPSS Modeler
Also, you can use Watson Studio to run these notebooks faster with bigger datasets. Watson Studio is IBM's leading cloud solution for data scientists, built by data scientists. With Jupyter notebooks, RStudio, Apache Spark and popular libraries pre-packaged in the cloud, Watson Studio enables data scientists to collaborate on their projects without having to install anything. Join the fast-growing community of Watson Studio users today with a free account at Watson Studio
Saeed Aghabozorgi, PhD is a Data Scientist in IBM with a track record of developing enterprise level applications that substantially increases clients’ ability to turn data into actionable knowledge. He is a researcher in data mining field and expert in developing advanced analytic methods like machine learning and statistical modelling on large datasets.