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Multiple Regression

Multiple regression is like , but with more than one independent value, meaning that we try to predict a value based on two or more variables.

Take a look at the data set below, it contains some information about cars.

Car Model Volume Weight CO2
Toyota Aygo 1000 790 99
Mitsubishi Space Star 1200 1160 95
Skoda Citigo 1000 929 95
Fiat 500 900 865 90
Mini Cooper 1500 1140 105
VW Up! 1000 929 105
Skoda Fabia 1400 1109 90
Mercedes A-Class 1500 1365 92
Ford Fiesta 1500 1112 98
Audi A1 1600 1150 99
Hyundai I20 1100 980 99
Suzuki Swift 1300 990 101
Ford Fiesta 1000 1112 99
Honda Civic 1600 1252 94
Hundai I30 1600 1326 97
Opel Astra 1600 1330 97
BMW 1 1600 1365 99
Mazda 3 2200 1280 104
Skoda Rapid 1600 1119 104
Ford Focus 2000 1328 105
Ford Mondeo 1600 1584 94
Opel Insignia 2000 1428 99
Mercedes C-Class 2100 1365 99
Skoda Octavia 1600 1415 99
Volvo S60 2000 1415 99
Mercedes CLA 1500 1465 102
Audi A4 2000 1490 104
Audi A6 2000 1725 114
Volvo V70 1600 1523 109
BMW 5 2000 1705 114
Mercedes E-Class 2100 1605 115
Volvo XC70 2000 1746 117
Ford B-Max 1600 1235 104
BMW 2 1600 1390 108
Opel Zafira 1600 1405 109
Mercedes SLK 2500 1395 120

We can predict the CO2 emission of a car based on the size of the engine, but with multiple regression we can throw in more variables, like the weight of the car, to make the prediction more accurate.

How Does it Work?

In Python we have modules that will do the work for us. Start by importing the Pandas module.

import pandas

Learn about the Pandas module in our .

The Pandas module allows us to read csv files and return a DataFrame object.

The file is meant for testing purposes only, you can download it here:

df = pandas.read_csv("data.csv")

Then make a list of the independent values and call this variable X.

Put the dependent values in a variable called y.

X = df[['Weight', 'Volume']]
y = df['CO2']

Tip: It is common to name the list of independent values with a upper case X, and the list of dependent values with a lower case y.

We will use some methods from the sklearn module, so we will have to import that module as well:

from sklearn import linear_model

From the sklearn module we will use the LinearRegression() method to create a linear regression object.

This object has a method called fit() that takes the independent and dependent values as parameters and fills the regression object with data that describes the relationship:

regr = linear_model.LinearRegression()
regr.fit(X, y)

Now we have a regression object that are ready to predict CO2 values based on a car's weight and volume:

#predict the CO2 emission of a car where the weight is 2300kg, and the volume is 1300cm3:
predictedCO2 = regr.predict([[2300, 1300]])

Example

See the whole example in action:

import pandas
from sklearn import linear_model

df = pandas.read_csv("data.csv")

X = df[['Weight', 'Volume']]
y = df['CO2']

regr = linear_model.LinearRegression()
regr.fit(X, y)

#predict the CO2 emission of a car where the weight is 2300kg, and the volume is 1300cm3:
predictedCO2 = regr.predict([[2300, 1300]])

print(predictedCO2)

Result:

[107.2087328]

We have predicted that a car with 1.3 liter engine, and a weight of 2300 kg, will release approximately 107 grams of CO2 for every kilometer it drives.

Coefficient

The coefficient is a factor that describes the relationship with an unknown variable.

Example: if x is a variable, then 2x is x two times. x is the unknown variable, and the number 2 is the coefficient.

In this case, we can ask for the coefficient value of weight against CO2, and for volume against CO2. The answer(s) we get tells us what would happen if we increase, or decrease, one of the independent values.

Example

Print the coefficient values of the regression object:

import pandas
from sklearn import linear_model

df = pandas.read_csv("data.csv")

X = df[['Weight', 'Volume']]
y = df['CO2']

regr = linear_model.LinearRegression()
regr.fit(X, y)

print(regr.coef_)

Result:

[0.00755095 0.00780526]

Result Explained

The result array represents the coefficient values of weight and volume.

Weight: 0.00755095
Volume: 0.00780526

These values tell us that if the weight increase by 1kg, the CO2 emission increases by 0.00755095g.

And if the engine size (Volume) increases by 1 cm3, the CO2 emission increases by 0.00780526 g.

I think that is a fair guess, but let test it!

We have already predicted that if a car with a 1300cm3 engine weighs 2300kg, the CO2 emission will be approximately 107g.

What if we increase the weight with 1000kg?

Example

Copy the example from before, but change the weight from 2300 to 3300:

import pandas
from sklearn import linear_model

df = pandas.read_csv("data.csv")

X = df[['Weight', 'Volume']]
y = df['CO2']

regr = linear_model.LinearRegression()
regr.fit(X, y)

predictedCO2 = regr.predict([[3300, 1300]])

print(predictedCO2)

Result:

[114.75968007]

We have predicted that a car with 1.3 liter engine, and a weight of 3300 kg, will release approximately 115 grams of CO2 for every kilometer it drives.

Which shows that the coefficient of 0.00755095 is correct:

107.2087328 + (1000 * 0.00755095) = 114.75968

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