Machine Learning - Multiple Regression
Multiple Regression
Multiple regression is likelinear regression, but with more than one independent value, meaning that we try to predict a value based ontwo 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 ourPandas Tutorial.
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:data.csv
df = pandas.read_csv("data.csv")
Then make a list of the independent values and call this variableX.
Put the dependent values in a variable calledy.
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 theLinearRegression() method to create a linear regression object.
This object has a method calledfit() 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:
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: ifx is a variable, then2x isx two times.x is the unknown variable, and the number2 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:
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 1cm3, the CO2 emission increases by 0.00780526g.
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:
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
