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# Machine Learning - Multiple Regression

## Multiple Regression

Multiple regression is like linear regression, 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`

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: cars.csv

`df = pandas.read_csv("cars.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

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

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

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