# Simple Linear Regression（简单线性回归）

## Objectives（目标）

• Use scikit-learn to implement simple Linear Regression
• Create a model, train it, test it and use the model

### Importing Needed packages（导入必要的包）

``````import matplotlib.pyplot as plt
import pandas as pd
import pylab as pl
import numpy as np
%matplotlib inline
``````

FuelConsumption(点我下载）

## Understanding the Data（理解数据）

### `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

``````# df = pd.read_csv("FuelConsumption.csv")
# 自己改个路径
# take a look at the dataset
``````
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

### Data Exploration（数据探索）

Let’s first have a descriptive exploration on our data.

``````# summarize the data
df.describe()
``````
MODELYEAR ENGINESIZE CYLINDERS FUELCONSUMPTION_CITY FUELCONSUMPTION_HWY FUELCONSUMPTION_COMB FUELCONSUMPTION_COMB_MPG CO2EMISSIONS
count 1067.0 1067.000000 1067.000000 1067.000000 1067.000000 1067.000000 1067.000000 1067.000000
mean 2014.0 3.346298 5.794752 13.296532 9.474602 11.580881 26.441425 256.228679
std 0.0 1.415895 1.797447 4.101253 2.794510 3.485595 7.468702 63.372304
min 2014.0 1.000000 3.000000 4.600000 4.900000 4.700000 11.000000 108.000000
25% 2014.0 2.000000 4.000000 10.250000 7.500000 9.000000 21.000000 207.000000
50% 2014.0 3.400000 6.000000 12.600000 8.800000 10.900000 26.000000 251.000000
75% 2014.0 4.300000 8.000000 15.550000 10.850000 13.350000 31.000000 294.000000
max 2014.0 8.400000 12.000000 30.200000 20.500000 25.800000 60.000000 488.000000

Let’s select some features to explore more.

``````cdf = df[['ENGINESIZE','CYLINDERS','FUELCONSUMPTION_COMB','CO2EMISSIONS']]
``````
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

We can plot each of these features:

``````viz = cdf[['CYLINDERS','ENGINESIZE','CO2EMISSIONS','FUELCONSUMPTION_COMB']]
viz.hist()
plt.show()
``````

Now, let’s plot each of these features against the Emission, to see how linear their relationship is:

``````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()
``````

## Practice（进一步练习）

Plot CYLINDER vs the Emission, to see how linear is their relationship is:

``````plt.scatter(cdf.CYLINDERS, cdf.CO2EMISSIONS,  color='blue')
plt.xlabel("Cylinders")
plt.ylabel("Emission")
plt.show()

``````

#### Creating train and test dataset（创建训练集和测试集）

Train/Test Split involves splitting the dataset into training and testing sets that 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 model. Therefore, it gives us a better understanding of how well our model generalizes on new data.

This means that we know the outcome of each data point in the testing dataset, making it great to test with! 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 is truly an out-of-sample testing.

Let’s split our dataset into train and test sets. 80% of the entire dataset will be used for training and 20% for testing. We create a mask to select random rows using np.random.rand() function:

``````msk = np.random.rand(len(df)) < 0.8
train = cdf[msk]
test = cdf[~msk]
``````
``````# msk是一串布尔值
# train就是作为训练集的那个部分
print(msk)
# print(msk)
train
``````
``````[ True  True  True ...  True  True  True]
``````
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
5 3.5 6 10.0 230
... ... ... ... ...
1062 3.0 6 11.8 271
1063 3.2 6 11.5 264
1064 3.0 6 11.8 271
1065 3.2 6 11.3 260
1066 3.2 6 12.8 294

844 rows × 4 columns

### Simple Regression Model（简单线性回归模型）

Linear Regression fits a linear model with coefficients B = (B1, …, Bn) to minimize the ‘residual sum of squares’ between the actual value y in the dataset, and the predicted value yhat using linear approximation.

#### Train data distribution（训练集的分布）

``````plt.scatter(train.ENGINESIZE, train.CO2EMISSIONS,  color='blue')
plt.xlabel("Engine size")
plt.ylabel("Emission")
plt.show()
``````

#### Modeling（建立模型）

Using sklearn package to model data.

``````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:  [[39.92902717]]
Intercept:  [123.15528139]
``````

As mentioned before, Coefficient and Intercept in the simple linear regression, are the parameters of the fit line.

Given that it is a simple linear regression, with only 2 parameters, and knowing that the parameters are the intercept and slope of the line, sklearn can estimate them directly from our data.

Notice that all of the data must be available to traverse and calculate the parameters.

#### Plot outputs（结果可视化）

We can plot the fit line over the data:

``````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')
``````

#### Evaluation（评估）

We compare the actual values and predicted values to calculate the accuracy of a regression model. Evaluation metrics provide a key role in the development of a model, as it provides insight to areas that require improvement.

There are different model evaluation metrics, lets use MSE here to calculate the accuracy of our model based on the test set:

• Mean Absolute Error: It is the mean of the absolute value of the errors. This is the easiest of the metrics to understand since it’s just average error.

• Mean Squared Error (MSE): Mean Squared Error (MSE) is the mean of the squared error. It’s more popular than Mean Absolute Error because the focus is geared more towards large errors. This is due to the squared term exponentially increasing larger errors in comparison to smaller ones.

• Root Mean Squared Error (RMSE).

• R-squared is not an error, but rather a popular metric to measure the performance of your regression model. It represents how close the data points are to the fitted regression line. The higher the R-squared value, the better the model fits your data. The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse).

``````from sklearn.metrics import r2_score

test_x = np.asanyarray(test[['ENGINESIZE']])
test_y = np.asanyarray(test[['CO2EMISSIONS']])
test_y_ = regr.predict(test_x)

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: 21.90
Residual sum of squares (MSE): 800.10
R2-score: 0.78
``````

## Exercise（进一步的练习）

Lets see what the evaluation metrics are if we trained a regression model using the `FUELCONSUMPTION_COMB` feature.

Start by selecting `FUELCONSUMPTION_COMB` as the train_x data from the `train` dataframe, then select `FUELCONSUMPTION_COMB` as the test_x data from the `test` dataframe

``````train_x = np.asanyarray(train[['FUELCONSUMPTION_COMB']])
test_x = test[["FUELCONSUMPTION_COMB"]]
# train_y = np.asanyarray(train[['CO2EMISSIONS']])
# train_y之前已经有了
``````

``````train_x = train[["FUELCONSUMPTION_COMB"]]

test_x = test[["FUELCONSUMPTION_COMB"]]

``````

Now train a Logistic Regression Model using the `train_x` you created and the `train_y` created previously

``````regr = linear_model.LinearRegression()
regr.fit(train_x, train_y)
# The coefficients
print ('Coefficients: ', regr.coef_)
print ('Intercept: ',regr.intercept_)
``````
``````Coefficients:  [[16.66965333]]
Intercept:  [63.78904974]
``````
``````plt.scatter(train.FUELCONSUMPTION_COMB, train.CO2EMISSIONS,  color='blue')
plt.plot(train_x, regr.coef_[0][0]*train_x + regr.intercept_[0], '-r')
plt.xlabel("FUELCONSUMPTION_COMB")
plt.ylabel("Emission")
``````
``````Text(0, 0.5, 'Emission')
``````

Find the predictions using the model’s `predict` function and the `test_x` data

``````predictions = regr.predict(test_x)
predictions
``````

Finally use the `predictions` and the `test_y` data and find the Mean Absolute Error value using the `np.absolute` and `np.mean` function like done previously

``````#ADD CODE
# test_y = np.asanyarray(test[['CO2EMISSIONS']])
predictions = regr.predict(test_x)

print("Mean absolute error: %.2f" % np.mean(np.absolute(predictions - test_y)))
print("Residual sum of squares (MSE): %.2f" % np.mean((predictions - test_y) ** 2))
print("R2-score: %.2f" % r2_score(test_y , predictions) )
``````
``````Mean absolute error: 136.65
Residual sum of squares (MSE): 20277.89
R2-score: -4.62
``````

We can see that the MAE is much worse when we train using `ENGINESIZE` than `FUELCONSUMPTION_COMB`

THE END

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