Linear regression is the process of fitting a linear equation to a set of sample data, in order to predict the output.
In order to do this, we assume that the input
X, and the output
Y have a linear relationship.
Ymay or may not have a linear relationship. We just want to find the closest linear relationship between them, in order to explain the data that we observe.
We can get a better understanding of linear regression from the following chart:
The line is the linear relationship that we predicted based on the points which we observed.
If you want to get a brief recap of the theory behind linear regression, you can see my notes here
In order to perform linear regression with python, we will need to:
- Generate the sample data, and divide it into training and testing data.
- Create a linear regression model
- Fit our model using the training data
- Test our model using the testing data
Simple linear regression using “scikit learn”
In order to use scikit learn to perform linear regression, you will have to install it first
Generating our data
Instead of using a popular sample dataset, let’s generate our own data instead. This will help us understand the values of the sample data better than if we took a real life dataset, and will also help us judge the accuracy of our model better, as you will see in the later sections.
Let’s assume there is only one predictor variable. In that case the linear relationship will be of the form:
$$ y = \beta_0 + \beta_1x_1 + \epsilon $$
If we normalize our data, so that $ \beta_0=0 $ , we will get the simplified form of the above equation:
$$ y = \beta x + \epsilon $$
Here, $\epsilon$ is a random value that represents the irreducible error that occurs with each measurement of $y$
Lets write a function to generate this data for us:
def generate_dataset_simple(beta, n, std_dev): # Generate x as an array of `n` samples which can take a value between 0 and 100 x = np.zeros(np.random() * 100) # Generate the random error of n samples, with a random value from a normal distribution, with a standard # deviation provided in the function argument e = np.random.randn(n) * std_dev # Calculate `y` according to the equation discussed y = x * beta + e
We can then create the required number of samples, and then separate them into training and testing sets:
x, y = generate_dataset_simple(10, 50, 100) # Take the first 40 samples to train, and the last 10 to test x_train = x[:-10] y_train = y[:-10] x_test = x[-10:] y_test = y[-10:]
Estimating the coefficient from the data
Now that we have our data, let’s use scikit learn’s
LinearRegression model to predict the coefficients from the raw data using the ordinary least squares method of regression:
# Import, and create an instance of a simple least squares regression model from sklearn import linear_model from sklearn.metrics import mean_squared_error, r2_score model = linear_model.LinearRegression() # Train the model using the training data that we created model.fit(x_train, y_train) # Now that we have trained the model, we can print the coefficient of x that it has predicted print('Coefficient: \n', model.coef_) # We then use the model to make predictions based on the test values of x y_pred = model.predict(x_test) # Now, we can calculate the models accuracy metrics based on what the actual value of y was print("Mean squared error: %.2f" % mean_squared_error(y_test, y_pred)) print('r_2 statistic: %.2f' % r2_score(y_test, y_pred))
Running this on a sample set of data gave us:
Coefficients: [11.04668525] Mean squared error: 6455.56 r_2 statistic: 0.95
We can see that the $\beta$ coefficient obtained from regression (
11.04668525) is very close to the actual value that we used to generate our data.
We have also displayed the $R^2$ statistic, which indicates, on a scale of 0 to 1, how much of our data can be explained by our model. We can see that the score is close to 1, and so our model has done quite well in this regard.
Plotting our results
To get a better sense of our data and predictions, we can use the “matplotlib” library for visualization:
from matplotlib import pyplot as plt plt.scatter(x_train, y_train) plt.plot(x_test, y_pred, color='red') x_actual = np.array([0, 100]) y_actual = x_actual*beta plt.plot(x_actual, y_actual, color='green') plt.show()
Which will give us the following plot:
The blue points show us the testing data that we generated. The red line is the predicted relationship between $x$ and $y$ as determined by linear regression. The green line is the actual relationship, with the $\beta$ value that we used to generate the data.
Working with multiple predictor variables
In the previous sections, we used just one predictor. Let’s generate data with more than one predictor variable to estimate $y$. In this case, the relationship between $X$ and $y$ (ignoring the $\beta_0$ term) would be of the form:
$$ y = \beta_1x_1 + \beta_2x_2 + … + \beta_px_p + \epsilon $$
For this, we will have to make a new data generation function:
def generate_dataset(coeffs, n, std_dev): # We calculate the number of predictors, and create a coefficient matrix # With `p` rows and 1 column, for matrix multiplication p = len(coeffs) coeff_mat = np.array(coeffs).reshape(p, 1) # Similar as before, but with `n` rows and `p` columns this time x = np.random.random_sample((n, p))* 100 e = np.random.randn(n) * std_dev # Since x is a n*p matrix, and coefficients is a p*1 matrix # we can use matrix multiplication to get the value of y for each # set of values x1, x2 .. xp # We need to transpose it to get a 1*n array from a n*1 matrix to use in the regression model y = np.matmul(x, coeff_mat).transpose() + e return x, y
We can then generate our data:
# We now have an array of coefficients, instead of a single one x, y = generate_dataset_simple([10, 5], 50, 100)
The rest of the process is the same as the single predictor case. The only difference is that this time, there will be multiple predicted coefficients, which we can compare to the coefficients we chose.
In this case, since we have 2 predictor variables, we can generate some sample data and generate a 3d plot which plots $x_1$ and $x_2$ against $y$, along with independent 2d plots for each variable against $y$
from mpl_toolkits.mplot3d import Axes3D from generate_dataset import generate_dataset as gd from matplotlib import pyplot as plt predictor_coeffs =[1, 1] std_dev = 10 n = 100 X, Y = gd(predictor_coeffs, n, std_dev) f, [[p1, p2], [p3, p4]] = plt.subplots(2,2) p1.scatter(X[:,0], Y) p1.set_title("x1") p2.scatter(X[:,1], Y) p2.set_title("x2") p3 = f.add_subplot(223, projection='3d') p3.scatter(X[:,0], X[:,1], Y) p3.set_title("x1, x2, y") plt.show()
Which will give us:
An interesting observation is that when we plot each individual variable against $y$, it appears to have a lot of variance. However, when we visualize the combined effects in the form of a 3d plot, we can see the result actually forms a plane, which when viewed at a skewed angle (to see the edge of the plane formed), looks like it has a lot less variance.
This same concept applies for models with greater than 2 predictor variables, although we cannot visualize in more than 3 dimensions.
You can find the source code for all examples as well as the Jupyter notebook here.