Linear Regression

This article is the first of a series covering fundamental machine learning algorithms. Each post will be split into two parts

1. The idea and key concepts - how the algorithm works.
2. The maths - derivations followed by an implementation in Python.

The idea and key concepts

Regression is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables.

Let’s say we are trying to predict how much a house will sell for in a desirable area of North London. We may know a few features or independent variables about the house e.g. the square footage of the house, the square footage of the garden and whether there is a garage.

Linear regression uses a linear combination of the features to predict the output. This just means summing up each feature value multiplied by a coefficient to represent how important that feature is e.g

$$ \begin{aligned} \text{House Price} &= (£1000 * \text{🏠}) \\ & + (£50 * \text{🌳}) \\ & + (£1000 * \text{🚙?}) \end{aligned} $$

In this example the house price is calculated as $£1000$ for each square foot of the house plus $£50$ for each foot in the garden plus $£1000$ if there is a garage. If we took a nice $600$ square foot property with a small $500$ square foot garden and no garage our linear regression model would predict the house is worth $£625,000$.

Now that’s how the linear regression model works! The question now is how do you choose the coefficients for the features, in our example the $1000$ the $50$ and the $1000$?

In supervised machine learning algorithms these parameters are estimated using training data. Training data is just examples where you already know the answer. In our case it is a list of houses where we know both the house features and the house prices in advance. Here is an example of three training examples with the model predictions:

We want to chooses the coefficients to minimise the average error when predicting the house price for all the training examples. This average error is also known as the cost and can be defined in different ways.

To better visualise the error associated with a prediction it is easier to see in a graph. To make the visualisation easier let’s assume that we only know one feature - the house size - for each training example.

The linear regression model in this case can be written as:

$$ \text{House Price} = \text{Price per Sq. ft.} * \text{Size} $$

Below I have plotted 20 example houses in green showing the price on the y-axis and the house size (in square feet) on the x-axis. I have also plotted the line of best fit (the linear regression model) with the predictions marked as black circles. The red lines shows the absolute error between the predictions and the true house price.

You can vary the price per square foot (the coefficient in the model) to see how this impacts the average error.

The best coefficients - that minimise the error - can be found by solving an equation called the normal equation or by gradient descent.

Gradient descent works by iteratively changing the coefficients to reduce the error between the predictions and the true house prices. The specifics of this process is detailed in the next section. I haven’t gone into the normal equations in this post.

In short the linear regression algorithm chooses the coefficients to minimise the average error when predicting the house prices for all the training examples.

The maths

The model

If we let $y$ represent a single continuous target variable and $x_1,\dots,x_n$ (where $n \in \mathbb{N}$¹ and $x_0 = 1$) represent the feature values. Then the linear model can be written as

$$ \begin{align} \hat{y}&=\beta_0x_0+\cdots+\beta_nx_n \\ \hat{y}&=\sum^{n}_{i=0}\beta_ix_i \end{align} $$

Here the y hat represents the linear model prediction.

The cost function

We define below the cost (a.k.a. error or loss) function $J$ as half of the mean square error for the $m$ training samples where $m \in \mathbb{N}$. We use a superscript to represent each training example so $y^j$ is the $j$th training data target value and $x_i^j$ is the $i$th feature value of the $j$th training example.

$$ \begin{align} J(\boldsymbol{\beta}) &= \frac{1}{2m}\sum^{m}_{j=1}\left( y^j-\hat{y}^j \right)^2\\ &= \frac{1}{2m}\sum^{m}_{j=1}\left( y^j-\sum^{n}_{i=0}\beta_ix_i^j \right)^2 \end{align} $$

Gradient descent

In order to find the coefficients that lead to the minimal cost we use gradient descent. The gradient of the cost function tells you in which direction to change your coefficients in order to reduce the value of the cost function the most. The gradient is defined as the vector of partial derivatives with respect to each coefficient so let’s first calculate these:

$$ \begin{align} \frac{\partial J}{\partial\beta_k}\left(\boldsymbol{\beta}\right) &= \frac{\partial}{\partial\beta_k}\left( \frac{1}{2m}\sum^{m}_{j=1} \left( y^j-\sum^{n}_{i=0}\beta_ix_i^j \right)^2 \right)\\ &= \frac{1}{m}\sum^{m}_{j=1} \left( y^j-\sum^{n}_{i=0}\beta_ix_i^j \right)(-x^j_k)\\ \end{align} $$

Now we can write the gradient as:

$$ \begin{aligned} \nabla_{\boldsymbol{\beta}} J &= \begin{bmatrix} \frac{\partial J}{\partial\beta_0} \\ \vdots \\ \frac{\partial J}{\partial\beta_n} \end{bmatrix} \\ &= \begin{bmatrix} -\frac{1}{m}\sum^{m}_{j=1} \left(y^j-\sum^{n}_{i=0}\beta_ix_i^j\right)x^j_0\\ \vdots \\ -\frac{1}{m}\sum^{m}_{j=1} \left(y^j-\sum^{n}_{i=0}\beta_ix_i^j\right)x^j_n\\ \end{bmatrix} \end{aligned} $$

We could calculate the above gradient using the sums defined but it is more efficient for implementing if we vectorise the calculation.

Vectorise

For this we define the design matrix $\mathbf{X}$ by stacking the $m$ training examples on top of each other, so each row of $\mathbf{X}$ represents one training example and the columns represent the different features. We also define $\mathbf{y}$ the vector of target values by stacking the $m$ target variables on top of each other. Finally we also define the vector of $n+1$ coefficients $\boldsymbol{\beta}$. Where:²

$$ \mathbf{X}\in\mathbb{R}^{m\times (n+1)}, \quad \mathbf{y}\in\mathbb{R}^{m\times 1}, \quad \boldsymbol{\beta}\in\mathbb{R}^{(n+1)\times1} $$

Or more visually

$$ \mathbf{X}=\begin{bmatrix} 1 & x_1^1 & x_2^1 & \dots & x_n^1 \\ 1 & x_1^2 & x_2^2 & \dots & x_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_1^m & x_2^m & \dots & x_n^m \\ \end{bmatrix} $$

and

$$ \mathbf{y}=\begin{bmatrix} y_1\\y_2\\\vdots\\y_m \end{bmatrix} \quad \boldsymbol{\beta} = \begin{bmatrix} \beta_0\\\beta_1\\\vdots\\\beta_n \end{bmatrix} $$

Now if we take the above gradient we derived above and re-write it splitting the terms we see

$$ \nabla_{\boldsymbol{\beta}} J =-\frac{1}{m} \begin{bmatrix} \sum^{m}_{j=1}y^jx^j_0\\ \vdots \\ \sum^{m}_{j=1}y^jx^j_n\\ \end{bmatrix}+ \frac{1}{m} \begin{bmatrix} \sum^{m}_{j=1}\sum^{n}_{i=0}\beta_ix_i^jx^j_0\\ \vdots \\ \sum^{m}_{j=1}\sum^{n}_{i=0}\beta_ix_i^jx^j_n \end{bmatrix}\\ $$

From this we can write

$$ \begin{align} \nabla_{\boldsymbol{\beta}} J &=\frac{1}{m}\left( \mathbf{X}^T\mathbf{X}\mathbf{\boldsymbol{\beta}}-\mathbf{X}^T\mathbf{y} \right)\\ &=\frac{1}{m}\mathbf{X}^T\left( \mathbf{X}\mathbf{\boldsymbol{\beta}}-\mathbf{y} \right)\\ &=\frac{1}{m}\mathbf{X}^T\left( \mathbf{\hat{y}}-\mathbf{y} \right) \end{align} $$

Where $\mathbf{\hat{y}} = \mathbf{X}\mathbf{\boldsymbol{\beta}}$ are the predictions of the linear model.

Now that we have derived the gradient formula let’s implement gradient descent in python to iteratively step towards the optimal coefficients.

Python implementation

We will build the implementation in an object oriented fashion defining a class for Linear regression. For the full code (with doc strings) it’s on github here.

class LinearRegression():

Next we define the __init__ method on the class setting the learning rate. The gradient tells you in which direction to change the coefficients. The gradient descent algorithm repeatedly updates the coefficients by stepping in the direction of negative gradient. The size of the step is governed by the learning rate.

def __init__(self, learning_rate=0.05):
    self.learning_rate = learning_rate
    print('Creating linear model instance')

Next let’s define a method for the cost function as defined above

def cost(self, y, y_pred):
    m = y.shape[0]
    cost = (1 / (2 * m)) * \
        (y - y_pred).T @ (y - y_pred)
    return cost

Next let’s define a method to calculate the gradient of the cost function

def gradient(self, y, y_pred, X):
    m = X.shape[0]
    gradient = (1 / m) * \
         X.T @ (y_pred - y)
    return gradient

Before we define the fit method to implement gradient descent let’s define one last method which predicts the target variable $y$ given the current coefficients and features $X$

def predict(self, X):
    y_pred = X @ self.beta
    return y_pred

And finally here is the fit method implementing n_iter iterations of gradient descent.

def fit(self, X, y, n_iter=1000):
    m, n = X.shape
    print(f'fitting with m={m} samples with n={n-1} features\n')
    self.beta = np.zeros(shape=(n, 1))
    self.costs = []
    self.betas = [self.beta]
    for iteration in range(n_iter):
        y_pred = self.predict(X)
        cost = self.cost(y, y_pred)
        self.costs.append(cost[0][0])
        gradient = self.gradient(y, y_pred, X)
        self.beta = self.beta - (
            self.learning_rate * gradient)
        self.betas.append(self.beta)

Here’s an example use of the class:

linear_regression = LinearRegression()
linear_regression.fit(X, y)

linear_regression.predict(X_new)

Thanks for reading! Please get in touch with any questions, mistakes or improvements.