Logistic Regression

This article is the second in 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.

Click here if you missed Linear Regression.

The idea and key concepts

In regression (for example linear regression - as discussed in the previous article) we predict a continuous output such as house prices where the correct answer can be any number. Logistic regression is part of a family of machine learning algorithms called classification algorithms. These are used to predict the outcome from a discrete set of categories. Logistic regression in particular is used when we are trying to predict a binary outcome, by this we mean a $1$ or a $0$, think success or failure.

Again it is much easier to explain with an example. Let’s say we are trying to predict whether a student is going to pass an exam. As with linear regression we use the input features to predict the result. In this case let’s say we know the students I.Q. and how many hours of revision they have done. We use these features to make a prediction.

In linear regression we used a linear combination of the features to make a prediction, but this doesn’t work when trying to predict a binary outcome.

Logistic regression works instead by predicting a probability of success but a probability has to be between $0$ and $1$! In order that the prediction is between $0$ and $1$ logistic regression takes the linear combination and transforms it to be between 0 and 1.

Logistic regression takes the combination of the features and coefficients and transforms them to be between $0$ and $1$ so that large values have a high probability and small values have a low probability (in case of predicting whether a student will pass an exam we expect students with a high combined I.Q. and revision time to have a high chance of passing). The transformation uses the sigmoid function as plotted below (notice the large $x$ values have high probability and vice versa).

For our example we would likely predict a student has passed if the logistic model says they have more than a $0.5$ probability of passing. This step of converting the probability to an outcome is called a decision rule.

Sigmoid Function

The Logistic regression algorithm chooses the coefficients to minimise the average error between the predicted results and the true results for the training data:

It does this by iteratively changing the coefficients to reduce the error.

The maths

The model

If we let $y$ represent the discrete target variable where $y\in0,1$ and $x_1,\dots,x_n$ represent the feature values (where $n \in \mathbb{N}$¹ and $x_0 = 1$). Then the logistic model can be written as

$$ \begin{aligned} \hat{y} &= \sigma\left(\beta_0x_0+\cdots+\beta_nx_n\right) \\ \hat{y} &= \sigma\left(\sum^{n}_{i=0}\beta_ix_i\right) \end{aligned} $$

Where $\sigma$ is the sigmoid function defined as:

$$ \sigma(x) = \frac{1}{1+e^{-x}} $$

So the logistic model is:

$$ \hat{y} = \frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}} $$

Here the y hat represents the logistic model prediction of the probability of success.

Different formulation

The odds of an event with probability $p$ is defined as the chance of the event happening divided by the chance of the event not happening:

$$ \frac{p}{1-p} $$

Taking the above formula for $\hat{y}$ we can re-arrange to show that it is equivalent to modelling the log of the odds as a linear combination of the features.

The logarithm of the odds is also know as the logit. This is why logistic regression is also know as the logit model.

$$ \begin{aligned} \hat{y} &= \frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}} \\ \hat{y} + \hat{y}e^{-\sum^{n}_{i=0}\beta_ix_i} &= 1\\ \hat{y}e^{-\sum^{n}_{i=0}\beta_ix_i} &= 1 - \hat{y}\\ \frac{\hat{y}}{1 - \hat{y}} &= e^{\sum^{n}_{i=0}\beta_ix_i} \\ \ln\left(\frac{\hat{y}}{1 - \hat{y}}\right) &= \sum^{n}_{i=0}\beta_ix_i \end{aligned} $$

The cost function

We define below the cost function denoted as $J$ (a.k.a. error or loss) as the cross entropy which is also known as the log loss. For one sample $\mathbf{x}$ and corresponding $y$:

$$ \begin{align} J(\mathbf{x}) &= \begin{cases} -\log\left(\hat{y}\right) &\text{if y=1}\\ -\log\left(1-\hat{y}\right) &\text{if y=0}\\ \end{cases} \end{align} $$

We define the full cost for the $m$ training samples where $m \in \mathbb{N}$ below. 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.

$$ \begin{aligned} J(\mathbf{x}) &= -\frac{1}{m}\sum_{j=1}^my^j\log\left(\hat{y}^j\right) +(1-y^j)\log\left(1-\hat{y}^j\right)\\ \end{aligned} $$

$$ \begin{aligned} J(\mathbf{x}) &= -\frac{1}{m}\sum_{j=1}^m\left(y^j\log\left(\frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i^j}}\right)\right. \\ & + \left.(1-y^j)\log\left(1-\frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i^j}} \right)\right) \end{aligned} $$

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.

From above we know

$$ \hat{y} = \frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}} \\ $$

So differentiating with respect to the coefficient $\beta_k$ we see

$$ \begin{align} \frac{\partial\hat{y}}{\partial\beta_k} &= -\left(1+e^{-\sum^{n}_{i=0}\beta_ix_i}\right)^{-2}e^{-\sum^{n}_{i=0}\beta_ix_i} (-x_k^j)\\ &=\frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}} \frac{e^{-\sum^{n}_{i=0}\beta_ix_i} (x_k^j)}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}}\\ &=\frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}} \frac{(1-1+e^{-\sum^{n}_{i=0}\beta_ix_i})(x_k^j)}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}}\\ &=\frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}} \frac{(1+e^{-\sum^{n}_{i=0}\beta_ix_i}-1)(x_k^j)}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}}\\ &=\frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}} \left(1 - \frac{1}{1+e^{-\sum^{n}_{i=0}\beta_ix_i}}\right)(x_k^j)\\ &=\hat{y}(1-\hat{y})x_k^j \end{align} $$

Now we differentiate the cost function:

$$ \frac{\partial J}{\partial\beta_k} = \frac{\partial}{\partial\beta_k}\left( -\frac{1}{m}\sum_{j=1}^my^j\log\left(\hat{y}^j\right) +(1-y^j)\log\left(1-\hat{y}^j\right) \right) $$

$$ \begin{aligned} &= -\frac{1}{m}\sum_{j=1}^m \frac{y^j}{\hat{y}^j}\frac{\partial\hat{y}}{\partial\beta_k} + \frac{y^j-1}{1-\hat{y}^j}\frac{\partial\hat{y}}{\partial\beta_k}\\ &= -\frac{1}{m}\sum_{j=1}^m \left( \frac{y^j}{\hat{y}^j} + \frac{y^j-1}{1-\hat{y}^j} \right) \frac{\partial\hat{y}}{\partial\beta_k} \\ &= -\frac{1}{m}\sum_{j=1}^m \left( \frac{y^j}{\hat{y}^j} + \frac{y^j-1}{1-\hat{y}^j} \right) (\hat{y}^j(1-\hat{y}^j)x^j_k) \\ &= -\frac{1}{m}\sum_{j=1}^m \left( y^j(1-\hat{y}^j)x^j_k + (y^j-1)\hat{y}^jx^j_k \right) \\ &= -\frac{1}{m}\sum_{j=1}^m (y^j - \hat{y}^j)x^j_k \\ &= \frac{1}{m}\sum_{j=1}^m (\hat{y}^j - y^j)x^j_k \\ \end{aligned} $$

Now we can write the gradient as:

$$ \begin{aligned} \nabla_{\boldsymbol{\beta}} J &= \begin{bmatrix} \frac{\partial J}{\partial\beta_0} \\ \vdots \\ \frac{\partial J}{\partial\beta_n} \end{bmatrix} \\ &= \begin{bmatrix} \frac{1}{m}\sum_{j=1}^m (\hat{y}^j - y^j)x^j_0 \\ \vdots \\ \frac{1}{m}\sum_{j=1}^m (\hat{y}^j - y^j)x^j_n \end{bmatrix} \end{aligned} $$

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

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

From this we can write

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

where

$$ \mathbf{\hat{y}} = \sigma(\mathbf{X}\mathbf{\boldsymbol{\beta}}) $$

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 Logistic regression. For the full code (with doc strings) it’s on github here.

class LogisticRegression():

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 logistic model instance')

We also define the sigmoid function.

def sigmoid(self, x):
    return 1 / (1 + np.exp(-x))

Next we define a method predict_proba to predict the probability of success given the samples $X$.

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

Similarly we define a predict method to predict the outcome target variable $y$ using a the decision rule described above.

def predict(self, X, descision_prob=0.5):
    y_pred = self.predict_proba(X)
    return (y_pred > descision_prob) * 1

Next let’s define a method fit to implement gradient descent. This function calculates the cost and gradient at each iteration of gradient descent.

def fit(self, X, y, n_iter=1000):
    m, n = X.shape
    print(f'fitting with m={m} samples'
          f' 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_proba(X)
        cost = (-1 / m) * (
            (y.T @ np.log(y_pred)) +
            ((np.ones(shape=y.shape) - y).T @ np.log(
                np.ones(shape=y_pred.shape) - y_pred))
        )
        self.costs.append(cost[0][0])
        gradient = (1 / m) * X.T @ (y_pred - y)
        self.beta = self.beta - (
            self.learning_rate * gradient)
        self.betas.append(self.beta)

Here’s an example use of the class:

logistic_regression = LogisticRegression()
logistic_regression.fit(X, y)

logistic_regression.predict(example_X)

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