Linear Regression

4 things to take in consideration:

1. Dataset:

   ($x$, $y$) with $x$ being the feature and $y$ being the target.

2. Model:

   represented by the function $f(x) = ax + b$, $a$ and $b$ being the parameters.

3. Cost Function:

   A function to compute the error between the predicted values and the targets.

   In our case, $J(a,b) = {{1 \over{2m}} \sum_{i=1}^{m} (f(x^{(i)}) - y^{(i)})^2 }$, $m$ being the number of rows.

4. Minimisation Algorithm:

   In our case, we are going to use the Gradient Descent.

   Basically, for each parameter $a_{i}$, we compute $a_{i}$ = $a_{i}$ - $\alpha$ $\frac{\partial J}{\partial a_i}(a)$.

   $\alpha$ being the learning rate.

Transform this to a problem using matrices

Let's $n$ is the number of features, $m$ is the number of rows.

$x$ = $\begin{pmatrix} x_{1}^{(1)} & x_{2}^{(1)} & x_{3}^{(1)} & \dots & x_{n}^{(1)} \\ x_{1}^{(2)} & x_{2}^{(2)} & x_{3}^{(2)} & \dots & x_{n}^{(2)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{1}^{(m)} & x_{2}^{(m)} & x_{3}^{(m)} & \dots & x_{n}^{(m)} \end{pmatrix}$ thus $X$ = $\begin{pmatrix} x_{1}^{(1)} & x_{2}^{(1)} & x_{3}^{(1)} & \dots & x_{n}^{(1)} & 1 \\ x_{1}^{(2)} & x_{2}^{(2)} & x_{3}^{(2)} & \dots & x_{n}^{(2)} & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ x_{1}^{(m)} & x_{2}^{(m)} & x_{3}^{(m)} & \dots & x_{n}^{(m)} & 1 \end{pmatrix}$ $\in {\rm I\!R}^{m \times (n+1)}$

$y$ = $\begin{pmatrix} y^{(1)} \\ y^{(2)} \\ \vdots \\ y^{(m)} \end{pmatrix}$ $\in {\rm I\!R}^{m \times 1}$

And the parameters $\theta$ = $\begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \\ b \end{pmatrix}$ $\in {\rm I\!R}^{(n+1) \times 1}$

Finally we get the model $F = X\theta \in {\rm I\!R}^{m \times 1}$

As for the cost function $J(\theta) = {{1 \over{2m}} \sum_{i=1}^{m} (F - y)^2 } = {{1 \over{2m}} \sum_{i=1}^{m} (X\theta - y)^2 } \in {\rm I\!R}$

As for the gradient $\frac{\partial J}{\partial \theta}(\theta) = {{1 \over{m}} X^T(X\theta - y) } \in {\rm I\!R^{(n+1) \times 1}}$

Thus $\theta = \theta - \alpha \frac{\partial J}{\partial \theta}$

import numpy as np
from sklearn.datasets import make_regression

x, y =  make_regression(n_samples=100, n_features=1, noise=10) # Here m=100 and n=1
y = y.reshape(y.shape[0], 1)
x.shape, y.shape

((100, 1), (100, 1))

import matplotlib.pyplot as plt

plt.scatter(x, y)

X = np.hstack((x, np.ones((x.shape[0], 1))))
X.shape

(100, 2)

theta = np.random.randn(X.shape[1], 1)
theta

array([[-1.11140959],[ 0.15718568]])

def model(X, theta):
    return X.dot(theta)

plt.scatter(x, y, c='g')
plt.plot(x, model(X, theta), c='orange')

def cost_function(theta, X, y):
    return 1 / (2 * len(y)) * np.sum((X.dot(theta) - y) ** 2)
cost_function(theta, X, y)

2466.3199193827004

def gradients(theta, X, y):
    return 1 / len(y) * X.T.dot(X.dot(theta) - y)
gradients(theta, X, y)

array([[-76.34230311], [ -4.87819021]])

def gradient_descent(theta, X, y, learning_rate, n):
    for i in range(n):
        theta = theta - learning_rate * gradients(theta, X, y)
    return theta
final_theta =  gradient_descent(theta, X, y, 0.01, 1000)

plt.scatter(x, y, c='g')
plt.plot(x, model(X, final_theta), c='orange')

And there you have it, we have created a linear model that represents our data!

Now, let's study the learning process.

Let's visualize the cost function's variation, and the linear model variation!

def gradient_descent_v2(theta, X, y, learning_rate, n):
    costs = []
    thetas = []
    for i in range(n):
        theta = theta - learning_rate * gradients(theta, X, y)
        costs.append(cost_function(theta, X, y))
        thetas.append(theta)
    return {"theta": theta, "costs": costs, "thetas": thetas}

result = gradient_descent_v2(theta, X, y, 0.01, 1000)

plt.plot(range(1000), result["costs"])

plt.scatter(x, y, c='g')
color = iter(plt.cm.rainbow(np.linspace(0, 1, 10)))
for i in range(10):
   c = next(color)
   plt.plot(x, model(X, result["thetas"][i * 100]), c=c)

What we can conclude is $n = 200$ is sufficient to train the model. Let's test it!

result = gradient_descent_v2(theta, X, y, 0.01, 200)

plt.plot(range(200), result["costs"])

plt.scatter(x, y, c='g')
color = iter(plt.cm.rainbow(np.linspace(0, 1, 10)))
for i in range(10):
   c = next(color)
   plt.plot(x, model(X, result["thetas"][i * 20 + 19]), c=c)