Linear Regression

Today, we’re diving into the math behind one of the most fundamental models in machine learning: linear regression. This is the first model you’ll typically learn when starting out in the field.

DEFINATION —

Linear Regression is a way to find a straight line that best matches a set of data points. It helps predict one value based on another by showing the relationship between them. The formula of linear regression –

? = c + mx

where, ? -> predicted value

m -> slope; it tells how much the predicted value changes for each unit in the input value

c -> Intercept; it tells you the predicted value when the input variable is ‘0’

x -> data points

Notations used in the Research paper are –

In case of more than 1 input feature, the formula will be -

Linear regression is used for:

• Supervised Learning?—?It’s a type of machine learning where the model is trained on labeled data.
• Regression Problems?—?It predicts a continuous outcome, like forecasting sales or estimating prices.

IMPORTANT TERMINOLOGIES –

1. Residual Error: It’s the difference between the actual value and what the model predicts. If the residual error is ‘0’, it may indicate that the model is overfitting.
2. Cost Function: It measures the error between predicted and actual values. The cost function varies depending on the model. In linear regression, the cost function used is Mean Squared Error.

Where J(θ) → cost function,

m -> number of training samples,

h(θ) -> predicted value,

y -> actual value

3. Repeat Convergence Algorithm?—?It is an iterative process that repeatedly updates the model parameters (θ)?to reduce the cost function until it reaches the global minimum.

where α is the learning rate
      ?/?θ1  J(θ1)   is derivate of cost function, i.e., slope        

CASE I?—?In this case, the slope is +ve ->

θ?:= θ1 – α(+ve)

So, the value of θ1 decreases.

CASE II?—?In this case, the slope is -ve ->

θ1?:= θ1 – α(-ve)

θ1?:= θ1 + α(+ve)

So, the value of θ1 increases.

CASE III?—?In case, the slope is nearly 0 ->

θ1?:= θ1 – α(0)

θ1?:= θ1 – 0

So, the value of θ1 will be unchanged.

4. Learning Rate?—?It is a hyperparameter that controls the step size at each iteration while moving toward a minimum of the cost function. It determines how quickly or slowly the model updates its parameters (weights) during gradient descent.

CASE I?—?Learning Rate is too high?-?The model may overshoot the minimum, leading to divergence.

CASE II?—?Learning Rate is low?-?The model converges very slowly, taking more time to reach the optimal solution.

How θ value is calculated?

Let’s break down the calculation of θ step by step.

To keep it simple, we’ll use basic data points: (1,1), (2,2), and (3,3), with an initial θ0 = 0 and α = 0.1

STEP 1?—?Define the hypothesis function and cost function for simple linear regression:

• Hypothesis Function:

• Cost Function:

STEP 2?—?Assume a Random Value for θ1 and Calculate the Cost Function

Let’s assume θ1 = 0:

STEP 3?—?Minimize the Cost Function Using Gradient Descent:

Iteration I -?θ1 = 0:

Iteration II?-?θ1 = 0.467:

Iteration III?-?θ1 = 0.716:

STEP 4?—?Continue repeating Step 5 until the cost function hits its lowest value.

Interesting Questions —

Q?—?Why do we divide the MSE by 2 in the cost function?

A?—?Let’s look at both scenarios:

Situation I?—?Cost function without 1/2:

When calculating the gradient, an extra factor of 2 (which appears after taking the derivative) makes the math a bit messier.

Situation II— Cost function with 1/2:

When we divide the cost function by 2, the extra factor of 2 (which appears after taking the derivative) cancels out, simplifying the gradient descent process and making calculations easier.

Q?—?Why do we take the derivative of the cost function while updating θ?

A?—?We take the derivative to find the slope of the cost function, which helps us adjust the model’s parameters to minimize errors and improve the model.

Reference —

1. https://www.youtube.com/watch?v=jerPVDaHbEA&list=PLTDARY42LDV7WGmlzZtY-w9pemyPrKNUZ&index=2

Finally —

I hope this blog clarifies linear regression for you!

Got a particular ML topic you’re curious about? Drop your suggestions in the comments, and I’ll do my best to cover them. Thanks for reading!

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