Unpacking Linear Regression

"Machine learning" and "Predictive models" often sound like magical tools that foresee trends and patterns. Yet, behind this magic lies solid, well-understood mathematics.

Let's explore one of the fundamental building blocks of machine learning: Linear Regression.

Linear regression models the relationship between the input features (x) and the target variable (y) using the line equation:

? = w · x + b

Here:

·?????? w: weight (or slope of the line)

·?????? b: bias (or y-intercept)

·?????? x: input feature(s)

·?????? ?: predicted value of the target variable

The goal is to find the optimal values for w and b that minimize the difference between the predicted output ? and the actual y.?

Steps in Linear Regression: From Concept to Calculation

1.???? Initialize Parameters

·?????? Start with initial values for w (weights) and b (bias). Usually they are initialized to small random values or zeros.

·?????? Choose a learning rate (α) that controls how much the parameters are updated/adjusted during training.

2.???? Predict Outputs (?)

·?????? Using the current values of w and b, calculate predictions:

? = w · x + b

3.???? Calculate the Loss (Cost Function)

·?????? To assess how well the model fits, we measure the difference between the predicted (?) and actual y values.

·?????? Loss measures the difference between predicted (?) and actual (y) values.

·?????? A common cost function is the Mean Squared Error (MSE):

? ???????????????????????? J(w, b) = (1 / 2m) Σ (?? - y?)2

?where m is the number of data points.

This loss function quantifies how well the model is performing by providing a numerical value representing how far off the predictions are.

4.???? Compute the Gradient (Rate of Change of the Cost Function)

·?????? To reduce the loss, compute the gradients (partial derivatives) with respect to w and b: ??????????????????????? ??J/?w = (1 / m) Σ (?? - y?) · x? ??????????????????????? ?

?J/?b = (1 / m) Σ (?? - y?)

·?????? These gradients indicate the direction and rate of change of the cost function for each parameter.

5.???? Update Parameters Using Gradient Descent

·?????? Apply the Gradient Descent Algorithm to optimize w and b in the direction that minimizes the cost: ??????????????????

????????????????? ?w = w - α · ?J/?w ? ???????????????????????

?????????? ?b = b - α · ?J/?b

·?????? The learning rate (α) determines the size of the steps taken towards minimizing the cost.

6.???? Iterate Until Convergence

·?????? Repeat steps 2–5 until convergence (i.e., when the changes in w and b become very small or the cost function stops decreasing significantly.

Type of Loss Function:

While Mean Squared Error (MSE) is the most common, other loss functions like Mean Absolute Error (MAE) can also be used.

Purpose of Gradients:

Gradients represent the slope of the cost function with respect to each parameter (w and b). Gradients guide the model's optimization journey, ensuring it moves toward the best-fitting line.

Linear Regression Assumption:

Linear regression assumes a linear relationship between x and y. If this assumption is not met, the model may not perform well.

Bringing It All Together: Here's a high-level look at the linear regression process,

1.???? Initialization:

Choose initial w, b and learning rate α.

2.???? Prediction:

Compute ? = w · x + b.

3.???? Loss Calculation:

Compute the cost function J(w, b) = (1 / 2m) Σ (?? - y?)2.

4.???? Gradient Calculation:

Compute ?J/?w and ?J/?b.

5.???? Parameter Update:

Update w and b using: ? w = w - α · ?J/?w,?? b = b - α · ?J/?b.

6.???? Repeat Until Convergence:

Iterate until w and b stabilize, or the cost function stops improving.

The Science behind Magic

Linear regression is a simple, but very fundamental technique in machine learning. By understanding the mathematics behind it, we can understand the logical process that drives predictions.

Stay tuned for more insights!