📊 Pure Mathematics Behind Regression

Regression is the backbone of predictive modeling in machine learning. In this blog, we will dive deep into regression techniques—simple linear, multi-linear, and logistic regression—and learn how to create our own predictive models from scratch. 🧠✨

🚀 What you will learn:

• How data is interpreted and calculated to form predictive models
• How to create a model that works accurately using mathematical principles
• Step-by-step explanation of regression techniques

🔍 Let's start with Linear Regression!

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📈 Linear Regression

Before we dive into calculations, let's understand what regression really means. In simple terms, regression helps machines learn from given data so that they can predict future values based on past trends. 📊

🔹 What is Linear Regression?

Linear regression is a technique that predicts continuous values using a straight-line equation:

📌 Mathematical equation of a straight line:

$$y = mx + c$$

Where:

• y = dependent variable (e.g., salary)
• x = independent variable (e.g., years of experience)
• m = slope of the line
• c = intercept (where the line crosses the y-axis)

Let's break it down with an example. 👇

📊 Example Dataset

🏆 Years of Experience (X)	💰 Salary (Y) (in $1000)
1	35
2	40
3	45
4	50
5	55

📌 Our goal: Find a pattern so that given a new input (years of experience), we can predict the salary.

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🛠 Finding the Best Fit Line

🔍 Step 1: Start with the General Equation

We assume the relationship follows:

$$y = mx + c$$

where $y$ represents salary, $x$ represents years of experience, $m$ is the slope, and $c$ is the intercept.

📌 Step 2: Forming Two Main Summation Equations

To generalize for multiple data points, we sum up the equations for all data points:

📍 Summing the original equation over all data points:

$$\sum Y = m \sum X + Nc$$

📍 Multiplying each equation by $x$ and summing:

$$\sum XY = m \sum X^2 + c \sum X$$

Now, these two equations contain summations that can be computed for any dataset. 🧮

📌 Step 3: Solving for $m$ and $c$

Rearranging the equations:

$$m = \frac{N \sum XY - \sum X \sum Y}{N \sum X^2 - (\sum X)^2}$$ $$c = \frac{\sum Y - m \sum X}{N}$$

✅ With these formulas, we can compute $m$ and $c$ for any dataset, whether small or large!

📌 Once we have $m$ and $c$, we can use the equation y = mx + c to predict salary for new experience values.

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🚀 What's Next?

Now that we've built an understanding of simple linear regression, we will explore multi-linear regression and logistic regression in the next sections. Stay tuned! 🔥📊