Linear regression in most simple way with proper examples in detail

 Sure! Linear regression is a simple and widely used statistical method for predicting a numeric value (target variable) based on one or more input features. It assumes a linear relationship between the input features and the target variable.

The "linear" in linear regression refers to the fact that the relationship can be represented by a straight line equation, which is defined as:

y = mx + b

Where:

- y is the target variable (the value we want to predict).

- x is the input feature(s) (the independent variable(s)).

- m is the slope (also known as the coefficient), representing the change in y with respect to a unit change in x.

- b is the intercept, representing the value of y when x is zero.

The main goal of linear regression is to find the best-fitting line that minimizes the difference between the predicted values and the actual target values in the training data.

Let's illustrate this with a simple example using a single input feature and target variable:

Example: Predicting House Prices

Suppose we want to predict the price of a house based on its size (in square feet). We have some historical data on house sizes and their corresponding prices:

| House Size (x) | Price (y)  |

|----------------|------------|

| 1000           | 200,000    |

| 1500           | 250,000    |

| 1200           | 220,000    |

| 1800           | 280,000    |

| 1350           | 240,000    |

To use linear regression, we need to find the best-fitting line that represents this data. The line will have the form: y = mx + b.

Step 1: Calculate the slope (m) and intercept (b).

To calculate the slope (m) and intercept (b), we use formulas derived from the method of least squares.

```

m = (N * Σ(xy) - Σx * Σy) / (N * Σ(x^2) - (Σx)^2)

b = (Σy - m * Σx) / N

```

where N is the number of data points, Σ denotes summation, and xy represents the product of x and y values.

Step 2: Plug the values of m and b into the equation y = mx + b.

```

m = (5 * 1371500000 - 8000 * 990000) / (5 * 10350000 - 8000^2) ≈ 29.545

b = (990000 - 29.545 * 8000) / 5 ≈ 122727.27

```

So, the equation of the line is: y ≈ 29.545x + 122727.27

Step 3: Make predictions.

Now, we can use the equation to make predictions on new data. For example, if we have a house with a size of 1250 square feet:

```

Predicted Price (y) ≈ 29.545 * 1250 + 122727.27 ≈ 159545.45

```

In this example, we used a simple linear regression model to predict house prices based on house sizes. In real-world scenarios, linear regression can have multiple input features, and the process remains fundamentally the same.

Keep in mind that linear regression is a basic model and may not always be suitable for complex relationships in the data. For more complex relationships, you might need to consider other regression techniques or use polynomial regression.