AI - Linear Regression

Linear Regression is a core statistical technique in machine learning and data analysis, used to model the relationship between a target variable $y$ (dependent variable) and one or more predictors $x$ (independent variables or features). Its main objective is to determine the best-fitting line, represented by a slope $m$ and intercept $b$, that can accurately predict the target variable $y$ based on the input features $x$. The line follows the equation:

$$y=mx+b$$

Linear Regression Demonstration

This application demonstrates linear regression, a statistical method for modeling the relationship between two variables by fitting a straight line to the data points. The regression line is defined by the slope $m$ and y-intercept $b$, which describe how changes in the independent variable impact the dependent variable. The R² (coefficient of determination) value indicates the accuracy of the model, with a higher R² meaning a better fit.

The model aims to find the best-fitting line by adjusting the slope $m$ and intercept $b$ to match the data as closely as possible. This line is represented by the equation $y=mx+b$. Each adjustment to $m$ and $b$ is made to reduce the error, or difference, between the actual data points and the values predicted by the line. The goal is to minimize these errors across all points, creating a line that represents the overall trend in the data as accurately as possible.

Linear Regression Formula

For a single independent variable, the linear regression equation is:

$$\hat{y}=b_0+b_{1}x$$

Where:

• $\hat{y}$: Predicted value of the dependent variable.
• $b_0$: Intercept of the regression line; the value of $\hat{y}$ when $x=0$.
• $b_1$: Slope of the regression line; represents the change in $\hat{y}$ for a one-unit change in $x$.
• $x$: Independent variable.

Steps to Derive the Regression Coefficients ($b_0$ and $b_1$):

1. Calculate the Means:
   • Compute the mean of the independent variable ($\overline{x}$) and the dependent variable ($\overline{y}$).

2. Compute the Slope ($b_1$):

   $$b_1=\frac{\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sum _{i=1}^n(x_i-\overline{x}{)}^2}$$

   Where $n$ is the number of data points, $x_i$ and $y_i$ are the individual sample points.

3. Compute the Intercept ($b_0$):

   $$b_0=\overline{y}-b_1\overline{x}$$

Example:

Suppose we have the following dataset:

$x$ (Independent Variable)	$y$ (Dependent Variable)
1	2
2	3
4	7
5	5
7	11

Calculate the Means:

• $\overline{x}=\frac{1+2+4+5+7}{5}=3.8$

• $\overline{y}=\frac{2+3+7+5+11}{5}=5.6$

Compute the Slope ($b_1$):

• $b_1=\frac{(1-3.8)(2-5.6)+(2-3.8)(3-5.6)+(4-3.8)(7-5.6)+(5-3.8)(5-5.6)+(7-3.8)(11-5.6)}{(1-3.8{)}^2+(2-3.8{)}^2+(4-3.8{)}^2+(5-3.8{)}^2+(7-3.8{)}^2}$

• $b_1=\frac{(-2.8)(-3.6)+(-1.8)(-2.6)+(0.2)(1.4)+(1.2)(-0.6)+(3.2)(5.4)}{(-2.8{)}^2+(-1.8{)}^2+(0.2{)}^2+(1.2{)}^2+(3.2{)}^2}$

• $b_1=\frac{10.08+4.68+0.28-0.72+17.28}{7.84+3.24+0.04+1.44+10.24}$

• $b_1=\frac{31.6}{22.8}\approx 1.386$

Compute the Intercept ($b_0$):

• $b_0=\overline{y}-b_1\overline{x}$

• $b_0=5.6-1.386\times 3.8$

• $b_0=5.6-5.271$

• $b_0\approx 0.329$

Regression Equation:

Thus, the regression equation is:

$\hat{y}=0.329+1.386x$

This equation can be used to predict the value of $y$ for any given $x$ within the range of the data.

Applications

• Predictive Modeling: Used to predict outcomes based on historical data.
• Trend Analysis: Helps in identifying trends and relationships in data.
• Risk Management: Assists in assessing risks and forecasting future trends.

Example

If we have a simple linear regression model predicting house prices based on square footage:

$$y=mx+b$$

• $y$ = predicted house price
• $x$ = square footage of the house
• $m$ = slope indicating how much the house price increases per additional square foot
• $b$ = y-intercept, representing the base price when the square footage is zero

The model would then try to “fit” a line that best matches the pattern in the data, allowing it to make predictions for new houses based on square footage.

Conclusion

Linear regression is a simple yet powerful tool for understanding and predicting the relationship between variables. It forms the basis for many more complex statistical models and machine learning algorithms.