README.md

https://SenatorovAI.com

Normal Equations Solver for Simple Linear Regression

Research-oriented implementation of the Normal Equations method for solving Simple Linear Regression in closed form.

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Overview

This repository provides a mathematically transparent implementation of the closed-form solution to the ordinary least squares (OLS) problem for simple linear regression.

We consider the model:

$$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i $$

where:

• $x_i$ — scalar input
• $y_i$ — scalar target
• $\beta_0$ — intercept
• $\beta_1$ — slope
• $\varepsilon_i$ — noise term

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Least Squares Formulation

The objective is to minimize the residual sum of squares:

$$ \min_{\beta_0, \beta_1} \sum_{i=1}^{n} \left( y_i - (\beta_0 + \beta_1 x_i) \right)^2 $$

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Matrix Form

Define the design matrix:

$$ X = \begin{bmatrix} 1 & x_1 \\ 1 & x_2 \\ \vdots & \vdots \\ 1 & x_n \end{bmatrix} $$

Parameter vector:

$$ \beta = \begin{bmatrix} \beta_0 \\ \beta_1 \end{bmatrix} $$

Target vector:

$$ y = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix} $$

Then the least squares problem becomes:

$$ \min_{\beta} |X\beta - y|_2^2 $$

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Closed-Form Solution (Scalar form)

For simple linear regression, the coefficients can also be written explicitly:

$$ \beta_1 = \frac {\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})} {\sum_{i=1}^{n} (x_i - \bar{x})^2} $$

$$ \beta_0 = \bar{y} - \beta_1 \bar{x} $$

where:

$$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i $$

$$ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i $$

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Numerical Considerations

The matrix $X^T X$ must be non-singular.
If multicollinearity or degeneracy occurs, numerical instability may arise.

For improved stability in high-dimensional settings, QR decomposition or SVD-based solvers are preferred.

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Implementation

The solver is implemented using NumPy and performs:

1. Construction of $X$
2. Computation of $X^T X$
3. Matrix inversion
4. Coefficient estimation

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Example Usage

from solver import normal_equation

beta_0, beta_1 = normal_equation(x, y)