Linear regression as a basic model

Case 1. Apartment valuation based on parameters

Implementation in pure PHP

(analytical solution – via normal equations)

Now we will solve the same task without iterative training — analytically, using normal equations. This approach gives us the model weights directly from the formula $w^* = (X^T X)^{-1} X^T y$. The idea is simple: build the feature matrix $X$, transpose it, multiply it by itself, invert $X^T X$, and obtain the final weights directly.

Example of use


                
<?php

include('formula-code.php');

use app\classes\Matrix;

// Normal equation formula: w* = (X^T X)^-1 X^T y
// Prepare data
$X = new Matrix([
    [50, 3, 5, 10, 1],
    [80, 12, 2, 3, 1],
    [45, 5, 7, 20, 1],
    [60, 8, 4, 15, 1],
    [30, 1, 10, 40, 1],
]);

$y = new Matrix([
    [60_000],
    [120_000],
    [55_000],
    [80_000],
    [30_000],
]);

// Compute weights
$Xt = $X->transpose();                          // X^T
$XtX = $Xt->multiply($X);                       // X^T * X
$XtX_inv = $XtX->regularizedInverse(0.01, 4);   // (X^T * X + λI)^-1
$Xt_y = $Xt->multiply($y);                      // X^T y
$w = $XtX_inv->multiply($Xt_y);                 // final weights

function predict(array $x, Matrix $w): float {
    $sum = 0.0;

    foreach ($x as $i => $value) {
        $sum += $value * $w->data[$i][0];
    }

    return $sum;
}

$newApartment = [60, 5, 4, 12, 1];
$price = predict($newApartment, $w);

$bias = array_pop($w->data);
$weights = $w->data;

echo 'Apartment valuation: $' . number_format($price) . PHP_EOL . PHP_EOL;

echo 'Weights: ' . implode(', ', array_map(fn ($weight) => number_format($weight[0], 2, '.', ''), $weights)) . PHP_EOL;
echo 'Bias: ' . number_format(array_pop($bias), 2, '.', '') . PHP_EOL;

Charts:

Chart Type:

Result: Memory: 0.034 Mb Time running: 0.001 sec.


                Apartment valuation: $80,003

Weights: 2326.89, 6.71, 3286.87, 3.65
Bias: -72834.76

The block above shows the result of the script: the computed weights and the predicted apartment price for the following features:

Apartment area: 60 m²
Number of rooms: 5
Number of bathrooms: 4
Number of floors: 12