Use cosh function to build a mathematical approximation of normal distribution

gitbox 2025-06-06

In data modeling, image processing, or machine learning, normal distribution (also known as Gaussian distribution) is a very common probability distribution model. Although PHP is not the preferred language for mathematical modeling, it can still be used to build basic mathematical models, especially in the web environment. This article will introduce how to use PHP's built-in cosh() function (hyperbolic cosine function) to build a simplified normal distribution approximation model.

Why choose cosh() ?

The mathematical form of a standard normal distribution function is:

 f(x) = (1 / (σ√2π)) * e^(-(x - μ)^2 / (2σ^2))

However, because PHP has limited support for complex mathematical functions, we can build an approximate function similar to a normal distribution curve by using the symmetry of the cosh() function and its rapidly growing characteristics. For example:

 f(x) = A / cosh(B * (x - μ))

Where A is the scaling factor, B controls the width of the distribution, and μ is the mean.

This function produces a bell curve on the graph, which, although not exactly equivalent to a normal distribution, can be used for visualization or rough modeling purposes.

PHP implementation

The following is a PHP script example that uses the cosh() function to generate data points with an approximate normal distribution and output it to JSON format, which is convenient for front-end chart library to use:

 <?php

// Parameter settings
$mu = 0;        // average value
$b = 0.5;       // Control width
$a = 1;         // Scaling factor
$range = 5;     // x Value range
$step = 0.1;    // Increased step length each time

$data = [];

for ($x = -$range; $x <= $range; $x += $step) {
    $y = $a / cosh($b * ($x - $mu));
    $data[] = ['x' => $x, 'y' => $y];
}

// Output JSON
header('Content-Type: application/json');
echo json_encode($data);
?>

You can save the above code as distribution.php and deploy it to the server, and then load the generated data through AJAX using front-end tools such as Chart.js or ECharts. For example:

 fetch("https://gitbox.net/distribution.php")
  .then(response => response.json())
  .then(data => {
    // use data Draw a chart
    console.log(data);
  });

Curve effect and adjustment

Adjusting parameters A and B can control the height and width of the graph to fit different needs. For example:

Increase B value: The curve is steeper and the shape is closer to the "spike"
Reduce B value: The curve is smoother and closer to the true normal distribution

Although this method cannot replace a true normal distribution function (such as using Gaussian function library or C extension), it still has practical application value in Web scenarios.

Application scenarios

Simple data visualization model : can be used to graph data fluctuations.
UI Animated Rhythm Control : Curves control gradient rhythms to make the visual experience more natural.
Web simulation calculation : Provides fast numerical simulation results when communicating front and back ends.

Summarize

Although PHP is not a strong point in numerical computing, its flexibility and wide application scenarios make it shine in situations where mathematical models are deployed quickly. Approximate construction of normal distributions through the cosh() function is a lightweight, easy-to-implement method for web projects that require rapid visualization and simulation. Combined with modern JavaScript charting tools, it is easy to build an interactive data display interface.

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