understanding chart stack generation randomly and smoothening.

Question

I'm tring to understand the below logic.

function bumps(m) {
        var values = [], i, j, w, x, y, z;
        for (var i = 0; i < m; ++i) {
            values[i] = 0.1 + 0.1 * Math.random();
        }

        for (var j = 0; j < 5; j++) {
            x = 1 / (0.1 + Math.random());
            y = 2 * Math.random() - 0.5;
            z = 10 / (0.1 + Math.random());
            for (var i = 0; i < m; i++) {
                w = (i / m - y) * z;
                values[i] += x * Math.exp(-w * w);
            }
        }
        for (var i = 0; i < m; i++) {
            values[i] = Math.max(0, values[i]);
        }
        return values;
    }

The above code is snippet from http://bl.ocks.org/mbostock/3943967 and the above function is used to display the random series stack on the y axis in the graph displayed url. I'm trying to understand what they are trying to do in the below

x = 1 / (0.1 + Math.random());
                y = 2 * Math.random() - 0.5;
                z = 10 / (0.1 + Math.random());

How to interpret or understand mathematically / logically the above operations in bumps method ? Please help me in understanding the same.

Answer 1

As the given reference indicates:

// Add five random bumps.
for (j = 0; j < 5; ++j) {
    x = 1 / (0.1 + Math.random());
    y = 2 * Math.random() - 0.5;
    z = 10 / (0.1 + Math.random());
    for (i = 0; i < m; i++) {
        w = (i / m - y) * z;
        values[i] += x * Math.exp(-w * w);
    }
}

This is generating $m=4$ histogram series with five gaussian lobes each: $$ v_j=\sum_{i=1}^{m} z_j \exp^{-\left(\frac{iz_j}{m-y_j}\right)^2} $$

This function do not have any special significance. This is not a distribution, because this do not integrate 1. It is just a pictorical sample for displaying purposes.

They could truly be interpreted as histogram data, if truncated to the nearest integer, or as is, considering an histogram built in discrete ranges from a continuous variable.

The values of $x,y,z$ neither do possess a special significance; they are only random numbers generated from the Math.random() function, which outputs a uniform random number in the $[0,1]$ range, and operated in order to give different values, in the $[0.909,10]$, $[-0.5,1.5]$ and $[90.90,100]$ ranges, respectively.

As pictorical relevance, they are related to the height, spread, and center of each lobe.

Please note the similarities with the gaussian distribution, which do integrate 1 as per any proper probability density function: $$ v=\frac{1}{\sqrt{2\pi}\sigma}\exp^{-(\frac{x-\mu}{2\sigma})^2} $$