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copied!<h2>Logarithmic Least Squares</h2> <p>Since we can convert a logarithmic function into a line by taking the <code>log</code> of the <code>x</code> values, we can perform a <a href="http://en.wikipedia.org/wiki/Linear_least_squares" rel="nofollow noreferrer">linear least squares</a> curve fitting. In fact, the work has been done for us and a solution is presented at <a href="http://mathworld.wolfram.com/LeastSquaresFittingLogarithmic.html" rel="nofollow noreferrer">Math World</a>.</p> <p>In brief, we're given <code>$X</code> and <code>$Y</code> values that are from a distribution like <code>y = a + b * log(x)</code>. The least squares method will give some values <code>aFit</code> and <code>bFit</code> that minimize the distance from the parametric curve to the data points given.</p> <p><strong>Here is an example implementation in PHP:</strong></p> <p>First I'll generate some random data with known underlying distribution given by <code>$a</code> and <code>$b</code></p> <pre><code> // True parameter valaues $a = 10; $b = 5; // Range of x values to generate $x_min = 1; $x_max = 10; $nPoints = 50; // Generate some random points on y = a * log(x) + b $X = array(); $Y = array(); for($p = 0; $p < $nPoints; $p++){ $x = $p / $nPoints * ($x_max - $x_min) + $x_min; $y = $a + $b * log($x); $X[] = $x + rand(0, 200) / ($nPoints * $x_max); $Y[] = $y + rand(0, 200) / ($nPoints * $x_max); } </code></pre> <p>Now, here's how to use the equations given to estimate <code>$a</code> and <code>$b</code>.</p> <pre><code> // Now convert to log-scale for X $logX = array_map('log', $X); // Now estimate $a and $b using equations from Math World $n = count($X); $square = create_function('$x', 'return pow($x,2);'); $x_squared = array_sum(array_map($square, $logX)); $xy = array_sum(array_map(create_function('$x,$y', 'return $x*$y;'), $logX, $Y)); $bFit = ($n * $xy - array_sum($Y) * array_sum($logX)) / ($n * $x_squared - pow(array_sum($logX), 2)); $aFit = (array_sum($Y) - $bFit * array_sum($logX)) / $n; </code></pre> <p>You may then generate points for your Javascript as densely as you like:</p> <pre><code> $Yfit = array(); foreach($X as $x) { $Yfit[] = $aFit + $bFit * log($x); } </code></pre> <p>In this case, the code estimates <code>bFit = 5.17</code> and <code>aFit = 9.7</code>, which is quite close for only <code>50</code> data points.</p> <p><img src="https://i.stack.imgur.com/U6Ily.png" alt="alt text"></p> <p>For the example data given in the comment below, a logarithmic function does not fit well.</p> <p><img src="https://i.stack.imgur.com/mMq0I.png" alt="alt text"></p> <p>The least squares solution is <code>y = -514.734835478 + 2180.51562281 * log(x)</code> which is essentially a line in this domain.</p>

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