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Find a “best fit” equation#1
It's been a while since I was in college and knew how to calculate a best fit line, but I find myself needing to. Suppose I have a set of points, and I want to find the line that is the best of those points.

What is the equation to determine a best fit line?
How would I do that with PHP?

posted date: 2008-12-12 16:25:00

Re: Find a “best fit” equation#2
I had made out the solution of this problem. click to view my topic...

hope that hepls.

posted date: 2008-12-12 16:25:01

Re: Find a “best fit” equation#3
An often used approach is to iteratively minimize the sum of squared y-differences between your points and the fit function.

posted date: 2008-12-12 16:29:00

Re: Find a “best fit” equation#4
Although you can use an iterative approach, you can directly calculate the slope and intercept of a line given a set of observations using a least-squares approach. See the "Univariate Linear Case" section of the Wikipedia article on linear regression for how to calculate the coefficients a and b in y = a + bx given sets of (x,y) points.

posted date: 2008-12-12 16:35:00

Re: Find a “best fit” equation#5
You may want to check out linear regression, or more generally, curve fitting.

posted date: 2008-12-12 16:39:00

Re: Find a “best fit” equation#6
Here's an article comparing two ways to fit a line to data. One thing to watch out for is that there is a direct solution that is correct in theory but can have numerical problems. The article shows why that method can fail and gives another method that is better.

posted date: 2008-12-12 16:40:00

Re: Find a “best fit” equation#7
Implemented from wiki page, untested.$sx = 0;$sy = 0;$sxy = 0;$sx2 = 0;$n = count($data);foreach ($data as $x => $y){ $sx += $x; $sy += $y; $sxy += $x * $y; $sx2 += $x * $x;}$beta = ($n*$sxy - $sx*$sy) / ($n*$sx2 - $sx*$sx);$alpha = $sy/$n - $sx*$beta/$n;echo "y = $alpha + $beta x";

posted date: 2008-12-12 16:43:00

Re: Find a “best fit” equation#8
Of additional interest is probably how good of a fit the line is.For that, use the Pearson correlation, here in a PHP function:/** * returns the pearson correlation coefficient (least squares best fit line) * * @param array $x array of all x vals * @param array $y array of all y vals */function pearson(array $x, array $y){ // number of values $n = count($x); $keys = array_keys(array_intersect_key($x, $y)); // get all needed values as we step through the common keys $x_sum = 0; $y_sum = 0; $x_sum_sq = 0; $y_sum_sq = 0; $prod_sum = 0; foreach($keys as $k) { $x_sum += $x[$k]; $y_sum += $y[$k]; $x_sum_sq += pow($x[$k], 2); $y_sum_sq += pow($y[$k], 2); $prod_sum += $x[$k] * $y[$k]; } $numerator = $prod_sum - ($x_sum * $y_sum / $n); $denominator = sqrt( ($x_sum_sq - pow($x_sum, 2) / $n) * ($y_sum_sq - pow($y_sum, 2) / $n) ); return $denominator == 0 ? 0 : $numerator / $denominator;}

posted date: 2008-12-12 16:45:00

Re: Find a “best fit” equation#9
btw, the Pearson coefficient ranges from 0 (no correlation) to 1.0 (points lie on a straight line)

posted date: 2008-12-12 16:46:00

Re: Find a “best fit” equation#10
+1 This by far the best answer, the other method is vastly inferior, albeit more popular.

posted date: 2011-06-08 10:02:00

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