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POCurve fitting: Find the smoothest function that satisfies a list of constraints
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copied!<p>Consider the set of non-decreasing <a href="http://mathworld.wolfram.com/Surjection.html" rel="nofollow noreferrer">surjective</a> (onto) functions from (-inf,inf) to [0,1]. (Typical <a href="http://mathworld.wolfram.com/DistributionFunction.html" rel="nofollow noreferrer">CDF</a>s satisfy this property.) In other words, for any real number x, 0 <= f(x) <= 1. The <a href="http://en.wikipedia.org/wiki/Logistic_function" rel="nofollow noreferrer">logistic function</a> is perhaps the most well-known example.</p> <p>We are now given some constraints in the form of a list of x-values and for each x-value, a pair of y-values that the function must lie between. We can represent that as a list of {x,ymin,ymax} triples such as</p> <pre><code>constraints = {{0, 0, 0}, {1, 0.00311936, 0.00416369}, {2, 0.0847077, 0.109064}, {3, 0.272142, 0.354692}, {4, 0.53198, 0.646113}, {5, 0.623413, 0.743102}, {6, 0.744714, 0.905966}} </code></pre> <p>Graphically that looks like this:</p> <p><a href="http://yootles.com/outbox/cdffit1.png" rel="nofollow noreferrer">constraints on a cdf http://yootles.com/outbox/cdffit1.png</a></p> <p>We now seek a curve that respects those constraints. For example:</p> <p><a href="http://yootles.com/outbox/cdffit2.png" rel="nofollow noreferrer">fitted cdf http://yootles.com/outbox/cdffit2.png</a></p> <p>Let's first try a simple interpolation through the midpoints of the constraints:</p> <pre><code>mids = ({#1, Mean[{#2,#3}]}&) @@@ constraints f = Interpolation[mids, InterpolationOrder->0] </code></pre> <p>Plotted, f looks like this:</p> <p><a href="http://yootles.com/outbox/cdffit3.png" rel="nofollow noreferrer">interpolated cdf http://yootles.com/outbox/cdffit3.png</a></p> <p>That function is not surjective. Also, we'd like it to be smoother. We can increase the interpolation order but now it violates the constraint that its range is [0,1]:</p> <p><a href="http://yootles.com/outbox/cdffit4.png" rel="nofollow noreferrer">interpolated cdf with higher interpolation order http://yootles.com/outbox/cdffit4.png</a></p> <p>The goal, then, is to find the <a href="http://mathworld.wolfram.com/SmoothFunction.html" rel="nofollow noreferrer">smoothest function</a> that satisfies the constraints:</p> <ol> <li>Non-decreasing.</li> <li>Tends to 0 as x approaches negative infinity and tends to 1 as x approaches infinity.</li> <li>Passes through a given list of y-error-bars.</li> </ol> <p>The first example I plotted above seems to be a good candidate but I did that with Mathematica's <a href="http://reference.wolfram.com/mathematica/ref/FindFit.html" rel="nofollow noreferrer">FindFit</a> function assuming a <a href="http://mathworld.wolfram.com/LogNormalDistribution.html" rel="nofollow noreferrer">lognormal CDF</a>. That works well in this specific example but in general there need not be a lognormal CDF that satisfies the constraints.</p>

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