720 строки
18 KiB
JavaScript
720 строки
18 KiB
JavaScript
(function(){science.stats = {};
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// Bandwidth selectors for Gaussian kernels.
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// Based on R's implementations in `stats.bw`.
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science.stats.bandwidth = {
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// Silverman, B. W. (1986) Density Estimation. London: Chapman and Hall.
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nrd0: function(x) {
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var hi = Math.sqrt(science.stats.variance(x));
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if (!(lo = Math.min(hi, science.stats.iqr(x) / 1.34)))
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(lo = hi) || (lo = Math.abs(x[1])) || (lo = 1);
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return .9 * lo * Math.pow(x.length, -.2);
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},
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// Scott, D. W. (1992) Multivariate Density Estimation: Theory, Practice, and
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// Visualization. Wiley.
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nrd: function(x) {
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var h = science.stats.iqr(x) / 1.34;
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return 1.06 * Math.min(Math.sqrt(science.stats.variance(x)), h)
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* Math.pow(x.length, -1/5);
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}
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};
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science.stats.distance = {
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euclidean: function(a, b) {
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var n = a.length,
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i = -1,
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s = 0,
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x;
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while (++i < n) {
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x = a[i] - b[i];
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s += x * x;
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}
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return Math.sqrt(s);
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},
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manhattan: function(a, b) {
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var n = a.length,
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i = -1,
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s = 0;
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while (++i < n) s += Math.abs(a[i] - b[i]);
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return s;
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},
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minkowski: function(p) {
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return function(a, b) {
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var n = a.length,
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i = -1,
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s = 0;
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while (++i < n) s += Math.pow(Math.abs(a[i] - b[i]), p);
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return Math.pow(s, 1 / p);
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};
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},
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chebyshev: function(a, b) {
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var n = a.length,
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i = -1,
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max = 0,
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x;
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while (++i < n) {
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x = Math.abs(a[i] - b[i]);
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if (x > max) max = x;
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}
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return max;
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},
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hamming: function(a, b) {
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var n = a.length,
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i = -1,
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d = 0;
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while (++i < n) if (a[i] !== b[i]) d++;
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return d;
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},
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jaccard: function(a, b) {
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var n = a.length,
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i = -1,
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s = 0;
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while (++i < n) if (a[i] === b[i]) s++;
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return s / n;
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},
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braycurtis: function(a, b) {
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var n = a.length,
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i = -1,
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s0 = 0,
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s1 = 0,
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ai,
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bi;
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while (++i < n) {
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ai = a[i];
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bi = b[i];
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s0 += Math.abs(ai - bi);
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s1 += Math.abs(ai + bi);
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}
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return s0 / s1;
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}
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};
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// Based on implementation in http://picomath.org/.
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science.stats.erf = function(x) {
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var a1 = 0.254829592,
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a2 = -0.284496736,
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a3 = 1.421413741,
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a4 = -1.453152027,
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a5 = 1.061405429,
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p = 0.3275911;
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// Save the sign of x
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var sign = x < 0 ? -1 : 1;
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if (x < 0) {
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sign = -1;
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x = -x;
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}
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// A&S formula 7.1.26
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var t = 1 / (1 + p * x);
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return sign * (
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1 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1)
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* t * Math.exp(-x * x));
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};
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science.stats.phi = function(x) {
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return .5 * (1 + science.stats.erf(x / Math.SQRT2));
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};
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// See <http://en.wikipedia.org/wiki/Kernel_(statistics)>.
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science.stats.kernel = {
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uniform: function(u) {
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if (u <= 1 && u >= -1) return .5;
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return 0;
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},
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triangular: function(u) {
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if (u <= 1 && u >= -1) return 1 - Math.abs(u);
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return 0;
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},
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epanechnikov: function(u) {
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if (u <= 1 && u >= -1) return .75 * (1 - u * u);
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return 0;
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},
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quartic: function(u) {
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if (u <= 1 && u >= -1) {
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var tmp = 1 - u * u;
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return (15 / 16) * tmp * tmp;
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}
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return 0;
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},
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triweight: function(u) {
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if (u <= 1 && u >= -1) {
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var tmp = 1 - u * u;
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return (35 / 32) * tmp * tmp * tmp;
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}
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return 0;
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},
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gaussian: function(u) {
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return 1 / Math.sqrt(2 * Math.PI) * Math.exp(-.5 * u * u);
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},
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cosine: function(u) {
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if (u <= 1 && u >= -1) return Math.PI / 4 * Math.cos(Math.PI / 2 * u);
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return 0;
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}
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};
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// http://exploringdata.net/den_trac.htm
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science.stats.kde = function() {
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var kernel = science.stats.kernel.gaussian,
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sample = [],
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bandwidth = science.stats.bandwidth.nrd;
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function kde(points, i) {
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var bw = bandwidth.call(this, sample);
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return points.map(function(x) {
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var i = -1,
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y = 0,
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n = sample.length;
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while (++i < n) {
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y += kernel((x - sample[i]) / bw);
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}
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return [x, y / bw / n];
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});
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}
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kde.kernel = function(x) {
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if (!arguments.length) return kernel;
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kernel = x;
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return kde;
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};
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kde.sample = function(x) {
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if (!arguments.length) return sample;
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sample = x;
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return kde;
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};
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kde.bandwidth = function(x) {
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if (!arguments.length) return bandwidth;
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bandwidth = science.functor(x);
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return kde;
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};
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return kde;
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};
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// Based on figue implementation by Jean-Yves Delort.
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// http://code.google.com/p/figue/
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science.stats.kmeans = function() {
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var distance = science.stats.distance.euclidean,
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maxIterations = 1000,
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k = 1;
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function kmeans(vectors) {
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var n = vectors.length,
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assignments = [],
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clusterSizes = [],
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repeat = 1,
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iterations = 0,
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centroids = science_stats_kmeansRandom(k, vectors),
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newCentroids,
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i,
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j,
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x,
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d,
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min,
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best;
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while (repeat && iterations < maxIterations) {
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// Assignment step.
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j = -1; while (++j < k) {
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clusterSizes[j] = 0;
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}
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i = -1; while (++i < n) {
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x = vectors[i];
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min = Infinity;
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j = -1; while (++j < k) {
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d = distance.call(this, centroids[j], x);
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if (d < min) {
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min = d;
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best = j;
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}
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}
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clusterSizes[assignments[i] = best]++;
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}
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// Update centroids step.
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newCentroids = [];
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i = -1; while (++i < n) {
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x = assignments[i];
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d = newCentroids[x];
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if (d == null) newCentroids[x] = vectors[i].slice();
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else {
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j = -1; while (++j < d.length) {
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d[j] += vectors[i][j];
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}
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}
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}
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j = -1; while (++j < k) {
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x = newCentroids[j];
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d = 1 / clusterSizes[j];
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i = -1; while (++i < x.length) x[i] *= d;
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}
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// Check convergence.
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repeat = 0;
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j = -1; while (++j < k) {
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if (!science_stats_kmeansCompare(newCentroids[j], centroids[j])) {
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repeat = 1;
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break;
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}
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}
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centroids = newCentroids;
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iterations++;
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}
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return {assignments: assignments, centroids: centroids};
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}
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kmeans.k = function(x) {
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if (!arguments.length) return k;
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k = x;
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return kmeans;
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};
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kmeans.distance = function(x) {
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if (!arguments.length) return distance;
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distance = x;
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return kmeans;
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};
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return kmeans;
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};
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function science_stats_kmeansCompare(a, b) {
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if (!a || !b || a.length !== b.length) return false;
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var n = a.length,
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i = -1;
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while (++i < n) if (a[i] !== b[i]) return false;
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return true;
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}
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// Returns an array of k distinct vectors randomly selected from the input
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// array of vectors. Returns null if k > n or if there are less than k distinct
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// objects in vectors.
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function science_stats_kmeansRandom(k, vectors) {
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var n = vectors.length;
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if (k > n) return null;
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var selected_vectors = [];
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var selected_indices = [];
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var tested_indices = {};
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var tested = 0;
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var selected = 0;
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var i,
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vector,
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select;
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while (selected < k) {
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if (tested === n) return null;
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var random_index = Math.floor(Math.random() * n);
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if (random_index in tested_indices) continue;
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tested_indices[random_index] = 1;
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tested++;
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vector = vectors[random_index];
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select = true;
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for (i = 0; i < selected; i++) {
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if (science_stats_kmeansCompare(vector, selected_vectors[i])) {
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select = false;
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break;
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}
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}
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if (select) {
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selected_vectors[selected] = vector;
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selected_indices[selected] = random_index;
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selected++;
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}
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}
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return selected_vectors;
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}
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science.stats.hcluster = function() {
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var distance = science.stats.distance.euclidean,
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linkage = "simple"; // simple, complete or average
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function hcluster(vectors) {
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var n = vectors.length,
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dMin = [],
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cSize = [],
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distMatrix = [],
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clusters = [],
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c1,
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c2,
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c1Cluster,
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c2Cluster,
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p,
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root,
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i,
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j;
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// Initialise distance matrix and vector of closest clusters.
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i = -1; while (++i < n) {
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dMin[i] = 0;
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distMatrix[i] = [];
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j = -1; while (++j < n) {
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distMatrix[i][j] = i === j ? Infinity : distance(vectors[i] , vectors[j]);
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if (distMatrix[i][dMin[i]] > distMatrix[i][j]) dMin[i] = j;
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}
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}
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// create leaves of the tree
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i = -1; while (++i < n) {
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clusters[i] = [];
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clusters[i][0] = {
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left: null,
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right: null,
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dist: 0,
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centroid: vectors[i],
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size: 1,
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depth: 0
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};
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cSize[i] = 1;
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}
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// Main loop
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for (p = 0; p < n-1; p++) {
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// find the closest pair of clusters
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c1 = 0;
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for (i = 0; i < n; i++) {
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if (distMatrix[i][dMin[i]] < distMatrix[c1][dMin[c1]]) c1 = i;
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}
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c2 = dMin[c1];
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// create node to store cluster info
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c1Cluster = clusters[c1][0];
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c2Cluster = clusters[c2][0];
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newCluster = {
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left: c1Cluster,
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right: c2Cluster,
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dist: distMatrix[c1][c2],
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centroid: calculateCentroid(c1Cluster.size, c1Cluster.centroid,
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c2Cluster.size, c2Cluster.centroid),
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size: c1Cluster.size + c2Cluster.size,
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depth: 1 + Math.max(c1Cluster.depth, c2Cluster.depth)
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};
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clusters[c1].splice(0, 0, newCluster);
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cSize[c1] += cSize[c2];
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// overwrite row c1 with respect to the linkage type
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for (j = 0; j < n; j++) {
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switch (linkage) {
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case "single":
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if (distMatrix[c1][j] > distMatrix[c2][j])
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distMatrix[j][c1] = distMatrix[c1][j] = distMatrix[c2][j];
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break;
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case "complete":
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if (distMatrix[c1][j] < distMatrix[c2][j])
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distMatrix[j][c1] = distMatrix[c1][j] = distMatrix[c2][j];
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break;
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case "average":
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distMatrix[j][c1] = distMatrix[c1][j] = (cSize[c1] * distMatrix[c1][j] + cSize[c2] * distMatrix[c2][j]) / (cSize[c1] + cSize[j]);
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break;
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}
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}
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distMatrix[c1][c1] = Infinity;
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// infinity out old row c2 and column c2
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for (i = 0; i < n; i++)
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distMatrix[i][c2] = distMatrix[c2][i] = Infinity;
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// update dmin and replace ones that previous pointed to c2 to point to c1
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for (j = 0; j < n; j++) {
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if (dMin[j] == c2) dMin[j] = c1;
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if (distMatrix[c1][j] < distMatrix[c1][dMin[c1]]) dMin[c1] = j;
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}
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// keep track of the last added cluster
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root = newCluster;
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}
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return root;
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}
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hcluster.distance = function(x) {
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if (!arguments.length) return distance;
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distance = x;
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return hcluster;
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};
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return hcluster;
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};
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function calculateCentroid(c1Size, c1Centroid, c2Size, c2Centroid) {
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var newCentroid = [],
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newSize = c1Size + c2Size,
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n = c1Centroid.length,
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i = -1;
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while (++i < n) {
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newCentroid[i] = (c1Size * c1Centroid[i] + c2Size * c2Centroid[i]) / newSize;
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}
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return newCentroid;
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}
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science.stats.iqr = function(x) {
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var quartiles = science.stats.quantiles(x, [.25, .75]);
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return quartiles[1] - quartiles[0];
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};
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// Based on org.apache.commons.math.analysis.interpolation.LoessInterpolator
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// from http://commons.apache.org/math/
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science.stats.loess = function() {
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var bandwidth = .3,
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robustnessIters = 2,
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accuracy = 1e-12;
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function smooth(xval, yval, weights) {
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var n = xval.length,
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i;
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if (n !== yval.length) throw {error: "Mismatched array lengths"};
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if (n == 0) throw {error: "At least one point required."};
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if (arguments.length < 3) {
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weights = [];
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i = -1; while (++i < n) weights[i] = 1;
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}
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science_stats_loessFiniteReal(xval);
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science_stats_loessFiniteReal(yval);
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science_stats_loessFiniteReal(weights);
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science_stats_loessStrictlyIncreasing(xval);
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if (n == 1) return [yval[0]];
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if (n == 2) return [yval[0], yval[1]];
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var bandwidthInPoints = Math.floor(bandwidth * n);
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if (bandwidthInPoints < 2) throw {error: "Bandwidth too small."};
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var res = [],
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residuals = [],
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robustnessWeights = [];
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// Do an initial fit and 'robustnessIters' robustness iterations.
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// This is equivalent to doing 'robustnessIters+1' robustness iterations
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// starting with all robustness weights set to 1.
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i = -1; while (++i < n) {
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res[i] = 0;
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residuals[i] = 0;
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robustnessWeights[i] = 1;
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}
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var iter = -1;
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while (++iter <= robustnessIters) {
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var bandwidthInterval = [0, bandwidthInPoints - 1];
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// At each x, compute a local weighted linear regression
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var x;
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i = -1; while (++i < n) {
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x = xval[i];
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// Find out the interval of source points on which
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// a regression is to be made.
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if (i > 0) {
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science_stats_loessUpdateBandwidthInterval(xval, weights, i, bandwidthInterval);
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}
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var ileft = bandwidthInterval[0],
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iright = bandwidthInterval[1];
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// Compute the point of the bandwidth interval that is
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// farthest from x
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var edge = (xval[i] - xval[ileft]) > (xval[iright] - xval[i]) ? ileft : iright;
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// Compute a least-squares linear fit weighted by
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// the product of robustness weights and the tricube
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// weight function.
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// See http://en.wikipedia.org/wiki/Linear_regression
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// (section "Univariate linear case")
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// and http://en.wikipedia.org/wiki/Weighted_least_squares
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// (section "Weighted least squares")
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var sumWeights = 0,
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sumX = 0,
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sumXSquared = 0,
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sumY = 0,
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sumXY = 0,
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denom = Math.abs(1 / (xval[edge] - x));
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|
||
for (var k = ileft; k <= iright; ++k) {
|
||
var xk = xval[k],
|
||
yk = yval[k],
|
||
dist = k < i ? x - xk : xk - x,
|
||
w = science_stats_loessTricube(dist * denom) * robustnessWeights[k] * weights[k],
|
||
xkw = xk * w;
|
||
sumWeights += w;
|
||
sumX += xkw;
|
||
sumXSquared += xk * xkw;
|
||
sumY += yk * w;
|
||
sumXY += yk * xkw;
|
||
}
|
||
|
||
var meanX = sumX / sumWeights,
|
||
meanY = sumY / sumWeights,
|
||
meanXY = sumXY / sumWeights,
|
||
meanXSquared = sumXSquared / sumWeights;
|
||
|
||
var beta = (Math.sqrt(Math.abs(meanXSquared - meanX * meanX)) < accuracy)
|
||
? 0 : ((meanXY - meanX * meanY) / (meanXSquared - meanX * meanX));
|
||
|
||
var alpha = meanY - beta * meanX;
|
||
|
||
res[i] = beta * x + alpha;
|
||
residuals[i] = Math.abs(yval[i] - res[i]);
|
||
}
|
||
|
||
// No need to recompute the robustness weights at the last
|
||
// iteration, they won't be needed anymore
|
||
if (iter === robustnessIters) {
|
||
break;
|
||
}
|
||
|
||
// Recompute the robustness weights.
|
||
|
||
// Find the median residual.
|
||
var sortedResiduals = residuals.slice();
|
||
sortedResiduals.sort();
|
||
var medianResidual = sortedResiduals[Math.floor(n / 2)];
|
||
|
||
if (Math.abs(medianResidual) < accuracy)
|
||
break;
|
||
|
||
var arg,
|
||
w;
|
||
i = -1; while (++i < n) {
|
||
arg = residuals[i] / (6 * medianResidual);
|
||
robustnessWeights[i] = (arg >= 1) ? 0 : ((w = 1 - arg * arg) * w);
|
||
}
|
||
}
|
||
|
||
return res;
|
||
}
|
||
|
||
smooth.bandwidth = function(x) {
|
||
if (!arguments.length) return x;
|
||
bandwidth = x;
|
||
return smooth;
|
||
};
|
||
|
||
smooth.robustnessIterations = function(x) {
|
||
if (!arguments.length) return x;
|
||
robustnessIters = x;
|
||
return smooth;
|
||
};
|
||
|
||
smooth.accuracy = function(x) {
|
||
if (!arguments.length) return x;
|
||
accuracy = x;
|
||
return smooth;
|
||
};
|
||
|
||
return smooth;
|
||
};
|
||
|
||
function science_stats_loessFiniteReal(values) {
|
||
var n = values.length,
|
||
i = -1;
|
||
|
||
while (++i < n) if (!isFinite(values[i])) return false;
|
||
|
||
return true;
|
||
}
|
||
|
||
function science_stats_loessStrictlyIncreasing(xval) {
|
||
var n = xval.length,
|
||
i = 0;
|
||
|
||
while (++i < n) if (xval[i - 1] >= xval[i]) return false;
|
||
|
||
return true;
|
||
}
|
||
|
||
// Compute the tricube weight function.
|
||
// http://en.wikipedia.org/wiki/Local_regression#Weight_function
|
||
function science_stats_loessTricube(x) {
|
||
return (x = 1 - x * x * x) * x * x;
|
||
}
|
||
|
||
// Given an index interval into xval that embraces a certain number of
|
||
// points closest to xval[i-1], update the interval so that it embraces
|
||
// the same number of points closest to xval[i], ignoring zero weights.
|
||
function science_stats_loessUpdateBandwidthInterval(
|
||
xval, weights, i, bandwidthInterval) {
|
||
|
||
var left = bandwidthInterval[0],
|
||
right = bandwidthInterval[1];
|
||
|
||
// The right edge should be adjusted if the next point to the right
|
||
// is closer to xval[i] than the leftmost point of the current interval
|
||
var nextRight = science_stats_loessNextNonzero(weights, right);
|
||
if ((nextRight < xval.length) && (xval[nextRight] - xval[i]) < (xval[i] - xval[left])) {
|
||
var nextLeft = science_stats_loessNextNonzero(weights, left);
|
||
bandwidthInterval[0] = nextLeft;
|
||
bandwidthInterval[1] = nextRight;
|
||
}
|
||
}
|
||
|
||
function science_stats_loessNextNonzero(weights, i) {
|
||
var j = i + 1;
|
||
while (j < weights.length && weights[j] === 0) j++;
|
||
return j;
|
||
}
|
||
// Welford's algorithm.
|
||
science.stats.mean = function(x) {
|
||
var n = x.length;
|
||
if (n === 0) return NaN;
|
||
var m = 0,
|
||
i = -1;
|
||
while (++i < n) m += (x[i] - m) / (i + 1);
|
||
return m;
|
||
};
|
||
science.stats.median = function(x) {
|
||
return science.stats.quantiles(x, [.5])[0];
|
||
};
|
||
science.stats.mode = function(x) {
|
||
x = x.slice().sort(science.ascending);
|
||
var mode,
|
||
n = x.length,
|
||
i = -1,
|
||
l = i,
|
||
last = null,
|
||
max = 0,
|
||
tmp,
|
||
v;
|
||
while (++i < n) {
|
||
if ((v = x[i]) !== last) {
|
||
if ((tmp = i - l) > max) {
|
||
max = tmp;
|
||
mode = last;
|
||
}
|
||
last = v;
|
||
l = i;
|
||
}
|
||
}
|
||
return mode;
|
||
};
|
||
// Uses R's quantile algorithm type=7.
|
||
science.stats.quantiles = function(d, quantiles) {
|
||
d = d.slice().sort(science.ascending);
|
||
var n_1 = d.length - 1;
|
||
return quantiles.map(function(q) {
|
||
if (q === 0) return d[0];
|
||
else if (q === 1) return d[n_1];
|
||
|
||
var index = 1 + q * n_1,
|
||
lo = Math.floor(index),
|
||
h = index - lo,
|
||
a = d[lo - 1];
|
||
|
||
return h === 0 ? a : a + h * (d[lo] - a);
|
||
});
|
||
};
|
||
// Unbiased estimate of a sample's variance.
|
||
// Also known as the sample variance, where the denominator is n - 1.
|
||
science.stats.variance = function(x) {
|
||
var n = x.length;
|
||
if (n < 1) return NaN;
|
||
if (n === 1) return 0;
|
||
var mean = science.stats.mean(x),
|
||
i = -1,
|
||
s = 0;
|
||
while (++i < n) {
|
||
var v = x[i] - mean;
|
||
s += v * v;
|
||
}
|
||
return s / (n - 1);
|
||
};
|
||
})() |