Pseudo = { initialRandomSeed: 49734321, randomSeed: 49734321, resetRandomSeed: function() { Pseudo.randomSeed = Pseudo.initialRandomSeed; }, random: function() { var randomSeed = Pseudo.randomSeed; randomSeed = ((randomSeed + 0x7ed55d16) + (randomSeed << 12)) & 0xffffffff; randomSeed = ((randomSeed ^ 0xc761c23c) ^ (randomSeed >>> 19)) & 0xffffffff; randomSeed = ((randomSeed + 0x165667b1) + (randomSeed << 5)) & 0xffffffff; randomSeed = ((randomSeed + 0xd3a2646c) ^ (randomSeed << 9)) & 0xffffffff; randomSeed = ((randomSeed + 0xfd7046c5) + (randomSeed << 3)) & 0xffffffff; randomSeed = ((randomSeed ^ 0xb55a4f09) ^ (randomSeed >>> 16)) & 0xffffffff; Pseudo.randomSeed = randomSeed; return (randomSeed & 0xfffffff) / 0x10000000; } }; Statistics = { sampleMean: function(numberOfSamples, sum) { if (numberOfSamples < 1) return 0; return sum / numberOfSamples; }, // With sum and sum of squares, we can compute the sample standard deviation in O(1). // See https://rniwa.com/2012-11-10/sample-standard-deviation-in-terms-of-sum-and-square-sum-of-samples/ unbiasedSampleStandardDeviation: function(numberOfSamples, sum, squareSum) { if (numberOfSamples < 2) return 0; return Math.sqrt((squareSum - sum * sum / numberOfSamples) / (numberOfSamples - 1)); }, geometricMean: function(values) { if (!values.length) return 0; var roots = values.map(function(value) { return Math.pow(value, 1 / values.length); }) return roots.reduce(function(a, b) { return a * b; }); }, // Cumulative distribution function cdf: function(value, mean, standardDeviation) { return 0.5 * (1 + Statistics.erf((value - mean) / (Math.sqrt(2 * standardDeviation * standardDeviation)))); }, // Approximation of Gauss error function, Abramowitz and Stegun 7.1.26 erf: function(value) { var sign = (value >= 0) ? 1 : -1; value = Math.abs(value); var a1 = 0.254829592; var a2 = -0.284496736; var a3 = 1.421413741; var a4 = -1.453152027; var a5 = 1.061405429; var p = 0.3275911; var t = 1.0 / (1.0 + p * value); var y = 1.0 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-value * value); return sign * y; }, largestDeviationPercentage: function(low, mean, high) { return Math.max(Math.abs(low / mean - 1), (high / mean - 1)); } }; Experiment = Utilities.createClass( function(includeConcern) { if (includeConcern) this._maxHeap = Heap.createMaxHeap(Experiment.defaults.CONCERN_SIZE); this.reset(); }, { reset: function() { this._sum = 0; this._squareSum = 0; this._numberOfSamples = 0; if (this._maxHeap) this._maxHeap.init(); }, get sampleCount() { return this._numberOfSamples; }, sample: function(value) { this._sum += value; this._squareSum += value * value; if (this._maxHeap) this._maxHeap.push(value); ++this._numberOfSamples; }, mean: function() { return Statistics.sampleMean(this._numberOfSamples, this._sum); }, standardDeviation: function() { return Statistics.unbiasedSampleStandardDeviation(this._numberOfSamples, this._sum, this._squareSum); }, cdf: function(value) { return Statistics.cdf(value, this.mean(), this.standardDeviation()); }, percentage: function() { var mean = this.mean(); return mean ? this.standardDeviation() * 100 / mean : 0; }, concern: function(percentage) { if (!this._maxHeap) return this.mean(); var size = Math.ceil(this._numberOfSamples * percentage / 100); var values = this._maxHeap.values(size); return values.length ? values.reduce(function(a, b) { return a + b; }) / values.length : 0; }, score: function(percentage) { return Statistics.geometricMean([this.mean(), Math.max(this.concern(percentage), 1)]); } }); Experiment.defaults = { CONCERN: 5, CONCERN_SIZE: 100, }; Regression = Utilities.createClass( function(samples, getComplexity, getFrameLength, startIndex, endIndex, options) { var desiredFrameLength = options.desiredFrameLength || 1000/60; var bestProfile; if (!options.preferredProfile || options.preferredProfile == Strings.json.profiles.slope) { var slope = this._calculateRegression(samples, getComplexity, getFrameLength, startIndex, endIndex, { shouldClip: true, s1: desiredFrameLength, t1: 0 }); if (!bestProfile || slope.error < bestProfile.error) { bestProfile = slope; this.profile = Strings.json.profiles.slope; } } if (!options.preferredProfile || options.preferredProfile == Strings.json.profiles.flat) { var flat = this._calculateRegression(samples, getComplexity, getFrameLength, startIndex, endIndex, { shouldClip: true, s1: desiredFrameLength, t1: 0, t2: 0 }); if (!bestProfile || flat.error < bestProfile.error) { bestProfile = flat; this.profile = Strings.json.profiles.flat; } } this.startIndex = Math.min(startIndex, endIndex); this.endIndex = Math.max(startIndex, endIndex); this.complexity = bestProfile.complexity; this.s1 = bestProfile.s1; this.t1 = bestProfile.t1; this.s2 = bestProfile.s2; this.t2 = bestProfile.t2; this.stdev1 = bestProfile.stdev1; this.stdev2 = bestProfile.stdev2; this.n1 = bestProfile.n1; this.n2 = bestProfile.n2; this.error = bestProfile.error; }, { valueAt: function(complexity) { if (this.n1 == 1 || complexity > this.complexity) return this.s2 + this.t2 * complexity; return this.s1 + this.t1 * complexity; }, // A generic two-segment piecewise regression calculator. Based on Kundu/Ubhaya // // Minimize sum of (y - y')^2 // where y = s1 + t1*x // y = s2 + t2*x // y' = s1 + t1*x' = s2 + t2*x' if x_0 <= x' <= x_n // // Allows for fixing s1, t1, s2, t2 // // x is assumed to be complexity, y is frame length. Can be used for pure complexity-FPS // analysis or for ramp controllers since complexity monotonically decreases with time. _calculateRegression: function(samples, getComplexity, getFrameLength, startIndex, endIndex, options) { if (startIndex == endIndex) { // Only one sample point; we can't calculate any regression. var x = getComplexity(samples, startIndex); return { complexity: x, s1: x, t1: 0, s2: x, t2: 0, error1: 0, error2: 0 }; } // x is expected to increase in complexity var iterationDirection = endIndex > startIndex ? 1 : -1; var lowComplexity = getComplexity(samples, startIndex); var highComplexity = getComplexity(samples, endIndex); var a1 = 0, b1 = 0, c1 = 0, d1 = 0, h1 = 0, k1 = 0; var a2 = 0, b2 = 0, c2 = 0, d2 = 0, h2 = 0, k2 = 0; // Iterate from low to high complexity for (var i = startIndex; iterationDirection * (endIndex - i) > -1; i += iterationDirection) { var x = getComplexity(samples, i); var y = getFrameLength(samples, i); a2 += 1; b2 += x; c2 += x * x; d2 += y; h2 += y * x; k2 += y * y; } var s1_best, t1_best, s2_best, t2_best, n1_best, n2_best, error1_best, error2_best, x_best, x_prime; function setBest(s1, t1, error1, s2, t2, error2, splitIndex, x_prime, x) { s1_best = s1; t1_best = t1; error1_best = error1; s2_best = s2; t2_best = t2; error2_best = error2; // Number of samples included in the first segment, inclusive of splitIndex n1_best = iterationDirection * (splitIndex - startIndex) + 1; // Number of samples included in the second segment n2_best = iterationDirection * (endIndex - splitIndex); if (!options.shouldClip || (x_prime >= lowComplexity && x_prime <= highComplexity)) x_best = x_prime; else { // Discontinuous piecewise regression x_best = x; } } // Iterate from startIndex to endIndex - 1, inclusive for (var i = startIndex; iterationDirection * (endIndex - i) > 0; i += iterationDirection) { var x = getComplexity(samples, i); var y = getFrameLength(samples, i); var xx = x * x; var yx = y * x; var yy = y * y; // a1, b1, etc. is sum from startIndex to i, inclusive a1 += 1; b1 += x; c1 += xx; d1 += y; h1 += yx; k1 += yy; // a2, b2, etc. is sum from i+1 to endIndex, inclusive a2 -= 1; b2 -= x; c2 -= xx; d2 -= y; h2 -= yx; k2 -= yy; var A = c1*d1 - b1*h1; var B = a1*h1 - b1*d1; var C = a1*c1 - b1*b1; var D = c2*d2 - b2*h2; var E = a2*h2 - b2*d2; var F = a2*c2 - b2*b2; var s1 = options.s1 !== undefined ? options.s1 : (A / C); var t1 = options.t1 !== undefined ? options.t1 : (B / C); var s2 = options.s2 !== undefined ? options.s2 : (D / F); var t2 = options.t2 !== undefined ? options.t2 : (E / F); // Assumes that the two segments meet var x_prime = (s1 - s2) / (t2 - t1); var error1 = (k1 + a1*s1*s1 + c1*t1*t1 - 2*d1*s1 - 2*h1*t1 + 2*b1*s1*t1) || Number.MAX_VALUE; var error2 = (k2 + a2*s2*s2 + c2*t2*t2 - 2*d2*s2 - 2*h2*t2 + 2*b2*s2*t2) || Number.MAX_VALUE; if (i == startIndex) { setBest(s1, t1, error1, s2, t2, error2, i, x_prime, x); continue; } if (C == 0 || F == 0) continue; // Projected point is not between this and the next sample if (x_prime > getComplexity(samples, i + iterationDirection) || x_prime < x) { // Calculate lambda, which divides the weight of this sample between the two lines // These values remove the influence of this sample var I = c1 - 2*b1*x + a1*xx; var H = C - I; var G = A + B*x - C*y; var J = D + E*x - F*y; var K = c2 - 2*b2*x + a2*xx; var lambda = (G*F + G*K - H*J) / (I*J + G*K); if (lambda > 0 && lambda < 1) { var lambda1 = 1 - lambda; s1 = options.s1 !== undefined ? options.s1 : ((A - lambda1*(-h1*x + d1*xx + c1*y - b1*yx)) / (C - lambda1*I)); t1 = options.t1 !== undefined ? options.t1 : ((B - lambda1*(h1 - d1*x - b1*y + a1*yx)) / (C - lambda1*I)); s2 = options.s2 !== undefined ? options.s2 : ((D + lambda1*(-h2*x + d2*xx + c2*y - b2*yx)) / (F + lambda1*K)); t2 = options.t2 !== undefined ? options.t2 : ((E + lambda1*(h2 - d2*x - b2*y + a2*yx)) / (F + lambda1*K)); x_prime = (s1 - s2) / (t2 - t1); error1 = ((k1 + a1*s1*s1 + c1*t1*t1 - 2*d1*s1 - 2*h1*t1 + 2*b1*s1*t1) - lambda1 * Math.pow(y - (s1 + t1*x), 2)) || Number.MAX_VALUE; error2 = ((k2 + a2*s2*s2 + c2*t2*t2 - 2*d2*s2 - 2*h2*t2 + 2*b2*s2*t2) + lambda1 * Math.pow(y - (s2 + t2*x), 2)) || Number.MAX_VALUE; } else if (t1 != t2) continue; } if (error1 + error2 < error1_best + error2_best) setBest(s1, t1, error1, s2, t2, error2, i, x_prime, x); } return { complexity: x_best, s1: s1_best, t1: t1_best, stdev1: Math.sqrt(error1_best / n1_best), s2: s2_best, t2: t2_best, stdev2: Math.sqrt(error2_best / n2_best), error: error1_best + error2_best, n1: n1_best, n2: n2_best }; } }); Utilities.extendObject(Regression, { bootstrap: function(samples, iterationCount, processResample, confidencePercentage) { var sampleLength = samples.length; var resample = new Array(sampleLength); var bootstrapEstimator = new Experiment; var bootstrapData = new Array(iterationCount); Pseudo.resetRandomSeed(); for (var i = 0; i < iterationCount; ++i) { for (var j = 0; j < sampleLength; ++j) resample[j] = samples[Math.floor(Pseudo.random() * sampleLength)]; var resampleResult = processResample(resample); bootstrapEstimator.sample(resampleResult); bootstrapData[i] = resampleResult; } bootstrapData.sort(function(a, b) { return a - b; }); return { confidenceLow: bootstrapData[Math.round((iterationCount - 1) * (1 - confidencePercentage) / 2)], confidenceHigh: bootstrapData[Math.round((iterationCount - 1) * (1 + confidencePercentage) / 2)], median: bootstrapData[Math.round(iterationCount / 2)], mean: bootstrapEstimator.mean(), data: bootstrapData, confidencePercentage: confidencePercentage }; } });