1 /* 2 * Copyright (C) 2012, 2013 Apple Inc. All rights reserved. 3 * 4 * Redistribution and use in source and binary forms, with or without 5 * modification, are permitted provided that the following conditions 6 * are met: 7 * 1. Redistributions of source code must retain the above copyright 8 * notice, this list of conditions and the following disclaimer. 9 * 2. Redistributions in binary form must reproduce the above copyright 10 * notice, this list of conditions and the following disclaimer in the 11 * documentation and/or other materials provided with the distribution. 12 * 13 * THIS SOFTWARE IS PROVIDED BY APPLE INC. AND ITS CONTRIBUTORS ``AS IS'' 14 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, 15 * THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 16 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL APPLE INC. OR ITS CONTRIBUTORS 17 * BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 18 * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 19 * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 20 * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 21 * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 22 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF 23 * THE POSSIBILITY OF SUCH DAMAGE. 24 */ 25 26 var Statistics = new (function () { 27 28 this.max = function (values) { 29 var maxVal = values[0]; 30 for (var i = 1; i < values.length; i++) { 31 maxVal = Math.max(maxVal, values[i]); 32 } 33 return maxVal; 34 } 35 36 this.min = function (values) { 37 var minVal = values[0]; 38 for (var i = 1; i < values.length; i++) { 39 minVal = Math.min(minVal, values[i]); 40 } 41 return minVal; 42 } 43 44 this.sum = function (values) { 45 return values.reduce(function (a, b) { return a + b; }, 0); 46 } 47 48 this.squareSum = function (values) { 49 return values.reduce(function (sum, value) { return sum + value * value;}, 0); 50 } 51 52 // With sum and sum of squares, we can compute the sample standard deviation in O(1). 53 // See https://rniwa.com/2012-11-10/sample-standard-deviation-in-terms-of-sum-and-square-sum-of-samples/ 54 this.sampleStandardDeviation = function (numberOfSamples, sum, squareSum) { 55 if (numberOfSamples < 2) 56 return 0; 57 return Math.sqrt(squareSum / (numberOfSamples - 1) 58 - sum * sum / (numberOfSamples - 1) / numberOfSamples); 59 } 60 61 this.supportedConfidenceLevels = function () { 62 var supportedLevels = []; 63 for (var quantile in tDistributionInverseCDF) 64 supportedLevels.push((1 - (1 - quantile) * 2).toFixed(2)); 65 return supportedLevels; 66 } 67 68 this.quantile = function (confidenceLevel, numberOfSamples, opt_degreesOfFreedom) { 69 var probability = (1 - (1 - confidenceLevel) / 2); 70 if (!(probability in tDistributionInverseCDF)) { 71 console.warn('We only support ' + this.supportedConfidenceLevels().map( 72 function (level) { return level * 100 + '%'; } ).join(', ') + ' confidence intervals.'); 73 return NaN; 74 } 75 if (numberOfSamples < 2) 76 return Number.POSITIVE_INFINITY; 77 78 var cdfForProbability = tDistributionInverseCDF[probability]; 79 var degreesOfFreedom = opt_degreesOfFreedom; 80 if (degreesOfFreedom === undefined) 81 degreesOfFreedom = numberOfSamples - 1; 82 83 // tDistributionQuantile(degreesOfFreedom, confidenceLevel) * sampleStandardDeviation / sqrt(numberOfSamples) * S/sqrt(numberOfSamples) 84 if (degreesOfFreedom <= 100) 85 return cdfForProbability[degreesOfFreedom - 1]; // The first entry is for the one degree of freedom. 86 else if (degreesOfFreedom <= 300) 87 return cdfForProbability[Math.round(degreesOfFreedom / 10) + 100 - 10 - 1]; 88 else if (degreesOfFreedom <= 1300) 89 return cdfForProbability[Math.round(degreesOfFreedom / 100) + 120 - 3 - 1]; 90 else 91 return cdfForProbability[cdfForProbability.length - 1]; 92 } 93 94 // Computes the delta d s.t. (mean - d, mean + d) is the confidence interval with the specified confidence level in O(1). 95 this.confidenceIntervalDelta = function (confidenceLevel, numberOfSamples, sum, squareSum) { 96 var sampleStandardDeviation = this.sampleStandardDeviation(numberOfSamples, sum, squareSum); 97 return this.confidenceIntervalDeltaFromStd(confidenceLevel, numberOfSamples, sampleStandardDeviation); 98 } 99 100 this.confidenceIntervalDeltaFromStd = function (confidenceLevel, numberOfSamples, sampleStandardDeviation, opt_degreesOfFreedom) { 101 var quantile = this.quantile(confidenceLevel, numberOfSamples, opt_degreesOfFreedom); 102 return quantile * sampleStandardDeviation / Math.sqrt(numberOfSamples); 103 } 104 105 106 this.confidenceInterval = function (values, probability) { 107 var sum = this.sum(values); 108 var mean = sum / values.length; 109 var delta = this.confidenceIntervalDelta(probability || 0.95, values.length, sum, this.squareSum(values)); 110 return [mean - delta, mean + delta]; 111 } 112 113 // See http://en.wikipedia.org/wiki/Student's_t-distribution#Table_of_selected_values 114 // This table contains one sided (a.k.a. tail) values. 115 // Use TINV((1 - probability) * 2, df) in your favorite spreadsheet software to compute these. 116 // The spacing of the values with df greater than 100 maintains error less than 0.8%. 117 var tDistributionInverseCDF = { 118 0.9: [ 119 // 1 - 100 step 1 120 3.077684, 1.885618, 1.637744, 1.533206, 1.475884, 1.439756, 1.414924, 1.396815, 1.383029, 1.372184, 121 1.363430, 1.356217, 1.350171, 1.345030, 1.340606, 1.336757, 1.333379, 1.330391, 1.327728, 1.325341, 122 1.323188, 1.321237, 1.319460, 1.317836, 1.316345, 1.314972, 1.313703, 1.312527, 1.311434, 1.310415, 123 1.309464, 1.308573, 1.307737, 1.306952, 1.306212, 1.305514, 1.304854, 1.304230, 1.303639, 1.303077, 124 1.302543, 1.302035, 1.301552, 1.301090, 1.300649, 1.300228, 1.299825, 1.299439, 1.299069, 1.298714, 125 1.298373, 1.298045, 1.297730, 1.297426, 1.297134, 1.296853, 1.296581, 1.296319, 1.296066, 1.295821, 126 1.295585, 1.295356, 1.295134, 1.294920, 1.294712, 1.294511, 1.294315, 1.294126, 1.293942, 1.293763, 127 1.293589, 1.293421, 1.293256, 1.293097, 1.292941, 1.292790, 1.292643, 1.292500, 1.292360, 1.292224, 128 1.292091, 1.291961, 1.291835, 1.291711, 1.291591, 1.291473, 1.291358, 1.291246, 1.291136, 1.291029, 129 1.290924, 1.290821, 1.290721, 1.290623, 1.290527, 1.290432, 1.290340, 1.290250, 1.290161, 1.290075, 130 // 110 - 300 step 10 131 1.289295, 1.288646, 1.288098, 1.287628, 1.287221, 1.286865, 1.286551, 1.286272, 1.286023, 1.285799, 132 1.285596, 1.285411, 1.285243, 1.285089, 1.284947, 1.284816, 1.284695, 1.284582, 1.284478, 1.284380, 133 // 400 - 1300 step 100 134 1.283672, 1.283247, 1.282964, 1.282762, 1.282611, 1.282493, 1.282399, 1.282322, 1.282257, 1.282203, 135 // Infinity 136 1.281548], 137 0.95: [ 138 // 1 - 100 step 1 139 6.313752, 2.919986, 2.353363, 2.131847, 2.015048, 1.943180, 1.894579, 1.859548, 1.833113, 1.812461, 140 1.795885, 1.782288, 1.770933, 1.761310, 1.753050, 1.745884, 1.739607, 1.734064, 1.729133, 1.724718, 141 1.720743, 1.717144, 1.713872, 1.710882, 1.708141, 1.705618, 1.703288, 1.701131, 1.699127, 1.697261, 142 1.695519, 1.693889, 1.692360, 1.690924, 1.689572, 1.688298, 1.687094, 1.685954, 1.684875, 1.683851, 143 1.682878, 1.681952, 1.681071, 1.680230, 1.679427, 1.678660, 1.677927, 1.677224, 1.676551, 1.675905, 144 1.675285, 1.674689, 1.674116, 1.673565, 1.673034, 1.672522, 1.672029, 1.671553, 1.671093, 1.670649, 145 1.670219, 1.669804, 1.669402, 1.669013, 1.668636, 1.668271, 1.667916, 1.667572, 1.667239, 1.666914, 146 1.666600, 1.666294, 1.665996, 1.665707, 1.665425, 1.665151, 1.664885, 1.664625, 1.664371, 1.664125, 147 1.663884, 1.663649, 1.663420, 1.663197, 1.662978, 1.662765, 1.662557, 1.662354, 1.662155, 1.661961, 148 1.661771, 1.661585, 1.661404, 1.661226, 1.661052, 1.660881, 1.660715, 1.660551, 1.660391, 1.660234, 149 // 110 - 300 step 10 150 1.658824, 1.657651, 1.656659, 1.655811, 1.655076, 1.654433, 1.653866, 1.653363, 1.652913, 1.652508, 151 1.652142, 1.651809, 1.651506, 1.651227, 1.650971, 1.650735, 1.650517, 1.650314, 1.650125, 1.649949, 152 // 400 - 1300 step 100 153 1.648672, 1.647907, 1.647397, 1.647033, 1.646761, 1.646548, 1.646379, 1.646240, 1.646124, 1.646027, 154 // Infinity 155 1.644847], 156 0.975: [ 157 // 1 - 100 step 1 158 12.706205, 4.302653, 3.182446, 2.776445, 2.570582, 2.446912, 2.364624, 2.306004, 2.262157, 2.228139, 159 2.200985, 2.178813, 2.160369, 2.144787, 2.131450, 2.119905, 2.109816, 2.100922, 2.093024, 2.085963, 160 2.079614, 2.073873, 2.068658, 2.063899, 2.059539, 2.055529, 2.051831, 2.048407, 2.045230, 2.042272, 161 2.039513, 2.036933, 2.034515, 2.032245, 2.030108, 2.028094, 2.026192, 2.024394, 2.022691, 2.021075, 162 2.019541, 2.018082, 2.016692, 2.015368, 2.014103, 2.012896, 2.011741, 2.010635, 2.009575, 2.008559, 163 2.007584, 2.006647, 2.005746, 2.004879, 2.004045, 2.003241, 2.002465, 2.001717, 2.000995, 2.000298, 164 1.999624, 1.998972, 1.998341, 1.997730, 1.997138, 1.996564, 1.996008, 1.995469, 1.994945, 1.994437, 165 1.993943, 1.993464, 1.992997, 1.992543, 1.992102, 1.991673, 1.991254, 1.990847, 1.990450, 1.990063, 166 1.989686, 1.989319, 1.988960, 1.988610, 1.988268, 1.987934, 1.987608, 1.987290, 1.986979, 1.986675, 167 1.986377, 1.986086, 1.985802, 1.985523, 1.985251, 1.984984, 1.984723, 1.984467, 1.984217, 1.983972, 168 // 110 - 300 step 10 169 1.981765, 1.979930, 1.978380, 1.977054, 1.975905, 1.974902, 1.974017, 1.973231, 1.972528, 1.971896, 170 1.971325, 1.970806, 1.970332, 1.969898, 1.969498, 1.969130, 1.968789, 1.968472, 1.968178, 1.967903, 171 // 400 - 1300 step 100 172 1.965912, 1.964720, 1.963926, 1.963359, 1.962934, 1.962603, 1.962339, 1.962123, 1.961943, 1.961790, 173 // Infinity 174 1.959964], 175 0.99: [ 176 // 1 - 100 step 1 177 31.820516, 6.964557, 4.540703, 3.746947, 3.364930, 3.142668, 2.997952, 2.896459, 2.821438, 2.763769, 178 2.718079, 2.680998, 2.650309, 2.624494, 2.602480, 2.583487, 2.566934, 2.552380, 2.539483, 2.527977, 179 2.517648, 2.508325, 2.499867, 2.492159, 2.485107, 2.478630, 2.472660, 2.467140, 2.462021, 2.457262, 180 2.452824, 2.448678, 2.444794, 2.441150, 2.437723, 2.434494, 2.431447, 2.428568, 2.425841, 2.423257, 181 2.420803, 2.418470, 2.416250, 2.414134, 2.412116, 2.410188, 2.408345, 2.406581, 2.404892, 2.403272, 182 2.401718, 2.400225, 2.398790, 2.397410, 2.396081, 2.394801, 2.393568, 2.392377, 2.391229, 2.390119, 183 2.389047, 2.388011, 2.387008, 2.386037, 2.385097, 2.384186, 2.383302, 2.382446, 2.381615, 2.380807, 184 2.380024, 2.379262, 2.378522, 2.377802, 2.377102, 2.376420, 2.375757, 2.375111, 2.374482, 2.373868, 185 2.373270, 2.372687, 2.372119, 2.371564, 2.371022, 2.370493, 2.369977, 2.369472, 2.368979, 2.368497, 186 2.368026, 2.367566, 2.367115, 2.366674, 2.366243, 2.365821, 2.365407, 2.365002, 2.364606, 2.364217, 187 // 110 - 300 step 10 188 2.360726, 2.357825, 2.355375, 2.353278, 2.351465, 2.349880, 2.348483, 2.347243, 2.346134, 2.345137, 189 2.344236, 2.343417, 2.342670, 2.341985, 2.341356, 2.340775, 2.340238, 2.339739, 2.339275, 2.338842, 190 // 400 - 1300 step 100 191 2.335706, 2.333829, 2.332579, 2.331687, 2.331018, 2.330498, 2.330083, 2.329743, 2.329459, 2.329220, 192 // Infinity 193 2.326348], 194 }; 195 196 })(); 197 198 if (typeof module != 'undefined') { 199 for (var key in Statistics) 200 module.exports[key] = Statistics[key]; 201 } 202
