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1 (function(f){if(typeof exports==="object"&&typeof module!=="undefined"){module.exports=f()}else if(typeof define==="function"&&define.amd){define([],f)}else{var g;if(typeof window!=="undefined"){g=window}else if(typeof global!=="undefined"){g=global}else if(typeof self!=="undefined"){g=self}else{g=this}g.ss = f()}})(function(){var define,module,exports;return (function e(t,n,r){function s(o,u){if(!n[o]){if(!t[o]){var a=typeof require=="function"&&require;if(!u&&a)return a(o,!0);if(i)return i(o,!0);var f=new Error("Cannot find module '"+o+"'");throw f.code="MODULE_NOT_FOUND",f}var l=n[o]={exports:{}};t[o][0].call(l.exports,function(e){var n=t[o][1][e];return s(n?n:e)},l,l.exports,e,t,n,r)}return n[o].exports}var i=typeof require=="function"&&require;for(var o=0;o<r.length;o++)s(r[o]);return s})({1:[function(require,module,exports){

2 /* @flow */

3 'use strict';

5 // # simple-statistics

7 // A simple, literate statistics system.

9 var ss = module.exports = {};

11 // Linear Regression

12 ss.linearRegression = require(21);

13 ss.linearRegressionLine = require(22);

14 ss.standardDeviation = require(54);

15 ss.rSquared = require(43);

16 ss.mode = require(32);

17 ss.modeSorted = require(33);

18 ss.min = require(29);

19 ss.max = require(23);

20 ss.minSorted = require(30);

21 ss.maxSorted = require(24);

22 ss.sum = require(56);

23 ss.sumSimple = require(58);

24 ss.product = require(39);

25 ss.quantile = require(40);

26 ss.quantileSorted = require(41);

27 ss.iqr = ss.interquartileRange = require(19);

28 ss.medianAbsoluteDeviation = ss.mad = require(27);

29 ss.chunk = require(8);

30 ss.shuffle = require(51);

31 ss.shuffleInPlace = require(52);

32 ss.sample = require(45);

33 ss.ckmeans = require(9);

34 ss.uniqueCountSorted = require(61);

35 ss.sumNthPowerDeviations = require(57);

36 ss.equalIntervalBreaks = require(14);

38 // sample statistics

39 ss.sampleCovariance = require(47);

40 ss.sampleCorrelation = require(46);

41 ss.sampleVariance = require(50);

42 ss.sampleStandardDeviation = require(49);

43 ss.sampleSkewness = require(48);

45 // combinatorics

46 ss.permutationsHeap = require(36);

47 ss.combinations = require(10);

48 ss.combinationsReplacement = require(11);

50 // measures of centrality

51 ss.geometricMean = require(17);

52 ss.harmonicMean = require(18);

53 ss.mean = ss.average = require(25);

54 ss.median = require(26);

55 ss.medianSorted = require(28);

57 ss.rootMeanSquare = ss.rms = require(44);

58 ss.variance = require(62);

59 ss.tTest = require(59);

60 ss.tTestTwoSample = require(60);

61 // ss.jenks = require('./src/jenks');

63 // Classifiers

64 ss.bayesian = require(2);

65 ss.perceptron = require(35);

67 // Distribution-related methods

68 ss.epsilon = require(13); // We make ε available to the test suite.

69 ss.factorial = require(16);

70 ss.bernoulliDistribution = require(3);

71 ss.binomialDistribution = require(4);

72 ss.poissonDistribution = require(37);

73 ss.chiSquaredGoodnessOfFit = require(7);

75 // Normal distribution

76 ss.zScore = require(63);

77 ss.cumulativeStdNormalProbability = require(12);

78 ss.standardNormalTable = require(55);

79 ss.errorFunction = ss.erf = require(15);

80 ss.inverseErrorFunction = require(20);

81 ss.probit = require(38);

82 ss.mixin = require(31);

84 // Root-finding methods

85 ss.bisect = require(5);

87 },{"10":10,"11":11,"12":12,"13":13,"14":14,"15":15,"16":16,"17":17,"18":18,"19":19,"2":2,"20":20,"21":21,"22":22,"23":23,"24":24,"25":25,"26":26,"27":27,"28":28,"29":29,"3":3,"30":30,"31":31,"32":32,"33":33,"35":35,"36":36,"37":37,"38":38,"39":39,"4":4,"40":40,"41":41,"43":43,"44":44,"45":45,"46":46,"47":47,"48":48,"49":49,"5":5,"50":50,"51":51,"52":52,"54":54,"55":55,"56":56,"57":57,"58":58,"59":59,"60":60,"61":61,"62":62,"63":63,"7":7,"8":8,"9":9}],2:[function(require,module,exports){

88 'use strict';

89 /* @flow */

92 * [Bayesian Classifier](http://en.wikipedia.org/wiki/Naive_Bayes_classifier)

94 * This is a naïve bayesian classifier that takes

95 * singly-nested objects.

97 * @class

98 * @example

99 * var bayes = new BayesianClassifier();

100 * bayes.train({

101 * species: 'Cat'

102 * }, 'animal');

103 * var result = bayes.score({

104 * species: 'Cat'

105 * })

106 * // result

107 * // {

108 * // animal: 1

109 * // }

111 function BayesianClassifier() {

112 // The number of items that are currently

113 // classified in the model

114 this.totalCount = 0;

115 // Every item classified in the model

116 this.data = {};

120 * Train the classifier with a new item, which has a single

121 * dimension of Javascript literal keys and values.

123 * @param {Object} item an object with singly-deep properties

124 * @param {string} category the category this item belongs to

125 * @return {undefined} adds the item to the classifier

127 BayesianClassifier.prototype.train = function(item, category) {

128 // If the data object doesn't have any values

129 // for this category, create a new object for it.

130 if (!this.data[category]) {

131 this.data[category] = {};

132 }

134 // Iterate through each key in the item.

135 for (var k in item) {

136 var v = item[k];

137 // Initialize the nested object `data[category][k][item[k]]`

138 // with an object of keys that equal 0.

139 if (this.data[category][k] === undefined) {

140 this.data[category][k] = {};

141 }

142 if (this.data[category][k][v] === undefined) {

143 this.data[category][k][v] = 0;

144 }

146 // And increment the key for this key/value combination.

147 this.data[category][k][v]++;

148 }

150 // Increment the number of items classified

151 this.totalCount++;

155 * Generate a score of how well this item matches all

156 * possible categories based on its attributes

158 * @param {Object} item an item in the same format as with train

159 * @returns {Object} of probabilities that this item belongs to a

160 * given category.

162 BayesianClassifier.prototype.score = function(item) {

163 // Initialize an empty array of odds per category.

164 var odds = {}, category;

165 // Iterate through each key in the item,

166 // then iterate through each category that has been used

167 // in previous calls to `.train()`

168 for (var k in item) {

169 var v = item[k];

170 for (category in this.data) {

171 // Create an empty object for storing key - value combinations

172 // for this category.

173 odds[category] = {};

175 // If this item doesn't even have a property, it counts for nothing,

176 // but if it does have the property that we're looking for from

177 // the item to categorize, it counts based on how popular it is

178 // versus the whole population.

179 if (this.data[category][k]) {

180 odds[category][k + '_' + v] = (this.data[category][k][v] || 0) / this.totalCount;

181 } else {

182 odds[category][k + '_' + v] = 0;

183 }

184 }

185 }

187 // Set up a new object that will contain sums of these odds by category

188 var oddsSums = {};

190 for (category in odds) {

191 // Tally all of the odds for each category-combination pair -

192 // the non-existence of a category does not add anything to the

193 // score.

194 oddsSums[category] = 0;

195 for (var combination in odds[category]) {

196 oddsSums[category] += odds[category][combination];

197 }

198 }

200 return oddsSums;

203 module.exports = BayesianClassifier;

205 },{}],3:[function(require,module,exports){

206 'use strict';

207 /* @flow */

209 var binomialDistribution = require(4);

212 * The [Bernoulli distribution](http://en.wikipedia.org/wiki/Bernoulli_distribution)

213 * is the probability discrete

214 * distribution of a random variable which takes value 1 with success

215 * probability `p` and value 0 with failure

216 * probability `q` = 1 - `p`. It can be used, for example, to represent the

217 * toss of a coin, where "1" is defined to mean "heads" and "0" is defined

218 * to mean "tails" (or vice versa). It is

219 * a special case of a Binomial Distribution

220 * where `n` = 1.

222 * @param {number} p input value, between 0 and 1 inclusive

223 * @returns {number} value of bernoulli distribution at this point

224 * @example

225 * bernoulliDistribution(0.5); // => { '0': 0.5, '1': 0.5 }

227 function bernoulliDistribution(p/*: number */) {

228 // Check that `p` is a valid probability (0 ≤ p ≤ 1)

229 if (p < 0 || p > 1 ) { return NaN; }

231 return binomialDistribution(1, p);

234 module.exports = bernoulliDistribution;

236 },{"4":4}],4:[function(require,module,exports){

237 'use strict';

238 /* @flow */

240 var epsilon = require(13);

241 var factorial = require(16);

244 * The [Binomial Distribution](http://en.wikipedia.org/wiki/Binomial_distribution) is the discrete probability

245 * distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields

246 * success with probability `probability`. Such a success/failure experiment is also called a Bernoulli experiment or

247 * Bernoulli trial; when trials = 1, the Binomial Distribution is a Bernoulli Distribution.

249 * @param {number} trials number of trials to simulate

250 * @param {number} probability

251 * @returns {Object} output

253 function binomialDistribution(

254 trials/*: number */,

255 probability/*: number */)/*: ?Object */ {

256 // Check that `p` is a valid probability (0 ≤ p ≤ 1),

257 // that `n` is an integer, strictly positive.

258 if (probability < 0 || probability > 1 ||

259 trials <= 0 || trials % 1 !== 0) {

260 return undefined;

261 }

263 // We initialize `x`, the random variable, and `accumulator`, an accumulator

264 // for the cumulative distribution function to 0. `distribution_functions`

265 // is the object we'll return with the `probability_of_x` and the

266 // `cumulativeProbability_of_x`, as well as the calculated mean &

267 // variance. We iterate until the `cumulativeProbability_of_x` is

268 // within `epsilon` of 1.0.

269 var x = 0,

270 cumulativeProbability = 0,

271 cells = {};

273 // This algorithm iterates through each potential outcome,

274 // until the `cumulativeProbability` is very close to 1, at

275 // which point we've defined the vast majority of outcomes

276 do {

277 // a [probability mass function](https://en.wikipedia.org/wiki/Probability_mass_function)

278 cells[x] = factorial(trials) /

279 (factorial(x) * factorial(trials - x)) *

280 (Math.pow(probability, x) * Math.pow(1 - probability, trials - x));

281 cumulativeProbability += cells[x];

282 x++;

283 // when the cumulativeProbability is nearly 1, we've calculated

284 // the useful range of this distribution

285 } while (cumulativeProbability < 1 - epsilon);

287 return cells;

290 module.exports = binomialDistribution;

292 },{"13":13,"16":16}],5:[function(require,module,exports){

293 'use strict';

294 /* @flow */

296 var sign = require(53);

298 * [Bisection method](https://en.wikipedia.org/wiki/Bisection_method) is a root-finding

299 * method that repeatedly bisects an interval to find the root.

301 * This function returns a numerical approximation to the exact value.

303 * @param {Function} func input function

304 * @param {Number} start - start of interval

305 * @param {Number} end - end of interval

306 * @param {Number} maxIterations - the maximum number of iterations

307 * @param {Number} errorTolerance - the error tolerance

308 * @returns {Number} estimated root value

309 * @throws {TypeError} Argument func must be a function

311 * @example

312 * bisect(Math.cos,0,4,100,0.003); // => 1.572265625

314 function bisect(

315 func/*: (x: any) => number */,

316 start/*: number */,

317 end/*: number */,

318 maxIterations/*: number */,

319 errorTolerance/*: number */)/*:number*/ {

321 if (typeof func !== 'function') throw new TypeError('func must be a function');

322

323 for (var i = 0; i < maxIterations; i++) {

324 var output = (start + end) / 2;

326 if (func(output) === 0 || Math.abs((end - start) / 2) < errorTolerance) {

327 return output;

328 }

330 if (sign(func(output)) === sign(func(start))) {

331 start = output;

332 } else {

333 end = output;

334 }

335 }

336

337 throw new Error('maximum number of iterations exceeded');

340 module.exports = bisect;

342 },{"53":53}],6:[function(require,module,exports){

343 'use strict';

344 /* @flow */

347 * **Percentage Points of the χ2 (Chi-Squared) Distribution**

349 * The [χ2 (Chi-Squared) Distribution](http://en.wikipedia.org/wiki/Chi-squared_distribution) is used in the common

350 * chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two

351 * criteria of classification of qualitative data, and in confidence interval estimation for a population standard

352 * deviation of a normal distribution from a sample standard deviation.

354 * Values from Appendix 1, Table III of William W. Hines & Douglas C. Montgomery, "Probability and Statistics in

355 * Engineering and Management Science", Wiley (1980).

357 var chiSquaredDistributionTable = { '1':

358 { '0.995': 0,

359 '0.99': 0,

360 '0.975': 0,

361 '0.95': 0,

362 '0.9': 0.02,

363 '0.5': 0.45,

364 '0.1': 2.71,

365 '0.05': 3.84,

366 '0.025': 5.02,

367 '0.01': 6.63,

368 '0.005': 7.88 },

369 '2':

370 { '0.995': 0.01,

371 '0.99': 0.02,

372 '0.975': 0.05,

373 '0.95': 0.1,

374 '0.9': 0.21,

375 '0.5': 1.39,

376 '0.1': 4.61,

377 '0.05': 5.99,

378 '0.025': 7.38,

379 '0.01': 9.21,

380 '0.005': 10.6 },

381 '3':

382 { '0.995': 0.07,

383 '0.99': 0.11,

384 '0.975': 0.22,

385 '0.95': 0.35,

386 '0.9': 0.58,

387 '0.5': 2.37,

388 '0.1': 6.25,

389 '0.05': 7.81,

390 '0.025': 9.35,

391 '0.01': 11.34,

392 '0.005': 12.84 },

393 '4':

394 { '0.995': 0.21,

395 '0.99': 0.3,

396 '0.975': 0.48,

397 '0.95': 0.71,

398 '0.9': 1.06,

399 '0.5': 3.36,

400 '0.1': 7.78,

401 '0.05': 9.49,

402 '0.025': 11.14,

403 '0.01': 13.28,

404 '0.005': 14.86 },

405 '5':

406 { '0.995': 0.41,

407 '0.99': 0.55,

408 '0.975': 0.83,

409 '0.95': 1.15,

410 '0.9': 1.61,

411 '0.5': 4.35,

412 '0.1': 9.24,

413 '0.05': 11.07,

414 '0.025': 12.83,

415 '0.01': 15.09,

416 '0.005': 16.75 },

417 '6':

418 { '0.995': 0.68,

419 '0.99': 0.87,

420 '0.975': 1.24,

421 '0.95': 1.64,

422 '0.9': 2.2,

423 '0.5': 5.35,

424 '0.1': 10.65,

425 '0.05': 12.59,

426 '0.025': 14.45,

427 '0.01': 16.81,

428 '0.005': 18.55 },

429 '7':

430 { '0.995': 0.99,

431 '0.99': 1.25,

432 '0.975': 1.69,

433 '0.95': 2.17,

434 '0.9': 2.83,

435 '0.5': 6.35,

436 '0.1': 12.02,

437 '0.05': 14.07,

438 '0.025': 16.01,

439 '0.01': 18.48,

440 '0.005': 20.28 },

441 '8':

442 { '0.995': 1.34,

443 '0.99': 1.65,

444 '0.975': 2.18,

445 '0.95': 2.73,

446 '0.9': 3.49,

447 '0.5': 7.34,

448 '0.1': 13.36,

449 '0.05': 15.51,

450 '0.025': 17.53,

451 '0.01': 20.09,

452 '0.005': 21.96 },

453 '9':

454 { '0.995': 1.73,

455 '0.99': 2.09,

456 '0.975': 2.7,

457 '0.95': 3.33,

458 '0.9': 4.17,

459 '0.5': 8.34,

460 '0.1': 14.68,

461 '0.05': 16.92,

462 '0.025': 19.02,

463 '0.01': 21.67,

464 '0.005': 23.59 },

465 '10':

466 { '0.995': 2.16,

467 '0.99': 2.56,

468 '0.975': 3.25,

469 '0.95': 3.94,

470 '0.9': 4.87,

471 '0.5': 9.34,

472 '0.1': 15.99,

473 '0.05': 18.31,

474 '0.025': 20.48,

475 '0.01': 23.21,

476 '0.005': 25.19 },

477 '11':

478 { '0.995': 2.6,

479 '0.99': 3.05,

480 '0.975': 3.82,

481 '0.95': 4.57,

482 '0.9': 5.58,

483 '0.5': 10.34,

484 '0.1': 17.28,

485 '0.05': 19.68,

486 '0.025': 21.92,

487 '0.01': 24.72,

488 '0.005': 26.76 },

489 '12':

490 { '0.995': 3.07,

491 '0.99': 3.57,

492 '0.975': 4.4,

493 '0.95': 5.23,

494 '0.9': 6.3,

495 '0.5': 11.34,

496 '0.1': 18.55,

497 '0.05': 21.03,

498 '0.025': 23.34,

499 '0.01': 26.22,

500 '0.005': 28.3 },

501 '13':

502 { '0.995': 3.57,

503 '0.99': 4.11,

504 '0.975': 5.01,

505 '0.95': 5.89,

506 '0.9': 7.04,

507 '0.5': 12.34,

508 '0.1': 19.81,

509 '0.05': 22.36,

510 '0.025': 24.74,

511 '0.01': 27.69,

512 '0.005': 29.82 },

513 '14':

514 { '0.995': 4.07,

515 '0.99': 4.66,

516 '0.975': 5.63,

517 '0.95': 6.57,

518 '0.9': 7.79,

519 '0.5': 13.34,

520 '0.1': 21.06,

521 '0.05': 23.68,

522 '0.025': 26.12,

523 '0.01': 29.14,

524 '0.005': 31.32 },

525 '15':

526 { '0.995': 4.6,

527 '0.99': 5.23,

528 '0.975': 6.27,

529 '0.95': 7.26,

530 '0.9': 8.55,

531 '0.5': 14.34,

532 '0.1': 22.31,

533 '0.05': 25,

534 '0.025': 27.49,

535 '0.01': 30.58,

536 '0.005': 32.8 },

537 '16':

538 { '0.995': 5.14,

539 '0.99': 5.81,

540 '0.975': 6.91,

541 '0.95': 7.96,

542 '0.9': 9.31,

543 '0.5': 15.34,

544 '0.1': 23.54,

545 '0.05': 26.3,

546 '0.025': 28.85,

547 '0.01': 32,

548 '0.005': 34.27 },

549 '17':

550 { '0.995': 5.7,

551 '0.99': 6.41,

552 '0.975': 7.56,

553 '0.95': 8.67,

554 '0.9': 10.09,

555 '0.5': 16.34,

556 '0.1': 24.77,

557 '0.05': 27.59,

558 '0.025': 30.19,

559 '0.01': 33.41,

560 '0.005': 35.72 },

561 '18':

562 { '0.995': 6.26,

563 '0.99': 7.01,

564 '0.975': 8.23,

565 '0.95': 9.39,

566 '0.9': 10.87,

567 '0.5': 17.34,

568 '0.1': 25.99,

569 '0.05': 28.87,

570 '0.025': 31.53,

571 '0.01': 34.81,

572 '0.005': 37.16 },

573 '19':

574 { '0.995': 6.84,

575 '0.99': 7.63,

576 '0.975': 8.91,

577 '0.95': 10.12,

578 '0.9': 11.65,

579 '0.5': 18.34,

580 '0.1': 27.2,

581 '0.05': 30.14,

582 '0.025': 32.85,

583 '0.01': 36.19,

584 '0.005': 38.58 },

585 '20':

586 { '0.995': 7.43,

587 '0.99': 8.26,

588 '0.975': 9.59,

589 '0.95': 10.85,

590 '0.9': 12.44,

591 '0.5': 19.34,

592 '0.1': 28.41,

593 '0.05': 31.41,

594 '0.025': 34.17,

595 '0.01': 37.57,

596 '0.005': 40 },

597 '21':

598 { '0.995': 8.03,

599 '0.99': 8.9,

600 '0.975': 10.28,

601 '0.95': 11.59,

602 '0.9': 13.24,

603 '0.5': 20.34,

604 '0.1': 29.62,

605 '0.05': 32.67,

606 '0.025': 35.48,

607 '0.01': 38.93,

608 '0.005': 41.4 },

609 '22':

610 { '0.995': 8.64,

611 '0.99': 9.54,

612 '0.975': 10.98,

613 '0.95': 12.34,

614 '0.9': 14.04,

615 '0.5': 21.34,

616 '0.1': 30.81,

617 '0.05': 33.92,

618 '0.025': 36.78,

619 '0.01': 40.29,

620 '0.005': 42.8 },

621 '23':

622 { '0.995': 9.26,

623 '0.99': 10.2,

624 '0.975': 11.69,

625 '0.95': 13.09,

626 '0.9': 14.85,

627 '0.5': 22.34,

628 '0.1': 32.01,

629 '0.05': 35.17,

630 '0.025': 38.08,

631 '0.01': 41.64,

632 '0.005': 44.18 },

633 '24':

634 { '0.995': 9.89,

635 '0.99': 10.86,

636 '0.975': 12.4,

637 '0.95': 13.85,

638 '0.9': 15.66,

639 '0.5': 23.34,

640 '0.1': 33.2,

641 '0.05': 36.42,

642 '0.025': 39.36,

643 '0.01': 42.98,

644 '0.005': 45.56 },

645 '25':

646 { '0.995': 10.52,

647 '0.99': 11.52,

648 '0.975': 13.12,

649 '0.95': 14.61,

650 '0.9': 16.47,

651 '0.5': 24.34,

652 '0.1': 34.28,

653 '0.05': 37.65,

654 '0.025': 40.65,

655 '0.01': 44.31,

656 '0.005': 46.93 },

657 '26':

658 { '0.995': 11.16,

659 '0.99': 12.2,

660 '0.975': 13.84,

661 '0.95': 15.38,

662 '0.9': 17.29,

663 '0.5': 25.34,

664 '0.1': 35.56,

665 '0.05': 38.89,

666 '0.025': 41.92,

667 '0.01': 45.64,

668 '0.005': 48.29 },

669 '27':

670 { '0.995': 11.81,

671 '0.99': 12.88,

672 '0.975': 14.57,

673 '0.95': 16.15,

674 '0.9': 18.11,

675 '0.5': 26.34,

676 '0.1': 36.74,

677 '0.05': 40.11,

678 '0.025': 43.19,

679 '0.01': 46.96,

680 '0.005': 49.65 },

681 '28':

682 { '0.995': 12.46,

683 '0.99': 13.57,

684 '0.975': 15.31,

685 '0.95': 16.93,

686 '0.9': 18.94,

687 '0.5': 27.34,

688 '0.1': 37.92,

689 '0.05': 41.34,

690 '0.025': 44.46,

691 '0.01': 48.28,

692 '0.005': 50.99 },

693 '29':

694 { '0.995': 13.12,

695 '0.99': 14.26,

696 '0.975': 16.05,

697 '0.95': 17.71,

698 '0.9': 19.77,

699 '0.5': 28.34,

700 '0.1': 39.09,

701 '0.05': 42.56,

702 '0.025': 45.72,

703 '0.01': 49.59,

704 '0.005': 52.34 },

705 '30':

706 { '0.995': 13.79,

707 '0.99': 14.95,

708 '0.975': 16.79,

709 '0.95': 18.49,

710 '0.9': 20.6,

711 '0.5': 29.34,

712 '0.1': 40.26,

713 '0.05': 43.77,

714 '0.025': 46.98,

715 '0.01': 50.89,

716 '0.005': 53.67 },

717 '40':

718 { '0.995': 20.71,

719 '0.99': 22.16,

720 '0.975': 24.43,

721 '0.95': 26.51,

722 '0.9': 29.05,

723 '0.5': 39.34,

724 '0.1': 51.81,

725 '0.05': 55.76,

726 '0.025': 59.34,

727 '0.01': 63.69,

728 '0.005': 66.77 },

729 '50':

730 { '0.995': 27.99,

731 '0.99': 29.71,

732 '0.975': 32.36,

733 '0.95': 34.76,

734 '0.9': 37.69,

735 '0.5': 49.33,

736 '0.1': 63.17,

737 '0.05': 67.5,

738 '0.025': 71.42,

739 '0.01': 76.15,

740 '0.005': 79.49 },

741 '60':

742 { '0.995': 35.53,

743 '0.99': 37.48,

744 '0.975': 40.48,

745 '0.95': 43.19,

746 '0.9': 46.46,

747 '0.5': 59.33,

748 '0.1': 74.4,

749 '0.05': 79.08,

750 '0.025': 83.3,

751 '0.01': 88.38,

752 '0.005': 91.95 },

753 '70':

754 { '0.995': 43.28,

755 '0.99': 45.44,

756 '0.975': 48.76,

757 '0.95': 51.74,

758 '0.9': 55.33,

759 '0.5': 69.33,

760 '0.1': 85.53,

761 '0.05': 90.53,

762 '0.025': 95.02,

763 '0.01': 100.42,

764 '0.005': 104.22 },

765 '80':

766 { '0.995': 51.17,

767 '0.99': 53.54,

768 '0.975': 57.15,

769 '0.95': 60.39,

770 '0.9': 64.28,

771 '0.5': 79.33,

772 '0.1': 96.58,

773 '0.05': 101.88,

774 '0.025': 106.63,

775 '0.01': 112.33,

776 '0.005': 116.32 },

777 '90':

778 { '0.995': 59.2,

779 '0.99': 61.75,

780 '0.975': 65.65,

781 '0.95': 69.13,

782 '0.9': 73.29,

783 '0.5': 89.33,

784 '0.1': 107.57,

785 '0.05': 113.14,

786 '0.025': 118.14,

787 '0.01': 124.12,

788 '0.005': 128.3 },

789 '100':

790 { '0.995': 67.33,

791 '0.99': 70.06,

792 '0.975': 74.22,

793 '0.95': 77.93,

794 '0.9': 82.36,

795 '0.5': 99.33,

796 '0.1': 118.5,

797 '0.05': 124.34,

798 '0.025': 129.56,

799 '0.01': 135.81,

800 '0.005': 140.17 } };

802 module.exports = chiSquaredDistributionTable;

804 },{}],7:[function(require,module,exports){

805 'use strict';

806 /* @flow */

808 var mean = require(25);

809 var chiSquaredDistributionTable = require(6);

812 * The [χ2 (Chi-Squared) Goodness-of-Fit Test](http://en.wikipedia.org/wiki/Goodness_of_fit#Pearson.27s_chi-squared_test)

813 * uses a measure of goodness of fit which is the sum of differences between observed and expected outcome frequencies

814 * (that is, counts of observations), each squared and divided by the number of observations expected given the

815 * hypothesized distribution. The resulting χ2 statistic, `chiSquared`, can be compared to the chi-squared distribution

816 * to determine the goodness of fit. In order to determine the degrees of freedom of the chi-squared distribution, one

817 * takes the total number of observed frequencies and subtracts the number of estimated parameters. The test statistic

818 * follows, approximately, a chi-square distribution with (k − c) degrees of freedom where `k` is the number of non-empty

819 * cells and `c` is the number of estimated parameters for the distribution.

821 * @param {Array<number>} data

822 * @param {Function} distributionType a function that returns a point in a distribution:

823 * for instance, binomial, bernoulli, or poisson

824 * @param {number} significance

825 * @returns {number} chi squared goodness of fit

826 * @example

827 * // Data from Poisson goodness-of-fit example 10-19 in William W. Hines & Douglas C. Montgomery,

828 * // "Probability and Statistics in Engineering and Management Science", Wiley (1980).

829 * var data1019 = [

830 * 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

831 * 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

832 * 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

833 * 2, 2, 2, 2, 2, 2, 2, 2, 2,

834 * 3, 3, 3, 3

835 * ];

836 * ss.chiSquaredGoodnessOfFit(data1019, ss.poissonDistribution, 0.05)); //= false

838 function chiSquaredGoodnessOfFit(

839 data/*: Array<number> */,

840 distributionType/*: Function */,

841 significance/*: number */)/*: boolean */ {

842 // Estimate from the sample data, a weighted mean.

843 var inputMean = mean(data),

844 // Calculated value of the χ2 statistic.

845 chiSquared = 0,

846 // Degrees of freedom, calculated as (number of class intervals -

847 // number of hypothesized distribution parameters estimated - 1)

848 degreesOfFreedom,

849 // Number of hypothesized distribution parameters estimated, expected to be supplied in the distribution test.

850 // Lose one degree of freedom for estimating `lambda` from the sample data.

851 c = 1,

852 // The hypothesized distribution.

853 // Generate the hypothesized distribution.

854 hypothesizedDistribution = distributionType(inputMean),

855 observedFrequencies = [],

856 expectedFrequencies = [],

857 k;

859 // Create an array holding a histogram from the sample data, of

860 // the form `{ value: numberOfOcurrences }`

861 for (var i = 0; i < data.length; i++) {

862 if (observedFrequencies[data[i]] === undefined) {

863 observedFrequencies[data[i]] = 0;

864 }

865 observedFrequencies[data[i]]++;

866 }

868 // The histogram we created might be sparse - there might be gaps

869 // between values. So we iterate through the histogram, making

870 // sure that instead of undefined, gaps have 0 values.

871 for (i = 0; i < observedFrequencies.length; i++) {

872 if (observedFrequencies[i] === undefined) {

873 observedFrequencies[i] = 0;

874 }

875 }

877 // Create an array holding a histogram of expected data given the

878 // sample size and hypothesized distribution.

879 for (k in hypothesizedDistribution) {

880 if (k in observedFrequencies) {

881 expectedFrequencies[+k] = hypothesizedDistribution[k] * data.length;

882 }

883 }

885 // Working backward through the expected frequencies, collapse classes

886 // if less than three observations are expected for a class.

887 // This transformation is applied to the observed frequencies as well.

888 for (k = expectedFrequencies.length - 1; k >= 0; k--) {

889 if (expectedFrequencies[k] < 3) {

890 expectedFrequencies[k - 1] += expectedFrequencies[k];

891 expectedFrequencies.pop();

893 observedFrequencies[k - 1] += observedFrequencies[k];

894 observedFrequencies.pop();

895 }

896 }

898 // Iterate through the squared differences between observed & expected

899 // frequencies, accumulating the `chiSquared` statistic.

900 for (k = 0; k < observedFrequencies.length; k++) {

901 chiSquared += Math.pow(

902 observedFrequencies[k] - expectedFrequencies[k], 2) /

903 expectedFrequencies[k];

904 }

906 // Calculate degrees of freedom for this test and look it up in the

907 // `chiSquaredDistributionTable` in order to

908 // accept or reject the goodness-of-fit of the hypothesized distribution.

909 degreesOfFreedom = observedFrequencies.length - c - 1;

910 return chiSquaredDistributionTable[degreesOfFreedom][significance] < chiSquared;

913 module.exports = chiSquaredGoodnessOfFit;

915 },{"25":25,"6":6}],8:[function(require,module,exports){

916 'use strict';

917 /* @flow */

920 * Split an array into chunks of a specified size. This function

921 * has the same behavior as [PHP's array_chunk](http://php.net/manual/en/function.array-chunk.php)

922 * function, and thus will insert smaller-sized chunks at the end if

923 * the input size is not divisible by the chunk size.

925 * `sample` is expected to be an array, and `chunkSize` a number.

926 * The `sample` array can contain any kind of data.

928 * @param {Array} sample any array of values

929 * @param {number} chunkSize size of each output array

930 * @returns {Array<Array>} a chunked array

931 * @example

932 * chunk([1, 2, 3, 4, 5, 6], 2);

933 * // => [[1, 2], [3, 4], [5, 6]]

935 function chunk(sample/*:Array<any>*/, chunkSize/*:number*/)/*:?Array<Array<any>>*/ {

937 // a list of result chunks, as arrays in an array

938 var output = [];

940 // `chunkSize` must be zero or higher - otherwise the loop below,

941 // in which we call `start += chunkSize`, will loop infinitely.

942 // So, we'll detect and throw in that case to indicate

943 // invalid input.

944 if (chunkSize <= 0) {

945 throw new Error('chunk size must be a positive integer');

946 }

948 // `start` is the index at which `.slice` will start selecting

949 // new array elements

950 for (var start = 0; start < sample.length; start += chunkSize) {

952 // for each chunk, slice that part of the array and add it

953 // to the output. The `.slice` function does not change

954 // the original array.

955 output.push(sample.slice(start, start + chunkSize));

956 }

957 return output;

960 module.exports = chunk;

962 },{}],9:[function(require,module,exports){

963 'use strict';

964 /* @flow */

966 var uniqueCountSorted = require(61),

967 numericSort = require(34);

970 * Create a new column x row matrix.

972 * @private

973 * @param {number} columns

974 * @param {number} rows

975 * @return {Array<Array<number>>} matrix

976 * @example

977 * makeMatrix(10, 10);

979 function makeMatrix(columns, rows) {

980 var matrix = [];

981 for (var i = 0; i < columns; i++) {

982 var column = [];

983 for (var j = 0; j < rows; j++) {

984 column.push(0);

985 }

986 matrix.push(column);

987 }

988 return matrix;

992 * Generates incrementally computed values based on the sums and sums of

993 * squares for the data array

995 * @private

996 * @param {number} j

997 * @param {number} i

998 * @param {Array<number>} sums

999 * @param {Array<number>} sumsOfSquares

1000 * @return {number}

1001 * @example

1002 * ssq(0, 1, [-1, 0, 2], [1, 1, 5]);

1004 function ssq(j, i, sums, sumsOfSquares) {

1005 var sji; // s(j, i)

1006 if (j > 0) {

1007 var muji = (sums[i] - sums[j - 1]) / (i - j + 1); // mu(j, i)

1008 sji = sumsOfSquares[i] - sumsOfSquares[j - 1] - (i - j + 1) * muji * muji;

1009 } else {

1010 sji = sumsOfSquares[i] - sums[i] * sums[i] / (i + 1);

1011 }

1012 if (sji < 0) {

1013 return 0;

1014 }

1015 return sji;

1019 * Function that recursively divides and conquers computations

1020 * for cluster j

1022 * @private

1023 * @param {number} iMin Minimum index in cluster to be computed

1024 * @param {number} iMax Maximum index in cluster to be computed

1025 * @param {number} cluster Index of the cluster currently being computed

1026 * @param {Array<Array<number>>} matrix

1027 * @param {Array<Array<number>>} backtrackMatrix

1028 * @param {Array<number>} sums

1029 * @param {Array<number>} sumsOfSquares

1031 function fillMatrixColumn(iMin, iMax, cluster, matrix, backtrackMatrix, sums, sumsOfSquares) {

1032 if (iMin > iMax) {

1033 return;

1034 }

1036 // Start at midpoint between iMin and iMax

1037 var i = Math.floor((iMin + iMax) / 2);

1039 matrix[cluster][i] = matrix[cluster - 1][i - 1];

1040 backtrackMatrix[cluster][i] = i;

1042 var jlow = cluster; // the lower end for j

1044 if (iMin > cluster) {

1045 jlow = Math.max(jlow, backtrackMatrix[cluster][iMin - 1] || 0);

1046 }

1047 jlow = Math.max(jlow, backtrackMatrix[cluster - 1][i] || 0);

1049 var jhigh = i - 1; // the upper end for j

1050 if (iMax < matrix.length - 1) {

1051 jhigh = Math.min(jhigh, backtrackMatrix[cluster][iMax + 1] || 0);

1052 }

1054 var sji;

1055 var sjlowi;

1056 var ssqjlow;

1057 var ssqj;

1058 for (var j = jhigh; j >= jlow; --j) {

1059 sji = ssq(j, i, sums, sumsOfSquares);

1061 if (sji + matrix[cluster - 1][jlow - 1] >= matrix[cluster][i]) {

1062 break;

1063 }

1065 // Examine the lower bound of the cluster border

1066 sjlowi = ssq(jlow, i, sums, sumsOfSquares);

1068 ssqjlow = sjlowi + matrix[cluster - 1][jlow - 1];

1070 if (ssqjlow < matrix[cluster][i]) {

1071 // Shrink the lower bound

1072 matrix[cluster][i] = ssqjlow;

1073 backtrackMatrix[cluster][i] = jlow;

1074 }

1075 jlow++;

1077 ssqj = sji + matrix[cluster - 1][j - 1];

1078 if (ssqj < matrix[cluster][i]) {

1079 matrix[cluster][i] = ssqj;

1080 backtrackMatrix[cluster][i] = j;

1081 }

1082 }

1084 fillMatrixColumn(iMin, i - 1, cluster, matrix, backtrackMatrix, sums, sumsOfSquares);

1085 fillMatrixColumn(i + 1, iMax, cluster, matrix, backtrackMatrix, sums, sumsOfSquares);

1089 * Initializes the main matrices used in Ckmeans and kicks

1090 * off the divide and conquer cluster computation strategy

1092 * @private

1093 * @param {Array<number>} data sorted array of values

1094 * @param {Array<Array<number>>} matrix

1095 * @param {Array<Array<number>>} backtrackMatrix

1097 function fillMatrices(data, matrix, backtrackMatrix) {

1098 var nValues = matrix[0].length;

1100 // Shift values by the median to improve numeric stability

1101 var shift = data[Math.floor(nValues / 2)];

1103 // Cumulative sum and cumulative sum of squares for all values in data array

1104 var sums = [];

1105 var sumsOfSquares = [];

1107 // Initialize first column in matrix & backtrackMatrix

1108 for (var i = 0, shiftedValue; i < nValues; ++i) {

1109 shiftedValue = data[i] - shift;

1110 if (i === 0) {

1111 sums.push(shiftedValue);

1112 sumsOfSquares.push(shiftedValue * shiftedValue);

1113 } else {

1114 sums.push(sums[i - 1] + shiftedValue);

1115 sumsOfSquares.push(sumsOfSquares[i - 1] + shiftedValue * shiftedValue);

1116 }

1118 // Initialize for cluster = 0

1119 matrix[0][i] = ssq(0, i, sums, sumsOfSquares);

1120 backtrackMatrix[0][i] = 0;

1121 }

1123 // Initialize the rest of the columns

1124 var iMin;

1125 for (var cluster = 1; cluster < matrix.length; ++cluster) {

1126 if (cluster < matrix.length - 1) {

1127 iMin = cluster;

1128 } else {

1129 // No need to compute matrix[K-1][0] ... matrix[K-1][N-2]

1130 iMin = nValues - 1;

1131 }

1133 fillMatrixColumn(iMin, nValues - 1, cluster, matrix, backtrackMatrix, sums, sumsOfSquares);

1134 }

1138 * Ckmeans clustering is an improvement on heuristic-based clustering

1139 * approaches like Jenks. The algorithm was developed in

1140 * [Haizhou Wang and Mingzhou Song](http://journal.r-project.org/archive/2011-2/RJournal_2011-2_Wang+Song.pdf)

1141 * as a [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming) approach

1142 * to the problem of clustering numeric data into groups with the least

1143 * within-group sum-of-squared-deviations.

1145 * Minimizing the difference within groups - what Wang & Song refer to as

1146 * `withinss`, or within sum-of-squares, means that groups are optimally

1147 * homogenous within and the data is split into representative groups.

1148 * This is very useful for visualization, where you may want to represent

1149 * a continuous variable in discrete color or style groups. This function

1150 * can provide groups that emphasize differences between data.

1152 * Being a dynamic approach, this algorithm is based on two matrices that

1153 * store incrementally-computed values for squared deviations and backtracking

1154 * indexes.

1156 * This implementation is based on Ckmeans 3.4.6, which introduced a new divide

1157 * and conquer approach that improved runtime from O(kn^2) to O(kn log(n)).

1159 * Unlike the [original implementation](https://cran.r-project.org/web/packages/Ckmeans.1d.dp/index.html),

1160 * this implementation does not include any code to automatically determine

1161 * the optimal number of clusters: this information needs to be explicitly

1162 * provided.

1164 * ### References

1165 * _Ckmeans.1d.dp: Optimal k-means Clustering in One Dimension by Dynamic

1166 * Programming_ Haizhou Wang and Mingzhou Song ISSN 2073-4859

1168 * from The R Journal Vol. 3/2, December 2011

1169 * @param {Array<number>} data input data, as an array of number values

1170 * @param {number} nClusters number of desired classes. This cannot be

1171 * greater than the number of values in the data array.

1172 * @returns {Array<Array<number>>} clustered input

1173 * @example

1174 * ckmeans([-1, 2, -1, 2, 4, 5, 6, -1, 2, -1], 3);

1175 * // The input, clustered into groups of similar numbers.

1176 * //= [[-1, -1, -1, -1], [2, 2, 2], [4, 5, 6]]);

1178 function ckmeans(data/*: Array<number> */, nClusters/*: number */)/*: Array<Array<number>> */ {

1180 if (nClusters > data.length) {

1181 throw new Error('Cannot generate more classes than there are data values');

1182 }

1184 var sorted = numericSort(data),

1185 // we'll use this as the maximum number of clusters

1186 uniqueCount = uniqueCountSorted(sorted);

1188 // if all of the input values are identical, there's one cluster

1189 // with all of the input in it.

1190 if (uniqueCount === 1) {

1191 return [sorted];

1192 }

1194 // named 'S' originally

1195 var matrix = makeMatrix(nClusters, sorted.length),

1196 // named 'J' originally

1197 backtrackMatrix = makeMatrix(nClusters, sorted.length);

1199 // This is a dynamic programming way to solve the problem of minimizing

1200 // within-cluster sum of squares. It's similar to linear regression

1201 // in this way, and this calculation incrementally computes the

1202 // sum of squares that are later read.

1203 fillMatrices(sorted, matrix, backtrackMatrix);

1205 // The real work of Ckmeans clustering happens in the matrix generation:

1206 // the generated matrices encode all possible clustering combinations, and

1207 // once they're generated we can solve for the best clustering groups

1208 // very quickly.

1209 var clusters = [],

1210 clusterRight = backtrackMatrix[0].length - 1;

1212 // Backtrack the clusters from the dynamic programming matrix. This

1213 // starts at the bottom-right corner of the matrix (if the top-left is 0, 0),

1214 // and moves the cluster target with the loop.

1215 for (var cluster = backtrackMatrix.length - 1; cluster >= 0; cluster--) {

1217 var clusterLeft = backtrackMatrix[cluster][clusterRight];

1219 // fill the cluster from the sorted input by taking a slice of the

1220 // array. the backtrack matrix makes this easy - it stores the

1221 // indexes where the cluster should start and end.

1222 clusters[cluster] = sorted.slice(clusterLeft, clusterRight + 1);

1224 if (cluster > 0) {

1225 clusterRight = clusterLeft - 1;

1226 }

1227 }

1229 return clusters;

1232 module.exports = ckmeans;

1234 },{"34":34,"61":61}],10:[function(require,module,exports){

1235 /* @flow */

1236 'use strict';

1238 * Implementation of Combinations

1239 * Combinations are unique subsets of a collection - in this case, k elements from a collection at a time.

1240 * https://en.wikipedia.org/wiki/Combination

1241 * @param {Array} elements any type of data

1242 * @param {int} k the number of objects in each group (without replacement)

1243 * @returns {Array<Array>} array of permutations

1244 * @example

1245 * combinations([1, 2, 3], 2); // => [[1,2], [1,3], [2,3]]

1248 function combinations(elements /*: Array<any> */, k/*: number */) {

1249 var i;

1250 var subI;

1251 var combinationList = [];

1252 var subsetCombinations;

1253 var next;

1255 for (i = 0; i < elements.length; i++) {

1256 if (k === 1) {

1257 combinationList.push([elements[i]])

1258 } else {

1259 subsetCombinations = combinations(elements.slice( i + 1, elements.length ), k - 1);

1260 for (subI = 0; subI < subsetCombinations.length; subI++) {

1261 next = subsetCombinations[subI];

1262 next.unshift(elements[i]);

1263 combinationList.push(next);

1264 }

1265 }

1266 }

1267 return combinationList;

1270 module.exports = combinations;

1272 },{}],11:[function(require,module,exports){

1273 /* @flow */

1274 'use strict';

1277 * Implementation of [Combinations](https://en.wikipedia.org/wiki/Combination) with replacement

1278 * Combinations are unique subsets of a collection - in this case, k elements from a collection at a time.

1279 * 'With replacement' means that a given element can be chosen multiple times.

1280 * Unlike permutation, order doesn't matter for combinations.

1282 * @param {Array} elements any type of data

1283 * @param {int} k the number of objects in each group (without replacement)

1284 * @returns {Array<Array>} array of permutations

1285 * @example

1286 * combinationsReplacement([1, 2], 2); // => [[1, 1], [1, 2], [2, 2]]

1288 function combinationsReplacement(

1289 elements /*: Array<any> */,

1290 k /*: number */) {

1292 var combinationList = [];

1294 for (var i = 0; i < elements.length; i++) {

1295 if (k === 1) {

1296 // If we're requested to find only one element, we don't need

1297 // to recurse: just push `elements[i]` onto the list of combinations.

1298 combinationList.push([elements[i]])

1299 } else {

1300 // Otherwise, recursively find combinations, given `k - 1`. Note that

1301 // we request `k - 1`, so if you were looking for k=3 combinations, we're

1302 // requesting k=2. This -1 gets reversed in the for loop right after this

1303 // code, since we concatenate `elements[i]` onto the selected combinations,

1304 // bringing `k` back up to your requested level.

1305 // This recursion may go many levels deep, since it only stops once

1306 // k=1.

1307 var subsetCombinations = combinationsReplacement(

1308 elements.slice(i, elements.length),

1309 k - 1);

1311 for (var j = 0; j < subsetCombinations.length; j++) {

1312 combinationList.push([elements[i]]

1313 .concat(subsetCombinations[j]));

1314 }

1315 }

1316 }

1318 return combinationList;

1321 module.exports = combinationsReplacement;

1323 },{}],12:[function(require,module,exports){

1324 'use strict';

1325 /* @flow */

1327 var standardNormalTable = require(55);

1330 * **[Cumulative Standard Normal Probability](http://en.wikipedia.org/wiki/Standard_normal_table)**

1332 * Since probability tables cannot be

1333 * printed for every normal distribution, as there are an infinite variety

1334 * of normal distributions, it is common practice to convert a normal to a

1335 * standard normal and then use the standard normal table to find probabilities.

1337 * You can use `.5 + .5 * errorFunction(x / Math.sqrt(2))` to calculate the probability

1338 * instead of looking it up in a table.

1340 * @param {number} z

1341 * @returns {number} cumulative standard normal probability

1343 function cumulativeStdNormalProbability(z /*:number */)/*:number */ {

1345 // Calculate the position of this value.

1346 var absZ = Math.abs(z),

1347 // Each row begins with a different

1348 // significant digit: 0.5, 0.6, 0.7, and so on. Each value in the table

1349 // corresponds to a range of 0.01 in the input values, so the value is

1350 // multiplied by 100.

1351 index = Math.min(Math.round(absZ * 100), standardNormalTable.length - 1);

1353 // The index we calculate must be in the table as a positive value,

1354 // but we still pay attention to whether the input is positive

1355 // or negative, and flip the output value as a last step.

1356 if (z >= 0) {

1357 return standardNormalTable[index];

1358 } else {

1359 // due to floating-point arithmetic, values in the table with

1360 // 4 significant figures can nevertheless end up as repeating

1361 // fractions when they're computed here.

1362 return +(1 - standardNormalTable[index]).toFixed(4);

1363 }

1366 module.exports = cumulativeStdNormalProbability;

1368 },{"55":55}],13:[function(require,module,exports){

1369 'use strict';

1370 /* @flow */

1373 * We use `ε`, epsilon, as a stopping criterion when we want to iterate

1374 * until we're "close enough". Epsilon is a very small number: for

1375 * simple statistics, that number is **0.0001**

1377 * This is used in calculations like the binomialDistribution, in which

1378 * the process of finding a value is [iterative](https://en.wikipedia.org/wiki/Iterative_method):

1379 * it progresses until it is close enough.

1381 * Below is an example of using epsilon in [gradient descent](https://en.wikipedia.org/wiki/Gradient_descent),

1382 * where we're trying to find a local minimum of a function's derivative,

1383 * given by the `fDerivative` method.

1385 * @example

1386 * // From calculation, we expect that the local minimum occurs at x=9/4

1387 * var x_old = 0;

1388 * // The algorithm starts at x=6

1389 * var x_new = 6;

1390 * var stepSize = 0.01;

1392 * function fDerivative(x) {

1393 * return 4 * Math.pow(x, 3) - 9 * Math.pow(x, 2);

1394 * }

1396 * // The loop runs until the difference between the previous

1397 * // value and the current value is smaller than epsilon - a rough

1398 * // meaure of 'close enough'

1399 * while (Math.abs(x_new - x_old) > ss.epsilon) {

1400 * x_old = x_new;

1401 * x_new = x_old - stepSize * fDerivative(x_old);

1402 * }

1404 * console.log('Local minimum occurs at', x_new);

1406 var epsilon = 0.0001;

1408 module.exports = epsilon;

1410 },{}],14:[function(require,module,exports){

1411 'use strict';

1412 /* @flow */

1414 var max = require(23),

1415 min = require(29);

1418 * Given an array of data, this will find the extent of the

1419 * data and return an array of breaks that can be used

1420 * to categorize the data into a number of classes. The

1421 * returned array will always be 1 longer than the number of

1422 * classes because it includes the minimum value.

1424 * @param {Array<number>} data input data, as an array of number values

1425 * @param {number} nClasses number of desired classes

1426 * @returns {Array<number>} array of class break positions

1427 * @example

1428 * equalIntervalBreaks([1, 2, 3, 4, 5, 6], 4); //= [1, 2.25, 3.5, 4.75, 6]

1430 function equalIntervalBreaks(data/*: Array<number> */, nClasses/*:number*/)/*: Array<number> */ {

1432 if (data.length <= 1) {

1433 return data;

1434 }

1436 var theMin = min(data),

1437 theMax = max(data);

1439 // the first break will always be the minimum value

1440 // in the dataset

1441 var breaks = [theMin];

1443 // The size of each break is the full range of the data

1444 // divided by the number of classes requested

1445 var breakSize = (theMax - theMin) / nClasses;

1447 // In the case of nClasses = 1, this loop won't run

1448 // and the returned breaks will be [min, max]

1449 for (var i = 1; i < nClasses; i++) {

1450 breaks.push(breaks[0] + breakSize * i);

1451 }

1453 // the last break will always be the

1454 // maximum.

1455 breaks.push(theMax);

1457 return breaks;

1460 module.exports = equalIntervalBreaks;

1462 },{"23":23,"29":29}],15:[function(require,module,exports){

1463 'use strict';

1464 /* @flow */

1467 * **[Gaussian error function](http://en.wikipedia.org/wiki/Error_function)**

1469 * The `errorFunction(x/(sd * Math.sqrt(2)))` is the probability that a value in a

1470 * normal distribution with standard deviation sd is within x of the mean.

1472 * This function returns a numerical approximation to the exact value.

1474 * @param {number} x input

1475 * @return {number} error estimation

1476 * @example

1477 * errorFunction(1).toFixed(2); // => '0.84'

1479 function errorFunction(x/*: number */)/*: number */ {

1480 var t = 1 / (1 + 0.5 * Math.abs(x));

1481 var tau = t * Math.exp(-Math.pow(x, 2) -

1482 1.26551223 +

1483 1.00002368 * t +

1484 0.37409196 * Math.pow(t, 2) +

1485 0.09678418 * Math.pow(t, 3) -

1486 0.18628806 * Math.pow(t, 4) +

1487 0.27886807 * Math.pow(t, 5) -

1488 1.13520398 * Math.pow(t, 6) +

1489 1.48851587 * Math.pow(t, 7) -

1490 0.82215223 * Math.pow(t, 8) +

1491 0.17087277 * Math.pow(t, 9));

1492 if (x >= 0) {

1493 return 1 - tau;

1494 } else {

1495 return tau - 1;

1496 }

1499 module.exports = errorFunction;

1501 },{}],16:[function(require,module,exports){

1502 'use strict';

1503 /* @flow */

1506 * A [Factorial](https://en.wikipedia.org/wiki/Factorial), usually written n!, is the product of all positive

1507 * integers less than or equal to n. Often factorial is implemented

1508 * recursively, but this iterative approach is significantly faster

1509 * and simpler.

1511 * @param {number} n input

1512 * @returns {number} factorial: n!

1513 * @example

1514 * factorial(5); // => 120

1516 function factorial(n /*: number */)/*: number */ {

1518 // factorial is mathematically undefined for negative numbers

1519 if (n < 0) { return NaN; }

1521 // typically you'll expand the factorial function going down, like

1522 // 5! = 5 * 4 * 3 * 2 * 1. This is going in the opposite direction,

1523 // counting from 2 up to the number in question, and since anything

1524 // multiplied by 1 is itself, the loop only needs to start at 2.

1525 var accumulator = 1;

1526 for (var i = 2; i <= n; i++) {

1527 // for each number up to and including the number `n`, multiply

1528 // the accumulator my that number.

1529 accumulator *= i;

1530 }

1531 return accumulator;

1534 module.exports = factorial;

1536 },{}],17:[function(require,module,exports){

1537 'use strict';

1538 /* @flow */

1541 * The [Geometric Mean](https://en.wikipedia.org/wiki/Geometric_mean) is

1542 * a mean function that is more useful for numbers in different

1543 * ranges.

1545 * This is the nth root of the input numbers multiplied by each other.

1547 * The geometric mean is often useful for

1548 * **[proportional growth](https://en.wikipedia.org/wiki/Geometric_mean#Proportional_growth)**: given

1549 * growth rates for multiple years, like _80%, 16.66% and 42.85%_, a simple

1550 * mean will incorrectly estimate an average growth rate, whereas a geometric

1551 * mean will correctly estimate a growth rate that, over those years,

1552 * will yield the same end value.

1554 * This runs on `O(n)`, linear time in respect to the array

1556 * @param {Array<number>} x input array

1557 * @returns {number} geometric mean

1558 * @example

1559 * var growthRates = [1.80, 1.166666, 1.428571];

1560 * var averageGrowth = geometricMean(growthRates);

1561 * var averageGrowthRates = [averageGrowth, averageGrowth, averageGrowth];

1562 * var startingValue = 10;

1563 * var startingValueMean = 10;

1564 * growthRates.forEach(function(rate) {

1565 * startingValue *= rate;

1566 * });

1567 * averageGrowthRates.forEach(function(rate) {

1568 * startingValueMean *= rate;

1569 * });

1570 * startingValueMean === startingValue;

1572 function geometricMean(x /*: Array<number> */) {

1573 // The mean of no numbers is null

1574 if (x.length === 0) { return undefined; }

1576 // the starting value.

1577 var value = 1;

1579 for (var i = 0; i < x.length; i++) {

1580 // the geometric mean is only valid for positive numbers

1581 if (x[i] <= 0) { return undefined; }

1583 // repeatedly multiply the value by each number

1584 value *= x[i];

1585 }

1587 return Math.pow(value, 1 / x.length);

1590 module.exports = geometricMean;

1592 },{}],18:[function(require,module,exports){

1593 'use strict';

1594 /* @flow */

1597 * The [Harmonic Mean](https://en.wikipedia.org/wiki/Harmonic_mean) is

1598 * a mean function typically used to find the average of rates.

1599 * This mean is calculated by taking the reciprocal of the arithmetic mean

1600 * of the reciprocals of the input numbers.

1602 * This is a [measure of central tendency](https://en.wikipedia.org/wiki/Central_tendency):

1603 * a method of finding a typical or central value of a set of numbers.

1605 * This runs on `O(n)`, linear time in respect to the array.

1607 * @param {Array<number>} x input

1608 * @returns {number} harmonic mean

1609 * @example

1610 * harmonicMean([2, 3]).toFixed(2) // => '2.40'

1612 function harmonicMean(x /*: Array<number> */) {

1613 // The mean of no numbers is null

1614 if (x.length === 0) { return undefined; }

1616 var reciprocalSum = 0;

1618 for (var i = 0; i < x.length; i++) {

1619 // the harmonic mean is only valid for positive numbers

1620 if (x[i] <= 0) { return undefined; }

1622 reciprocalSum += 1 / x[i];

1623 }

1625 // divide n by the the reciprocal sum

1626 return x.length / reciprocalSum;

1629 module.exports = harmonicMean;

1631 },{}],19:[function(require,module,exports){

1632 'use strict';

1633 /* @flow */

1635 var quantile = require(40);

1638 * The [Interquartile range](http://en.wikipedia.org/wiki/Interquartile_range) is

1639 * a measure of statistical dispersion, or how scattered, spread, or

1640 * concentrated a distribution is. It's computed as the difference between

1641 * the third quartile and first quartile.

1643 * @param {Array<number>} sample

1644 * @returns {number} interquartile range: the span between lower and upper quartile,

1645 * 0.25 and 0.75

1646 * @example

1647 * interquartileRange([0, 1, 2, 3]); // => 2

1649 function interquartileRange(sample/*: Array<number> */) {

1650 // Interquartile range is the span between the upper quartile,

1651 // at `0.75`, and lower quartile, `0.25`

1652 var q1 = quantile(sample, 0.75),

1653 q2 = quantile(sample, 0.25);

1655 if (typeof q1 === 'number' && typeof q2 === 'number') {

1656 return q1 - q2;

1657 }

1660 module.exports = interquartileRange;

1662 },{"40":40}],20:[function(require,module,exports){

1663 'use strict';

1664 /* @flow */

1667 * The Inverse [Gaussian error function](http://en.wikipedia.org/wiki/Error_function)

1668 * returns a numerical approximation to the value that would have caused

1669 * `errorFunction()` to return x.

1671 * @param {number} x value of error function

1672 * @returns {number} estimated inverted value

1674 function inverseErrorFunction(x/*: number */)/*: number */ {

1675 var a = (8 * (Math.PI - 3)) / (3 * Math.PI * (4 - Math.PI));

1677 var inv = Math.sqrt(Math.sqrt(

1678 Math.pow(2 / (Math.PI * a) + Math.log(1 - x * x) / 2, 2) -

1679 Math.log(1 - x * x) / a) -

1680 (2 / (Math.PI * a) + Math.log(1 - x * x) / 2));

1682 if (x >= 0) {

1683 return inv;

1684 } else {

1685 return -inv;

1686 }

1689 module.exports = inverseErrorFunction;

1691 },{}],21:[function(require,module,exports){

1692 'use strict';

1693 /* @flow */

1696 * [Simple linear regression](http://en.wikipedia.org/wiki/Simple_linear_regression)

1697 * is a simple way to find a fitted line

1698 * between a set of coordinates. This algorithm finds the slope and y-intercept of a regression line

1699 * using the least sum of squares.

1701 * @param {Array<Array<number>>} data an array of two-element of arrays,

1702 * like `[[0, 1], [2, 3]]`

1703 * @returns {Object} object containing slope and intersect of regression line

1704 * @example

1705 * linearRegression([[0, 0], [1, 1]]); // => { m: 1, b: 0 }

1707 function linearRegression(data/*: Array<Array<number>> */)/*: { m: number, b: number } */ {

1709 var m, b;

1711 // Store data length in a local variable to reduce

1712 // repeated object property lookups

1713 var dataLength = data.length;

1715 //if there's only one point, arbitrarily choose a slope of 0

1716 //and a y-intercept of whatever the y of the initial point is

1717 if (dataLength === 1) {

1718 m = 0;

1719 b = data[0][1];

1720 } else {

1721 // Initialize our sums and scope the `m` and `b`

1722 // variables that define the line.

1723 var sumX = 0, sumY = 0,

1724 sumXX = 0, sumXY = 0;

1726 // Use local variables to grab point values

1727 // with minimal object property lookups

1728 var point, x, y;

1730 // Gather the sum of all x values, the sum of all

1731 // y values, and the sum of x^2 and (x*y) for each

1732 // value.

1733 //

1734 // In math notation, these would be SS_x, SS_y, SS_xx, and SS_xy

1735 for (var i = 0; i < dataLength; i++) {

1736 point = data[i];

1737 x = point[0];

1738 y = point[1];

1740 sumX += x;

1741 sumY += y;

1743 sumXX += x * x;

1744 sumXY += x * y;

1745 }

1747 // `m` is the slope of the regression line

1748 m = ((dataLength * sumXY) - (sumX * sumY)) /

1749 ((dataLength * sumXX) - (sumX * sumX));

1751 // `b` is the y-intercept of the line.

1752 b = (sumY / dataLength) - ((m * sumX) / dataLength);

1753 }

1755 // Return both values as an object.

1756 return {

1757 m: m,

1758 b: b

1759 };

1763 module.exports = linearRegression;

1765 },{}],22:[function(require,module,exports){

1766 'use strict';

1767 /* @flow */

1770 * Given the output of `linearRegression`: an object

1771 * with `m` and `b` values indicating slope and intercept,

1772 * respectively, generate a line function that translates

1773 * x values into y values.

1775 * @param {Object} mb object with `m` and `b` members, representing

1776 * slope and intersect of desired line

1777 * @returns {Function} method that computes y-value at any given

1778 * x-value on the line.

1779 * @example

1780 * var l = linearRegressionLine(linearRegression([[0, 0], [1, 1]]));

1781 * l(0) // = 0

1782 * l(2) // = 2

1783 * linearRegressionLine({ b: 0, m: 1 })(1); // => 1

1784 * linearRegressionLine({ b: 1, m: 1 })(1); // => 2

1786 function linearRegressionLine(mb/*: { b: number, m: number }*/)/*: Function */ {

1787 // Return a function that computes a `y` value for each

1788 // x value it is given, based on the values of `b` and `a`

1789 // that we just computed.

1790 return function(x) {

1791 return mb.b + (mb.m * x);

1792 };

1795 module.exports = linearRegressionLine;

1797 },{}],23:[function(require,module,exports){

1798 'use strict';

1799 /* @flow */

1802 * This computes the maximum number in an array.

1804 * This runs on `O(n)`, linear time in respect to the array

1806 * @param {Array<number>} x input

1807 * @returns {number} maximum value

1808 * @example

1809 * max([1, 2, 3, 4]);

1810 * // => 4

1812 function max(x /*: Array<number> */) /*:number*/ {

1813 var value;

1814 for (var i = 0; i < x.length; i++) {

1815 // On the first iteration of this loop, max is

1816 // NaN and is thus made the maximum element in the array

1817 if (value === undefined || x[i] > value) {

1818 value = x[i];

1819 }

1820 }

1821 if (value === undefined) {

1822 return NaN;

1823 }

1824 return value;

1827 module.exports = max;

1829 },{}],24:[function(require,module,exports){

1830 'use strict';

1831 /* @flow */

1834 * The maximum is the highest number in the array. With a sorted array,

1835 * the last element in the array is always the largest, so this calculation

1836 * can be done in one step, or constant time.

1838 * @param {Array<number>} x input

1839 * @returns {number} maximum value

1840 * @example

1841 * maxSorted([-100, -10, 1, 2, 5]); // => 5

1843 function maxSorted(x /*: Array<number> */)/*:number*/ {

1844 return x[x.length - 1];

1847 module.exports = maxSorted;

1849 },{}],25:[function(require,module,exports){

1850 'use strict';

1851 /* @flow */

1853 var sum = require(56);

1856 * The mean, _also known as average_,

1857 * is the sum of all values over the number of values.

1858 * This is a [measure of central tendency](https://en.wikipedia.org/wiki/Central_tendency):

1859 * a method of finding a typical or central value of a set of numbers.

1861 * This runs on `O(n)`, linear time in respect to the array

1863 * @param {Array<number>} x input values

1864 * @returns {number} mean

1865 * @example

1866 * mean([0, 10]); // => 5

1868 function mean(x /*: Array<number> */)/*:number*/ {

1869 // The mean of no numbers is null

1870 if (x.length === 0) { return NaN; }

1872 return sum(x) / x.length;

1875 module.exports = mean;

1877 },{"56":56}],26:[function(require,module,exports){

1878 'use strict';

1879 /* @flow */

1881 var quantile = require(40);

1884 * The [median](http://en.wikipedia.org/wiki/Median) is

1885 * the middle number of a list. This is often a good indicator of 'the middle'

1886 * when there are outliers that skew the `mean()` value.

1887 * This is a [measure of central tendency](https://en.wikipedia.org/wiki/Central_tendency):

1888 * a method of finding a typical or central value of a set of numbers.

1890 * The median isn't necessarily one of the elements in the list: the value

1891 * can be the average of two elements if the list has an even length

1892 * and the two central values are different.

1894 * @param {Array<number>} x input

1895 * @returns {number} median value

1896 * @example

1897 * median([10, 2, 5, 100, 2, 1]); // => 3.5

1899 function median(x /*: Array<number> */)/*:number*/ {

1900 return +quantile(x, 0.5);

1903 module.exports = median;

1905 },{"40":40}],27:[function(require,module,exports){

1906 'use strict';

1907 /* @flow */

1909 var median = require(26);

1912 * The [Median Absolute Deviation](http://en.wikipedia.org/wiki/Median_absolute_deviation) is

1913 * a robust measure of statistical

1914 * dispersion. It is more resilient to outliers than the standard deviation.

1916 * @param {Array<number>} x input array

1917 * @returns {number} median absolute deviation

1918 * @example

1919 * medianAbsoluteDeviation([1, 1, 2, 2, 4, 6, 9]); // => 1

1921 function medianAbsoluteDeviation(x /*: Array<number> */) {

1922 // The mad of nothing is null

1923 var medianValue = median(x),

1924 medianAbsoluteDeviations = [];

1926 // Make a list of absolute deviations from the median

1927 for (var i = 0; i < x.length; i++) {

1928 medianAbsoluteDeviations.push(Math.abs(x[i] - medianValue));

1929 }

1931 // Find the median value of that list

1932 return median(medianAbsoluteDeviations);

1935 module.exports = medianAbsoluteDeviation;

1937 },{"26":26}],28:[function(require,module,exports){

1938 'use strict';

1939 /* @flow */

1941 var quantileSorted = require(41);

1944 * The [median](http://en.wikipedia.org/wiki/Median) is

1945 * the middle number of a list. This is often a good indicator of 'the middle'

1946 * when there are outliers that skew the `mean()` value.

1947 * This is a [measure of central tendency](https://en.wikipedia.org/wiki/Central_tendency):

1948 * a method of finding a typical or central value of a set of numbers.

1950 * The median isn't necessarily one of the elements in the list: the value

1951 * can be the average of two elements if the list has an even length

1952 * and the two central values are different.

1954 * @param {Array<number>} sorted input

1955 * @returns {number} median value

1956 * @example

1957 * medianSorted([10, 2, 5, 100, 2, 1]); // => 52.5

1959 function medianSorted(sorted /*: Array<number> */)/*:number*/ {

1960 return quantileSorted(sorted, 0.5);

1963 module.exports = medianSorted;

1965 },{"41":41}],29:[function(require,module,exports){

1966 'use strict';

1967 /* @flow */

1970 * The min is the lowest number in the array. This runs on `O(n)`, linear time in respect to the array

1972 * @param {Array<number>} x input

1973 * @returns {number} minimum value

1974 * @example

1975 * min([1, 5, -10, 100, 2]); // => -10

1977 function min(x /*: Array<number> */)/*:number*/ {

1978 var value;

1979 for (var i = 0; i < x.length; i++) {

1980 // On the first iteration of this loop, min is

1981 // NaN and is thus made the minimum element in the array

1982 if (value === undefined || x[i] < value) {

1983 value = x[i];

1984 }

1985 }

1986 if (value === undefined) {

1987 return NaN;

1988 }

1989 return value;

1992 module.exports = min;

1994 },{}],30:[function(require,module,exports){

1995 'use strict';

1996 /* @flow */

1999 * The minimum is the lowest number in the array. With a sorted array,

2000 * the first element in the array is always the smallest, so this calculation

2001 * can be done in one step, or constant time.

2003 * @param {Array<number>} x input

2004 * @returns {number} minimum value

2005 * @example

2006 * minSorted([-100, -10, 1, 2, 5]); // => -100

2008 function minSorted(x /*: Array<number> */)/*:number*/ {

2009 return x[0];

2012 module.exports = minSorted;

2014 },{}],31:[function(require,module,exports){

2015 'use strict';

2016 /* @flow */

2019 * **Mixin** simple_statistics to a single Array instance if provided

2020 * or the Array native object if not. This is an optional

2021 * feature that lets you treat simple_statistics as a native feature

2022 * of Javascript.

2024 * @param {Object} ss simple statistics

2025 * @param {Array} [array=] a single array instance which will be augmented

2026 * with the extra methods. If omitted, mixin will apply to all arrays

2027 * by changing the global `Array.prototype`.

2028 * @returns {*} the extended Array, or Array.prototype if no object

2029 * is given.

2031 * @example

2032 * var myNumbers = [1, 2, 3];

2033 * mixin(ss, myNumbers);

2034 * console.log(myNumbers.sum()); // 6

2036 function mixin(ss /*: Object */, array /*: ?Array<any> */)/*: any */ {

2037 var support = !!(Object.defineProperty && Object.defineProperties);

2038 // Coverage testing will never test this error.

2039 /* istanbul ignore next */

2040 if (!support) {

2041 throw new Error('without defineProperty, simple-statistics cannot be mixed in');

2042 }

2044 // only methods which work on basic arrays in a single step

2045 // are supported

2046 var arrayMethods = ['median', 'standardDeviation', 'sum', 'product',

2047 'sampleSkewness',

2048 'mean', 'min', 'max', 'quantile', 'geometricMean',

2049 'harmonicMean', 'root_mean_square'];

2051 // create a closure with a method name so that a reference

2052 // like `arrayMethods[i]` doesn't follow the loop increment

2053 function wrap(method) {

2054 return function() {

2055 // cast any arguments into an array, since they're

2056 // natively objects

2057 var args = Array.prototype.slice.apply(arguments);

2058 // make the first argument the array itself

2059 args.unshift(this);

2060 // return the result of the ss method

2061 return ss[method].apply(ss, args);

2062 };

2063 }

2065 // select object to extend

2066 var extending;

2067 if (array) {

2068 // create a shallow copy of the array so that our internal

2069 // operations do not change it by reference

2070 extending = array.slice();

2071 } else {

2072 extending = Array.prototype;

2073 }

2075 // for each array function, define a function that gets

2076 // the array as the first argument.

2077 // We use [defineProperty](https://developer.mozilla.org/en-US/docs/JavaScript/Reference/Global_Objects/Object/defineProperty)

2078 // because it allows these properties to be non-enumerable:

2079 // `for (var in x)` loops will not run into problems with this

2080 // implementation.

2081 for (var i = 0; i < arrayMethods.length; i++) {

2082 Object.defineProperty(extending, arrayMethods[i], {

2083 value: wrap(arrayMethods[i]),

2084 configurable: true,

2085 enumerable: false,

2086 writable: true

2087 });

2088 }

2090 return extending;

2093 module.exports = mixin;

2095 },{}],32:[function(require,module,exports){

2096 'use strict';

2097 /* @flow */

2099 var numericSort = require(34),

2100 modeSorted = require(33);

2103 * The [mode](http://bit.ly/W5K4Yt) is the number that appears in a list the highest number of times.

2104 * There can be multiple modes in a list: in the event of a tie, this

2105 * algorithm will return the most recently seen mode.

2107 * This is a [measure of central tendency](https://en.wikipedia.org/wiki/Central_tendency):

2108 * a method of finding a typical or central value of a set of numbers.

2110 * This runs on `O(nlog(n))` because it needs to sort the array internally

2111 * before running an `O(n)` search to find the mode.

2113 * @param {Array<number>} x input

2114 * @returns {number} mode

2115 * @example

2116 * mode([0, 0, 1]); // => 0

2118 function mode(x /*: Array<number> */)/*:number*/ {

2119 // Sorting the array lets us iterate through it below and be sure

2120 // that every time we see a new number it's new and we'll never

2121 // see the same number twice

2122 return modeSorted(numericSort(x));

2125 module.exports = mode;

2127 },{"33":33,"34":34}],33:[function(require,module,exports){

2128 'use strict';

2129 /* @flow */

2132 * The [mode](http://bit.ly/W5K4Yt) is the number that appears in a list the highest number of times.

2133 * There can be multiple modes in a list: in the event of a tie, this

2134 * algorithm will return the most recently seen mode.

2136 * This is a [measure of central tendency](https://en.wikipedia.org/wiki/Central_tendency):

2137 * a method of finding a typical or central value of a set of numbers.

2139 * This runs in `O(n)` because the input is sorted.

2141 * @param {Array<number>} sorted input

2142 * @returns {number} mode

2143 * @example

2144 * modeSorted([0, 0, 1]); // => 0

2146 function modeSorted(sorted /*: Array<number> */)/*:number*/ {

2148 // Handle edge cases:

2149 // The mode of an empty list is NaN

2150 if (sorted.length === 0) { return NaN; }

2151 else if (sorted.length === 1) { return sorted[0]; }

2153 // This assumes it is dealing with an array of size > 1, since size

2154 // 0 and 1 are handled immediately. Hence it starts at index 1 in the

2155 // array.

2156 var last = sorted[0],

2157 // store the mode as we find new modes

2158 value = NaN,

2159 // store how many times we've seen the mode

2160 maxSeen = 0,

2161 // how many times the current candidate for the mode

2162 // has been seen

2163 seenThis = 1;

2165 // end at sorted.length + 1 to fix the case in which the mode is

2166 // the highest number that occurs in the sequence. the last iteration

2167 // compares sorted[i], which is undefined, to the highest number

2168 // in the series

2169 for (var i = 1; i < sorted.length + 1; i++) {

2170 // we're seeing a new number pass by

2171 if (sorted[i] !== last) {

2172 // the last number is the new mode since we saw it more

2173 // often than the old one

2174 if (seenThis > maxSeen) {

2175 maxSeen = seenThis;

2176 value = last;

2177 }

2178 seenThis = 1;

2179 last = sorted[i];

2180 // if this isn't a new number, it's one more occurrence of

2181 // the potential mode

2182 } else { seenThis++; }

2183 }

2184 return value;

2187 module.exports = modeSorted;

2189 },{}],34:[function(require,module,exports){

2190 'use strict';

2191 /* @flow */

2194 * Sort an array of numbers by their numeric value, ensuring that the

2195 * array is not changed in place.

2197 * This is necessary because the default behavior of .sort

2198 * in JavaScript is to sort arrays as string values

2200 * [1, 10, 12, 102, 20].sort()

2201 * // output

2202 * [1, 10, 102, 12, 20]

2204 * @param {Array<number>} array input array

2205 * @return {Array<number>} sorted array

2206 * @private

2207 * @example

2208 * numericSort([3, 2, 1]) // => [1, 2, 3]

2210 function numericSort(array /*: Array<number> */) /*: Array<number> */ {

2211 return array

2212 // ensure the array is not changed in-place

2213 .slice()

2214 // comparator function that treats input as numeric

2215 .sort(function(a, b) {

2216 return a - b;

2217 });

2220 module.exports = numericSort;

2222 },{}],35:[function(require,module,exports){

2223 'use strict';

2224 /* @flow */

2227 * This is a single-layer [Perceptron Classifier](http://en.wikipedia.org/wiki/Perceptron) that takes

2228 * arrays of numbers and predicts whether they should be classified

2229 * as either 0 or 1 (negative or positive examples).

2230 * @class

2231 * @example

2232 * // Create the model

2233 * var p = new PerceptronModel();

2234 * // Train the model with input with a diagonal boundary.

2235 * for (var i = 0; i < 5; i++) {

2236 * p.train([1, 1], 1);

2237 * p.train([0, 1], 0);

2238 * p.train([1, 0], 0);

2239 * p.train([0, 0], 0);

2240 * }

2241 * p.predict([0, 0]); // 0

2242 * p.predict([0, 1]); // 0

2243 * p.predict([1, 0]); // 0

2244 * p.predict([1, 1]); // 1

2246 function PerceptronModel() {

2247 // The weights, or coefficients of the model;

2248 // weights are only populated when training with data.

2249 this.weights = [];

2250 // The bias term, or intercept; it is also a weight but

2251 // it's stored separately for convenience as it is always

2252 // multiplied by one.

2253 this.bias = 0;

2257 * **Predict**: Use an array of features with the weight array and bias

2258 * to predict whether an example is labeled 0 or 1.

2260 * @param {Array<number>} features an array of features as numbers

2261 * @returns {number} 1 if the score is over 0, otherwise 0

2263 PerceptronModel.prototype.predict = function(features) {

2265 // Only predict if previously trained

2266 // on the same size feature array(s).

2267 if (features.length !== this.weights.length) { return null; }

2269 // Calculate the sum of features times weights,

2270 // with the bias added (implicitly times one).

2271 var score = 0;

2272 for (var i = 0; i < this.weights.length; i++) {

2273 score += this.weights[i] * features[i];

2274 }

2275 score += this.bias;

2277 // Classify as 1 if the score is over 0, otherwise 0.

2278 if (score > 0) {

2279 return 1;

2280 } else {

2281 return 0;

2282 }

2286 * **Train** the classifier with a new example, which is

2287 * a numeric array of features and a 0 or 1 label.

2289 * @param {Array<number>} features an array of features as numbers

2290 * @param {number} label either 0 or 1

2291 * @returns {PerceptronModel} this

2293 PerceptronModel.prototype.train = function(features, label) {

2294 // Require that only labels of 0 or 1 are considered.

2295 if (label !== 0 && label !== 1) { return null; }

2296 // The length of the feature array determines

2297 // the length of the weight array.

2298 // The perceptron will continue learning as long as

2299 // it keeps seeing feature arrays of the same length.

2300 // When it sees a new data shape, it initializes.

2301 if (features.length !== this.weights.length) {

2302 this.weights = features;

2303 this.bias = 1;

2304 }

2305 // Make a prediction based on current weights.

2306 var prediction = this.predict(features);

2307 // Update the weights if the prediction is wrong.

2308 if (prediction !== label) {

2309 var gradient = label - prediction;

2310 for (var i = 0; i < this.weights.length; i++) {

2311 this.weights[i] += gradient * features[i];

2312 }

2313 this.bias += gradient;

2314 }

2315 return this;

2318 module.exports = PerceptronModel;

2320 },{}],36:[function(require,module,exports){

2321 /* @flow */

2323 'use strict';

2326 * Implementation of [Heap's Algorithm](https://en.wikipedia.org/wiki/Heap%27s_algorithm)

2327 * for generating permutations.

2329 * @param {Array} elements any type of data

2330 * @returns {Array<Array>} array of permutations

2332 function permutationsHeap/*:: <T> */(elements /*: Array<T> */)/*: Array<Array<T>> */ {

2333 var indexes = new Array(elements.length);

2334 var permutations = [elements.slice()];

2336 for (var i = 0; i < elements.length; i++) {

2337 indexes[i] = 0;

2338 }

2340 for (i = 0; i < elements.length;) {

2341 if (indexes[i] < i) {

2343 // At odd indexes, swap from indexes[i] instead

2344 // of from the beginning of the array

2345 var swapFrom = 0;

2346 if (i % 2 !== 0) {

2347 swapFrom = indexes[i];

2348 }

2350 // swap between swapFrom and i, using

2351 // a temporary variable as storage.

2352 var temp = elements[swapFrom];

2353 elements[swapFrom] = elements[i];

2354 elements[i] = temp;

2356 permutations.push(elements.slice());

2357 indexes[i]++;

2358 i = 0;

2360 } else {

2361 indexes[i] = 0;

2362 i++;

2363 }

2364 }

2366 return permutations;

2369 module.exports = permutationsHeap;

2371 },{}],37:[function(require,module,exports){

2372 'use strict';

2373 /* @flow */

2375 var epsilon = require(13);

2376 var factorial = require(16);

2379 * The [Poisson Distribution](http://en.wikipedia.org/wiki/Poisson_distribution)

2380 * is a discrete probability distribution that expresses the probability

2381 * of a given number of events occurring in a fixed interval of time

2382 * and/or space if these events occur with a known average rate and

2383 * independently of the time since the last event.

2385 * The Poisson Distribution is characterized by the strictly positive

2386 * mean arrival or occurrence rate, `λ`.

2388 * @param {number} lambda location poisson distribution

2389 * @returns {number} value of poisson distribution at that point

2391 function poissonDistribution(lambda/*: number */) {

2392 // Check that lambda is strictly positive

2393 if (lambda <= 0) { return undefined; }

2395 // our current place in the distribution

2396 var x = 0,

2397 // and we keep track of the current cumulative probability, in

2398 // order to know when to stop calculating chances.

2399 cumulativeProbability = 0,

2400 // the calculated cells to be returned

2401 cells = {};

2403 // This algorithm iterates through each potential outcome,

2404 // until the `cumulativeProbability` is very close to 1, at

2405 // which point we've defined the vast majority of outcomes

2406 do {

2407 // a [probability mass function](https://en.wikipedia.org/wiki/Probability_mass_function)

2408 cells[x] = (Math.pow(Math.E, -lambda) * Math.pow(lambda, x)) / factorial(x);

2409 cumulativeProbability += cells[x];

2410 x++;

2411 // when the cumulativeProbability is nearly 1, we've calculated

2412 // the useful range of this distribution

2413 } while (cumulativeProbability < 1 - epsilon);

2415 return cells;

2418 module.exports = poissonDistribution;

2420 },{"13":13,"16":16}],38:[function(require,module,exports){

2421 'use strict';

2422 /* @flow */

2424 var epsilon = require(13);

2425 var inverseErrorFunction = require(20);

2428 * The [Probit](http://en.wikipedia.org/wiki/Probit)

2429 * is the inverse of cumulativeStdNormalProbability(),

2430 * and is also known as the normal quantile function.

2432 * It returns the number of standard deviations from the mean

2433 * where the p'th quantile of values can be found in a normal distribution.

2434 * So, for example, probit(0.5 + 0.6827/2) ≈ 1 because 68.27% of values are

2435 * normally found within 1 standard deviation above or below the mean.

2437 * @param {number} p

2438 * @returns {number} probit

2440 function probit(p /*: number */)/*: number */ {

2441 if (p === 0) {

2442 p = epsilon;

2443 } else if (p >= 1) {

2444 p = 1 - epsilon;

2445 }

2446 return Math.sqrt(2) * inverseErrorFunction(2 * p - 1);

2449 module.exports = probit;

2451 },{"13":13,"20":20}],39:[function(require,module,exports){

2452 'use strict';

2453 /* @flow */

2456 * The [product](https://en.wikipedia.org/wiki/Product_(mathematics)) of an array

2457 * is the result of multiplying all numbers together, starting using one as the multiplicative identity.

2459 * This runs on `O(n)`, linear time in respect to the array

2461 * @param {Array<number>} x input

2462 * @return {number} product of all input numbers

2463 * @example

2464 * product([1, 2, 3, 4]); // => 24

2466 function product(x/*: Array<number> */)/*: number */ {

2467 var value = 1;

2468 for (var i = 0; i < x.length; i++) {

2469 value *= x[i];

2470 }

2471 return value;

2474 module.exports = product;

2476 },{}],40:[function(require,module,exports){

2477 'use strict';

2478 /* @flow */

2480 var quantileSorted = require(41);

2481 var quickselect = require(42);

2484 * The [quantile](https://en.wikipedia.org/wiki/Quantile):

2485 * this is a population quantile, since we assume to know the entire

2486 * dataset in this library. This is an implementation of the

2487 * [Quantiles of a Population](http://en.wikipedia.org/wiki/Quantile#Quantiles_of_a_population)

2488 * algorithm from wikipedia.

2490 * Sample is a one-dimensional array of numbers,

2491 * and p is either a decimal number from 0 to 1 or an array of decimal

2492 * numbers from 0 to 1.

2493 * In terms of a k/q quantile, p = k/q - it's just dealing with fractions or dealing

2494 * with decimal values.

2495 * When p is an array, the result of the function is also an array containing the appropriate

2496 * quantiles in input order

2498 * @param {Array<number>} sample a sample from the population

2499 * @param {number} p the desired quantile, as a number between 0 and 1

2500 * @returns {number} quantile

2501 * @example

2502 * quantile([3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20], 0.5); // => 9

2504 function quantile(sample /*: Array<number> */, p /*: Array<number> | number */) {

2505 var copy = sample.slice();

2507 if (Array.isArray(p)) {

2508 // rearrange elements so that each element corresponding to a requested

2509 // quantile is on a place it would be if the array was fully sorted

2510 multiQuantileSelect(copy, p);

2511 // Initialize the result array

2512 var results = [];

2513 // For each requested quantile

2514 for (var i = 0; i < p.length; i++) {

2515 results[i] = quantileSorted(copy, p[i]);

2516 }

2517 return results;

2518 } else {

2519 var idx = quantileIndex(copy.length, p);

2520 quantileSelect(copy, idx, 0, copy.length - 1);

2521 return quantileSorted(copy, p);

2522 }

2525 function quantileSelect(arr, k, left, right) {

2526 if (k % 1 === 0) {

2527 quickselect(arr, k, left, right);

2528 } else {

2529 k = Math.floor(k);

2530 quickselect(arr, k, left, right);

2531 quickselect(arr, k + 1, k + 1, right);

2532 }

2535 function multiQuantileSelect(arr, p) {

2536 var indices = [0];

2537 for (var i = 0; i < p.length; i++) {

2538 indices.push(quantileIndex(arr.length, p[i]));

2539 }

2540 indices.push(arr.length - 1);

2541 indices.sort(compare);

2543 var stack = [0, indices.length - 1];

2545 while (stack.length) {

2546 var r = Math.ceil(stack.pop());

2547 var l = Math.floor(stack.pop());

2548 if (r - l <= 1) continue;

2550 var m = Math.floor((l + r) / 2);

2551 quantileSelect(arr, indices[m], indices[l], indices[r]);

2553 stack.push(l, m, m, r);

2554 }

2557 function compare(a, b) {

2558 return a - b;

2561 function quantileIndex(len /*: number */, p /*: number */)/*:number*/ {

2562 var idx = len * p;

2563 if (p === 1) {

2564 // If p is 1, directly return the last index

2565 return len - 1;

2566 } else if (p === 0) {

2567 // If p is 0, directly return the first index

2568 return 0;

2569 } else if (idx % 1 !== 0) {

2570 // If index is not integer, return the next index in array

2571 return Math.ceil(idx) - 1;

2572 } else if (len % 2 === 0) {

2573 // If the list has even-length, we'll return the middle of two indices

2574 // around quantile to indicate that we need an average value of the two

2575 return idx - 0.5;

2576 } else {

2577 // Finally, in the simple case of an integer index

2578 // with an odd-length list, return the index

2579 return idx;

2580 }

2583 module.exports = quantile;

2585 },{"41":41,"42":42}],41:[function(require,module,exports){

2586 'use strict';

2587 /* @flow */

2590 * This is the internal implementation of quantiles: when you know

2591 * that the order is sorted, you don't need to re-sort it, and the computations

2592 * are faster.

2594 * @param {Array<number>} sample input data

2595 * @param {number} p desired quantile: a number between 0 to 1, inclusive

2596 * @returns {number} quantile value

2597 * @example

2598 * quantileSorted([3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20], 0.5); // => 9

2600 function quantileSorted(sample /*: Array<number> */, p /*: number */)/*:number*/ {

2601 var idx = sample.length * p;

2602 if (p < 0 || p > 1) {

2603 return NaN;

2604 } else if (p === 1) {

2605 // If p is 1, directly return the last element

2606 return sample[sample.length - 1];

2607 } else if (p === 0) {

2608 // If p is 0, directly return the first element

2609 return sample[0];

2610 } else if (idx % 1 !== 0) {

2611 // If p is not integer, return the next element in array

2612 return sample[Math.ceil(idx) - 1];

2613 } else if (sample.length % 2 === 0) {

2614 // If the list has even-length, we'll take the average of this number

2615 // and the next value, if there is one

2616 return (sample[idx - 1] + sample[idx]) / 2;

2617 } else {

2618 // Finally, in the simple case of an integer value

2619 // with an odd-length list, return the sample value at the index.

2620 return sample[idx];

2621 }

2624 module.exports = quantileSorted;

2626 },{}],42:[function(require,module,exports){

2627 'use strict';

2628 /* @flow */

2630 module.exports = quickselect;

2633 * Rearrange items in `arr` so that all items in `[left, k]` range are the smallest.

2634 * The `k`-th element will have the `(k - left + 1)`-th smallest value in `[left, right]`.

2636 * Implements Floyd-Rivest selection algorithm https://en.wikipedia.org/wiki/Floyd-Rivest_algorithm

2638 * @private

2639 * @param {Array<number>} arr input array

2640 * @param {number} k pivot index

2641 * @param {number} left left index

2642 * @param {number} right right index

2643 * @returns {undefined}

2644 * @example

2645 * var arr = [65, 28, 59, 33, 21, 56, 22, 95, 50, 12, 90, 53, 28, 77, 39];

2646 * quickselect(arr, 8);

2647 * // = [39, 28, 28, 33, 21, 12, 22, 50, 53, 56, 59, 65, 90, 77, 95]

2649 function quickselect(arr /*: Array<number> */, k /*: number */, left /*: number */, right /*: number */) {

2650 left = left || 0;

2651 right = right || (arr.length - 1);

2653 while (right > left) {

2654 // 600 and 0.5 are arbitrary constants chosen in the original paper to minimize execution time

2655 if (right - left > 600) {

2656 var n = right - left + 1;

2657 var m = k - left + 1;

2658 var z = Math.log(n);

2659 var s = 0.5 * Math.exp(2 * z / 3);

2660 var sd = 0.5 * Math.sqrt(z * s * (n - s) / n);

2661 if (m - n / 2 < 0) sd *= -1;

2662 var newLeft = Math.max(left, Math.floor(k - m * s / n + sd));

2663 var newRight = Math.min(right, Math.floor(k + (n - m) * s / n + sd));

2664 quickselect(arr, k, newLeft, newRight);

2665 }

2667 var t = arr[k];

2668 var i = left;

2669 var j = right;

2671 swap(arr, left, k);

2672 if (arr[right] > t) swap(arr, left, right);

2674 while (i < j) {

2675 swap(arr, i, j);

2676 i++;

2677 j--;

2678 while (arr[i] < t) i++;

2679 while (arr[j] > t) j--;

2680 }

2682 if (arr[left] === t) swap(arr, left, j);

2683 else {

2684 j++;

2685 swap(arr, j, right);

2686 }

2688 if (j <= k) left = j + 1;

2689 if (k <= j) right = j - 1;

2690 }

2693 function swap(arr, i, j) {

2694 var tmp = arr[i];

2695 arr[i] = arr[j];

2696 arr[j] = tmp;

2699 },{}],43:[function(require,module,exports){

2700 'use strict';

2701 /* @flow */

2704 * The [R Squared](http://en.wikipedia.org/wiki/Coefficient_of_determination)

2705 * value of data compared with a function `f`

2706 * is the sum of the squared differences between the prediction

2707 * and the actual value.

2709 * @param {Array<Array<number>>} data input data: this should be doubly-nested

2710 * @param {Function} func function called on `[i][0]` values within the dataset

2711 * @returns {number} r-squared value

2712 * @example

2713 * var samples = [[0, 0], [1, 1]];

2714 * var regressionLine = linearRegressionLine(linearRegression(samples));

2715 * rSquared(samples, regressionLine); // = 1 this line is a perfect fit

2717 function rSquared(data /*: Array<Array<number>> */, func /*: Function */) /*: number */ {

2718 if (data.length < 2) { return 1; }

2720 // Compute the average y value for the actual

2721 // data set in order to compute the

2722 // _total sum of squares_

2723 var sum = 0, average;

2724 for (var i = 0; i < data.length; i++) {

2725 sum += data[i][1];

2726 }

2727 average = sum / data.length;

2729 // Compute the total sum of squares - the

2730 // squared difference between each point

2731 // and the average of all points.

2732 var sumOfSquares = 0;

2733 for (var j = 0; j < data.length; j++) {

2734 sumOfSquares += Math.pow(average - data[j][1], 2);

2735 }

2737 // Finally estimate the error: the squared

2738 // difference between the estimate and the actual data

2739 // value at each point.

2740 var err = 0;

2741 for (var k = 0; k < data.length; k++) {

2742 err += Math.pow(data[k][1] - func(data[k][0]), 2);

2743 }

2745 // As the error grows larger, its ratio to the

2746 // sum of squares increases and the r squared

2747 // value grows lower.

2748 return 1 - err / sumOfSquares;

2751 module.exports = rSquared;

2753 },{}],44:[function(require,module,exports){

2754 'use strict';

2755 /* @flow */

2758 * The Root Mean Square (RMS) is

2759 * a mean function used as a measure of the magnitude of a set

2760 * of numbers, regardless of their sign.

2761 * This is the square root of the mean of the squares of the

2762 * input numbers.

2763 * This runs on `O(n)`, linear time in respect to the array

2765 * @param {Array<number>} x input

2766 * @returns {number} root mean square

2767 * @example

2768 * rootMeanSquare([-1, 1, -1, 1]); // => 1

2770 function rootMeanSquare(x /*: Array<number> */)/*:number*/ {

2771 if (x.length === 0) { return NaN; }

2773 var sumOfSquares = 0;

2774 for (var i = 0; i < x.length; i++) {

2775 sumOfSquares += Math.pow(x[i], 2);

2776 }

2778 return Math.sqrt(sumOfSquares / x.length);

2781 module.exports = rootMeanSquare;

2783 },{}],45:[function(require,module,exports){

2784 'use strict';

2785 /* @flow */

2787 var shuffle = require(51);

2790 * Create a [simple random sample](http://en.wikipedia.org/wiki/Simple_random_sample)

2791 * from a given array of `n` elements.

2793 * The sampled values will be in any order, not necessarily the order

2794 * they appear in the input.

2796 * @param {Array} array input array. can contain any type

2797 * @param {number} n count of how many elements to take

2798 * @param {Function} [randomSource=Math.random] an optional source of entropy

2799 * instead of Math.random

2800 * @return {Array} subset of n elements in original array

2801 * @example

2802 * var values = [1, 2, 4, 5, 6, 7, 8, 9];

2803 * sample(values, 3); // returns 3 random values, like [2, 5, 8];

2805 function sample/*:: <T> */(

2806 array /*: Array<T> */,

2807 n /*: number */,

2808 randomSource /*: Function */) /*: Array<T> */ {

2809 // shuffle the original array using a fisher-yates shuffle

2810 var shuffled = shuffle(array, randomSource);

2812 // and then return a subset of it - the first `n` elements.

2813 return shuffled.slice(0, n);

2816 module.exports = sample;

2818 },{"51":51}],46:[function(require,module,exports){

2819 'use strict';

2820 /* @flow */

2822 var sampleCovariance = require(47);

2823 var sampleStandardDeviation = require(49);

2826 * The [correlation](http://en.wikipedia.org/wiki/Correlation_and_dependence) is

2827 * a measure of how correlated two datasets are, between -1 and 1

2829 * @param {Array<number>} x first input

2830 * @param {Array<number>} y second input

2831 * @returns {number} sample correlation

2832 * @example

2833 * sampleCorrelation([1, 2, 3, 4, 5, 6], [2, 2, 3, 4, 5, 60]).toFixed(2);

2834 * // => '0.69'

2836 function sampleCorrelation(x/*: Array<number> */, y/*: Array<number> */)/*:number*/ {

2837 var cov = sampleCovariance(x, y),

2838 xstd = sampleStandardDeviation(x),

2839 ystd = sampleStandardDeviation(y);

2841 return cov / xstd / ystd;

2844 module.exports = sampleCorrelation;

2846 },{"47":47,"49":49}],47:[function(require,module,exports){

2847 'use strict';

2848 /* @flow */

2850 var mean = require(25);

2853 * [Sample covariance](https://en.wikipedia.org/wiki/Sample_mean_and_sampleCovariance) of two datasets:

2854 * how much do the two datasets move together?

2855 * x and y are two datasets, represented as arrays of numbers.

2857 * @param {Array<number>} x first input

2858 * @param {Array<number>} y second input

2859 * @returns {number} sample covariance

2860 * @example

2861 * sampleCovariance([1, 2, 3, 4, 5, 6], [6, 5, 4, 3, 2, 1]); // => -3.5

2863 function sampleCovariance(x /*:Array<number>*/, y /*:Array<number>*/)/*:number*/ {

2865 // The two datasets must have the same length which must be more than 1

2866 if (x.length <= 1 || x.length !== y.length) {

2867 return NaN;

2868 }

2870 // determine the mean of each dataset so that we can judge each

2871 // value of the dataset fairly as the difference from the mean. this

2872 // way, if one dataset is [1, 2, 3] and [2, 3, 4], their covariance

2873 // does not suffer because of the difference in absolute values

2874 var xmean = mean(x),

2875 ymean = mean(y),

2876 sum = 0;

2878 // for each pair of values, the covariance increases when their

2879 // difference from the mean is associated - if both are well above

2880 // or if both are well below

2881 // the mean, the covariance increases significantly.

2882 for (var i = 0; i < x.length; i++) {

2883 sum += (x[i] - xmean) * (y[i] - ymean);

2884 }

2886 // this is Bessels' Correction: an adjustment made to sample statistics

2887 // that allows for the reduced degree of freedom entailed in calculating

2888 // values from samples rather than complete populations.

2889 var besselsCorrection = x.length - 1;

2891 // the covariance is weighted by the length of the datasets.

2892 return sum / besselsCorrection;

2895 module.exports = sampleCovariance;

2897 },{"25":25}],48:[function(require,module,exports){

2898 'use strict';

2899 /* @flow */

2901 var sumNthPowerDeviations = require(57);

2902 var sampleStandardDeviation = require(49);

2905 * [Skewness](http://en.wikipedia.org/wiki/Skewness) is

2906 * a measure of the extent to which a probability distribution of a

2907 * real-valued random variable "leans" to one side of the mean.

2908 * The skewness value can be positive or negative, or even undefined.

2910 * Implementation is based on the adjusted Fisher-Pearson standardized

2911 * moment coefficient, which is the version found in Excel and several

2912 * statistical packages including Minitab, SAS and SPSS.

2914 * @param {Array<number>} x input

2915 * @returns {number} sample skewness

2916 * @example

2917 * sampleSkewness([2, 4, 6, 3, 1]); // => 0.590128656384365

2919 function sampleSkewness(x /*: Array<number> */)/*:number*/ {

2920 // The skewness of less than three arguments is null

2921 var theSampleStandardDeviation = sampleStandardDeviation(x);

2923 if (isNaN(theSampleStandardDeviation) || x.length < 3) {

2924 return NaN;

2925 }

2927 var n = x.length,

2928 cubedS = Math.pow(theSampleStandardDeviation, 3),

2929 sumCubedDeviations = sumNthPowerDeviations(x, 3);

2931 return n * sumCubedDeviations / ((n - 1) * (n - 2) * cubedS);

2934 module.exports = sampleSkewness;

2936 },{"49":49,"57":57}],49:[function(require,module,exports){

2937 'use strict';

2938 /* @flow */

2940 var sampleVariance = require(50);

2943 * The [standard deviation](http://en.wikipedia.org/wiki/Standard_deviation)

2944 * is the square root of the variance.

2946 * @param {Array<number>} x input array

2947 * @returns {number} sample standard deviation

2948 * @example

2949 * sampleStandardDeviation([2, 4, 4, 4, 5, 5, 7, 9]).toFixed(2);

2950 * // => '2.14'

2952 function sampleStandardDeviation(x/*:Array<number>*/)/*:number*/ {

2953 // The standard deviation of no numbers is null

2954 var sampleVarianceX = sampleVariance(x);

2955 if (isNaN(sampleVarianceX)) { return NaN; }

2956 return Math.sqrt(sampleVarianceX);

2959 module.exports = sampleStandardDeviation;

2961 },{"50":50}],50:[function(require,module,exports){

2962 'use strict';

2963 /* @flow */

2965 var sumNthPowerDeviations = require(57);

2968 * The [sample variance](https://en.wikipedia.org/wiki/Variance#Sample_variance)

2969 * is the sum of squared deviations from the mean. The sample variance

2970 * is distinguished from the variance by the usage of [Bessel's Correction](https://en.wikipedia.org/wiki/Bessel's_correction):

2971 * instead of dividing the sum of squared deviations by the length of the input,

2972 * it is divided by the length minus one. This corrects the bias in estimating

2973 * a value from a set that you don't know if full.

2975 * References:

2976 * * [Wolfram MathWorld on Sample Variance](http://mathworld.wolfram.com/SampleVariance.html)

2978 * @param {Array<number>} x input array

2979 * @return {number} sample variance

2980 * @example

2981 * sampleVariance([1, 2, 3, 4, 5]); // => 2.5

2983 function sampleVariance(x /*: Array<number> */)/*:number*/ {

2984 // The variance of no numbers is null

2985 if (x.length <= 1) { return NaN; }

2987 var sumSquaredDeviationsValue = sumNthPowerDeviations(x, 2);

2989 // this is Bessels' Correction: an adjustment made to sample statistics

2990 // that allows for the reduced degree of freedom entailed in calculating

2991 // values from samples rather than complete populations.

2992 var besselsCorrection = x.length - 1;

2994 // Find the mean value of that list

2995 return sumSquaredDeviationsValue / besselsCorrection;

2998 module.exports = sampleVariance;

3000 },{"57":57}],51:[function(require,module,exports){

3001 'use strict';

3002 /* @flow */

3004 var shuffleInPlace = require(52);

3007 * A [Fisher-Yates shuffle](http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle)

3008 * is a fast way to create a random permutation of a finite set. This is

3009 * a function around `shuffle_in_place` that adds the guarantee that

3010 * it will not modify its input.

3012 * @param {Array} sample an array of any kind of element

3013 * @param {Function} [randomSource=Math.random] an optional entropy source

3014 * @return {Array} shuffled version of input

3015 * @example

3016 * var shuffled = shuffle([1, 2, 3, 4]);

3017 * shuffled; // = [2, 3, 1, 4] or any other random permutation

3019 function shuffle/*::<T>*/(sample/*:Array<T>*/, randomSource/*:Function*/) {

3020 // slice the original array so that it is not modified

3021 sample = sample.slice();

3023 // and then shuffle that shallow-copied array, in place

3024 return shuffleInPlace(sample.slice(), randomSource);

3027 module.exports = shuffle;

3029 },{"52":52}],52:[function(require,module,exports){

3030 'use strict';

3031 /* @flow */

3034 * A [Fisher-Yates shuffle](http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle)

3035 * in-place - which means that it **will change the order of the original

3036 * array by reference**.

3038 * This is an algorithm that generates a random [permutation](https://en.wikipedia.org/wiki/Permutation)

3039 * of a set.

3041 * @param {Array} sample input array

3042 * @param {Function} [randomSource=Math.random] an optional source of entropy

3043 * @returns {Array} sample

3044 * @example

3045 * var sample = [1, 2, 3, 4];

3046 * shuffleInPlace(sample);

3047 * // sample is shuffled to a value like [2, 1, 4, 3]

3049 function shuffleInPlace(sample/*:Array<any>*/, randomSource/*:Function*/)/*:Array<any>*/ {

3052 // a custom random number source can be provided if you want to use

3053 // a fixed seed or another random number generator, like

3054 // [random-js](https://www.npmjs.org/package/random-js)

3055 randomSource = randomSource || Math.random;

3057 // store the current length of the sample to determine

3058 // when no elements remain to shuffle.

3059 var length = sample.length;

3061 // temporary is used to hold an item when it is being

3062 // swapped between indices.

3063 var temporary;

3065 // The index to swap at each stage.

3066 var index;

3068 // While there are still items to shuffle

3069 while (length > 0) {

3070 // chose a random index within the subset of the array

3071 // that is not yet shuffled

3072 index = Math.floor(randomSource() * length--);

3074 // store the value that we'll move temporarily

3075 temporary = sample[length];

3077 // swap the value at `sample[length]` with `sample[index]`

3078 sample[length] = sample[index];

3079 sample[index] = temporary;

3080 }

3082 return sample;

3085 module.exports = shuffleInPlace;

3087 },{}],53:[function(require,module,exports){

3088 'use strict';

3089 /* @flow */

3092 * [Sign](https://en.wikipedia.org/wiki/Sign_function) is a function

3093 * that extracts the sign of a real number

3095 * @param {Number} x input value

3096 * @returns {Number} sign value either 1, 0 or -1

3097 * @throws {TypeError} if the input argument x is not a number

3098 * @private

3100 * @example

3101 * sign(2); // => 1

3103 function sign(x/*: number */)/*: number */ {

3104 if (typeof x === 'number') {

3105 if (x < 0) {

3106 return -1;

3107 } else if (x === 0) {

3108 return 0

3109 } else {

3110 return 1;

3111 }

3112 } else {

3113 throw new TypeError('not a number');

3114 }

3117 module.exports = sign;

3119 },{}],54:[function(require,module,exports){

3120 'use strict';

3121 /* @flow */

3123 var variance = require(62);

3126 * The [standard deviation](http://en.wikipedia.org/wiki/Standard_deviation)

3127 * is the square root of the variance. It's useful for measuring the amount

3128 * of variation or dispersion in a set of values.

3130 * Standard deviation is only appropriate for full-population knowledge: for

3131 * samples of a population, {@link sampleStandardDeviation} is

3132 * more appropriate.

3134 * @param {Array<number>} x input

3135 * @returns {number} standard deviation

3136 * @example

3137 * variance([2, 4, 4, 4, 5, 5, 7, 9]); // => 4

3138 * standardDeviation([2, 4, 4, 4, 5, 5, 7, 9]); // => 2

3140 function standardDeviation(x /*: Array<number> */)/*:number*/ {

3141 // The standard deviation of no numbers is null

3142 var v = variance(x);

3143 if (isNaN(v)) { return 0; }

3144 return Math.sqrt(v);

3147 module.exports = standardDeviation;

3149 },{"62":62}],55:[function(require,module,exports){

3150 'use strict';

3151 /* @flow */

3153 var SQRT_2PI = Math.sqrt(2 * Math.PI);

3155 function cumulativeDistribution(z) {

3156 var sum = z,

3157 tmp = z;

3159 // 15 iterations are enough for 4-digit precision

3160 for (var i = 1; i < 15; i++) {

3161 tmp *= z * z / (2 * i + 1);

3162 sum += tmp;

3163 }

3164 return Math.round((0.5 + (sum / SQRT_2PI) * Math.exp(-z * z / 2)) * 1e4) / 1e4;

3168 * A standard normal table, also called the unit normal table or Z table,

3169 * is a mathematical table for the values of Φ (phi), which are the values of

3170 * the cumulative distribution function of the normal distribution.

3171 * It is used to find the probability that a statistic is observed below,

3172 * above, or between values on the standard normal distribution, and by

3173 * extension, any normal distribution.

3175 * The probabilities are calculated using the

3176 * [Cumulative distribution function](https://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function).

3177 * The table used is the cumulative, and not cumulative from 0 to mean

3178 * (even though the latter has 5 digits precision, instead of 4).

3180 var standardNormalTable/*: Array<number> */ = [];

3182 for (var z = 0; z <= 3.09; z += 0.01) {

3183 standardNormalTable.push(cumulativeDistribution(z));

3186 module.exports = standardNormalTable;

3188 },{}],56:[function(require,module,exports){

3189 'use strict';

3190 /* @flow */

3193 * Our default sum is the [Kahan summation algorithm](https://en.wikipedia.org/wiki/Kahan_summation_algorithm) is

3194 * a method for computing the sum of a list of numbers while correcting

3195 * for floating-point errors. Traditionally, sums are calculated as many

3196 * successive additions, each one with its own floating-point roundoff. These

3197 * losses in precision add up as the number of numbers increases. This alternative

3198 * algorithm is more accurate than the simple way of calculating sums by simple

3199 * addition.

3201 * This runs on `O(n)`, linear time in respect to the array

3203 * @param {Array<number>} x input

3204 * @return {number} sum of all input numbers

3205 * @example

3206 * sum([1, 2, 3]); // => 6

3208 function sum(x/*: Array<number> */)/*: number */ {

3210 // like the traditional sum algorithm, we keep a running

3211 // count of the current sum.

3212 var sum = 0;

3214 // but we also keep three extra variables as bookkeeping:

3215 // most importantly, an error correction value. This will be a very

3216 // small number that is the opposite of the floating point precision loss.

3217 var errorCompensation = 0;

3219 // this will be each number in the list corrected with the compensation value.

3220 var correctedCurrentValue;

3222 // and this will be the next sum

3223 var nextSum;

3225 for (var i = 0; i < x.length; i++) {

3226 // first correct the value that we're going to add to the sum

3227 correctedCurrentValue = x[i] - errorCompensation;

3229 // compute the next sum. sum is likely a much larger number

3230 // than correctedCurrentValue, so we'll lose precision here,

3231 // and measure how much precision is lost in the next step

3232 nextSum = sum + correctedCurrentValue;

3234 // we intentionally didn't assign sum immediately, but stored

3235 // it for now so we can figure out this: is (sum + nextValue) - nextValue

3236 // not equal to 0? ideally it would be, but in practice it won't:

3237 // it will be some very small number. that's what we record

3238 // as errorCompensation.

3239 errorCompensation = nextSum - sum - correctedCurrentValue;

3241 // now that we've computed how much we'll correct for in the next

3242 // loop, start treating the nextSum as the current sum.

3243 sum = nextSum;

3244 }

3246 return sum;

3249 module.exports = sum;

3251 },{}],57:[function(require,module,exports){

3252 'use strict';

3253 /* @flow */

3255 var mean = require(25);

3258 * The sum of deviations to the Nth power.

3259 * When n=2 it's the sum of squared deviations.

3260 * When n=3 it's the sum of cubed deviations.

3262 * @param {Array<number>} x

3263 * @param {number} n power

3264 * @returns {number} sum of nth power deviations

3265 * @example

3266 * var input = [1, 2, 3];

3267 * // since the variance of a set is the mean squared

3268 * // deviations, we can calculate that with sumNthPowerDeviations:

3269 * var variance = sumNthPowerDeviations(input) / input.length;

3271 function sumNthPowerDeviations(x/*: Array<number> */, n/*: number */)/*:number*/ {

3272 var meanValue = mean(x),

3273 sum = 0;

3275 for (var i = 0; i < x.length; i++) {

3276 sum += Math.pow(x[i] - meanValue, n);

3277 }

3279 return sum;

3282 module.exports = sumNthPowerDeviations;

3284 },{"25":25}],58:[function(require,module,exports){

3285 'use strict';

3286 /* @flow */

3289 * The simple [sum](https://en.wikipedia.org/wiki/Summation) of an array

3290 * is the result of adding all numbers together, starting from zero.

3292 * This runs on `O(n)`, linear time in respect to the array

3294 * @param {Array<number>} x input

3295 * @return {number} sum of all input numbers

3296 * @example

3297 * sumSimple([1, 2, 3]); // => 6

3299 function sumSimple(x/*: Array<number> */)/*: number */ {

3300 var value = 0;

3301 for (var i = 0; i < x.length; i++) {

3302 value += x[i];

3303 }

3304 return value;

3307 module.exports = sumSimple;

3309 },{}],59:[function(require,module,exports){

3310 'use strict';

3311 /* @flow */

3313 var standardDeviation = require(54);

3314 var mean = require(25);

3317 * This is to compute [a one-sample t-test](https://en.wikipedia.org/wiki/Student%27s_t-test#One-sample_t-test), comparing the mean

3318 * of a sample to a known value, x.

3320 * in this case, we're trying to determine whether the

3321 * population mean is equal to the value that we know, which is `x`

3322 * here. usually the results here are used to look up a

3323 * [p-value](http://en.wikipedia.org/wiki/P-value), which, for

3324 * a certain level of significance, will let you determine that the

3325 * null hypothesis can or cannot be rejected.

3327 * @param {Array<number>} sample an array of numbers as input

3328 * @param {number} x expected value of the population mean

3329 * @returns {number} value

3330 * @example

3331 * tTest([1, 2, 3, 4, 5, 6], 3.385).toFixed(2); // => '0.16'

3333 function tTest(sample/*: Array<number> */, x/*: number */)/*:number*/ {

3334 // The mean of the sample

3335 var sampleMean = mean(sample);

3337 // The standard deviation of the sample

3338 var sd = standardDeviation(sample);

3340 // Square root the length of the sample

3341 var rootN = Math.sqrt(sample.length);

3343 // returning the t value

3344 return (sampleMean - x) / (sd / rootN);

3347 module.exports = tTest;

3349 },{"25":25,"54":54}],60:[function(require,module,exports){

3350 'use strict';

3351 /* @flow */

3353 var mean = require(25);

3354 var sampleVariance = require(50);

3357 * This is to compute [two sample t-test](http://en.wikipedia.org/wiki/Student's_t-test).

3358 * Tests whether "mean(X)-mean(Y) = difference", (

3359 * in the most common case, we often have `difference == 0` to test if two samples

3360 * are likely to be taken from populations with the same mean value) with

3361 * no prior knowledge on standard deviations of both samples

3362 * other than the fact that they have the same standard deviation.

3364 * Usually the results here are used to look up a

3365 * [p-value](http://en.wikipedia.org/wiki/P-value), which, for

3366 * a certain level of significance, will let you determine that the

3367 * null hypothesis can or cannot be rejected.

3369 * `diff` can be omitted if it equals 0.

3371 * [This is used to confirm or deny](http://www.monarchlab.org/Lab/Research/Stats/2SampleT.aspx)

3372 * a null hypothesis that the two populations that have been sampled into

3373 * `sampleX` and `sampleY` are equal to each other.

3375 * @param {Array<number>} sampleX a sample as an array of numbers

3376 * @param {Array<number>} sampleY a sample as an array of numbers

3377 * @param {number} [difference=0]

3378 * @returns {number} test result

3379 * @example

3380 * ss.tTestTwoSample([1, 2, 3, 4], [3, 4, 5, 6], 0); //= -2.1908902300206643

3382 function tTestTwoSample(

3383 sampleX/*: Array<number> */,

3384 sampleY/*: Array<number> */,

3385 difference/*: number */) {

3386 var n = sampleX.length,

3387 m = sampleY.length;

3389 // If either sample doesn't actually have any values, we can't

3390 // compute this at all, so we return `null`.

3391 if (!n || !m) { return null; }

3393 // default difference (mu) is zero

3394 if (!difference) {

3395 difference = 0;

3396 }

3398 var meanX = mean(sampleX),

3399 meanY = mean(sampleY),

3400 sampleVarianceX = sampleVariance(sampleX),

3401 sampleVarianceY = sampleVariance(sampleY);

3403 if (typeof meanX === 'number' &&

3404 typeof meanY === 'number' &&

3405 typeof sampleVarianceX === 'number' &&

3406 typeof sampleVarianceY === 'number') {

3407 var weightedVariance = ((n - 1) * sampleVarianceX +

3408 (m - 1) * sampleVarianceY) / (n + m - 2);

3410 return (meanX - meanY - difference) /

3411 Math.sqrt(weightedVariance * (1 / n + 1 / m));

3412 }

3415 module.exports = tTestTwoSample;

3417 },{"25":25,"50":50}],61:[function(require,module,exports){

3418 'use strict';

3419 /* @flow */

3422 * For a sorted input, counting the number of unique values

3423 * is possible in constant time and constant memory. This is

3424 * a simple implementation of the algorithm.

3426 * Values are compared with `===`, so objects and non-primitive objects

3427 * are not handled in any special way.

3429 * @param {Array} input an array of primitive values.

3430 * @returns {number} count of unique values

3431 * @example

3432 * uniqueCountSorted([1, 2, 3]); // => 3

3433 * uniqueCountSorted([1, 1, 1]); // => 1

3435 function uniqueCountSorted(input/*: Array<any>*/)/*: number */ {

3436 var uniqueValueCount = 0,

3437 lastSeenValue;

3438 for (var i = 0; i < input.length; i++) {

3439 if (i === 0 || input[i] !== lastSeenValue) {

3440 lastSeenValue = input[i];

3441 uniqueValueCount++;

3442 }

3443 }

3444 return uniqueValueCount;

3447 module.exports = uniqueCountSorted;

3449 },{}],62:[function(require,module,exports){

3450 'use strict';

3451 /* @flow */

3453 var sumNthPowerDeviations = require(57);

3456 * The [variance](http://en.wikipedia.org/wiki/Variance)

3457 * is the sum of squared deviations from the mean.

3459 * This is an implementation of variance, not sample variance:

3460 * see the `sampleVariance` method if you want a sample measure.

3462 * @param {Array<number>} x a population

3463 * @returns {number} variance: a value greater than or equal to zero.

3464 * zero indicates that all values are identical.

3465 * @example

3466 * variance([1, 2, 3, 4, 5, 6]); // => 2.9166666666666665

3468 function variance(x/*: Array<number> */)/*:number*/ {

3469 // The variance of no numbers is null

3470 if (x.length === 0) { return NaN; }

3472 // Find the mean of squared deviations between the

3473 // mean value and each value.

3474 return sumNthPowerDeviations(x, 2) / x.length;

3477 module.exports = variance;

3479 },{"57":57}],63:[function(require,module,exports){

3480 'use strict';

3481 /* @flow */

3484 * The [Z-Score, or Standard Score](http://en.wikipedia.org/wiki/Standard_score).

3486 * The standard score is the number of standard deviations an observation

3487 * or datum is above or below the mean. Thus, a positive standard score

3488 * represents a datum above the mean, while a negative standard score

3489 * represents a datum below the mean. It is a dimensionless quantity

3490 * obtained by subtracting the population mean from an individual raw

3491 * score and then dividing the difference by the population standard

3492 * deviation.

3494 * The z-score is only defined if one knows the population parameters;

3495 * if one only has a sample set, then the analogous computation with

3496 * sample mean and sample standard deviation yields the

3497 * Student's t-statistic.

3499 * @param {number} x

3500 * @param {number} mean

3501 * @param {number} standardDeviation

3502 * @return {number} z score

3503 * @example

3504 * zScore(78, 80, 5); // => -0.4

3506 function zScore(x/*:number*/, mean/*:number*/, standardDeviation/*:number*/)/*:number*/ {

3507 return (x - mean) / standardDeviation;

3510 module.exports = zScore;

3512 },{}]},{},[1])(1)

3514 //# sourceMappingURL=simple-statistics.js.map