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/**
* Functions to help with numerical computations.
*
* @module compute
*/
/**
* Perform subtraction on each pair from two sequences.
*
* Example:
*
* const seqA = [ 3.0, 3.0, 4.5 ];
* const seqB = [ 2.5, 2.6, 3.3 ];
* subtract( seqA, seqB );
* // [ 0.5, 0.4, 1.2 ]
*
* @param {number[]} seqA
* @param {number[]} seqB
* @return {number[]}
*/
function subtract( seqA, seqB ) {
return seqA.map( ( a, i ) => a - seqB[ i ] );
}
/**
* Compute statistics about a sequence of numbers.
*
* Example:
*
* stats( [ 3, 4, 5 ] );
* // mean: 4.0, stdev: 0.82
*
* @param {number[]} values
* @return {Object} An object holding the mean average (`mean`),
* standard deviation (`stdev`) and values (`values`).
*/
function stats( values ) {
// The mean average:
// - sum total
// - number of values
// - answer = sum / number
let sum = 0;
for ( const value of values ) {
sum += value;
}
const mean = sum / values.length;
// The standard deviation:
// - the mean average
// - distances from values to the mean, squared
// - sum of squared distances
// - answer = square root of (sum / number)
// Formula courtesy of Khan Academy
// https://www.khanacademy.org/math/probability/data-distributions-a1/summarizing-spread-distributions/a/calculating-standard-deviation-step-by-step
let sqDiffSum = 0;
for ( const value of values ) {
sqDiffSum += Math.pow( value - mean, 2 );
}
const stdev = Math.sqrt( sqDiffSum / values.length );
return { mean, stdev, values };
}
/**
* Compare two objects from `stats()`.
*
* Example:
*
* const a = stats( [ 3, 4, 5 ] ); // mean: 4.0, stdev: 0.82
* const b = stats( [ 1.0, 1.5, 2.0 ] ); // mean: 1.5, stdev: 0.41
* diffStdev( a, b );
* // -1.27
*
* This is computed by creating a range of 1 stdev aroud each mean,
* and if they don't overlap, the distance between them is returned.
*
* In the above example, the range for sequence A is `3.18 ... 4.82`,
* and the range for B is `1.09 ... 1.91`. The ranges don't overlap and
* the distance between 3.18 and 1.91 is -1.27.
*
* @param {Object} before
* @param {Object} after
* @return {number} The difference between the before and after mean averages,
* after having compensated for 1 standard deviation. If lower numbers are better
* for your metric, then a negative difference represents an improvement.
*/
function diffStdev( before, after ) {
const beforeStart = before.mean - before.stdev;
const beforeEnd = before.mean + before.stdev;
const afterStart = after.mean - after.stdev;
const afterEnd = after.mean + after.stdev;
if ( afterEnd < beforeStart ) {
// Got lower
return afterEnd - beforeStart;
}
if ( beforeEnd < afterStart ) {
// Got higher
return afterStart - beforeEnd;
}
// Unchanged
return 0;
}
/**
* Find the rank for each value, giving any tied values the mean of the ranks
* that they cover. The ranks are used to calculate the U score. Also find
* the adjustment constant, used for calculating the standard deviation of U.
*
* Example:
*
* values: [ 4, 9, 8, 7, 3, 6, 6 ]
* sorted: [ 3, 4, 6, 6, 7, 8, 9 ]
* place: [ 1, 6, 5, 4, 0, 2, 3 ]
* ranks: [ 2, 7, 6, 5, 1, 3.5, 3.5 ]
*
* @param {number[]} values
* @return {Array} ranks of the values, adjustment constant
*/
function ranks( values ) {
// Sort the values
const sorted = values.slice().sort( ( a, b ) => {
return a - b;
} );
// Find the index of each value in the sorted array
const startSearch = {};
const place = sorted.map( ( v, _i ) => {
const ret = values.indexOf( v, startSearch[ v ] );
startSearch[ v ] = ret + 1;
return ret;
} );
// Find the rank of each value
// The rank is usually the index + 1, except...
// For tied values, the rank is the mean of their indices + 1
let adjustment = 0;
let i = 0;
const order = [];
while ( i < values.length ) {
let j = i;
while (
j + 1 <= values.length &&
values[ place[ i ] ] === values[ place[ j + 1 ] ]
) {
j += 1;
}
if ( j > i ) {
// There are tied values, so find the mean of their ranks
const numTies = j - i + 1;
let meanRank = 0;
for ( let k = i; k < j + 1; k++ ) {
meanRank += k;
}
meanRank /= numTies;
for ( let k = i; k < j + 1; k++ ) {
order[ place[ k ] ] = meanRank + 1;
}
// Adjustment constant is t^3 - t, where t is the number of ties
adjustment += Math.pow( numTies, 3 ) - numTies;
} else {
// This value is unique
order[ place[ i ] ] = i + 1;
}
i = j + 1;
}
return [ order, adjustment ];
}
/**
* Perform an approximate Mann-Whitney U test on two sets of values to test
* whether the values in the second set are significantly higher. The test
* is a non-parametric test that compares the ranks of the values without
* assuming they are distributed in a particular way.
*
* For details of the test and calculations see:
* https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test
*
* This implementation is paraphrased from:
* https://github.com/JuliaStats/HypothesisTests.jl/blob/b28a4587fe/src/mann_whitney.jl
*
* Assumptions made by this implementation:
* - each set contains the same number of values
* - we are interested in whether values in the second set are higher
* - the sample is large enough to use the approximate test
*
* @param {Object} before
* @param {Object} after
* @return {number} The p-value representing the likelihood of getting the
* observed U score (or more extreme) under the null hypothesis, that a
* randomly-chosen value from either set is equally likely to be higher or
* lower than a randomly-chosen value from the other set.
*/
function mannWhitney( before, after ) {
const beforeLen = before.values.length;
const afterLen = after.values.length;
const values = before.values.concat( after.values );
const [ order, adjustment ] = ranks( values );
const sumBeforeRanks = order
.slice( 0, beforeLen )
.reduce( ( a, b ) => {
return a + b;
} );
const U = sumBeforeRanks - ( beforeLen * ( beforeLen + 1 ) / 2 );
// U statistic mean
const mu = U - ( beforeLen * afterLen / 2 );
// U statistic standard deviation
const sigma = Math.sqrt(
(
beforeLen * afterLen * (
beforeLen + afterLen + 1 - adjustment / (
( beforeLen + afterLen ) * ( beforeLen + afterLen - 1 )
)
)
) / 12
);
if ( mu === 0 && sigma === 0 ) {
// Values all equal
return 1;
} else {
const z = ( mu + 0.5 ) / sigma;
// Approximation to Normal distribution CDF from
// http://web2.uwindsor.ca/math/hlynka/zogheibhlynka.pdf
return 1 / ( 1 + Math.pow(
Math.E, ( 0.0054 - 1.6101 * z - 0.0674 * Math.pow( z, 3 ) )
) );
}
}
/**
* Compare two sets of values using the Mann-Witney U test.
*
* @see {@link module:compute~mannWhitney #mannWhitney}
* @param {Object} before
* @param {Object} after
* @return {number} Number between 0.0 and 1.0. A higher number may suggest the values have
* increased, and a lower number may suggest the values remained the same or got lower.
* It is computed as 1 minus the {@link module:compute~mannWhitney Mann-Whitney p-value}.
*/
function diffMannWhitney( before, after ) {
return 1 - mannWhitney( before, after );
}
module.exports = { subtract, stats, diffStdev, mannWhitney, diffMannWhitney };
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