@bunmchmark/stats

This provides the (robust) statistics primitives used in bunchmark.

They all present tradeoffs that favor speed and implementation simplicity, when doing so doesn't impair robustness for our use case. These limitations are systematically documented and could be lifted over time if there is demand.

• @bunmchmark/stats
  • Median, MAD, percentiles and quickselect
    • getStats(array: number[], bonferonni: number = 1): Stats
    • quickselect(array: number[], i: number, left: number = 0): number
    • quickselect(array: T[], i: number, left?: number = 0, compare: (a:T, b: T) => number): number
    • quickselectFloat(array, i, left): number
    • percentile(array, p, left)
    • median(array)
    • MAD(array, median = median(array))
  • Statistical tests
    • mwu(A: number[], B: number[]): {U: number, z: number, p:number}
    • wilcoxon(D: number[]): {T: number, z: number, p: number}
    • wilcoxon(A: number[], B: number): {T: number, z: number, p: number}
  • Probability helpers
    • pForZ(z)
    • zForP(p)

Median, MAD, percentiles and quickselect

getStats(array: number[], bonferonni: number = 1): Stats

where type Stats = {q1: number, lowCI: number, median: number, highCI: number, q3: number, MAD: number}.

• median is the median of the array
• MAD is the median absolute deviation (see the dedicated section).
• q1 and q3 are the first and third quartiles
• lowCI and highCI are the bounds of the p=.95 confidence interval over the median, taking into account the bonferonni factor (the total number of comparisons to make, i.e. n*(n-1)/2 for n categories).

Important: this is built on top of quickselect(). It will reorder the content of the array (see quickselect() for the details). If you want to do a Wilcoxon rank-sum test, it must be done before this step, or on an independant copy of the data set. This is because the Wilcoxon test relies on the order of the arrays to pair items.

quickselect(array: number[], i: number, left: number = 0): number

quickselect(array: T[], i: number, left?: number = 0, compare: (a:T, b: T) => number): number

i, left are integer indices between 0 and array.length - 1. i must be larger than or equal to left.

Reorders array such that

• array[i] is the value that we would get when doing array.sort((a, b)=> a - b)[i]
• all the elements smaller than that value end up to its left
• thus all the elements larger than array[i] end up to its right

Values with indices smaller than left are ignored.

A custom compare can be be provided. It must return a negative value when a < b, 0 when a === b and a positive value otherwise. It is mandatory for non-numeric arrays.

Returns array[i].

If several values are needed for a given array, it is strategic to get them from the smallest to the largest index, using the previous index as the left parameter. This takes advantage of the way quickselect() reoders its input array.

Suppose we want to get both i0 and i1 where i0 < i1, we can do

const v0 = quickselect(array, i0)
const v1 = quickselect(array, i1, i0)

When doing the second call, we know that the values at indices smaller than i0 are smaller than array[i0]. They can thus be ignored, saving us some cycles.

Our implementation is derived from Vladimir Agafonkin's https://github.com/mourner/quickselect with minute modifications.

quickselectFloat(array, i, left): number

Like quickselect but i, and left, can be floats. If the aren't integers, this will return a linear interpolation(†) between Math.floor(i) and ceil(i) proporional to the decimal part.

array ends up with array[ceil(i)] equals to array.sort((a, b) => a-b)[ceil(i)] (see quickselect() for how the rest of the array ends up).

Values at indices smaller than ceil(left) are left untouched (see quickselect for the details)

†. My implementation is quite naive, there are considerations in that section that are beyond my current understanding.

percentile(array, p, left)

Where p is a decimal value between 0 and 1.

Equivalent to quickselectFloat(array, p * (array.length - 1), left * (array.length - 1))

median(array)

Equivalent to percentile(array, .5).

MAD(array, median = median(array))

Returns the median absolute deviation of the array, a robust mesure of variablility that stands in for standard deviation.

It uses quickselectFloat(array, (a, b) => abs(a - median) - abs(b - median)) under the hood and reorders the values in array accordingly.

Statistical tests

We provide the Wilcoxon rank-sum test and the Mann-Whitne U test.

These are bare, limited versions of the corresponding SciPy routines. I've only implemented what I need for bunchmark, if you need more feel free to open an issue, or idealy a PR. Some things are easy to implement (one-tailed tests, optional continuity correction). Exact probability calculations and alternative zero-hanlding procedures for the Wilcoxon test would be a bit more involved.

mwu(A: number[], B: number[]): {U: number, z: number, p:number}

The Mann-Whitney U test

• two tailed
• asymptotic p-value estimation (can't be used when N<8), with continuity correction (more conservative)
• Assumes that A and B don't contain NaNs

Inspired by the second method found in https://en.wikipedia.org/w/index.php?title=Mann%E2%80%93Whitney_U_test&oldid=1188631305#Calculations and sci-py for the tie correction of the variance computation

wilcoxon(D: number[]): {T: number, z: number, p: number}

A Wilcoxon rank-sum test:

• two tailed
• Pratt signed-rank zero procedure
• asymptotic p-value estimation (can't be used when N < 10), with continuity correction (more conservative)
• Assumes that D doesn't contain NaNs

This is based on https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test#Ties and scipy.stats.wilcoxon

wilcoxon(A: number[], B: number): {T: number, z: number, p: number}

Equivalent to wilcoxon(D) where D is the pairwise difference between A and B.

• Assumes that A and B don't contain NaNs

Probability helpers

Hereunder, "variable" is meant in the statistical sense.

pForZ(z)

Gets the p-value corresponding to a z-value (the probability of finding a value smaller than z by chance in a normally distributied variable).

This is based on a z-table built with scipy.stats.sf() with 0.001 granularity in the z-domain, for z between 0 and 9.999. That level of precision is sufficient for our use case.

To get even more precision, we'd have to approximate the integral from -Infinity to z for the normal distribution using the sum of trapeze algorithm (there are no closed-form equations to compute these values).

Equivalent to scipy.stats.cdf(round(z, 3)) in SciPy or pnorm(round(z, 3)) in R.

zForP(p)

Find z such that the probability of finding a value smaller than than it in a normally distributed variable has a probability p.

Equivalent to scipy.stats.norm.ppf(p) in SciPy and qnorm(p) in R.

This is a direct port of the SciPy C++ version.