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* Averaging a track over a set of places, in the two shapes that are worth
* asking for.
*
* **Reference-point** takes a point on every feature — its start, its end, its
* centre — and reads a fixed flank either side. It is what you want when the
* features are points or when their edges are the thing: a TSS, a peak summit, a
* called boundary.
*
* **Scaled-region** resamples every feature's *body* to the same width and puts
* a flank on each end. It is what you want when the features have extent and
* that extent varies: genes, domains. A reference-point plot over genes of
* wildly different lengths averages the middle of a short gene against the start
* of a long one and produces a shape that belongs to neither.
*
* Four decisions in here are the ones that decide whether the answer means
* anything.
*
* **Strand.** A feature on the minus strand is read right to left, or every
* asymmetric signal cancels against itself and a promoter looks symmetric.
*
* **The average, and the band around it.** The mean of a bimodal set is a line
* that describes nothing, which is why the companion heatmap exists and why the
* median is offered. The band is a **percentile bootstrap over features**, not a
* standard error: signal over a feature set is not normal, its variance is not
* the interesting thing, and a ±2 SEM ribbon on a skewed distribution is a claim
* about a Gaussian nobody has.
*
* **Missing data.** A window that runs off the end of a sequence is clipped, and
* the slots outside it are `NaN` rather than zero — so a feature near a telomere
* contributes to the slots it has evidence for and to no others.
*
* **Comparing tracks.** Two tracks share a pair of axes only if they share a
* unit, and a CPM bigWig and a log2 ratio do not. {@link standardiseTrack} is
* the way out — each track centred and scaled by its *own* spread, which is the
* one unit they have in common — and its whole difficulty is which population
* the statistics are taken over. That is answered there, at length, because
* getting it wrong deletes the ranking the reader split the set to see.
*/
import type { TrackSamples } from '../annotations/types'
import { nanMean, nanMedian, nanTrimmedMean } from './stats'
import type { ScaledAxis } from './types'
/** Where on a feature a reference-point window is centred. */
export type Reference = 'start' | 'end' | 'centre'
/** How the per-feature rows are collapsed into a line. */
export type Average = 'mean' | 'median' | 'trimmed'
/** One place to read, already resolved to coordinates and an orientation. */
export interface Window {
contig: string
/** Half-open, clamped to the sequence. */
start: number
end: number
/** `-1` reads the window right to left. */
strand: number
/**
* Slots this window fills, and where it starts filling them.
*
* A window clipped at a sequence end covers fewer slots than the layout has,
* and this is which ones — so the row lands in the right columns rather than
* being stretched across all of them, which would shift a telomeric feature's
* signal inwards by however much was clipped.
*/
slotFrom: number
slotTo: number
/**
* The stretch the slots from {@link slotFrom} to {@link slotTo} stand for,
* where it is not `start` to `end`: a reference-point window clipped at a
* sequence end part of the way through a column. Its columns are still
* `step` base pairs each, counted from the anchor, and the one the sequence
* ends in is the mean over the bases it has — as deepTools' `computeMatrix`
* pads the bases past the end with gaps — rather than the rest of the window
* spread over a whole number of columns, which drew every column of a
* telomeric feature's row a few percent off its place.
*/
span?: { start: number; end: number }
/**
* The feature's own extent, and the columns it owns — a scaled-region window
* only.
*
* This is what makes *flank, body, flank* mean what it says. Without it
* {@link windowRow} spreads a window's base pairs evenly over its slots,
* which is exactly right for a reference-point window — its span is the same
* length for every feature, so every column is `step` base pairs — and wrong
* for a scaled one, whose columns are base pairs at the ends and a
* *proportion* in the middle. Spread evenly, each feature's body would begin
* wherever its own length happened to put it: column 15 for a 20 kb gene,
* column 4 for a 100 kb one, against a {@link Layout} that promises column 40
* for both.
*
* Left `undefined` by {@link referenceWindows}, which is what keeps that
* mapping the single linear one it should be.
*/
body?: {
/** The feature itself, clamped into the window. */
start: number
end: number
/** The slots the body fills, in reading order, clamped into the window's. */
slotFrom: number
slotTo: number
}
}
/** The column layout a run produces, and what each column means. */
export interface Layout {
/** Total columns. */
slots: number
/** Column at which the body begins; equals the upstream flank's width. */
bodyFrom: number
/** Column one past the body. Equals {@link slots} minus the downstream flank. */
bodyTo: number
/** Base pairs per column in the flanks; `null` for a scaled body. */
flankStep: number
}
/**
* The windows for a reference-point run.
*
* `slotFrom`/`slotTo` carry the clipping, so a feature 300 bp from the start of
* a chromosome with a 2 kb flank fills the columns it can reach and leaves the
* rest empty — rather than reading a 300 bp window and having it resampled
* across the full width, which would silently compress a promoter into a
* quarter of the plot.
*/
export function referenceWindows(
features: ReadonlyArray<{ contig: string; start: number; end: number; strand: number }>,
options: { at: Reference; flank: number; step: number; lengthOf(contig: string): number },
): { windows: Window[]; layout: Layout } {
const perSide = Math.max(1, Math.round(options.flank / options.step))
const slots = perSide * 2
const layout: Layout = { slots, bodyFrom: perSide, bodyTo: perSide, flankStep: options.step }
const windows: Window[] = []
for (const feature of features) {
const anchor =
options.at === 'start'
? feature.strand < 0
? feature.end
: feature.start
: options.at === 'end'
? feature.strand < 0
? feature.start
: feature.end
: // deepTools' centre (`chopRegionsFromMiddle`): half the length, rounded down, from the
// start. Rounding the midpoint instead put an odd-length feature one base to the right,
// which on a sharp peak in 50 bp columns moved its mean by a percent.
feature.start + Math.floor((feature.end - feature.start) / 2)
const length = options.lengthOf(feature.contig)
const wanted = { from: anchor - perSide * options.step, to: anchor + perSide * options.step }
const start = Math.max(0, wanted.from)
const end = Math.min(length, wanted.to)
if (!(end > start)) continue
// How much was clipped, in whole slots, on the side that is *upstream in
// reading order* — which is the far side for a minus-strand feature. A slot
// the sequence ends part of the way through is kept, its stretch the
// `span` past `start` or `end`.
const lostLow = Math.floor((start - wanted.from) / options.step)
const lostHigh = Math.floor((wanted.to - end) / options.step)
const slotFrom = feature.strand < 0 ? lostHigh : lostLow
const slotTo = slots - (feature.strand < 0 ? lostLow : lostHigh)
if (!(slotTo > slotFrom)) continue
const span = { start: wanted.from + lostLow * options.step, end: wanted.to - lostHigh * options.step }
windows.push({
contig: feature.contig,
start,
end,
strand: feature.strand,
slotFrom,
slotTo,
...(span.start === start && span.end === end ? {} : { span }),
})
}
return { windows, layout }
}
/**
* The windows for a scaled-region run: flank, body, flank.
*
* The body of every feature is resampled to the same number of columns, so the
* plot's x axis is *proportion through the feature* in the middle and base pairs
* at the ends. Features shorter than one column of body are dropped rather than
* stretched — a 200 bp gene resampled to fifty columns is fifty copies of two
* numbers, which flattens the average towards it for no evidence.
*/
export function scaledWindows(
features: ReadonlyArray<{ contig: string; start: number; end: number; strand: number }>,
options: {
body: number
flank: number
step: number
lengthOf(contig: string): number
},
): { windows: Window[]; layout: Layout; dropped: number } {
const perSide = Math.max(0, Math.round(options.flank / options.step))
const slots = options.body + perSide * 2
const layout: Layout = {
slots,
bodyFrom: perSide,
bodyTo: perSide + options.body,
flankStep: options.step,
}
const windows: Window[] = []
let dropped = 0
for (const feature of features) {
if (feature.end - feature.start < options.body) {
dropped++
continue
}
const length = options.lengthOf(feature.contig)
const wanted = {
from: feature.start - perSide * options.step,
to: feature.end + perSide * options.step,
}
const start = Math.max(0, wanted.from)
const end = Math.min(length, wanted.to)
if (!(end > start)) {
dropped++
continue
}
// A clipped flank costs the slots it could not reach, at the flank's own
// resolution; the body is never clipped because it is inside the feature.
const lostLow = Math.round((start - wanted.from) / options.step)
const lostHigh = Math.round((wanted.to - end) / options.step)
const slotFrom = feature.strand < 0 ? lostHigh : lostLow
const slotTo = slots - (feature.strand < 0 ? lostLow : lostHigh)
if (!(slotTo > slotFrom)) {
dropped++
continue
}
// The body's own columns, which is what lets {@link windowRow} resample it
// rather than smearing the whole window across the row uniformly. Clipping
// normally costs flank columns only, as above — but a feature that runs off
// the end of its own sequence loses body columns too, so both the extent
// and the columns are clamped into what this window actually reaches.
const bodyStart = Math.min(Math.max(feature.start, start), end)
const bodyEnd = Math.min(Math.max(feature.end, bodyStart), end)
const bodySlotFrom = Math.min(Math.max(layout.bodyFrom, slotFrom), slotTo)
const bodySlotTo = Math.min(Math.max(layout.bodyTo, bodySlotFrom), slotTo)
windows.push({
contig: feature.contig,
start,
end,
strand: feature.strand,
slotFrom,
slotTo,
body: { start: bodyStart, end: bodyEnd, slotFrom: bodySlotFrom, slotTo: bodySlotTo },
})
}
return { windows, layout, dropped }
}
/**
* One window's samples, area-weighted into its slots.
*
* Area-weighted rather than nearest: an interval that straddles two columns
* belongs partly to each, and taking whichever it starts in makes a coarse
* summary level jump between columns. It is the same rule `render/tracks.ts`
* uses to turn samples into pixels, for the same reason — and using a different
* one here would make a metaplot of a track disagree with the picture of it.
*
* A slot no interval reached is `NaN`, so the aggregation below skips it rather
* than averaging in a zero nothing measured.
*
* *Which* slot an interval reaches is {@link positionOf}, which is also where
* strand is turned round — so everything here reads in the row's own order and
* a minus-strand feature needs no second thought below.
*/
export function windowRow(
samples: TrackSamples,
window: Window,
slots: number,
): Float64Array {
const row = new Float64Array(slots).fill(Number.NaN)
const covered = window.slotTo - window.slotFrom
if (covered <= 0) return row
const span = window.end - window.start
if (span <= 0) return row
const totals = new Float64Array(covered)
const weights = new Float64Array(covered)
const edges = slotEdges(covered)
// Every interval of every feature passes through here — tens of millions
// over the RefSeq TSS at base resolution — so what does not change from one
// to the next is read once: the arrays, the window's ends, its direction. The
// arithmetic is `positionOf`'s and the slots' own, in the same order, so a
// row is the same to the last bit.
const { values, starts, ends } = samples
const lo = window.span?.start ?? window.start
const hi = window.span?.end ?? window.end
const back = window.strand < 0
const linear = window.body === undefined
for (let i = 0; i < values.length; i++) {
const value = values[i]
if (!Number.isFinite(value)) continue
const from = Math.max(lo, starts[i])
const to = Math.min(hi, ends[i])
if (!(to > from)) continue
// Where the interval falls, as a fraction of the slots this window covers.
// Already in reading order, so `a` is the low end of the row's span and `b`
// the high one whichever way the feature points.
let a: number
let b: number
if (linear) {
// `positionOf` for a window with no body, written out.
const one = (from - lo) / (hi - lo)
const two = (to - lo) / (hi - lo)
a = back ? 1 - two : one
b = back ? 1 - one : two
} else {
const one = positionOf(window, from)
const two = positionOf(window, to)
a = back ? two : one
b = back ? one : two
}
const first = Math.max(0, Math.floor(a * covered))
const last = Math.min(covered - 1, Math.ceil(b * covered) - 1)
for (let slot = first; slot <= last; slot++) {
const overlap = Math.min(b, edges[slot + 1]) - Math.max(a, edges[slot])
if (!(overlap > 0)) continue
totals[slot] += value * overlap
weights[slot] += overlap
}
}
for (let slot = 0; slot < covered; slot++) {
if (weights[slot] > 0) row[window.slotFrom + slot] = totals[slot] / weights[slot]
}
return row
}
/** `slot / covered` for every slot edge, `0 … covered`, by the slot count. */
const EDGES = new Map<number, Float64Array>()
/** The slot edges of a window covering `covered` slots, as fractions of it. */
function slotEdges(covered: number): Float64Array {
let edges = EDGES.get(covered)
if (edges === undefined) {
edges = new Float64Array(covered + 1)
for (let slot = 0; slot <= covered; slot++) edges[slot] = slot / covered
EDGES.set(covered, edges)
}
return edges
}
/**
* Where one base pair falls, as a fraction of the slots a window covers, in
* reading order.
*
* A reference-point window is **one** linear map, and has to be: its span is
* the same length for every feature, so spreading it evenly over the row is
* what puts every feature's anchor in the same column.
*
* A scaled-region window is **three** maps joined end to end — upstream flank,
* body, downstream flank — because that is the only way the body can be
* *resampled* to a fixed number of columns while the flanks stay in base pairs.
* One linear map over the whole of such a window (which is what this was before
* {@link Window.body} existed) hands each feature a share of the row set by its
* own length instead: with the shipped defaults a 20 kb gene's body occupied
* columns 15 to 165 and a 100 kb gene's columns 4 to 176, against a layout
* promising 40 to 140 for both. Nothing was resampled to a fixed width and no
* two features were aligned, so the average was the middle of a short gene
* against the start of a long one — the very failure the mode exists to
* prevent, and the companion heatmap carried it too.
*
* **Reading order, not coordinate order.** A minus-strand feature is read from
* its high coordinate down, so the base-pair knots reverse. The slot knots do
* not: `slotFrom`/`slotTo` and `body.slotFrom`/`body.slotTo` are already in
* reading order.
*/
function positionOf(window: Window, bp: number): number {
const body = window.body
if (body === undefined) {
const lo = window.span?.start ?? window.start
const hi = window.span?.end ?? window.end
const through = (bp - lo) / (hi - lo)
return window.strand < 0 ? 1 - through : through
}
const back = window.strand < 0
const opens = back ? window.end : window.start
const bodyOpens = back ? body.end : body.start
const bodyCloses = back ? body.start : body.end
const closes = back ? window.start : window.end
// Segment by segment rather than by a search for the one `bp` is in, which
// is what lets an empty segment fall out rather than being special-cased.
const slot =
window.slotFrom +
progress(bp, opens, bodyOpens) * (body.slotFrom - window.slotFrom) +
progress(bp, bodyOpens, bodyCloses) * (body.slotTo - body.slotFrom) +
progress(bp, bodyCloses, closes) * (window.slotTo - body.slotTo)
return (slot - window.slotFrom) / (window.slotTo - window.slotFrom)
}
/**
* How far `bp` is through one segment, in `[0, 1]`, whichever way it runs.
*
* A segment of no base pairs reads as entirely behind `bp`, so the segments
* after it still receive the whole coordinate. That is safe because a segment
* with no base pairs has no columns either — a run with no flank at all, or a
* flank clipped away down to nothing — so it awards a range of width zero.
*/
function progress(bp: number, from: number, to: number): number {
if (from === to) return 1
const through = (bp - from) / (to - from)
return through < 0 ? 0 : through > 1 ? 1 : through
}
/** A line and the band around it, over a matrix of one row per feature. */
export interface Aggregated {
mean: Float64Array
low: Float64Array
high: Float64Array
/** Rows that contributed at least one finite value. */
n: number
}
/**
* Collapse the per-feature rows into a line.
*
* `rows` is row-major, `count × slots`, with `NaN` wherever a feature had no
* evidence for a column. Every statistic here skips those rather than treating
* them as zero, so a column's average is over the features that reached it and
* the count is reported.
*/
export function aggregate(
rows: Float64Array,
count: number,
slots: number,
options: {
average: Average
trim: number
band: 'none' | 'bootstrap' | 'sd'
iterations: number
level: number
rng(): number
},
): Aggregated {
const mean = new Float64Array(slots).fill(Number.NaN)
const low = new Float64Array(slots).fill(Number.NaN)
const high = new Float64Array(slots).fill(Number.NaN)
if (count === 0) return { mean, low, high, n: 0 }
const column = new Float64Array(count)
// One block, reused: {@link nanSd} takes a list because the *other* caller has
// a track's rows spread across one array per group, and allocating a fresh
// wrapper inside the slot loop would be an allocation per column for nothing.
const columnBlock: Float64Array[] = [column]
const collapse = (values: Float64Array): number =>
options.average === 'median'
? nanMedian(values)
: options.average === 'trimmed'
? nanTrimmedMean(values, options.trim)
: nanMean(values)
for (let slot = 0; slot < slots; slot++) {
for (let row = 0; row < count; row++) column[row] = rows[row * slots + slot]
mean[slot] = collapse(column)
if (options.band === 'sd') {
const sd = nanSd(columnBlock, mean[slot])
if (Number.isFinite(sd)) {
low[slot] = mean[slot] - sd
high[slot] = mean[slot] + sd
}
}
}
if (options.band === 'bootstrap' && count > 1) {
// Resamples **features**, not columns, and the same resample across every
// column of one iteration — which is what makes the band a statement about
// the feature set rather than about each column independently.
const draws = new Float64Array(options.iterations * slots)
const pick = new Int32Array(count)
const scratch = new Float64Array(count)
for (let iteration = 0; iteration < options.iterations; iteration++) {
for (let row = 0; row < count; row++) pick[row] = Math.floor(options.rng() * count) % count
for (let slot = 0; slot < slots; slot++) {
for (let row = 0; row < count; row++) scratch[row] = rows[pick[row] * slots + slot]
draws[iteration * slots + slot] = collapse(scratch)
}
}
const alpha = (1 - options.level) / 2
const sample = new Float64Array(options.iterations)
for (let slot = 0; slot < slots; slot++) {
for (let i = 0; i < options.iterations; i++) sample[i] = draws[i * slots + slot]
const sorted = Array.from(sample).filter(Number.isFinite).sort((a, b) => a - b)
if (sorted.length === 0) continue
low[slot] = sorted[Math.floor(alpha * (sorted.length - 1))]
high[slot] = sorted[Math.ceil((1 - alpha) * (sorted.length - 1))]
}
}
let n = 0
for (let row = 0; row < count; row++) {
for (let slot = 0; slot < slots; slot++) {
if (Number.isFinite(rows[row * slots + slot])) {
n++
break
}
}
}
return { mean, low, high, n }
}
/**
* Centre and scale one track's rows by that track's own spread, in place.
*
* A metaplot of two tracks on one pair of axes is a comparison only if the two
* are in the same units, and a CPM bigWig and a log2 ratio are not: one runs
* over tens and the other over ±2, so drawn together the second is a flat line
* along the bottom of the frame and whatever shape it was loaded for cannot be
* seen. Standardising each track against itself puts both on an axis whose unit
* is *this track's own spread*, which is the only unit they have in common. The
* y axis then names z-scores rather than the raw quantity, and the caller has to
* say so — `run/metaplot.ts` relabels it.
*
* **The population is the track, over every one of its groups.** `blocks` is one
* array of rows per ranking group and the mean and the standard deviation are
* taken over all of them together, which is why this takes a list rather than
* being called once per group. Per group, each group would be re-centred on
* itself and every one of them would sit on zero — deleting exactly the thing
* the reader split the feature set to see, that the strongest quartile is
* *above* the weakest. The ranking would survive in the legend and nowhere in
* the picture, which is the worst way for a figure to be wrong.
*
* Gaps stay gaps. A `NaN` slot is no evidence, it enters neither statistic and
* it comes back `NaN` — `analysis/stats.ts` opens with the rule, and turning an
* uncovered slot into a 0 here would plant a value one standard deviation below
* the track's mean at every position no read reached.
*
* **A track with no spread is centred and not scaled.** A constant track, or one
* with a single covered slot in the whole run, has a standard deviation of zero
* or none at all; dividing by it would fill the plot with `±Infinity`, which is
* not a picture of anything and which propagates through the average and the
* bootstrap alike. Subtracting the mean and stopping there draws the flat line
* such a track honestly is. `scaled` reports which of the two happened, so a
* caller that wants to say so out loud can.
*/
export function standardiseTrack(
blocks: ReadonlyArray<Float64Array>,
): { mean: number; sd: number; scaled: boolean } {
const mean = nanMeanOf(blocks)
const sd = nanSd(blocks, mean)
// Nothing finite anywhere: there is nothing to centre and the rows are already
// all gaps. Rewriting them would only turn `NaN` into `NaN` more slowly.
if (!Number.isFinite(mean)) return { mean, sd, scaled: false }
const scaled = Number.isFinite(sd) && sd > 0
for (const rows of blocks) {
for (let i = 0; i < rows.length; i++) {
const value = rows[i]
if (!Number.isFinite(value)) continue
rows[i] = scaled ? (value - mean) / sd : value - mean
}
}
return { mean, sd, scaled }
}
/**
* The mean of the finite values across several blocks; `NaN` when there are
* none.
*
* `stats.ts`'s `nanMean` over one array at a time would need the counts carried
* between the calls to weight them, and a mean of per-group means is not the
* mean of the population unless the groups are the same size — which the top-N
* and bottom-N subsets are not.
*
* Skips everything non-finite rather than only `NaN`, which is where this parts
* company with `nanMean`: one infinity in one row would otherwise make the mean
* infinite and take every slot of every group down with it.
*/
function nanMeanOf(blocks: ReadonlyArray<ArrayLike<number>>): number {
let total = 0
let seen = 0
for (const values of blocks) {
for (let i = 0; i < values.length; i++) {
const value = values[i]
if (!Number.isFinite(value)) continue
total += value
seen++
}
}
return seen === 0 ? Number.NaN : total / seen
}
/**
* Standard deviation about a known mean, skipping gaps.
*
* Takes *blocks* because its two callers have their population in different
* shapes — one column of an aggregation, or a whole track's rows held as one
* array per ranking group — and they are one population each. The sample
* denominator (`n - 1`) needs a second observation, and one observation gets
* `NaN` rather than `0`: no spread measured is not a spread of none.
*/
function nanSd(blocks: ReadonlyArray<ArrayLike<number>>, mean: number): number {
if (!Number.isFinite(mean)) return Number.NaN
let total = 0
let seen = 0
for (const values of blocks) {
for (let i = 0; i < values.length; i++) {
const value = values[i]
if (!Number.isFinite(value)) continue
total += (value - mean) ** 2
seen++
}
}
return seen > 1 ? Math.sqrt(total / (seen - 1)) : Number.NaN
}
/**
* The x coordinate of every column, and what it is a coordinate *in*.
*
* A reference-point plot has one unit throughout — base pairs from the anchor —
* and says so. A scaled-region plot does not: its middle is *proportion through
* a feature* and its ends are base pairs, and there is no honest single number
* that is both. Rather than invent one, the axis for a scaled plot is the column
* index and the label says so; the meaning of the three regions belongs on the
* ticks, which is what every tool that draws this plot does.
*
* The alternative — pretending the body is some number of base pairs — puts a
* scale on the axis that is true of no feature in the set.
*
* `scaled` is where those three regions meet, in the axis's own units, for the
* ticks to be written from: `Start` and `End` at the body's edges, a share of
* the body between them, a distance from the nearer edge outside. Edges, not
* centres — column `bodyFrom` is the body's first and is drawn at its own
* index, so the body begins half a column before it.
*/
export function axisFor(layout: Layout): { x: Float64Array; label: string; scaled?: ScaledAxis } {
const out = new Float64Array(layout.slots)
const scaled = layout.bodyTo > layout.bodyFrom
for (let slot = 0; slot < layout.slots; slot++) {
out[slot] = scaled ? slot : (slot - layout.bodyFrom + 0.5) * layout.flankStep
}
if (!scaled) return { x: out, label: 'Distance from the anchor (bp)' }
return {
x: out,
label: 'Position (upstream · body · downstream)',
scaled: {
from: -0.5,
start: layout.bodyFrom - 0.5,
end: layout.bodyTo - 0.5,
to: layout.slots - 0.5,
flankStep: layout.flankStep,
},
}
}
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