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* pastis-NB: a count is a count, not a measurement of a distance (§25.6).
*
* `reconstruct.ts` takes the ShRec3D position — a contact count `c` is read as
* a distance `c^(−1/α)`, the missing pairs are filled so the matrix is
* complete, and beads are placed to honour those distances. That position has
* two costs it cannot pay off. A **zero is not a distance**: `0^(−1/α)` is
* infinite, so a pair the experiment did not sample has to be either dropped or
* invented, and the filling that invents it is the one step in that pipeline
* that is a guess rather than a reading. And a **count's precision is thrown
* away**: a pair seen ten thousand times and a pair seen once become two
* distances, and only the weighting bolted onto the refinement remembers which
* of them the map was sure about.
*
* The negative binomial takes the other position, which is pastis'
* (Varoquaux, Ay, Noble & Vert, *Bioinformatics* 2014 for the Poisson models;
* Varoquaux, Noble & Vert 2023 for the overdispersed one). A count is a
* **count**, drawn from a distribution whose mean falls off as a power of the
* distance between the two beads:
*
* ```
* c_ij ~ NegativeBinomial(mean μ = b_i · b_j · d_ij^(−α), dispersion r_ij)
* r_ij = b_i · b_j · exp(c₀ + c₁ · log λ_ij), λ_ij = d_ij^(−α)
* ```
*
* where `b` is each bead's coverage bias — pastis' own place for it, and the
* reason this takes **raw counts** and never a balanced matrix ({@link
* NegBinomModel.bias}) — and the model is the one that makes the counts most
* probable. Both costs go away at once. A zero is **evidence**, and weak
* evidence at that — it says *probably far*, it needs no filling, and past a
* certain distance it stops caring how far, which is exactly right for a pair a
* shallow experiment missed. And a count's precision is in the likelihood
* itself: the curvature of `−log P` at a pair seen ten thousand times is ten
* thousand times that of a pair seen once, so the near diagonal holds the beads
* without anybody weighting anything.
*
* **This is pastis-NB as pastis runs it**, function for function, so that a
* model made here and one made by `pastis.optimization.negative_binomial_
* structure.estimate_X` answer the same question in the same words:
*
* - **The dispersion is a function of the mean**, not of the separation. The
* counts' mean and variance are taken at each genomic separation
* ({@link momentsBySeparation}, pastis' `compute_mean_variance`), the
* dispersion each pair of moments implies, `r = m²/(v − m)`, is fitted
* against the mean as `r = exp(c₀ + c₁ log m)` ({@link fitDispersion}, its
* `ExponentialDispersion(degree=1)`), and during the fit every pair's `r` is
* read off that curve **at the model's own mean for the pair** — so it moves
* with the beads, and its derivative is part of the gradient.
* - **The likelihood is the whole negative binomial**, gamma functions and all
* ({@link negBinomObjective}), so its value is the number pastis reports.
* - **`β` is 1 and the biases are scaled to a mean of 1**, as pastis' pipeline
* scales them. The scale of the model is then fitted rather than chosen: a
* distance of 1 is where a pair of average coverage expects one contact.
* - **The start is pastis' MDS** ({@link mdsObjective}): the relative stress
* `Σ (d − δ)²/δ²` over the pairs the map counted, `δ = (c/(b_i b_j))^(−1/α)`,
* from beads placed uniformly at random in a cube by a seeded generator —
* {@link randomStart}, whose draws are numpy's, so seed `s` starts from the
* very coordinates pastis' `random_state=s` does.
* - **Both are solved by L-BFGS to pastis' own tolerances** (`structure/
* lbfgs.ts`, with SciPy's line-search constants): the MDS to
* `factr = 10¹²`, the likelihood to `factr = 10` and a largest gradient
* component of 1 — or at the reader's step budget, whichever comes first.
* The default budget, 150, is about where pastis' own L-BFGS-B stops on
* the validation's yeast map (125 to 148 steps, by its relative-reduction
* test); run on to the gradient, the likelihood keeps climbing slowly for
* several hundred more.
*
* One thing differs, and is named in §25.6: a separation whose counts are not
* overdispersed beyond sampling noise is left out of the dispersion's fit
* ({@link fitDispersion}), and if fewer than three remain the counts are read
* as Poisson — pastis-PM2, the `r = ∞` corner of what is written here. On a
* real map every separation is in, and the curve is pastis'.
*
* **What is optimised.** Three coordinates per bead. `α` is not optimised: it
* is the reader's control, the same number *Model from the map* and *Model
* versus map* already share, and estimating it here would make the one
* parameter a reader sets mean something different depending on which method
* they picked. pastis fixes it too, at the 3 its pipeline is run with.
*
* Arrays in and arrays out — no worker, no store, no DOM (§18.9).
*/
import { answered, lbfgsWalk, type LbfgsAsk, type LbfgsProgress, type LbfgsResult, type LbfgsStop, type Objective } from './lbfgs'
import type { Coordinates, DenseMatrix } from './reconstruct'
/** The machine epsilon SciPy's L-BFGS-B multiplies `factr` by. */
const EPSILON = 2.220446049250313e-16
/**
* How each phase stops: pastis' arguments to SciPy, translated.
*
* `tolerance` is `factr · ε`, the share of the value an iteration must still
* buy; `gradient` is `pgtol`, the largest gradient component at which the
* problem counts as solved; `iterations` is `maxiter`. The MDS stops early on
* purpose — `factr = 10¹²` is a part in five thousand — because it only has to
* land the likelihood in the right basin; the likelihood itself runs until an
* iteration buys nothing a double can hold.
*/
export const PASTIS_MDS = { tolerance: 1e12 * EPSILON, gradient: 1e-5, iterations: 10000 } as const
export const PASTIS_NB = { tolerance: 10 * EPSILON, gradient: 1 } as const
/** Curvature pairs L-BFGS keeps: SciPy's `maxcor`, which pastis leaves at its default. */
const HISTORY = 10
/** Overdispersed separations a curve of two coefficients is fitted through, at the fewest. */
const MIN_SEPARATIONS = 3
/** Of the longest sequence, the share of separations the moments are taken over: pastis' `f`. */
const SEPARATION_SHARE = 2 / 3
/** Everything the likelihood is a function of, besides the coordinates. */
export interface NegBinomModel {
/**
* The counts, `n × n`, **read above the diagonal only**.
*
* **`NaN` and `0` are different here**, and the difference is the whole point
* of the method: `0` is a pair the map looked at and counted nothing for,
* which is evidence that the two beads are far apart, and `NaN` is a pair the
* map could not look at — a filtered bin, a gap — which is no evidence at
* all. A `NaN` pair is skipped; a zero pair pushes. Below the diagonal is
* free for {@link storeWishes} to use.
*/
counts: DenseMatrix
n: number
/**
* Each bead's coverage bias, scaled to a mean of 1, or `null` where the file
* has no weights.
*
* A Hi-C bin is not read as often as its neighbour: mappability, restriction
* density and copy number all scale how many pairs land on it, by a factor
* that has nothing to do with where the bin is in space. `cooler balance`
* measures that factor as a weight `w` per bin and the balanced matrix is
* `raw·w_i·w_j`.
*
* **A likelihood cannot be handed the balanced matrix**, which is why this
* field exists rather than the run simply reading balanced values. A balanced
* value is not a count: it is not an integer, it is scaled by an arbitrary
* constant, and — fatally for {@link momentsBySeparation} — *overdispersion
* is a statement about a count*. `variance > mean` compares a squared quantity
* with a linear one, so scaling the data by `k` scales the two sides
* differently and a balanced map is never found overdispersed, whatever it
* actually is. Handed balanced values the negative binomial silently becomes
* a Poisson.
*
* So the counts stay raw and the bias goes where pastis puts it — **into the
* mean**, `μ_ij = b_i·b_j·d^(−α)` with `b_i = 1/w_i` — and into the
* dispersion, which pastis scales by the same product. The counts are then
* counts, the dispersion is measurable, and the coverage is still corrected.
*/
bias?: Float64Array | null
/** The dispersion's curve against the mean, or `null` for none: a Poisson. */
dispersion: Dispersion | null
/**
* The reader's exponent, positive, as in `d = c^(−1/α)`.
*
* The mean falls as `d^(−α)`, so the exponent this module works in is `−α`.
* Kept in the reader's sign so that one control means one thing across both
* methods and both directions of the law.
*/
alpha: number
}
/**
* The dispersion as a curve against the mean: `r(λ) = exp(c₀ + c₁ log λ)`.
*
* pastis' `ExponentialDispersion(degree=1)`. Large `r` is Poisson — variance
* equal to mean; small `r` is heavy overdispersion. `λ` is the mean a pair of
* average coverage would have; a pair's own `r` is this times `b_i·b_j`.
*/
export interface Dispersion {
c0: number
c1: number
}
/** The moments of the counts at each genomic separation, as pastis takes them. */
export interface SeparationMoments {
/** The separation, in bins along a sequence. */
separation: Int32Array
/** The mean of `c / (b_i b_j)` over the pairs that far apart. */
mean: Float64Array
/** Their unbiased variance, over the mean of `1 / (b_i b_j)`. */
variance: Float64Array
/** How many pairs were seen at it, zeros included. */
pairs: Float64Array
}
/** What a fit ends with. */
export interface NegBinomFit {
coords: Coordinates
/** The negative log-likelihood, less `Σ log c!` — the number pastis' objective returns. */
value: number
iterations: number
evaluations: number
stop: LbfgsStop
}
/** What the MDS start ends with: the same shape, its value the relative stress. */
export type MdsFit = NegBinomFit
/**
* pastis' `compute_mean_variance`: the counts' mean and variance at each
* genomic separation within a sequence.
*
* Over the separations up to two-thirds of the longest sequence, and over the
* pairs within one sequence only. Each count is divided by its pair's bias
* product before the moments are taken, and the variance is then divided by the
* mean of `1/(b_i b_j)`: if a pair's count is negative binomial with mean
* `bλ` and dispersion `b·r`, its corrected count has variance `(λ + λ²/r)/b`,
* and averaging `1/b` out is what leaves the `λ + λ²/r` the dispersion is read
* from. **Zeros are counted** — they are observations of the count at that
* separation, and leaving them out would overstate the mean and understate the
* dispersion. A `NaN`, a pair the map could not see, is not.
*
* The separation is counted **in beads**, as pastis counts it: it is handed
* the beads and their sequences' lengths and nothing else, so two beads either
* side of a filtered bin are one apart to it. `index` orders the beads along
* their sequence and is not otherwise read.
*/
export function momentsBySeparation(
counts: DenseMatrix,
n: number,
group: Int32Array,
index: Int32Array,
bias: Float64Array | null = null,
): SeparationMoments {
// Each sequence's beads in their order along it, so a separation is a walk
// down one diagonal rather than a pass over every pair — and counted in
// beads, as pastis counts it, a filtered bin being no bead at all.
const byGroup = new Map<number, number[]>()
for (let i = 0; i < n; i++) {
const held = byGroup.get(group[i])
if (held === undefined) byGroup.set(group[i], [i])
else held.push(i)
}
let longest = 0
const along: Int32Array[] = []
for (const beads of byGroup.values()) {
beads.sort((a, b) => index[a] - index[b])
along.push(Int32Array.from(beads))
longest = Math.max(longest, beads.length)
}
const separation: number[] = []
const mean: number[] = []
const variance: number[] = []
const pairs: number[] = []
// `range(1, int(min_size))`: up to, and not including, two-thirds of the longest.
const last = Math.floor(longest * SEPARATION_SHARE)
for (let s = 1; s < last; s++) {
let seen = 0
let sum = 0
let inverse = 0
for (const beads of along) {
for (let p = 0; p + s < beads.length; p++) {
const i = beads[p]
const j = beads[p + s]
const c = counts[i < j ? i * n + j : j * n + i]
if (!Number.isFinite(c)) continue
const b = bias === null ? 1 : bias[i] * bias[j]
seen++
sum += c / b
inverse += 1 / b
}
}
if (seen < 2) continue
const m = sum / seen
// The second pass, about the mean, as numpy takes it: a sum of squares
// less `n·m²` would lose to cancellation exactly where a map is nearly
// Poisson and the variance is a small difference of large numbers.
let squares = 0
for (const beads of along) {
for (let p = 0; p + s < beads.length; p++) {
const i = beads[p]
const j = beads[p + s]
const c = counts[i < j ? i * n + j : j * n + i]
if (!Number.isFinite(c)) continue
const deviation = c / (bias === null ? 1 : bias[i] * bias[j]) - m
squares += deviation * deviation
}
}
separation.push(s)
mean.push(m)
variance.push(squares / (seen - 1) / (inverse / seen))
pairs.push(seen)
}
return {
separation: Int32Array.from(separation),
mean: Float64Array.from(mean),
variance: Float64Array.from(variance),
pairs: Float64Array.from(pairs),
}
}
/**
* pastis' `ExponentialDispersion(degree=1).fit`, weighted as its pipeline
* weights it: the curve `r = exp(c₀ + c₁ log m)` through the dispersion each
* separation's moments imply, `r = m²/(v − m)`, by least squares with each
* residual weighted by the square root of the separation's pair count.
*
* Least squares **on `r` itself**, not on its logarithm, and from `(0, 0)`, as
* SciPy's `least_squares` is called there — so the separations with the most
* pairs and the largest `r` steer it, which is pastis' choice and is kept. The
* solver is Levenberg–Marquardt on the two coefficients, which lands on the
* minimum SciPy's trust region does wherever there is one.
*
* A separation whose counts are **not overdispersed** is left out: one whose
* variance is not above its mean by more than three standard errors of a
* Poisson's sample variance, `√((m + 2m²)/N)`. A real Hi-C map clears that by
* orders of magnitude at every separation — the validation's yeast map by a
* factor of hundreds — so there it changes nothing. Poisson counts clear it one
* time in a thousand, and pastis, entering every separation as it comes, reads
* them otherwise: `v ≤ m` gives a negative `r` that drags the curve towards
* zero, and `v` a hair above `m` an `r` in the thousands, and a curve forced
* through both calls the most Poisson counts on the map the most overdispersed
* and loses the model. With fewer than three separations left there is no
* curve worth drawing, and the answer is `null`: no overdispersion, a Poisson.
*/
export function fitDispersion(moments: SeparationMoments): Dispersion | null {
const logs: number[] = []
const targets: number[] = []
const weights: number[] = []
for (let k = 0; k < moments.mean.length; k++) {
const m = moments.mean[k]
const v = moments.variance[k]
// Above the mean by more than three standard errors of a Poisson's sample
// variance, `√((m + 2m²)/N)`: a real map clears it by orders of magnitude
// at every separation, and Poisson counts clear it one time in a thousand.
if (!(m > 0) || !(v > m + 3 * Math.sqrt((m + 2 * m * m) / moments.pairs[k]))) continue
const r = (m * m) / (v - m)
if (!Number.isFinite(r)) continue
logs.push(Math.log(m))
targets.push(r)
weights.push(moments.pairs[k])
}
if (logs.length < MIN_SEPARATIONS) return null
const cost = (c0: number, c1: number): number => {
let sum = 0
for (let k = 0; k < logs.length; k++) {
const residual = Math.exp(c0 + c1 * logs[k]) - targets[k]
sum += weights[k] * residual * residual
}
return sum
}
let c0 = 0
let c1 = 0
let current = cost(c0, c1)
let damping = 1e-3
for (let iteration = 0; iteration < 500 && Number.isFinite(current); iteration++) {
// The normal equations of the linearised residuals, `JᵀJ δ = −Jᵀρ`, two by two.
let a00 = 0
let a01 = 0
let a11 = 0
let g0 = 0
let g1 = 0
for (let k = 0; k < logs.length; k++) {
const fitted = Math.exp(c0 + c1 * logs[k])
const residual = fitted - targets[k]
const w = weights[k]
a00 += w * fitted * fitted
a01 += w * fitted * fitted * logs[k]
a11 += w * fitted * fitted * logs[k] * logs[k]
g0 += w * fitted * residual
g1 += w * fitted * residual * logs[k]
}
let moved = false
while (damping < 1e16) {
const d00 = a00 * (1 + damping)
const d11 = a11 * (1 + damping)
const det = d00 * d11 - a01 * a01
if (!(Math.abs(det) > 0)) {
damping *= 10
continue
}
const step0 = -(d11 * g0 - a01 * g1) / det
const step1 = -(d00 * g1 - a01 * g0) / det
const tried = cost(c0 + step0, c1 + step1)
if (Number.isFinite(tried) && tried < current) {
c0 += step0
c1 += step1
const gain = current - tried
current = tried
damping = Math.max(damping / 10, 1e-12)
moved = gain > 1e-15 * current && Math.abs(step0) + Math.abs(step1) > 1e-15 * (1 + Math.abs(c0) + Math.abs(c1))
break
}
damping *= 10
}
if (!moved) break
}
return Number.isFinite(c0) && Number.isFinite(c1) ? { c0, c1 } : null
}
/** `r` at a mean of `λ`, for a pair of average coverage. */
export function dispersionAt(dispersion: Dispersion, lambda: number): number {
return Math.exp(dispersion.c0 + dispersion.c1 * Math.log(lambda))
}
/**
* pastis' random start: every coordinate uniform in `[−1, 1)`, drawn as
* `1 − 2·rand(3n)` and read three to a bead.
*
* Handed numpy's own generator (`analysis/random.ts`'s `numpyRandom`), seed `s`
* gives the coordinates `RandomState(s)` gives pastis, to the last bit — so the
* one place a run of pastis and a run here could part at the very first step is
* closed, and what is left between them is the optimiser's path.
*/
export function randomStart(n: number, rng: () => number): Coordinates {
const x = new Float64Array(n)
const y = new Float64Array(n)
const z = new Float64Array(n)
for (let i = 0; i < n; i++) {
x[i] = 1 - 2 * rng()
y[i] = 1 - 2 * rng()
z[i] = 1 - 2 * rng()
}
return { x, y, z }
}
/**
* The reciprocal of each counted pair's wish distance, `1/δ = (c/(b_i b_j))^(1/α)`,
* written **below the diagonal** of the counts, `0` where the map counted nothing.
*
* pastis' `compute_wish_distances` on the bias-corrected counts with `β = 1`.
* Below the diagonal because the likelihood reads only above it: the matrix is
* symmetric, its lower half holds nothing the counts above do not, and storing
* the wishes there costs the MDS no matrix of its own and no `pow` per pair per
* evaluation. Singles, which hold a wish to a part in ten million — far past
* where an MDS that stops at a part in five thousand can tell.
*/
export function storeWishes(counts: DenseMatrix, n: number, bias: Float64Array | null, alpha: number): void {
const power = 1 / alpha
for (let i = 0; i < n; i++) {
for (let j = i + 1; j < n; j++) {
const c = counts[i * n + j]
const wish = Number.isFinite(c) && c > 0 ? c / (bias === null ? 1 : bias[i] * bias[j]) : 0
counts[j * n + i] = wish > 0 ? (alpha === 3 ? Math.cbrt(wish) : Math.pow(wish, power)) : 0
}
}
}
/**
* pastis' MDS objective: the relative stress over the counted pairs.
*
* ```
* f = Σ_{i≠j} (d_ij − δ_ij)² / δ_ij² = 2 Σ_{i>j} (d_ij · w_ij − 1)², w = 1/δ
* ```
*
* Over ordered pairs, as pastis' dense objective sums it, so the value is
* pastis'. Relative rather than absolute because a near pair's wish is small and
* a far pair's large, and an absolute stress would spend the model on the far
* pairs' errors; this one asks every pair to be right to the same share. The
* gradient is the exact one — pastis' dense gradient is half of it, which moves
* SciPy's path but not the minimum. A pair whose beads have met contributes its
* constant and no gradient, as pastis' does.
*
* Summed as {@link negBinomObjective} is, in bands of rows ({@link mdsBands}),
* each band's value and gradient added in band order, so that it is the same
* bits on one thread and on several (`analysis/likelihoodThreads.ts`).
*/
export function mdsObjective(wishes: DenseMatrix, n: number): Objective {
const edges = mdsBands(n)
const parts = Array.from({ length: edges.length - 1 }, (_, b) => new Float64Array(3 * edges[b + 1]))
return (vector, gradient) => {
gradient.fill(0)
let value = 0
for (let b = 0; b + 1 < edges.length; b++) {
value += mdsBand(wishes, null, n, edges[b], edges[b + 1], vector, parts[b])
addBand(gradient, parts[b], n, 0, edges[b + 1])
}
return value
}
}
/**
* The bands of rows pastis' MDS is summed over: `[edges[b], edges[b + 1])`,
* each holding about the same share of the pairs **below** the diagonal, where
* {@link storeWishes} keeps them.
*
* The likelihood's layout turned over ({@link likelihoodBands}): a function of
* the bead count alone, so the sum is the same bits on any number of threads.
* Row `i` carries `i` pairs, so the rows before `s` carry `(s/n)²` of them and
* the edges sit at `n√(t/B)`.
*/
export function mdsBands(n: number): Int32Array {
const count = bandCount(n)
const edges = [0]
for (let t = 1; t < count; t++) {
const edge = Math.round(n * Math.sqrt(t / count))
if (edge > edges[edges.length - 1] && edge < n) edges.push(edge)
}
edges.push(n)
return Int32Array.from(edges)
}
/**
* One band's share of {@link mdsObjective}: its value returned, its gradient
* written into `out`.
*
* Rows `[r0, r1)`, each with every bead before it. `wishes[offsets[i − r0] + j]`
* is pair `(i, j)`'s wish — `null` for a whole `n × n` matrix, where it is
* `i·n + j`, and a row's start in a helper's packed copy otherwise — and `out`
* holds the gradient of beads `0` to `r1`, which are the only ones a band's
* pairs touch: `x`, then `y`, then `z`, `r1` of each. It is written over.
*/
export function mdsBand(
wishes: DenseMatrix,
offsets: ArrayLike<number> | null,
n: number,
r0: number,
r1: number,
vector: Float64Array,
out: Float64Array,
): number {
const len = r1
out.fill(0)
let value = 0
for (let i = Math.max(1, r0); i < r1; i++) {
const row = offsets === null ? i * n : offsets[i - r0]
const xi = vector[i]
const yi = vector[n + i]
const zi = vector[2 * n + i]
let gx = 0
let gy = 0
let gz = 0
for (let j = 0; j < i; j++) {
const w = wishes[row + j]
if (!(w > 0)) continue
const dx = xi - vector[j]
const dy = yi - vector[n + j]
const dz = zi - vector[2 * n + j]
const d = Math.sqrt(dx * dx + dy * dy + dz * dz)
const e = d * w - 1
value += 2 * e * e
if (!(d > 0)) continue
const coefficient = (4 * e * w) / d
const fx = coefficient * dx
const fy = coefficient * dy
const fz = coefficient * dz
gx += fx
gy += fy
gz += fz
out[j] -= fx
out[len + j] -= fy
out[2 * len + j] -= fz
}
out[i] += gx
out[len + i] += gy
out[2 * len + i] += gz
}
return value
}
// `lgamma(c + r) − lgamma(r) − c·log r` and `r·(ψ(c + r) − ψ(r))`, written by
// `countTerms` for the objective's inner loop to read: two numbers out of a
// function called a hundred million times an evaluation, without an object
// allocated for each.
let termG = 0
let termH = 0
/** Below this a gamma function is shifted up by its recurrence before its series is summed. */
const STIRLING = 7
/** Up to this a whole count's terms are summed exactly, factor by factor. */
const SMALL_COUNT = 16
/** …while `r` is below this, so that `(r + c)^c` stays far inside a double. */
const SMALL_RATE = 1e9
/**
* The two parts of a count's likelihood that need the gamma function, into
* `termG` and `termH`:
*
* ```
* G = lgamma(c + r) − lgamma(r) − c·log r (the log-likelihood's count term)
* H = r · (ψ(c + r) − ψ(r)) (its derivative, times r)
* ```
*
* A whole count up to {@link SMALL_COUNT} never comes here: the objective sums
* it exactly, inline, as `G = log Π_{k<c} (r + k) − c·log r` and
* `H = r Σ_{k<c} 1/(r + k)` — most pairs of most maps, whose far counts are a
* handful. Everything else is Stirling's series for both, each argument first lifted to {@link STIRLING}
* by the recurrence, written so that the `c·log r` cancels on paper rather
* than in rounding — `G` is a few units where `lgamma(r)` is thousands, and the
* difference of two large numbers is where a likelihood's digits go. To seven
* terms at seven the series is good to a part in 10¹³.
*/
function countTerms(c: number, r: number, logR: number): void {
// ψ(r) = ψ(r + m) − Σ 1/(r + k) and lgamma(r) = lgamma(r + m) − log Π (r + k).
let x = r
let px = 1
let sx = 0
while (x < STIRLING) {
px *= x
sx += 1 / x
x += 1
}
let y = c + r
let py = 1
let sy = 0
while (y < STIRLING) {
py *= y
sy += 1 / y
y += 1
}
const lx = x === r ? logR : Math.log(x)
const ratio = Math.log1p((y - x) / x) // log y − log x, without losing it to their difference
const ix = 1 / x
const iy = 1 / y
const ix2 = ix * ix
const iy2 = iy * iy
// lgamma(z) = (z − ½) log z − z + ½ log 2π + gammaTail(z)
const tailX = ix * (1 / 12 - ix2 * (1 / 360 - ix2 * (1 / 1260 - ix2 * (1 / 1680 - ix2 * (1 / 1188 - (ix2 * 691) / 360360)))))
const tailY = iy * (1 / 12 - iy2 * (1 / 360 - iy2 * (1 / 1260 - iy2 * (1 / 1680 - iy2 * (1 / 1188 - (iy2 * 691) / 360360)))))
// ψ(z) = log z − 1/(2z) − psiTail(z)
const psiX = ix2 * (1 / 12 - ix2 * (1 / 120 - ix2 * (1 / 252 - ix2 * (1 / 240 - ix2 * (1 / 132 - (ix2 * 691) / 32760)))))
const psiY = iy2 * (1 / 12 - iy2 * (1 / 120 - iy2 * (1 / 252 - iy2 * (1 / 240 - iy2 * (1 / 132 - (iy2 * 691) / 32760)))))
const shift = px === py ? 0 : Math.log(px / py)
// (y − ½) log y − (x − ½) log x = (y − ½)(log y − log x) + (y − x) log x, and
// where neither was lifted `(y − x) log x` is the `c log r` taken off after:
// the two are dropped together rather than subtracted.
const carried = px === 1 && py === 1 ? 0 : (y - x) * lx - c * logR
termG = (y - 0.5) * ratio - (px === 1 && py === 1 ? c : y - x) + tailY - tailX + shift + carried
termH = r * (ratio - 0.5 * (iy - ix) - psiY + psiX - sy + sx)
}
/**
* Everything a band of the likelihood reads besides the counts and the
* coordinates, prepared once: the same numbers on every thread that computes a
* band, so that a band is the same sum wherever it is computed.
*/
export interface LikelihoodTerms {
n: number
alpha: number
bias: Float64Array | null
/** `log bᵢ`, taken once rather than once a pair. */
logBias: Float64Array | null
dispersion: Dispersion | null
}
export function likelihoodTerms(model: Pick<NegBinomModel, 'n' | 'alpha' | 'bias' | 'dispersion'>): LikelihoodTerms {
const bias = model.bias ?? null
return {
n: model.n,
alpha: model.alpha,
bias,
logBias: bias === null ? null : Float64Array.from(bias, (b) => Math.log(b)),
dispersion: model.dispersion,
}
}
/** Beads to a band, roughly: the layout's grain. */
const BAND_BEADS = 32
/** Bands at most: enough to share among eight threads with room to balance them. */
const MAX_BANDS = 64
/** How many bands `n` beads are summed in: one a run of {@link BAND_BEADS}, at most {@link MAX_BANDS}. */
function bandCount(n: number): number {
return Math.max(1, Math.min(MAX_BANDS, Math.round(n / BAND_BEADS)))
}
/**
* The bands of rows the likelihood is summed over: `[edges[b], edges[b + 1])`,
* each holding about the same share of the pairs above the diagonal.
*
* **A function of the bead count alone**, and that is the property that
* matters: a band's sum is computed whole, wherever it is computed, and the
* bands are added in this order whatever the threads, so the likelihood and
* its gradient are the same bits on one thread and on eight
* (`analysis/likelihoodThreads.ts`). Row `i` carries `n − 1 − i` pairs, so the
* rows before `s` carry `1 − (1 − s/n)²` of them and the edges sit at
* `n(1 − √(1 − t/B))`.
*/
export function likelihoodBands(n: number): Int32Array {
const count = bandCount(n)
const edges = [0]
for (let t = 1; t < count; t++) {
const edge = Math.round(n * (1 - Math.sqrt(1 - t / count)))
if (edge > edges[edges.length - 1] && edge < n) edges.push(edge)
}
edges.push(n)
return Int32Array.from(edges)
}
/**
* One band's share of the negative log-likelihood and its gradient, in one pass
* over its pairs — pastis' `negative_binomial_obj` and
* `negative_binomial_gradient`, with every pair of beads the map could see,
* zeros included (`use_zero_entries=True`).
*
* Rows `[r0, r1)`, each with every bead after it. `counts[offsets[i − r0] + j]`
* is pair `(i, j)`'s count — `null` for a whole `n × n` matrix, where it is
* `i·n + j`, and a row's start in a helper's packed copy otherwise — and `out`
* holds the gradient of beads `r0` onward, which are the only ones a band's
* pairs touch: `x`, then `y`, then `z`, `n − r0` of each. It is written over,
* and the band's value is returned.
*
* Per pair, with `λ = d^(−α)`, `s = b_i b_j`, `μ = s·λ`, `r = s·exp(c₀ + c₁ log λ)`
* and `q = μ/r = exp((1 − c₁) log λ − c₀)`, which the bias cancels out of:
*
* ```
* f = −G(c, r) + (c + r)·log(1 + q) − c·log μ
* ∂f/∂log λ = (1 − c₁)·(μ − c)/(1 + q) + c₁·(r·log(1 + q) − H(c, r))
* ∂f/∂x_i = −α · ∂f/∂log λ · (x_i − x_j)/d²
* ```
*
* That `f` is pastis' `−lgamma(c + r) + lgamma(r) − c log μ − r log r +
* (c + r) log(r + μ)` rearranged so that nothing large is subtracted from
* anything large: `G` is {@link countTerms}', and the `r log r` that would be
* a billion at a near-Poisson pair is folded into `log(1 + q)`.
*
* **What the derivative says.** At `c₁ = 0` the dispersion is one number and
* the pull on a pair is the Poisson's `μ − c` shrunk by `r/(r + μ)`: the same
* force, turned down where the counts scatter. pastis-NB's `r` grows with the
* mean, `c₁` of it per unit of `log λ`, so moving the beads also moves how much
* the map is believed there — that is the second term, and it is why a
* dispersion read off a curve against the mean cannot be held fixed per pair.
*
* `dispersion: null` is the Poisson, `f = μ − c log μ` and `∂f/∂log λ = μ − c`:
* pastis-PM2, and the limit of the above as `r → ∞`.
*
* **One logarithm per pair, and no square root but at `α = 3`.** `log d²` gives
* `log λ`, `log μ` and `log r` with the biases' logarithms taken once; at
* `α = 3`, `λ` itself is a square root and a division. A pair with a
* non-finite count is skipped: the map could not look at it. A pair whose beads
* have collided has an infinite mean, and the band is returned as infinite so
* the line search shortens instead of stepping into the pole.
*/
export function likelihoodBand(
terms: LikelihoodTerms,
counts: DenseMatrix,
offsets: ArrayLike<number> | null,
r0: number,
r1: number,
vector: Float64Array,
out: Float64Array,
): number {
const { n, dispersion, alpha, bias, logBias } = terms
const half = alpha / 2
const cube = alpha === 3
const nb = dispersion !== null
const c0 = nb ? dispersion.c0 : 0
const c1 = nb ? dispersion.c1 : 0
const slope = 1 - c1
const len = n - r0
out.fill(0)
let value = 0
for (let i = r0; i < r1; i++) {
const row = offsets === null ? i * n : offsets[i - r0]
const xi = vector[i]
const yi = vector[n + i]
const zi = vector[2 * n + i]
const bi = bias === null ? 1 : bias[i]
const li = logBias === null ? 0 : logBias[i]
// Accumulated for `i` and flushed once, rather than read and written
// `n` times through the gradient array.
let gx = 0
let gy = 0
let gz = 0
for (let j = i + 1; j < n; j++) {
const c = counts[row + j]
if (!(c >= 0)) continue
const dx = xi - vector[j]
const dy = yi - vector[n + j]
const dz = zi - vector[2 * n + j]
const d2 = dx * dx + dy * dy + dz * dz
if (!(d2 > 0)) return Number.POSITIVE_INFINITY
const logLambda = -half * Math.log(d2)
const lambda = cube ? 1 / (d2 * Math.sqrt(d2)) : Math.exp(logLambda)
const mu = (bias === null ? 1 : bi * bias[j]) * lambda
let k: number
if (nb) {
const logQ = slope * logLambda - c0
const q = Math.exp(logQ)
const r = mu / q
// `log1p` only where it is needed: above a thousandth `log(1 + q)`
// loses nothing a double holds, and costs half as much.
const spread = q > 1e-3 ? Math.log(1 + q) : Math.log1p(q)
value += (c + r) * spread
k = (slope * (mu - c)) / (1 + q) + c1 * r * spread
if (c > 0) {
const logMu = li + (logBias === null ? 0 : logBias[j]) + logLambda
const logR = logMu - logQ
// The gamma terms, `G` and `H` ({@link countTerms}). A whole count
// up to {@link SMALL_COUNT} — most pairs of most maps — is summed
// exactly here rather than in a call: `Π_{k<c} (r + k)` and
// `Σ 1/(r + k)` as one running fraction over it, four factors at a
// time so the chain of products each waits on is a quarter as long.
let g = 0
let h = 1
if (c > 1 && c <= SMALL_COUNT && c === Math.floor(c) && r < SMALL_RATE) {
let product = 1
let numerator = 0
let m = 0
for (; m + 4 <= c; m += 4) {
const a = r + m
const b = a + 1
const e = a + 2
const f = a + 3
const ab = a * b
const ef = e * f
const block = ab * ef
numerator = numerator * block + ((a + b) * ef + (e + f) * ab) * product
product *= block
}
for (; m < c; m++) {
const a = r + m
numerator = numerator * a + product
product *= a
}
g = Math.log(product) - c * logR
h = (r * numerator) / product
} else if (c !== 1) {
countTerms(c, r, logR)
g = termG
h = termH
}
value -= g + c * logMu
k -= c1 * h
}
} else {
value += mu
k = mu - c
if (c > 0) value -= c * (li + (logBias === null ? 0 : logBias[j]) + logLambda)
}
const coefficient = (-alpha * k) / d2
const fx = coefficient * dx
const fy = coefficient * dy
const fz = coefficient * dz
gx += fx
gy += fy
gz += fz
out[j - r0] -= fx
out[len + j - r0] -= fy
out[2 * len + j - r0] -= fz
}
out[i - r0] += gx
out[len + i - r0] += gy
out[2 * len + i - r0] += gz
}
return value
}
/**
* A band's gradient added into the whole one: `part` holds beads `from` to
* `from + length`, `x` then `y` then `z` — `[r0, n)` for a band of the
* likelihood, `[0, r1)` for one of the MDS.
*/
export function addBand(gradient: Float64Array, part: Float64Array, n: number, from: number, length: number): void {
for (let k = 0; k < length; k++) {
gradient[from + k] += part[k]
gradient[n + from + k] += part[length + k]
gradient[2 * n + from + k] += part[2 * length + k]
}
}
/**
* The negative log-likelihood and its gradient, as the sum of its bands
* ({@link likelihoodBands}), each band's value and gradient added in band
* order — the same sums `analysis/likelihoodThreads.ts` makes of bands it had
* computed on other threads, so the two agree to the bit.
*/
export function negBinomObjective(model: NegBinomModel): Objective {
const terms = likelihoodTerms(model)
const { counts, n } = model
const edges = likelihoodBands(n)
const parts = Array.from({ length: edges.length - 1 }, (_, b) => new Float64Array(3 * (n - edges[b])))
return (vector, gradient) => {
gradient.fill(0)
let value = 0
for (let b = 0; b + 1 < edges.length; b++) {
const part = parts[b]
const share = likelihoodBand(terms, counts, null, edges[b], edges[b + 1], vector, part)
if (!Number.isFinite(share)) return Number.POSITIVE_INFINITY
value += share
addBand(gradient, part, n, edges[b], n - edges[b])
}
return Number.isFinite(value) ? value : Number.POSITIVE_INFINITY
}
}
/** The optimiser's vector: `x`, then `y`, then `z`. */
export function packCoordinates(coords: Coordinates): Float64Array {
const n = coords.x.length
const out = new Float64Array(3 * n)
out.set(coords.x, 0)
out.set(coords.y, n)
out.set(coords.z, 2 * n)
return out
}
/** The vector back as coordinates, in new arrays. */
export function readCoordinates(vector: Float64Array, n: number): Coordinates {
return {
x: Float64Array.from(vector.subarray(0, n)),
y: Float64Array.from(vector.subarray(n, 2 * n)),
z: Float64Array.from(vector.subarray(2 * n, 3 * n)),
}
}
/** The likelihood at a model, without moving it — for a caller that wants the number rather than the fit. */
export function negBinomValue(model: NegBinomModel, coords: Coordinates): number {
return negBinomObjective(model)(packCoordinates(coords), new Float64Array(3 * model.n))
}
/**
* pastis' MDS from `start`, an evaluation at a time: L-BFGS on
* {@link mdsObjective} to pastis' `factr = 10¹²`. `wishes` is the matrix
* {@link storeWishes} wrote below the diagonal of.
*/
export function fitMds(
wishes: DenseMatrix,
n: number,
start: Coordinates,
options: { iterations?: number } = {},
): Generator<LbfgsProgress, MdsFit, void> {
return answered(mdsWalk(n, start, options), mdsObjective(wishes, n))
}
/**
* pastis' MDS from `start`, asking for each evaluation rather than making it,
* as {@link likelihoodWalk} does, so whoever drives it can share each among
* threads. {@link fitMds} is it answered here.
*/
export function mdsWalk(
n: number,
start: Coordinates,
options: { iterations?: number } = {},
): Generator<LbfgsAsk, MdsFit, number> {
const walk = lbfgsWalk(packCoordinates(start), {
iterations: options.iterations ?? PASTIS_MDS.iterations,
history: HISTORY,
tolerance: PASTIS_MDS.tolerance,
gradientTolerance: PASTIS_MDS.gradient,
})
return fitOf(walk, n)
}
/**
* The likelihood's fit from `start`, to pastis' `factr = 10` and `pgtol = 1`,
* or until `iterations` run out — asking for each evaluation rather than
* making it (`lbfgsWalk`), so that whoever drives it can share each among
* threads and `await` them. {@link fitNegativeBinomial} is it answered here, by
* {@link negBinomObjective}; the two give the same steps and the same model.
*
* `start` is not written over — a cancelled fit leaves the MDS model it began
* from intact.
*/
export function likelihoodWalk(
n: number,
start: Coordinates,
options: { iterations?: number } = {},
): Generator<LbfgsAsk, NegBinomFit, number> {
const walk = lbfgsWalk(packCoordinates(start), {
iterations: options.iterations ?? 100000,
history: HISTORY,
tolerance: PASTIS_NB.tolerance,
gradientTolerance: PASTIS_NB.gradient,
})
return fitOf(walk, n)
}
/**
* Fit the model from `start`, an evaluation at a time, on this thread.
*
* The generator is the caller's way to breathe: at ten thousand beads one
* evaluation is fifty million pair visits, a line search is one or more of
* them, and a run that did not yield between them would freeze the tab for the
* whole of a fit and leave a cancel nowhere to land.
*/
export function fitNegativeBinomial(
model: NegBinomModel,
start: Coordinates,
options: { iterations?: number } = {},
): Generator<LbfgsProgress, NegBinomFit, void> {
return answered(likelihoodWalk(model.n, start, options), negBinomObjective(model))
}
/** A walk's result as coordinates. */
function* fitOf(walk: Generator<LbfgsAsk, LbfgsResult, number>, n: number): Generator<LbfgsAsk, NegBinomFit, number> {
const result = yield* walk
return {
coords: readCoordinates(result.point, n),
value: result.value,
iterations: result.iterations,
evaluations: result.evaluations,
stop: result.stop,
}
}
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