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* IDR — the irreproducible discovery rate — as a port (§18.9).
*
* The test yapc puts its candidates through, and the one the field puts every
* peak set through: two replicates rank the same candidates, and a copula
* mixture model asks how far down the two rankings go on agreeing (Li, Brown,
* Huang & Bickel 2011, *Ann. Appl. Stat.* 5:1752). What it reports per peak is
* the posterior probability that the peak belongs to the *irreproducible*
* component — the **local** IDR — and, from those, the **global** IDR of a
* threshold: the expected share of irreproducible peaks among everything at
* least as reproducible. The global IDR is what is thresholded, and it is to a
* ranked list what a q-value is to a list of p-values.
*
* ## The upstream, exactly
*
* Nathan Boley's `idr` 2.0.3, at the commit yapc's own `yapc/idr.py` cites
* (`74665e73`), through the function yapc calls: `fit_model_and_calc_local_idr`
* with its defaults, then `calc_global_IDR`. It is the implementation ENCODE's
* pipelines use. Everything is here in its order, with its constants and its
* convergence tests, because a fit is a path and not only a place: a coordinate
* ascent started one step differently ends a hair apart, and a peak on the
* threshold is then on the other side of it.
*
* - **Pseudo-values by bisection**, `inv_cdf.pyx`: each rank is mapped to the
* point where the mixture's marginal CDF reaches `(rank + 1) / (n + 1)`,
* bisected to a tolerance **in CDF space**, with the bracket carried from one
* rank to the next. The fit uses 10⁻⁸ and the final local IDRs 10⁻¹².
* - **`erf` is fdlibm's**, the one glibc's is built from, since the bisection
* stops on a comparison of it and a different `erf` stops somewhere else.
* - **Sums are numpy's**, pairwise (`numpySum`), for the same reason.
* - **EM, falling back to coordinate ascent** with SciPy's bounded Brent
* (`fminbound`) whenever an EM step leaves the parameter box — which is what
* happens when the two tracks do not behave like replicates, and is reported.
* - **The asserts are kept.** `idr` asserts that no step lowers the likelihood
* by more than 10⁻⁶; where it would stop with an `AssertionError`, this stops
* with an {@link IdrError} saying which step, rather than going on to produce
* numbers the reference never would.
*
* What is deliberately *not* reproduced is its tie-breaking: `idr` ranks with
* `np.lexsort` on an unseeded `np.random.random` key, so two equal scores are
* ordered by chance and a rerun is a different fit. Here a tie is ordered by
* position in the list, which is one of the orders `idr` could have drawn, and a
* rerun is the same fit (§18.4). Ties between curvature scores are rare enough
* that the golden has none.
*/
import { numpyMean, numpySum } from './stats'
/** `idr/__init__.py`'s constants, by the names it gives them. */
export const IDR_DEFAULTS = {
maxIter: 3000,
convergenceEps: 1e-6,
/** Starting point: μ, σ, ρ, and the mixing proportion. */
start: [0.1, 1.0, 0.2, 0.5] as const,
/** `utility.DEFAULT_PV_COVERGE_EPS`, the fit's pseudo-value tolerance. */
pseudoValueEps: 1e-8,
/** The tolerance `calc_local_IDR` recomputes the pseudo-values to. */
finalPseudoValueEps: 1e-12,
/** EM steps per round before the pseudo-values are recomputed. */
emStepsPerRound: 30,
} as const
/** The box the model's parameters are kept in: μ, σ, ρ, p. */
const MIN_THETA = [0.0, 0.2, 0.1, 0.01] as const
const MAX_THETA = [20.0, 20, 0.99, 0.99] as const
/** {@link MIN_THETA} and {@link MAX_THETA}, for saying where a fit ended against them. */
export const IDR_BOX = { min: MIN_THETA, max: MAX_THETA } as const
/** μ, σ, ρ and the mixing proportion `p` of the reproducible component. */
export type IdrTheta = [mu: number, sigma: number, rho: number, p: number]
/** Raised where `idr` itself would stop on an assertion. */
export class IdrError extends Error {
constructor(message: string) {
super(message)
this.name = 'IdrError'
}
}
// ---------------------------------------------------------------------------
// erf, as fdlibm computes it
// ---------------------------------------------------------------------------
const scratch = new Float64Array(1)
const words = new Int32Array(scratch.buffer)
/** Which of the two words is the high one, on this machine. */
const HIGH = new Uint8Array(Float64Array.of(1).buffer)[7] === 0x3f ? 1 : 0
const LOW = 1 - HIGH
const ERX = 8.45062911510467529297e-1
const EFX = 1.28379167095512586316e-1
const EFX8 = 1.02703333676410069053e0
const PP = [
1.28379167095512558561e-1, -3.25042107247001499370e-1, -2.84817495755985104766e-2,
-5.77027029648944159157e-3, -2.37630166566501626084e-5,
] as const
const QQ = [
3.97917223959155352819e-1, 6.50222499887672944485e-2, 5.08130628187576562776e-3,
1.32494738004321644526e-4, -3.96022827877536812320e-6,
] as const
const PA = [
-2.36211856075265944077e-3, 4.14856118683748331666e-1, -3.72207876035701323847e-1,
3.18346619901161753674e-1, -1.10894694282396677476e-1, 3.54783043256182359371e-2,
-2.16637559486879084300e-3,
] as const
const QA = [
1.06420880400844228286e-1, 5.40397917702171048937e-1, 7.18286544141962662868e-2,
1.26171219808761642112e-1, 1.36370839120290507362e-2, 1.19844998467991074170e-2,
] as const
const RA = [
-9.86494403484714822705e-3, -6.93858572707181764372e-1, -1.05586262253232909814e1,
-6.23753324503260060396e1, -1.62396669462573470355e2, -1.84605092906711035994e2,
-8.12874355063065934246e1, -9.81432934416914548592e0,
] as const
const SA = [
1.96512716674392571292e1, 1.37657754143519042600e2, 4.34565877475229228821e2,
6.45387271733267880336e2, 4.29008140027567833386e2, 1.08635005541779435134e2,
6.57024977031928170135e0, -6.04244152148580987438e-2,
] as const
const RB = [
-9.86494292470009928597e-3, -7.99283237680523006574e-1, -1.77579549177547519889e1,
-1.60636384855821916062e2, -6.37566443368389627722e2, -1.02509513161107724954e3,
-4.83519191608651397019e2,
] as const
const SB = [
3.03380607434824582924e1, 3.25792512996573918826e2, 1.53672958608443695994e3,
3.19985821950859553908e3, 2.55305040643316442583e3, 4.74528541206955367215e2,
-2.24409524465858183362e1,
] as const
/**
* The error function, as fdlibm's `s_erf.c` computes it — the code glibc's `erf`
* and so the C `erf` that `idr`'s Cython calls are both built from.
*
* The polynomial forms are written out rather than looped over, because a loop
* that evaluated the same Horner scheme in another association would be another
* function in the last place, which is the one thing this is here to not be.
*/
export function erf(x: number): number {
scratch[0] = x
const hx = words[HIGH]
const ix = hx & 0x7fffffff
if (ix >= 0x7ff00000) {
// erf(NaN) is NaN, and erf(±inf) is ±1.
const i = (hx >>> 31) << 1
return 1 - i + 1 / x
}
if (ix < 0x3feb0000) {
if (ix < 0x3e300000) {
if (ix < 0x00800000) return 0.125 * (8.0 * x + EFX8 * x)
return x + EFX * x
}
const z = x * x
const r = PP[0] + z * (PP[1] + z * (PP[2] + z * (PP[3] + z * PP[4])))
const s = 1 + z * (QQ[0] + z * (QQ[1] + z * (QQ[2] + z * (QQ[3] + z * QQ[4]))))
const y = r / s
return x + x * y
}
if (ix < 0x3ff40000) {
const s = Math.abs(x) - 1
const p = PA[0] + s * (PA[1] + s * (PA[2] + s * (PA[3] + s * (PA[4] + s * (PA[5] + s * PA[6])))))
const q = 1 + s * (QA[0] + s * (QA[1] + s * (QA[2] + s * (QA[3] + s * (QA[4] + s * QA[5])))))
return hx >= 0 ? ERX + p / q : -ERX - p / q
}
if (ix >= 0x40180000) return hx >= 0 ? 1 - 1e-300 : 1e-300 - 1
const ax = Math.abs(x)
const s = 1 / (ax * ax)
let r: number
let q: number
if (ix < 0x4006db6e) {
r = RA[0] + s * (RA[1] + s * (RA[2] + s * (RA[3] + s * (RA[4] + s * (RA[5] + s * (RA[6] + s * RA[7]))))))
q =
1 +
s * (SA[0] + s * (SA[1] + s * (SA[2] + s * (SA[3] + s * (SA[4] + s * (SA[5] + s * (SA[6] + s * SA[7])))))))
} else {
r = RB[0] + s * (RB[1] + s * (RB[2] + s * (RB[3] + s * (RB[4] + s * (RB[5] + s * RB[6])))))
q = 1 + s * (SB[0] + s * (SB[1] + s * (SB[2] + s * (SB[3] + s * (SB[4] + s * (SB[5] + s * SB[6]))))))
}
// `SET_LOW_WORD(z, 0)`: |x| with its low 32 bits cleared.
scratch[0] = ax
words[LOW] = 0
const z = scratch[0]
const e = Math.exp(-z * z - 0.5625) * Math.exp((z - ax) * (z + ax) + r / q)
return hx >= 0 ? 1 - e / ax : e / ax - 1
}
// ---------------------------------------------------------------------------
// Pseudo-values
// ---------------------------------------------------------------------------
/** `c_cdf`: the marginal CDF of the two-component mixture. */
function mixtureCdf(x: number, mu: number, sigma: number, lamda: number): number {
const normX = (x - mu) / sigma
return 0.5 * lamda * erf(normX / Math.SQRT2) + 0.5 * (1 - lamda) * erf(x / Math.SQRT2) + 0.5
}
/** `c_cdf_i`: bisection for the point whose CDF is `r`, to `eps` in CDF space. */
function inverseCdf(
r: number,
mu: number,
sigma: number,
lamda: number,
lb: number,
ub: number,
eps: number,
): number {
let mid = 0
for (let i = 0; i < 1000; i++) {
mid = lb + (ub - lb) / 2
const guess = mixtureCdf(mid, mu, sigma, lamda)
if (Math.abs(guess - r) < eps) return mid
if (guess < r) lb = mid
else ub = mid
}
return mid
}
/**
* `c_compute_pseudo_values`: each rank's point on the mixture's marginal.
*
* The ranks are 0-based and a permutation. The points are found in rank order,
* each bracketed from the one before — the lower end is the previous answer and
* the upper end twice the last step past it — which is what makes this linear
* in the number of peaks rather than a fresh bisection each.
*/
export function pseudoValues(
ranks: Int32Array,
mu: number,
sigma: number,
lamda: number,
eps: number,
): Float64Array {
return byRank(ranks, rankPoints(ranks.length, mu, sigma, lamda, eps))
}
/**
* Both replicates' pseudo-values, from one bisection.
*
* `idr` bisects once per replicate, but what it bisects for — the point of each
* rank `0 … n − 1` — depends on the number of peaks and the parameters and not
* on the ranks, which only say which peak takes which point. Two replicates of
* the same peaks are the same `n`, so the points are found once and each
* replicate's ranks read them: the same numbers, for half the `erf`s that were
* most of a fit's time.
*/
export function pseudoValuePair(
r1: Int32Array,
r2: Int32Array,
mu: number,
sigma: number,
lamda: number,
eps: number,
): [Float64Array, Float64Array] {
const points = rankPoints(r1.length, mu, sigma, lamda, eps)
const other = r2.length === r1.length ? points : rankPoints(r2.length, mu, sigma, lamda, eps)
return [byRank(r1, points), byRank(r2, other)]
}
/** `points[ranks[i]]` for each peak `i`. */
function byRank(ranks: Int32Array, points: Float64Array): Float64Array {
const out = new Float64Array(ranks.length)
for (let i = 0; i < ranks.length; i++) out[i] = points[ranks[i]]
return out
}
/** The point on the mixture's marginal of each rank `0 … n − 1`, in rank order. */
function rankPoints(n: number, mu: number, sigma: number, lamda: number, eps: number): Float64Array {
const ordered = new Float64Array(n)
if (n === 0) return ordered
const pseudoN = n + 1
const smallest = 1 / pseudoN
let lb = 0 <= mu ? 0 : mu
while (mixtureCdf(lb, mu, sigma, lamda) > smallest) lb -= 1
let ub = 0 >= mu ? 0 : mu
while (mixtureCdf(ub, mu, sigma, lamda) < 1 - smallest) ub += 1
lb = inverseCdf(smallest, mu, sigma, lamda, lb, ub, eps)
ordered[0] = lb
for (let i = 1; i < n; i++) {
const r = (i + 1) / pseudoN
if (mixtureCdf(ub, mu, sigma, lamda) < r) ub += 10
const res = inverseCdf(r, mu, sigma, lamda, lb, ub, eps)
ordered[i] = res
lb = res
ub = lb + 2 * (res - ordered[i - 1])
}
return ordered
}
// ---------------------------------------------------------------------------
// The model
// ---------------------------------------------------------------------------
/**
* The two-component model over one set of pseudo-values.
*
* Built once per set and asked many times: an EM round evaluates the likelihood
* after every step, and a coordinate-ascent round a few hundred times inside
* `fminbound`, all at the same pseudo-values. The **noise** component's density
* depends on nothing else — it is a standard bivariate normal — so it is
* computed once here, where `idr` recomputes it on every call; the values are
* identical, since the same operations are done on the same numbers, and a
* coordinate-ascent fit is two to three times faster for it.
*/
class Mixture {
readonly n: number
/** `exp(calc_gaussian_lhd(0, 0, 1, 1, 0, z1, z2))`, per peak. */
private readonly noise: Float64Array
/** The signal component's density at the parameters last asked about. */
private readonly signal: Float64Array
private readonly terms: Float64Array
/** The μ, σ and ρ {@link signal} was computed at; `p` does not enter it. */
private at: [number, number, number] = [Number.NaN, Number.NaN, Number.NaN]
constructor(
readonly z1: Float64Array,
readonly z2: Float64Array,
) {
this.n = z1.length
this.noise = new Float64Array(this.n)
this.signal = new Float64Array(this.n)
this.terms = new Float64Array(this.n)
density(0, 1, 0, z1, z2, this.noise)
}
/** `calc_gaussian_mix_log_lhd`: the log-likelihood, summed numpy's way. */
logLik(theta: Readonly<IdrTheta>): number {
this.signalAt(theta)
const p = theta[3]
for (let i = 0; i < this.n; i++) {
this.terms[i] = Math.log(p * this.signal[i] + (1 - p) * this.noise[i])
}
return numpySum(this.terms)
}
/** `calc_post_membership_prbs`: each peak's probability of being reproducible. */
membership(theta: Readonly<IdrTheta>): Float64Array {
this.signalAt(theta)
const p = theta[3]
const out = new Float64Array(this.n)
for (let i = 0; i < this.n; i++) {
const s = p * this.signal[i]
out[i] = s / (s + (1 - p) * this.noise[i])
}
return out
}
/**
* The signal density at `theta`, recomputed only when μ, σ or ρ moved: a
* coordinate-ascent step on the mixing proportion evaluates the likelihood
* thirty times at one density, and it is the same numbers each time.
*/
private signalAt(theta: Readonly<IdrTheta>): void {
const at = this.at
if (theta[0] === at[0] && theta[1] === at[1] && theta[2] === at[2]) return
density(theta[0], theta[1], theta[2], this.z1, this.z2, this.signal)
this.at = [theta[0], theta[1], theta[2]]
}
}
/**
* `numpy.exp(calc_gaussian_lhd(mu, mu, sigma, sigma, rho, z1, z2))`, into `out`.
*
* The log density of a bivariate normal with equal means, equal spreads and
* correlation `rho`, with `idr`'s own `-1.837877` for `−log 2π`, in `idr`'s
* order of operations, and then its exponential.
*/
function density(
mu: number,
sigma: number,
rho: number,
z1: Float64Array,
z2: Float64Array,
out: Float64Array,
): void {
const constant = -1.837877 - Math.log(sigma) - Math.log(sigma) - 0.5 * Math.log(1 - rho * rho)
const twoRho = 2 * rho
const denominator = 2 * (1 - rho * rho)
for (let i = 0; i < z1.length; i++) {
const a = (z1[i] - mu) / sigma
const b = (z2[i] - mu) / sigma
out[i] = Math.exp(constant - (a * a - twoRho * a * b + b * b) / denominator)
}
}
/** `EM_step`: one expectation–maximisation update, with the pseudo-values held. */
function emStep(model: Mixture, theta: Readonly<IdrTheta>): IdrTheta {
const { n, z1, z2 } = model
const ez = model.membership(theta)
const ezSum = numpySum(ez)
const work = new Float64Array(n)
for (let i = 0; i < n; i++) work[i] = ez[i] * z1[i]
const mu1 = numpySum(work) / ezSum
for (let i = 0; i < n; i++) work[i] = ez[i] * z2[i]
const mu2 = numpySum(work) / ezSum
const mu = (mu1 + mu2) / 2
for (let i = 0; i < n; i++) {
const d = z1[i] - mu
work[i] = ez[i] * (d * d)
}
const squares1 = numpySum(work)
for (let i = 0; i < n; i++) {
const d = z2[i] - mu
work[i] = ez[i] * (d * d)
}
const squares2 = numpySum(work)
const sigma = Math.sqrt((squares1 + squares2) / (2 * ezSum))
for (let i = 0; i < n; i++) work[i] = ez[i] * (z1[i] - mu) * (z2[i] - mu)
const rho = (2 * numpySum(work)) / (squares1 + squares2)
return [mu, sigma, rho, ezSum / n]
}
/** `clip_model_params`: the parameters put back in their box, and whether they left it. */
function clipTheta(theta: Readonly<IdrTheta>): { theta: IdrTheta; changed: boolean } {
const out: IdrTheta = [theta[0], theta[1], theta[2], theta[3]]
let changed = false
if (out[0] < MIN_THETA[0]) {
out[0] = MIN_THETA[0]
changed = true
}
if (out[1] < MIN_THETA[1]) {
out[1] = MIN_THETA[1]
changed = true
}
if (out[2] < MIN_THETA[2]) {
out[2] = MIN_THETA[2]
changed = true
} else if (out[2] > MAX_THETA[2]) {
out[2] = MAX_THETA[2]
changed = true
}
if (out[3] < MIN_THETA[3]) {
out[3] = MIN_THETA[3]
changed = true
} else if (out[3] > MAX_THETA[3]) {
out[3] = MAX_THETA[3]
changed = true
}
return { theta: out, changed }
}
/** `EM_iteration`: EM steps until the likelihood stops rising or a step leaves the box. */
function emIteration(
model: Mixture,
start: Readonly<IdrTheta>,
maxIter: number,
eps: number,
): { theta: IdrTheta; changed: boolean } {
let previous: Readonly<IdrTheta> = start
let previousLhd = model.logLik(previous)
let theta: IdrTheta = [start[0], start[1], start[2], start[3]]
for (let i = 0; i < maxIter; i++) {
const clipped = clipTheta(emStep(model, previous))
theta = clipped.theta
const lhd = model.logLik(theta)
if (clipped.changed) return { theta, changed: true }
if (!(lhd + 1e-6 >= previousLhd)) {
throw new IdrError('An EM step lowered the likelihood, which stops idr’s own fit.')
}
if (lhd - previousLhd < eps) return { theta, changed: false }
previous = theta
previousLhd = lhd
}
return { theta, changed: false }
}
/**
* `fminbound`: SciPy's bounded Brent minimisation (`_minimize_scalar_bounded`,
* SciPy 1.10), with its defaults — `xatol = 1e-5`, 500 evaluations.
*/
export function fminbound(
f: (x: number) => number,
x1: number,
x2: number,
xatol = 1e-5,
maxfun = 500,
): number {
if (!Number.isFinite(x1) || !Number.isFinite(x2)) {
throw new IdrError('Optimization bounds must be finite scalars.')
}
if (x1 > x2) throw new IdrError('The lower bound exceeds the upper bound.')
const sqrtEps = Math.sqrt(2.2e-16)
const goldenMean = 0.5 * (3.0 - Math.sqrt(5.0))
let a = x1
let b = x2
let fulc = a + goldenMean * (b - a)
let nfc = fulc
let xf = fulc
let rat = 0.0
let e = 0.0
let x = xf
let fx = f(x)
let num = 1
let ffulc = fx
let fnfc = fx
let xm = 0.5 * (a + b)
let tol1 = sqrtEps * Math.abs(xf) + xatol / 3.0
let tol2 = 2.0 * tol1
while (Math.abs(xf - xm) > tol2 - 0.5 * (b - a)) {
let golden = true
if (Math.abs(e) > tol1) {
golden = false
let r = (xf - nfc) * (fx - ffulc)
let q = (xf - fulc) * (fx - fnfc)
let p = (xf - fulc) * q - (xf - nfc) * r
q = 2.0 * (q - r)
if (q > 0.0) p = -p
q = Math.abs(q)
r = e
e = rat
if (Math.abs(p) < Math.abs(0.5 * q * r) && p > q * (a - xf) && p < q * (b - xf)) {
rat = (p + 0.0) / q
x = xf + rat
if (x - a < tol2 || b - x < tol2) {
const si = Math.sign(xm - xf) + (xm - xf === 0 ? 1 : 0)
rat = tol1 * si
}
} else {
golden = true
}
}
if (golden) {
e = xf >= xm ? a - xf : b - xf
rat = goldenMean * e
}
const si = Math.sign(rat) + (rat === 0 ? 1 : 0)
x = xf + si * Math.max(Math.abs(rat), tol1)
const fu = f(x)
num += 1
if (fu <= fx) {
if (x >= xf) a = xf
else b = xf
fulc = nfc
ffulc = fnfc
nfc = xf
fnfc = fx
xf = x
fx = fu
} else {
if (x < xf) a = x
else b = x
if (fu <= fnfc || nfc === xf) {
fulc = nfc
ffulc = fnfc
nfc = x
fnfc = fu
} else if (fu <= ffulc || fulc === xf || fulc === nfc) {
fulc = x
ffulc = fu
}
}
xm = 0.5 * (a + b)
tol1 = sqrtEps * Math.abs(xf) + xatol / 3.0
tol2 = 2.0 * tol1
if (num >= maxfun) break
}
return xf
}
/** `CA_step`: one coordinate moved to its bounded optimum, if that helps. */
function caStep(model: Mixture, theta: IdrTheta, index: number): number {
const inner: IdrTheta = [theta[0], theta[1], theta[2], theta[3]]
const fixed = theta[index]
const loss = (alpha: number): number => {
inner[index] = fixed + alpha
return -model.logLik(inner)
}
if (!(fixed >= MIN_THETA[index]) || !(fixed <= MAX_THETA[index])) {
throw new IdrError('A coordinate-ascent step started outside idr’s parameter box.')
}
const alpha = fminbound(loss, MIN_THETA[index] - fixed, MAX_THETA[index] - fixed)
const previous = -loss(0)
const next = -loss(alpha)
if (next > previous) {
theta[index] += alpha
return next
}
return previous
}
/** `CA_iteration`: coordinate ascent until the likelihood stops rising. */
function caIteration(
model: Mixture,
start: Readonly<IdrTheta>,
maxIter: number,
eps: number,
): IdrTheta {
let previousLhd = model.logLik(start)
let theta: IdrTheta = [start[0], start[1], start[2], start[3]]
let lhd = previousLhd
for (let i = 0; i < maxIter; i++) {
for (let index = 0; index < 4; index++) lhd = caStep(model, theta, index)
const clipped = clipTheta(theta)
if (clipped.changed) {
throw new IdrError('Coordinate ascent left idr’s parameter box, which stops idr’s own fit.')
}
theta = clipped.theta
if (!(lhd + 1e-6 >= previousLhd)) {
throw new IdrError('A coordinate-ascent round lowered the likelihood, which stops idr’s own fit.')
}
if (lhd - previousLhd < eps) return theta
previousLhd = lhd
}
return theta
}
/** What a fit found, and how it got there. */
export interface IdrFit {
theta: IdrTheta
/** Rounds of EM (or coordinate ascent) and pseudo-value updates. */
rounds: number
/** Whether any round fell back to coordinate ascent — an EM step left the box. */
coordinateAscent: boolean
}
/**
* `estimate_model_params` → `EMP_with_pseudo_value_algorithm`, with `idr`'s
* defaults: rounds of EM on fixed pseudo-values, the pseudo-values recomputed
* after each, until both the parameters and the pseudo-values move by less than
* the tolerance — and not before the fifth round.
*
* `between` is called after every round with the number of rounds done, and is
* awaited: it is where a caller in a worker hands the thread back and checks for
* a cancel, since a fit over a hundred thousand peaks is minutes rather than a
* moment. It is not given anything it could change the fit with.
*/
export async function fitIdr(
r1: Int32Array,
r2: Int32Array,
options: {
start?: Readonly<IdrTheta>
maxIter?: number
eps?: number
between?: (rounds: number) => void | Promise<void>
} = {},
): Promise<IdrFit> {
const eps = options.eps ?? IDR_DEFAULTS.convergenceEps
const maxIter = options.maxIter ?? IDR_DEFAULTS.maxIter
const pvEps = IDR_DEFAULTS.pseudoValueEps
const s = options.start ?? IDR_DEFAULTS.start
let theta: IdrTheta = [s[0], s[1], s[2], s[3]]
let [z1, z2] = pseudoValuePair(r1, r2, theta[0], theta[1], theta[3], pvEps)
let coordinateAscent = false
let rounds = 0
const change = new Float64Array(r1.length)
for (let i = 0; i < maxIter; i++) {
rounds = i + 1
const previous = theta
const model = new Mixture(z1, z2)
const em = emIteration(model, previous, IDR_DEFAULTS.emStepsPerRound, eps / 10)
theta = em.theta
if (em.changed) {
coordinateAscent = true
theta = caIteration(model, previous, IDR_DEFAULTS.emStepsPerRound, eps / 10)
}
const paramChange = numpySum([
Math.abs(theta[0] - previous[0]),
Math.abs(theta[1] - previous[1]),
Math.abs(theta[2] - previous[2]),
Math.abs(theta[3] - previous[3]),
])
const previousZ1 = z1
const previousZ2 = z2
;[z1, z2] = pseudoValuePair(r1, r2, theta[0], theta[1], theta[3], pvEps)
for (let k = 0; k < change.length; k++) change[k] = Math.abs(previousZ1[k] - z1[k])
const moved1 = numpyMean(change)
for (let k = 0; k < change.length; k++) change[k] = Math.abs(previousZ2[k] - z2[k])
const moved = moved1 + numpyMean(change)
if (options.between !== undefined) await options.between(rounds)
if (i > 3 && paramChange < eps && moved < eps) break
}
return { theta, rounds, coordinateAscent }
}
/**
* `calc_local_IDR`: one minus each peak's reproducible membership, at pseudo-values
* recomputed to 10⁻¹².
*
* Two of `idr`'s rules are applied after the model, and both are kept. A peak
* whose two pseudo-values sum below zero — below the noise component's mean in
* both replicates, taken together — is set to 1 (`FILTER_PEAKS_BELOW_NOISE_MEAN`),
* and every value is clipped to `[1e-6, 1]`, because "it doesn't make sense for
* the IDR values to be smaller than the optimization tolerance".
*/
export function localIdr(theta: Readonly<IdrTheta>, r1: Int32Array, r2: Int32Array): Float64Array {
const eps = IDR_DEFAULTS.finalPseudoValueEps
const [z1, z2] = pseudoValuePair(r1, r2, theta[0], theta[1], theta[3], eps)
const ez = new Mixture(z1, z2).membership(theta)
const floor = IDR_DEFAULTS.convergenceEps
const out = new Float64Array(ez.length)
for (let i = 0; i < out.length; i++) {
let value = 1 - ez[i]
if (z1[i] + z2[i] < 0) value = 1
out[i] = value < floor ? floor : value > 1 ? 1 : value
}
return out
}
/**
* `calc_global_IDR`: for each peak, the mean local IDR of every peak at least as
* reproducible — ties included, since `rankdata(…, method='max')` gives a tie
* the rank of its last member.
*
* `idr` takes each of those means afresh, which is quadratic in the number of
* peaks; a running total gives the same means to within a few units in the last
* place, which is below anything a threshold at 10⁻³ can see.
*/
export function globalIdr(local: Float64Array): Float64Array {
const n = local.length
const order = Array.from({ length: n }, (_, i) => i)
order.sort((a, b) => local[a] - local[b] || a - b)
const out = new Float64Array(n)
let total = 0
let at = 0
while (at < n) {
let end = at
while (end < n && local[order[end]] === local[order[at]]) {
total += local[order[end]]
end++
}
const mean = total / end
for (let k = at; k < end; k++) out[order[k]] = mean
at = end
}
return out
}
/** `-math.log10(max(1e-5, IDR))`: the scaled IDR yapc and `idr` both report. */
export function scaledIdr(value: number): number {
return -Math.log10(Math.max(1e-5, value))
}
/**
* Ranks in `idr`'s sense: 0 for the lowest score, `n − 1` for the highest.
*
* `np.lexsort((np.random.random(n), scores)).argsort()` — ties ordered by the
* random key. Here by position; see the module docstring.
*/
export function idrRanks(scores: Float64Array): Int32Array {
const n = scores.length
const order = Array.from({ length: n }, (_, i) => i)
order.sort((a, b) => scores[a] - scores[b] || a - b)
const out = new Int32Array(n)
for (let k = 0; k < n; k++) out[order[k]] = k
return out
}
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