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lib/evstats.js

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  1. // Extreme-value / robust statistical thresholds for per-customer behavioral
  2. // baselines (quantities today, amounts later). Pure JS, zero deps.
  3. // ---------------------------------------------------------------------------
  4. // Companions to the deterministic class caps in lib/item-classes.js. Where
  5. // `qtyCap` uses the crude "3× observed max" heuristic, the estimators here fit
  6. // an upper tail to the customer's own sample:
  7. //
  8. // mad — robust z-scale: median + k·(1.4826·MAD). Cheap, honest for small n,
  9. // useless when the sample is degenerate (all-equal, e.g. all qty 1).
  10. // gpd — Peaks-over-threshold (Pickands–Balkema–de Haan): exceedances over
  11. // the empirical q75 fitted with a Generalized Pareto (L-moments),
  12. // threshold = POT return level "once per returnPeriodK·n purchases".
  13. // The statistically recommended tail method for thresholds.
  14. // gev — Block-maxima (Fisher–Tippett): GEV fitted by L-moments to maxima of
  15. // consecutive blocks; threshold = high quantile of the fitted GEV.
  16. //
  17. // Every method has a minimum-sample floor below which it refuses to speak
  18. // (small-sample EV fits are noise, not signal). `statThreshold` combines the
  19. // methods that pass their floors (max = conservative: the higher honest
  20. // estimate wins) and clamps the result:
  21. // • never below `floor` (the deterministic class cap) — safety stays;
  22. // • never below the observed maximum — never flag what the customer did;
  23. // • never above `upperBound(floor, observedMax)` — a wild fit cannot blind
  24. // the check to fraud.
  25. //
  26. // This module is engine-side enforcement only. The trained feature 14
  27. // (qty_over_class_cap) and the Python trainer's mirrored qty_cap rule stay on
  28. // the stable `qtyCap` heuristic — see lib/item-classes.js lockstep note.
  30. const MAX_SAMPLES = 64;
  32. export const METHOD_FLOORS = { mad: 4, gpd: 10, gev: 12 };
  34. /** Lanczos approximation of Γ(z), z > 0 (g=7, n=9 coefficients). */
  35. export function gammaFn(z) {
  36. const g = 7;
  37. const C = [
  38. 0.99999999999980993, 676.5203681218851, -1259.1392167224028,
  39. 771.32342877765313, -176.61502916214059, 12.507343278686905,
  40. -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7,
  41. ];
  42. if (z < 0.5) return Math.PI / (Math.sin(Math.PI * z) * gammaFn(1 - z));
  43. z -= 1;
  44. let x = C[0];
  45. for (let i = 1; i < g + 2; i++) x += C[i] / (z + i);
  46. const t = z + g + 0.5;
  47. return Math.sqrt(2 * Math.PI) * Math.pow(t, z + 0.5) * Math.exp(-t) * x;
  48. }
  50. const EULER_GAMMA = 0.5772156649015329;
  52. /** Clean a raw sample: finite positive numbers, deduped order preserved,
  53. * most recent last, bounded length. Returns a sorted ascending copy too. */
  54. export function cleanSamples(raw, maxLen = MAX_SAMPLES) {
  55. const xs = (Array.isArray(raw) ? raw : [])
  56. .map(Number)
  57. .filter((x) => Number.isFinite(x) && x > 0)
  58. .slice(-maxLen);
  59. return { xs, sorted: [...xs].sort((a, b) => a - b) };
  60. }
  62. export function median(sorted) {
  63. if (!sorted.length) return NaN;
  64. const m = sorted.length >> 1;
  65. return sorted.length % 2 ? sorted[m] : (sorted[m - 1] + sorted[m]) / 2;
  66. }
  68. /** Empirical quantile (linear interpolation), p in [0,1], sorted input. */
  69. export function quantile(sorted, p) {
  70. if (!sorted.length) return NaN;
  71. const pos = (sorted.length - 1) * Math.min(1, Math.max(0, p));
  72. const lo = Math.floor(pos), hi = Math.ceil(pos);
  73. return sorted[lo] + (sorted[hi] - sorted[lo]) * (pos - lo);
  74. }
  76. /** First three unbiased L-moments (l1, l2, l3) of a sample. */
  77. export function lmoments(xs) {
  78. const n = xs.length;
  79. const sorted = [...xs].sort((a, b) => a - b);
  80. const b0 = sorted.reduce((s, x) => s + x, 0) / n;
  81. let b1 = 0, b2 = 0;
  82. if (n >= 2) for (let i = 0; i < n; i++) b1 += ((i) / (n - 1)) * sorted[i];
  83. b1 /= n;
  84. if (n >= 3) for (let i = 0; i < n; i++) b2 += ((i * (i - 1)) / ((n - 1) * (n - 2))) * sorted[i];
  85. b2 /= n;
  86. return { l1: b0, l2: 2 * b1 - b0, l3: 6 * b2 - 6 * b1 + b0 };
  87. }
  89. /** GEV (block-maxima) L-moment fit → {mu, sigma, xi} or null when degenerate. */
  90. export function gevLmomFit(xs) {
  91. if (xs.length < 4) return null;
  92. const { l1, l2, l3 } = lmoments(xs);
  93. if (!(l2 > 0) || !Number.isFinite(l1)) return null;
  94. const t3 = l3 / l2;
  95. const c = 2 / (3 + t3) - Math.log(2) / Math.log(3); // Hosking c-substitution (no Math.LN3!)
  96. let xi = 7.859 * c + 2.9554 * c * c; // Hosking–Wallis–Wood approx.
  97. let sigma, mu;
  98. if (Math.abs(xi) < 1e-8) {
  99. xi = 0; // Gumbel limit
  100. sigma = l2 / Math.LN2;
  101. mu = l1 - EULER_GAMMA * sigma;
  102. } else {
  103. const g = gammaFn(1 + xi);
  104. if (!Number.isFinite(g) || g === 0) return null;
  105. sigma = (l2 * xi) / ((1 - Math.pow(2, -xi)) * g);
  106. mu = l1 - (sigma * (1 - g)) / xi;
  107. }
  108. if (!Number.isFinite(mu) || !Number.isFinite(sigma) || sigma <= 0) return null;
  109. if (Math.abs(xi) >= 0.99) return null; // wild shape — refuse to speak
  110. return { mu, sigma, xi };
  111. }
  113. /** GEV quantile (inverse CDF) at probability F. */
  114. export function gevQuantile(F, { mu, sigma, xi }) {
  115. if (F <= 0 || F >= 1) return NaN;
  116. if (xi === 0) return mu - sigma * Math.log(-Math.log(F));
  117. return mu + (sigma / xi) * (1 - Math.pow(-Math.log(F), xi));
  118. }
  120. /** GPD L-moment fit to exceedances (over threshold) → {sigma, xi} or null.
  121. * Parameterization: F(x) = 1 - (1 - xi·x/sigma)^(1/xi), x ≥ 0. */
  122. export function gpdLmomFit(exceed) {
  123. if (exceed.length < 3) return null;
  124. const { l2, l3 } = lmoments(exceed);
  125. if (!(l2 > 0)) return null;
  126. const t3 = l3 / l2;
  127. // τ3 = (1-ξ)/(3+ξ) ⇒ ξ = (1-3τ3)/(1+τ3); σ = l2(1+ξ)(2+ξ)
  128. const xi = (1 - 3 * t3) / (1 + t3);
  129. const sigma = l2 * (1 + xi) * (2 + xi);
  130. if (!Number.isFinite(sigma) || sigma <= 0) return null;
  131. if (xi >= 0.99) return null; // unbounded wild tail — refuse
  132. return { sigma, xi };
  133. }
  135. /** POT return level: value exceeded with probability 1/n·(1/returnPeriodK)
  136. * given n total samples and Nu exceedances over threshold u. */
  137. export function potQuantile(u, n, nExceed, returnPeriodK, { sigma, xi }) {
  138. const ratio = n / Math.max(1, nExceed);
  139. const zeta = 1 - 1 / (returnPeriodK * n);
  140. if (Math.abs(xi) < 1e-8) return u + sigma * Math.log(ratio * zeta);
  141. return u + (sigma / xi) * (Math.pow(ratio * zeta, -xi) - 1);
  142. }
  144. /** Robust MAD baseline: median + k·(1.4826·MAD). Null when degenerate
  145. * (MAD = 0 — e.g. every observed quantity is 1). */
  146. export function madBaseline(xs, k = 6) {
  147. const sorted = [...xs].sort((a, b) => a - b);
  148. const med = median(sorted);
  149. const dev = sorted.map((x) => Math.abs(x - med)).sort((a, b) => a - b);
  150. const mad = median(dev);
  151. if (!(mad > 0)) return null;
  152. return med + k * 1.4826 * mad;
  153. }
  155. /** Combined statistical threshold from one customer's sample.
  156. * floor — deterministic lower bound (class base cap)
  157. * observedMax — customer's own observed maximum (never flag the past)
  158. * alpha — GEV tail probability for the block-maxima quantile
  159. * returnPeriodK — POT return period in units of n purchases
  160. * upperMult — hard ceiling multiplier (see header)
  161. * Returns {value, used, n, detail} or null when no method has enough data. */
  162. export function statThreshold(raw, {
  163. floor = 0,
  164. observedMax = 0,
  165. heuristicFloor = null,
  166. alpha = 1e-4,
  167. returnPeriodK = 10,
  168. upperMult = 10,
  169. } = {}) {
  170. const { xs, sorted } = cleanSamples(raw);
  171. const n = xs.length;
  172. if (!n) return null;
  173. const obsMax = Math.max(observedMax, sorted[n - 1]);
  174. const incumbent = Math.max(0, Number(heuristicFloor) || 0); // current rule's cap
  175. const ceiling = Math.max(upperMult * floor, upperMult * obsMax);
  176. const used = [];
  177. const detail = {};
  178. let best = 0;
  180. // mad — cheap and honest, runs first on any non-degenerate sample
  181. if (n >= METHOD_FLOORS.mad) {
  182. const m = madBaseline(xs);
  183. if (m != null && Number.isFinite(m)) {
  184. detail.mad = m;
  185. used.push('mad');
  186. best = Math.max(best, m);
  187. }
  188. }
  190. // gpd — POT over the empirical q75
  191. if (n >= METHOD_FLOORS.gpd) {
  192. const u = quantile(sorted, 0.75);
  193. const exceed = xs.filter((x) => x > u);
  194. if (exceed.length >= 3) {
  195. const fit = gpdLmomFit(exceed);
  196. if (fit) {
  197. const v = potQuantile(u, n, exceed.length, returnPeriodK, fit);
  198. if (Number.isFinite(v) && v > 0) {
  199. detail.gpd = v;
  200. used.push('gpd');
  201. best = Math.max(best, v);
  202. }
  203. }
  204. }
  205. }
  207. // gev — block maxima of consecutive blocks of 3
  208. if (n >= METHOD_FLOORS.gev) {
  209. const blocks = [];
  210. for (let i = 0; i + 3 <= n; i += 3) {
  211. blocks.push(Math.max(xs[i], xs[i + 1], xs[i + 2]));
  212. }
  213. if (blocks.length >= 4) {
  214. const fit = gevLmomFit(blocks);
  215. if (fit) {
  216. const v = gevQuantile(1 - alpha, fit);
  217. if (Number.isFinite(v) && v > 0) {
  218. detail.gev = v;
  219. used.push('gev');
  220. best = Math.max(best, v);
  221. }
  222. }
  223. }
  224. }
  226. if (!used.length && !(incumbent > 0)) return null;
  227. // The incumbent rule participates as a floor: statistics may only LOOSEN a
  228. // cap when the customer's own history justifies it — never tighten below
  229. // what the deterministic rule already allows (no tail-underestimation FPs).
  230. const value = Math.min(ceiling, Math.max(floor, obsMax, incumbent, best));
  231. return { value, used, n, detail: { ...detail, ceiling, floor, incumbent, observedMax: obsMax } };
  232. }