Distributions & Tests

Math Utility Functions (Distributions)

Collection of numerical distribution functions (normal, χ², t, F) — the foundation of all hypothesis tests.

Statusvalidated

Version1.0.0

Description

math-utils.js bundles the numerical primitives that every downstream engine relies on: error function (erf), log-gamma, normal CDF/quantile (Abramowitz & Stegun 7.1.26 with corrected scaling), χ² CDF/inverse (regularised incomplete gamma), t CDF/inverse (regularised incomplete beta), F CDF/p-value/quantile. The evaluate(inputs) dispatcher takes a {function, ...args} object and returns {result}, so every function can be validated against SciPy fixtures. The 'validated' status refers to that fixture suite — it was introduced after the normalCDF bug (A&S 7.1.26 argument-scaling error) and protects against regressions.

Formulas

erf

erf (x) = \frac{2}{π} \int_{0}^{x} e^{- t^{2}} d t

Gaussian error function (Abramowitz & Stegun 7.1.26)

normalCDF

Φ (x; μ, σ) = \frac{1}{2} [1 + erf (\frac{x - μ}{σ 2})]

Cumulative normal distribution

normalQuantile

Φ^{- 1} (p) via Beasley-Springer-Moro rationalen Approximation

Inverse standard normal distribution

lnGamma

ln Γ (x) via Lanczos-Approximation

Logarithm of the gamma function

chi2CDF

F_{χ_{ν}^{2}} (x) = P (\frac{ν}{2}, \frac{x}{2})

χ² distribution via the regularised incomplete gamma function

tCDF

F_{t_{ν}} (t) = 1 - \frac{1}{2} I_{x (t)} (\frac{ν}{2}, \frac{1}{2}), x (t) = \frac{ν}{ν + t ^{2}}

Student-t CDF via the regularised incomplete beta function

fCDF

F_{F_{d_{1}, d_{2}}} (x) = 1 - I_{y} (\frac{d _{2}}{2}, \frac{d _{1}}{2}), y = \frac{d _{2}}{d _{2} + d _{1} x}

F distribution via the regularised incomplete beta function

Assumptions

Arguments lie within each function's valid domain (e.g. df > 0, 0 ≤ p ≤ 1)

Limitations

Relative accuracy of the approximations is ~1e-10 to 1e-12 — sufficient for every use case in DMAIC.io, but not bit-identical to SciPy
fQuantile and chi2Inv/tInv use bisection — not suitable for extreme tail probabilities (p < 1e-10 or p > 1 − 1e-10)

References

Abramowitz & Stegun, Handbook of Mathematical Functions — Ch. 7 (erf), Ch. 26 (Normal CDF)
Press, Teukolsky, Vetterling, Flannery — Numerical Recipes, 3rd Ed.
Lanczos, C. (1964), A Precision Approximation of the Gamma Function