Distributional (Location-Scale) Models

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A { "mu": ..., "sigma": ... } formula fits a 2-parameter location-scale distribution (a GAMLSS [Rigby and Stasinopoulos, 2005]) as a schedule that composes two sub-atoms, no new solver.

The two-stage schedule

  1. Location. Fit \(\mu(x)\) with an L2 atom on the (log-)target \(t=\log y\).

  2. Scale. Read the squared residuals \(r^2=(t-\hat\mu)^2\) and fit \(\sigma(x)\) as a Gamma-GLM atom on \(r^2\) (or an L2 atom on \(\log r^2\) with a \(\log\chi^2_1\) bias correction).

The tail law and quantiles

The standardized residual law \(F\) (Normal or Student-t) is selected from the residual excess kurtosis. Conditional quantiles are then, by construction non-crossing:

\[ Q_\tau(x) = \exp\!\Big(\hat\mu(x) + \hat\sigma(x)\,F^{-1}(\tau)\Big). \]

An optional convex scale_shrinkage \(\lambda\in[0,1]\) blends the conditional variance toward the global one. The probability integral transform \(F\big((t-\hat\mu)/\hat\sigma\big)\) gives the CDF, a two-sided anomaly score, and the CRPS (closed form for the Normal, else a Gauss-Legendre quantile integral).