point-process.md

Point Process Models

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PointProcessModel supports peaks-over-threshold frequency analysis. Instead of modeling annual maxima, it models threshold exceedances and their occurrence rate.

Mathematical Form

BestFit implements the nonhomogeneous extreme-value point-process likelihood using GEV-compatible location, scale, and shape parameters, not a standalone generalized-Pareto excess model. Let u be the threshold, Ny the number of observation years, and let the fitted GEV parameters be location $\mu$, scale $\sigma&gt;0$, and Coles shape $\xi$. For an exceedance $x_i &gt; u$:

$$z_i = 1 + \xi\frac{x_i-\mu}{\sigma}, \qquad z_i > 0$$

For $\xi \ne 0$, BestFit uses:

$$\log L = \sum_i \left[-\log \sigma - \left(1+\frac{1}{\xi}\right)\log z_i\right] - N_y \left(1+\xi\frac{u-\mu}{\sigma}\right)^{-1/\xi}$$

For $|\xi| &lt; 10^{-4}$, the implementation switches to the Gumbel limit:

$$\log L = \sum_i \left[-\log\sigma-\frac{x_i-\mu}{\sigma}\right] - N_y\exp\left[-\frac{u-\mu}{\sigma}\right].$$

Numerics stores the GEV shape with Hosking's Kappa; BestFit converts it to the Coles sign convention internally by using $\xi=-\kappa$.

Public API

API	Purpose
PointProcessModel	Peaks-over-threshold model
PointProcessAnalysis	Bayesian analysis workflow
DataLogLikelihood(...)	Exceedance likelihood
PointwiseDataLogLikelihood(...)	WAIC/LOO-CV support
GenerateRandomValues(...)	Simulates threshold exceedance behavior

Usage Pattern

using RMC.BestFit.Analyses;
using RMC.BestFit.Models;

var model = new PointProcessModel();
var analysis = new PointProcessAnalysis(model);

analysis.BayesianAnalysis.Iterations = 5000;
analysis.BayesianAnalysis.WarmupIterations = 2500;

if (analysis.Validate().IsValid)
{
    await analysis.RunAsync();
}

Threshold Diagnostics

Before fitting a point-process model, inspect threshold stability with ThresholdDiagnostics, MeanResidualLifeResult, and ParameterStabilityResult from the data-frame API.

Seasonal Option

The seasonal implementation uses two GEV components and two day-of-year change points. BestFit validates 1 <= k1 < k2 <= 366, assigns observations outside [k1,k2) to season 1 and observations inside [k1,k2) to season 2, and scales the Poisson rate term by each season's fraction of the year.

Implementation Sources

Primary source paths: src/RMC.BestFit/Models/UnivariateDistribution/PointProcessModel.cs, src/RMC.BestFit/Analyses/Univariate/PointProcessAnalysis.cs, and src/RMC.BestFit/Models/DataFrame/ThresholdDiagnostics.cs.

References

[1] S. Coles, An Introduction to Statistical Modeling of Extreme Values. London, U.K.: Springer, 2001.

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