index.md

Univariate Functions

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The Numerics.Functions namespace provides the uncertain-function toolkit: univariate functional forms with optional stochastic residuals, sampled by confidence level, plus serialization, a factory, and posterior-ensemble sampling. These are the shared math behind risk input functions (rating curves, transforms, damage functions) in the consuming engineering applications.

The IUnivariateFunction contract

Every function implements IUnivariateFunction:

Member	Meaning
Function(x) / InverseFunction(y)	Forward and inverse evaluation
SetParameters(values) / ValidateParameters(...)	The parameter-vector surface (fitted parameter sets apply directly)
Minimum / Maximum	The evaluation support (clamped); some types derive Minimum from a location parameter
IsDeterministic	Whether the stochastic residual is active
ConfidenceLevel	The runtime draw in [0, 1] driving the residual; values outside [0, 1] evaluate the mean/median form. Runtime state — never serialized

Available functions

Function	Class	Form	Residual
Linear	LinearFunction	$Y = \alpha + \beta X$	Additive Gaussian, $\epsilon \sim N(0, \sigma)$
Power	PowerFunction	$Y = \alpha (X - \xi)^\beta$, optional inverse form	Log-space Gaussian (multiplicative)
Tabular	TabularFunction	Interpolation over UncertainOrderedPairedData with axis transforms	Co-monotonic percentile of the ordinate distributions
Segmented power	SegmentedPowerFunction	BaRatin addition mode: $Q(h) = \sum_k 10^{\log_{10}\alpha_k} (h - h_k)^{\beta_k} \cdot \mathbb{1}{h > h_k}$	Log₁₀-space Gaussian
Composite	CompositeFunction	Weighted average $\sum w_i f_i(x)$, or a mixture selected by one uniform	Inherited from the children

Segmented power (BaRatin addition mode)

SegmentedPowerFunction uses a parameter vector that can be applied directly to fitted draws. The parameter vector is [h₁, log₁₀α₁, β₁, h₂, log₁₀α₂, β₂, …, σ] (length 3·segments + 1), so a fitted posterior ParameterSet.Values applies directly through SetParameters. Breakpoints must be strictly ordered, exponents positive (the monotonicity constraint behind the numeric Brent inverse), discharge is zero at and below the cease-to-flow stage h₁, and one segment degenerates deterministically to the plain PowerFunction (the stochastic residual spaces differ: log₁₀ here versus natural log in PowerFunction).

using Numerics.Functions;

// Two controls: main channel from stage 1, overbank activating at stage 3.
var rating = new SegmentedPowerFunction(new[] { 1.0, 1.5, 2.0, 3.0, 1.2, 1.5, 0.1 });
double q = rating.Function(5.0);            // deterministic (median) curve: ConfidenceLevel
                                            // defaults to -1, and any value outside [0, 1]
                                            // selects the deterministic curve
rating.ConfidenceLevel = 0.75;              // 75th-percentile curve via the log₁₀-space
                                            // residual: multiplies by 10^(z·σ)
double q75 = rating.Function(5.0);

Composite (weighted average and mixture)

CompositeFunction combines child functions with non-negative weights summing to one. The weighted-average mode evaluates $\sum w_i f_i(x)$ with the composite's confidence level driving every child co-monotonically. The mixture mode composes with a single uniform: the draw selects a child by cumulative weight and re-scales the remainder as the child's own draw — deterministic composition sampling with no internal random source. Outside [0, 1] both modes evaluate the weighted average of each child evaluated at its own configured confidence level.

Serialization and the factory

Every concrete function serializes with an instance ToXElement() and a static FromXElement(XElement). UnivariateFunctionFactory dispatches on the UnivariateFunctionType enum or on a serialized element's local name, and maps live instances back to their enum type:

var element = rating.ToXElement();
IUnivariateFunction restored = UnivariateFunctionFactory.CreateFromXElement(element);
UnivariateFunctionType kind = UnivariateFunctionFactory.GetFunctionType(restored); // SegmentedPower

ConfidenceLevel is runtime sampling state and never serializes; PowerFunction.Minimum derives from ξ and never serializes. The serialization surface lives on the concrete classes (not the interface) so external IUnivariateFunction implementations remain source-compatible.

Posterior ensembles

EnsembleFunction carries a template function plus an array of posterior ParameterSet draws for use inside a simulation engine. Both sampling surfaces return a fresh clone of the template configured with the selected draw, so concurrent realizations share no mutable state.

var ensemble = new EnsembleFunction(rating, posteriorParameterSets);
IUnivariateFunction draw = ensemble.SampleAt(index);      // posterior draw by index
IUnivariateFunction pDraw = ensemble.Sample(0.37);        // min(⌊u·N⌋, N − 1)

Link functions

The sibling Link Functions family (ILinkFunction: identity, log, logit, probit, complementary log-log, Fisher-z, Yeo-Johnson) serves regression and machine-learning transformations and has its own factory, LinkFunctionFactory.

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