Sigma-point propagation

Use this page when a nonlinear map makes first-order propagation inadequate but a small deterministic set of representative input points is more practical than a large ensemble.

The scientific question¶

Can selected points reproduce chosen input moments and approximate the output mean and covariance after a nonlinear transformation? Sigma-point methods answer this by replacing an input law with a finite weighted rule. The rule is deterministic once the construction, scaling parameters, matrix factorization, component order, and input moments are fixed.

This does not make the finite rule a set of random draws. It is a quadrature-like representation designed to match specified moments under a convention. Different conventions can use different point counts, locations, scaling rules, and weights, so the convention name and parameters are part of the scientific record.

Mathematical objects¶

Let $\boldsymbol{\chi}_{i}\in\mathbb{R}^{d}$ for $i=0,\ldots,L-1$ be sigma points constructed from input mean $\boldsymbol{\mu}_{x}$ and covariance $\mathbf{C}_{x}$ . Let $w_i^{(m)}$ denote mean weights and $w_i^{(c)}$ covariance weights. The mean and covariance weights may differ.

Typically the weighted input points satisfy $\sum_i w_i^{(m)}\boldsymbol{\chi}_i=\boldsymbol{\mu}_x$ and a weighted centered outer-product condition approximating or reproducing $\mathbf{C}_x$ . Construction often requires a square root of a scaled covariance. Cholesky, eigendecomposition, and other roots can produce rotated point sets with different finite-precision behavior. The covariance entries retain units $[X_i][X_j]$ , so a factorization of mixed-unit components requires an explicit scaling convention.

The transformed points are $\mathbf{y}_i=f(\boldsymbol{\chi}_i)$ . Their ordering and weights remain static convention data; their values may be dynamic JAX leaves in a future representation.

Core derivation¶

Approximate transformed moments by applying the weighted rule to $f$ and to the centered outer product:

\widehat{\boldsymbol{\mu}}_{y}=\sum_i w_i^{(m)}f(\boldsymbol{\chi}_i), \qquad \widehat{\mathbf{C}}_y=\sum_i w_i^{(c)} \boldsymbol{\delta}_i\boldsymbol{\delta}_i^{\mathsf{T}}, \qquad \boldsymbol{\delta}_i=f(\boldsymbol{\chi}_i)- \widehat{\boldsymbol{\mu}}_y.

(1)

Equation (1) is a weighted quadrature approximation, not a sampling identity. If the transformed function belongs to a polynomial class integrated exactly by the chosen rule under the assumed input law, selected moments may be exact in exact arithmetic. Outside that class, omitted higher moments and the point geometry control the error.

Sigma-point construction and weight conventions vary. Scaled unscented rules, for example, may use different mean and covariance weights for the central point. Some conventions allow negative weights to reproduce target moments. Negative covariance weights mean positive-semidefinite output is not automatic under every finite-precision choice, even though each outer product is symmetric.

Failure modes and interpretation limits¶

A point rule that matches mean and covariance does not encode arbitrary tails, support boundaries, skewness, or multimodality.
Negative weights can amplify cancellation and roundoff, producing a covariance with small negative eigenvalues or worse failures under poor scaling.
A failed or regularized covariance factorization changes the represented input; silent jitter is not a harmless implementation detail.
Strong nonlinearity between sigma points can remain invisible to the rule.
Point locations can leave bounded physical support even when the intended input law does not, requiring a documented transformed-space construction rather than clipping.
Branches and discontinuities can make results depend abruptly on scaling parameters or point orientation.
Agreement among two closely related sigma-point conventions is not independent validation of the input probability model.

What Jaxstro may add¶

JAX owns transformations and random primitives. NumPyro and BlackJAX own probabilistic inference and sampling mechanics. Informax owns inference-aware scientific workflows in the Jaxstro ecosystem. A future jaxstro.uncertainty would own only domain-agnostic propagation representations, unit and shape conventions, deterministic key policy, provenance, and evidence contracts. Covariance propagation does not perform inference and does not validate a probability model.

A future surface may store point and weight conventions, verify weighted input moments, apply a map with fixed-shape batching, and report factorization, symmetry, rank, and positive-semidefinite diagnostics. It must never infer scaling parameters, clip points without declaring a new representation, or present one convention as a universal sigma-point rule.

Evidence required before implementation¶

Analytic evidence must include constant, affine, and low-order polynomial maps with known transformed moments. Tests must cover distinct mean and covariance weights, negative-weight conventions, diagonal and correlated covariances, singular and nearly singular inputs, mixed component scales, deterministic point ordering, and float32 versus float64 behavior. The generated point set must reconstruct the moments promised by its named convention within explicit tolerances.

Nonlinear comparisons should use independent high-sample ensembles and controlled input laws, reporting error in mean, covariance, and scientifically relevant tail events separately. JIT, VMAP, shape, dtype, unit metadata, and provenance behavior must be tested. No comparison can substitute for validating the input probability model or the downstream scientific assumptions.

Claim boundary¶

Sigma points approximate selected transformed moments under a specified finite rule. They are neither posterior samples nor proof that the input law is correct. This page does not choose a sigma-point convention, guarantee positive-semidefinite output for every weighted finite-precision computation, or promise a runtime module. The proposed owner is limited to explicit propagation representations and evidence contracts.

Connected representations, foundations, and methods¶

Read What uncertainty represents before treating moments as a complete uncertainty description.
Return to Scientific representations for current representation owners.
Review Linear algebra as the language of change for covariance factorizations and quadratic forms.
Compare Linearized propagation, Ensemble propagation, and Fixed and weighted quadrature to separate local, weighted-rule, sampled, and integration viewpoints.