Add monomial<->Hermite/Chebyshev basis conversion and statistical moments - #33
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Add monomial<->Hermite/Chebyshev basis conversion and statistical moments#33andreapasquale94 wants to merge 6 commits into
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…ents Adds tax::toHermite/fromHermite and tax::toChebyshev/fromChebyshev for dense, isotropic expansions, via a shared separable connection-coefficient driver (tax/core/basis/). Also adds tax::la::mean/covariance/skewnessTensor/ kurtosisTensor/excessKurtosisTensor (tax/la/moments.hpp), which extract statistical moments of a polynomial map assuming i.i.d. standard-normal expansion variables, computed directly from monomial coefficients via closed-form Gaussian raw moments (Isserlis'/Wick's theorem). Motivated by Michelotti, Burnett & Topputo, "Analytical Confidence Boundaries for Non-Gaussian Uncertainty in Perturbed Spacecraft Dynamics" (arXiv:2607.10095), which uses a monomial-to-Hermite transform and Isserlis' theorem for the same purpose; the moments implementation here takes an independently-verifiable route (direct Gaussian raw moments) rather than porting the paper's specific Hermite-tensor formulas, since arXiv was unreachable from this environment. Includes unit tests (round-trips, textbook polynomial identities, and hand-verified chi-square(1) moments) and internals/guide/reference docs.
Change skewnessTensor / kurtosisTensor / excessKurtosisTensor to return Eigen::Tensor<T,3> (D×D×D) and Eigen::Tensor<T,4> (D×D×D×D) from Eigen's unsupported/Eigen/CXX11/Tensor module, instead of std::vector-of-matrices slices. Access is now the natural S(i,j,k) / K(i,j,k,l). The tensors are fully symmetric, so each value is computed once over sorted index tuples and scattered to every distinct permutation via std::next_permutation (which also collapses repeated indices). Since every tuple is a permutation of exactly one sorted tuple, this writes every entry, so no zero-initialization pass is needed. Adds a multivariate symmetry test exercising the permutation scatter and off-diagonal cross-moments; updates guide/internals/reference docs.
Switch skewnessTensor / kurtosisTensor / excessKurtosisTensor from dynamic-size Eigen::Tensor<T,R> to fixed-size Eigen::TensorFixedSize<T, Eigen::Sizes<D,...>>, and covariance from a dynamic Eigen::MatrixXd to a fixed Eigen::Matrix<T,D,D>. The map dimension D is taken from Derived::SizeAtCompileTime; a dynamic-size input map is now rejected with a static_assert. Fixed-size returns are stack-allocated with compile-time shape, matching the library's fixed-shape, allocation-free convention. Tests gain static_asserts pinning the exact fixed-size return types; docs updated accordingly.
Add vector-returning tax::la::skewness / kurtosis / excessKurtosis, each a fixed-size D x 1 Eigen vector of the standardized marginal coefficient for every output component: Fisher skewness E[(F_i-mu_i)^3]/sigma_i^3, Pearson kurtosis E[(F_i-mu_i)^4]/sigma_i^4 (3 for a Gaussian marginal), and its excess (-3) form (0 for a Gaussian marginal). These are the diagonal of the corresponding central-moment tensors, normalized by the appropriate power of sigma_i, but computed directly per component (O(D) evaluations instead of building the O(D^3)/O(D^4) tensor). They complement the existing skewnessTensor/kurtosisTensor joint-moment tensors. Tests and guide/internals/reference docs updated.
Remove tax/la/moments.hpp from the <tax/la.hpp> include list so it is no
longer pulled in transitively by <tax/tax.hpp>. moments.hpp depends on
Eigen's heavy unsupported/Eigen/CXX11/Tensor module, and there is no reason
for every umbrella consumer to pay that compile-time cost.
Consumers that want statistical moments now include the header explicitly:
#include <tax/la/moments.hpp>
Update the moments test to include it directly, and note the opt-in include
requirement in the header comment and the guide/internals/reference docs.
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Add tax::la::correlation(F), the fixed-size D x D Pearson correlation matrix Corr(F_i, F_j) = Cov(F_i, F_j) / (sigma_i sigma_j) with a unit diagonal, and tax::la::correlationFromCovariance(C), a standalone conversion that normalizes any plain-scalar covariance matrix to a correlation matrix (returning the same Eigen shape). correlation(F) is a thin wrapper composing covariance(F) with that conversion. The conversion is constrained to arithmetic-scalar matrices so it never overload-collides with the map-taking correlation. A zero-variance component yields a non-finite row/column (correlation undefined). Adds tests (correlated and uncorrelated maps, plus a direct hand-built cov->corr conversion) and updates the guide/internals/reference docs.
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Adds tax::toHermite/fromHermite and tax::toChebyshev/fromChebyshev for
dense, isotropic expansions, via a shared separable connection-coefficient
driver (tax/core/basis/). Also adds tax::la::mean/covariance/skewnessTensor/
kurtosisTensor/excessKurtosisTensor (tax/la/moments.hpp), which extract
statistical moments of a polynomial map assuming i.i.d. standard-normal
expansion variables, computed directly from monomial coefficients via
closed-form Gaussian raw moments (Isserlis'/Wick's theorem).
Motivated by Michelotti, Burnett & Topputo, "Analytical Confidence Boundaries
for Non-Gaussian Uncertainty in Perturbed Spacecraft Dynamics" (arXiv:2607.10095),
which uses a monomial-to-Hermite transform and Isserlis' theorem for the same
purpose; the moments implementation here takes an independently-verifiable
route (direct Gaussian raw moments) rather than porting the paper's specific
Hermite-tensor formulas, since arXiv was unreachable from this environment.
Includes unit tests (round-trips, textbook polynomial identities, and
hand-verified chi-square(1) moments) and internals/guide/reference docs.