Hadamard Matrices, Quaternions, and the Pearson Chi-square Statistic

Abbas Alhakim

Home/Archive/arXiv

arXiv · 1806.03948

Hadamard Matrices, Quaternions, and the Pearson Chi-square Statistic

Abbas Alhakim

View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a symbolic decomposition of the Pearson chi-square statistic with unequal cell probabilities, by presenting Hadamard-type matrices whose columns are eigenvectors of the variance-covariance matrix of the cell counts. All of the eigenvectors have non-zero values so each component test uses all cell probabilities in a way that makes it intuitively interpretable. When all cell probabilities are distinct and unrelated we establish that such decomposition is only possible when the number of multinomial cells is a small power of 2. For higher powers of 2, we show, using the theory of orthogonal designs, that the targeted decomposition is possible when appropriate relations are imposed on the cell probabilities, the simplest of which is when the probabilities are equal and the decomposition is reduced to the one obtained by Hadamard matrices. Simulations are given to illustrate the sensitivity of various components to changes in location, scale skewness and tail probability, as well as to illustrate the potential improvement in power when the cell probabilities are changed.

Explore related subjects

stat.CO math.ST stat.TH

Keep this discovery

BibTeX RIS

Abbas Alhakim. 2018-06-08. Hadamard Matrices, Quaternions, and the Pearson Chi-square Statistic. https://arxiv.org/abs/1806.03948

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extended One-Liners for the Gamma, Poisson, and Binomial Distributions

We demonstrate explicit transformations of three independent uniforms with exact Gamma$(a,1)$, $a>0$, Poisson$(\lambda)$, $\lambda>0$, and Binomial$(N,p)$, $N\geq 1$, $0\leq p \leq 1$ laws using only elementary operations.

stat.CO↗

arXiv2026

Estimating Hierarchically Rank Structured Covariance Matrices

We consider the problem of estimating a high-dimensional covariance matrix from a very limited number of samples. This problem is ubiquitous in computational fluid dynamics, where a small number of fluid snapshots must be used to construct a Gramian matrix determining a reduced-order model, as well as in computational geoscience, where a small ensemble of Earth system forecasts must be used to estimate the covariance matrix associated with the forecast uncertainty. It is common practice to regularize the small-sample covariance by imposing a "localization" structure that enforces a physically realistic correlation length scale, imposing a sparsity constraint, "shrinking" towards a prescribed target, or attenuating small correlations. We propose an alternate technique that regularizes the small-sample covariance by imposing hierarchical rank structure. Compared to regularization methods that assume sparsity such as spatial localization, hierarchical rank structure accommodates a wider range of covariance matrices, roughly corresponding to situations where long-range correlations vary more smoothly than short-range ones. It also results in a data-sparse matrix format that permits highly efficient matrix-vector products. We present theory and algorithms which show how to efficiently estimate a high-dimensional, hierarchically rank structured covariance matrix from limited samples. Through an error analysis and numerical experiments with a variety of model problems, we demonstrate that these techniques are effective at reducing sampling errors, and that in many cases they achieve smaller estimation error than conventional techniques.

stat.CO↗

arXiv2026

Optimal Slice-Adaptive Tuning of Hybrid Slice Sampling

Slice sampling is a Markov chain Monte Carlo algorithm that draws its next state uniformly from a "slice"---a super-level set of the target density function---at each iteration, thereby providing automatic local adaptivity to the scale of the target. In practice the exact slice is not known, so general-purpose implementations use an approximate slice that is grown from a starting interval of length $w>0$, with a computational cost that depends on $w$. This work presents an analysis of the average per-iteration number of target density evaluations, as a function of $w$, of hybrid slice sampling with various slice-finding schemes for targets with contiguous slices. The paper uses the results of the analysis to develop automated, slice-adaptive tuning schemes along with suboptimality bounds and asymptotic convergence guarantees. Simulations demonstrate that the tuning schemes reliably yield near-optimal slice-adaptive tuning with essentially no dependence on the initial setting of $w$.

stat.CO↗