arXiv subjects

Abbas Alhakim

Publications and source records attributed to Abbas Alhakim.

10 recordsLinked to original sources

arXiv2026

Block-hierarchical covariance decompositions for finite-block additive functionals

We study additive functionals of stationary Markov chains whose observables depend on a fixed finite block of consecutive states. Such block observables arise naturally in sliding-window statistics, pattern counts, and local dependence analysis. In the independent setting, additive functionals of overlapping finite blocks are known to have a covariance operator with integer spectrum $0,1,\ldots,k$, and the eigenvalue-one component represents the information first detectable at block length $k$. We study the corresponding problem when the underlying sequence is a stationary Markov chain. For a fixed block observable $f(X_t,\ldots,X_{t+k-1})$, we introduce a Hilbert-space decomposition that separates information already contained in shorter consecutive blocks from the genuinely new block-$k$ component, called the incremental component. We show that this component persists as an eigenvalue-one component under Markovian dependence: on this space the Green--Kubo covariance converges trivially and the covariance operator acts as the identity. More generally, the integers $1,\ldots,k-1$ are shown to arise hierarchically as eigenvalues of the Green--Kubo operator. In the reversible case, the remaining covariance structure is described through a boundary-corrected one-coordinate marginal spectrum determined by the base transition operator. Reversible two-state chains and Gaussian AR(1) models illustrate the theory through concrete spectral formulas.

math.PR↗

arXiv2026

Overlapping window tests for correlation and trend

We develop a general framework for constructing and analyzing overlapping sliding-window statistics for dependence and trend detection. For a fixed window size, the overlapping blocks form a Markov chain, and the asymptotic variance of any centered window statistic is determined by the covariance operator of this chain. Using its spectral structure, we obtain an orthogonal decomposition of the space of window functions into components associated with different overlap levels. This leads to a natural notion of incremental dependence information: the part of a statistic that captures exactly the new information introduced by enlarging the window. We give an explicit procedure for extracting these components and apply the method to two classes of examples. For correlation detection, we study symmetric polynomial window functions and identify their informative projected part. For trend detection, we analyze localized rank-based statistics and isolate the contribution of the largest newly introduced lag. The examples also show that different statistics may exhibit different local detection scales, including nonclassical ones. The same viewpoint leads to a natural quantitative measure of incremental information, which can be used to assess how much new dependence structure is captured as the window size increases, and to guide scale selection. Overall, the paper provides a systematic method for designing overlapping-window tests and deriving their asymptotic normalization and local behavior.

math.ST↗

arXiv2024

Subdivisions of Oriented Cycles in Digraphs with Hamiltonian directed path

Cohen et al. conjectured that for every oriented cycle $C$ there exist an integer $f(C)$ such that every strong $f(C)$-chromatic digraph contains a subdivision of $C$. El Joubbeh confirmed this conjecture for Hamiltonian digraphs. Indeed, he showed that every $3n$-chromatic Hamiltonian digraph contains a subdivision of every oriented cycle of order $n$. In this article, we improve this bound to $2n$. Furthermore, we show that, if $D$ is a digraph containing a Hamiltonian directed path with chromatic number at least $12n-5$, then $D$ contains a subdivision of every oriented cycle of order $n$. Note that a digraph containing a Hamiltonian directed path need not be strongly connected. Thus, our current result provides a deeper understanding of the condition that may be needed to fully solve the conjecture.

math.CO↗

arXiv2024

Orientable sequences over non-binary alphabets

We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either direction. In particular we establish how two variants of a generalised Lempel homomorphism can be used to recursively construct such sequences, generalising previous work on the binary case. We also derive an upper bound on the period of an orientable sequence.

math.CO↗

arXiv2022

Nonbinary Counterparts of the Prefer-Same and Prefer-Opposite de Bruijn Sequences

The well known prefer-one, prefer-opposite, and prefer-same binary de Bruijn sequences are all constructed using simple preference rules. We apply the technique of preference functions of span one to define q-ary sequences that generalize the prefer-opposite and prefer-same sequences and we present some of their basic properties that are shared with their binary versions. In particular, we show that the prefer-higher sequence (the nonbinary counter-part of the prefer-one sequence) is obtained from a homomorphic image of the proposed prefer-opposite, when repetitions are cleaned up. This mirrors a known relationship between the binary versions. We also perform calculations that demonstrate that the discrepancy profile of the proposed sequences is similar to that of the binary case.

math.CO↗

arXiv2020

Efficient constructions of the Prefer-same and Prefer-opposite de Bruijn sequences

The greedy Prefer-same de Bruijn sequence construction was first presented by Eldert et al.[AIEE Transactions 77 (1958)]. As a greedy algorithm, it has one major downside: it requires an exponential amount of space to store the length $2^n$ de Bruijn sequence. Though de Bruijn sequences have been heavily studied over the last 60 years, finding an efficient construction for the Prefer-same de Bruijn sequence has remained a tantalizing open problem. In this paper, we unveil the underlying structure of the Prefer-same de Bruijn sequence and solve the open problem by presenting an efficient algorithm to construct it using $O(n)$ time per bit and only $O(n)$ space. Following a similar approach, we also present an efficient algorithm to construct the Prefer-opposite de Bruijn sequence.

cs.DM↗

arXiv2020

Hamiltonicity of the Cross-Join Graph of de Bruijn Sequences

A generalized de Bruijn digraph generalizes a de Bruijn digraph to the case where the number of vertices need not be a pure power of an integer. Hamiltonian cycles in these digraphs thus generalize regular de~Bruijn cycles, and we will thus refer to them simply as de Bruijn cycles. We define the cross-join to be the graph with all de Bruijn cycles as vertices, there is an edge between two of these vertices if one can be obtained from the other via a cross-join operation. We show that the cross-join graph is connected. This in particular means that any regular de Bruijn cycle can be cross-joined repeatedly to reach any other de Bruijn cycle, generalizing a result about regular binary de Bruijn cycles by Mykkeltveit and Szmidt in 2014. Furthermore, we present an algorithm that produces a Hamiltonian path across the cross-join graph, one that we may call a de~Bruijn sequence of de Bruijn sequences.

math.CO↗

arXiv2018

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

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.

stat.CO↗

arXiv2010

Spans of Preference Functions for De Bruijn Sequences

A nonbinary Ford sequence is a de Bruijn sequence generated by simple rules that determine the priorities of what symbols are to be tried first, given an initial word of size $n$ which is the order of the sequence being generated. This set of rules is generalized by the concept of a preference function of span $n-1$, which gives the priorities of what symbols to appear after a substring of size $n-1$ is encountered. In this paper we characterize preference functions that generate full de Bruijn sequences. More significantly, We establish that any preference function that generates a de Bruijn sequence of order $n$ also generates de Bruijn sequences of all orders higher than $n$, thus making the Ford sequence no special case. Consequently, we define the preference function complexity of a de Bruijn sequence to be the least possible span of a preference function that generates this de Bruijn sequence.

math.CO↗

arXiv2008

De Bruijn Graph Homomorphisms and Recursive De Bruijn Sequences

This paper presents a method to find new De Bruijn cycles based on ones of lesser order. This is done by mapping a De Bruijn cycle to several vertex disjoint cycles in a De Bruijn digraph of higher order and connecting these cycles into one full cycle. We characterize homomorphisms between De Bruijn digraphs of different orders that allow this construction. These maps generalize the well-known D-morphism of Lempel between De Bruijn digraphs of consecutive orders. Also, an efficient recursive algorithm that yields an exponential number of nonbinary De Bruijn cycles is implemented.

math.CO↗