Patterns in Random Permutations

A consecutive pattern in a permutation π is another permutation \(\sigma\) determined by the relative order of a subsequence of contiguous entries of π. Traditional notions such as descents, runs, and peaks can be viewed as particular examples of consecutive patterns in permutations, but the systematic study of these patterns has flourished in the last 15 years, during which a variety of different techniques have been used. We survey some interesting developments in the subject, focusing on exact and asymptotic enumeration results, the classification of consecutive patterns into equivalence classes, and their applications to the study of one-dimensional dynamical systems.

Recently introduced in analyzing data from functional MRI (fMRI) and other neuroimaging techniques, Bayesian networks (BN) is a method to characterize effective connectivity patterns among multiple brain regions. So far, interests of using BN have been primarily on learning the connectivity pattern for each single group with well investigated computational algorithms. Examination of the connectivity pattern differences between groups, on the other hand, lacks rigorous statistical inference procedure. In this study, we propose using random permutation, a type of non-parametric statistical significance test in which a reference distribution is obtained by calculating all possible values of the test statistic under re-arrangements of the group labels on the observed data points, to infer whether the difference is significant. Two different approaches to perform the permutation test are introduced, compared to each other and both compared to the routinely used parametric t-test. Permutation approach 1 permutes the group labels first followed by learning BN pattern for each of the newly formed groups. Approach 2 learns BN pattern for each individual and connection parameters are then subjected to the group label permutations. Synthetic data generated under varying signal-to-noise ratios are used to investigate the performances of the proposed methods. Our results demonstrated that permutation approach 1 in detecting the effective connectivity pattern difference between two groups is superior to permutation approach 2 and to the common-sense two sample t-test.

Let $I_n(\pi)$ denote the number of involutions in the symmetric group ${\cal S}_{n}$ which avoid the permutation $\pi$. We say that two permutations $\alpha,\beta\in{\cal S}_{j}$ may be exchanged if for every $n$, $k$, and ordering $\tau$ of $j+1,\ldots,k$, we have $I_n(\alpha\tau)=I_n(\beta\tau)$. Here we prove that $12$ and $21$ may be exchanged and that $123$ and $321$ may be exchanged. The ability to exchange $123$ and $321$ implies a conjecture of Guibert, thus completing the classification of ${\cal S}_{4}$ with respect to pattern avoidance by involutions; both of these results also have consequences for longer patterns. Pattern avoidance by involutions may be generalized to rook placements on Ferrers boards which satisfy certain symmetry conditions. Here we provide sufficient conditions for the corresponding generalization of the ability to exchange two prefixes and show that these conditions are satisfied by $12$ and $21$ and by $123$ and $321$. Our results and approach parallel work by Babson and West on analogous problems for pattern avoidance by general (not necessarily involutive) permutations, with some modifications required by the symmetry of the current problem.

Shannon Entropy is the preeminent tool for measuring the level of uncertainty (and conversely, information content) in a random variable. In the field of communications, entropy can be used to express the information content of given signals (represented as time series) by considering random variables which sample from specified subsequences. In this paper, we will discuss how an entropy variant, the \textit{permutation entropy} can be used to study and classify radio frequency signals in a noisy environment. The permutation entropy is the entropy of the random variable which samples occurrences of permutation patterns from time series given a fixed window length, making it a function of the distribution of permutation patterns. Since the permutation entropy is a function of the relative order of data, it is (global) amplitude agnostic and thus allows for comparison between signals at different scales. This article is intended to describe a permutation patterns approach to a data driven problem in radio frequency communications research, and includes a primer on all non-permutation pattern specific background. An empirical analysis of the methods herein on radio frequency data is included. No prior knowledge of signals analysis is assumed, and permutation pattern specific notation will be included. This article serves as a self-contained introduction to the relationship between permutation patterns, entropy, and signals analysis for studying radio frequency signals and includes results on a classification task.

Some Thoughts About Data Type, Distribution, and Statistical Significance

In this paper the authors present a statistical test for testing the strict avalanche criterion (SAC), a property that cryptographic primitives such as block ciphers and hash functions must have. Random permutations should also behave as good random number generators when, given any initial input, its output is considered part of a pseudorandom stream and then used as an input block to produce more output bits. Using these two ideal properties, we construct a test framework for cyptographic primitives that is shown at work on the block cipher TEA. In this way, we are able to distinguish reduced round versions of it from a random permutation.

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For nonnormal data we suggest a test of location based on a broader family of distributions than normality. Such a test will in a sense fall between the standard parametric and non parametric tests. We see that the Wald tests based on this family of distributions have some advantages over the score tests and that they perform well in comparison to standard parametric and nonparametric tests in a variety of situations. We also consider when and how to apply such tests in practice.

This thesis investigates three distinct questions concerning the asymptotic average order of certain group theoretic objects. The three main chapters can be read independently. In chapter zero, we give a high level overview of our results. For a polynomial f [is an element of] [finite field]q[x] with nonzero constant term, define the order of f to be the order of x in the quotient ring [finite field]q[x]/(f). In Chapter 1, we consider the average order of several classes of polynomials as the degree n [right arrow] [infinity]. We show that the average order of an irreducible polynomial is asymptotically [theta]([phi](q^n - 1)) with [phi] the Euler Totient function. Building on work of Stong [32], we show that the asymptotic average order of both squarefree and nonsquarefree polynomials is q^n / n^(1+o(1)). As an application we relate this to the expected orbit size of a vector under iteration by a random invertible matrix. In chapter two, we count the number of compositions of n with k parts whose pairwise gcds are coprime to a given finite set P of primes. We obtain asymptotic results when n [right arrow] [infinity], showing that the number of such compositions is ([product](p[is an element of]P|(n)) h[k](p) + (-1)^(k-1)(k-1)/p^(k-1) (1-1/p)) x ([product](p[is an element of P[crossed pipe](n)) h[k](p) + (-1)^k(k-1)/p^k) n^(k-1)/(k-1)! + O(n^(k-2)) where h[k](p) := (1-1/p)^(k-1)(1+(k-1)/p). Building on work of Bubbolini, Luca and Spiga [6], we also consider the set of compositions whose parts are pairwise coprime. We use a geometric argument to strengthen their result when k = 3. We apply their results to obtain bounds on the average order of a permutation with a given number of cycles. Chapter three is joint work with Huseyin Acan, Charles Burnette, Sean Eberhard, and Eric Schmutz. Motivated by a question of Thibault Godin [17], we show that the probability P₁ that two random permutations in the symmetric group S[n] have the same order is asymptotically n^(-2+o(1)). We compare this to the probability P₂ that two permutations are conjugate, showing by construction that lim sup(n) P₁/P₂ = [infinity]. Additionally, we show by more direct methods that with probability O(log log n / log n) there exists some prime dividing the order of one of the permutations but not the other.

Given a set of high dimensional binary vectors and a similarity function (such as Jaccard and Cosine), we study the problem of finding all pairs of vectors whose similarity exceeds a given threshold. The solution to this problem is a key component in many applications with feature-rich objects, such as text, images, music, videos, or social networks. In particular, there are many important emerging applications that require the use of relatively low similarity thresholds.We propose ATLAS, a probabilistic similarity search algorithm that in expectation finds a 1 - δ fraction of all similar vector pairs. ATLAS uses truly random permutations both to filter candidate pairs of vectors and to estimate the similarity between vectors. At a 97.5% recall rate, ATLAS consistently outperforms all state-of-the-art approaches and achieves a speed-up of up to two orders of magnitude over both exact and approximate algorithms.

Ordinal patterns can be used to construct non-parametric hypothesis tests that aim to discover (possibly non-linear) serial dependence in a real-valued time series. We derive the asymptotic distribution of the vector of sample frequencies of ordinal patterns and that of various corresponding tests statistics such that the targeted tests for serial dependence are easily implemented based on asymptotic approximations. Simulations are used to check the finite-sample performance of these tests as well as their power properties with respect to various alternative scenarios. The application and interpretation of the tests in practice are illustrated by an environmental data example.

Inspection of differential item functioning (DIF) in translated test items can be informed by graphical comparisons of item response functions (IRFs) across translated forms. Due to the many forms of DIF that can emerge in such analyses, it is important to develop statistical tests that can confirm various characteristics of DIF when present. Traditional nonparametric tests of DIF (Mantel‐Haenszel, SIBTEST) are not designed to test for the presence of nonuniform or local DIF, while common probability difference (P‐DIF) tests (e.g., SIBTEST) do not optimize power in testing for uniform DIF, and thus may be less useful in the context of graphical DIF analyses. In this article, modifications of three alternative nonparametric statistical tests for DIF, Fisher's χ 2 test, Cochran's Z test, and Goodman's U test (Marascuilo & Slaughter, 1981), are investigated for these purposes. A simulation study demonstrates the effectiveness of a regression correction procedure in improving the statistical performance of the tests when using an internal test score as the matching criterion. Simulation power and real data analyses demonstrate the unique information provided by these alternative methods compared to SIBTEST and Mantel‐Haenszel in confirming various forms of DIF in translated tests.

The objectives of this study were to compare nonparametric stability measures, and to identify promising high-yield and stable bread wheat (Triticum aestivum L.) genotypes in 7 environments during 2003-2005 in the central Black Sea region of Turkey. The bread wheat genotypes (20 advanced lines and 5 cultivars) were grown in a randomized complete block design with 4 replications in 7 different environments. Three nonparametric statistical tests of significance for genotype × environment (GE) interaction and 10 nonparametric measures of stability were used to identify stable genotypes in 7 environments. Combined ANOVA and nonparametric tests (Kubinger, Hildebrand, and De Kroon/Van der Laan) of genotype × environment interaction indicated the presence of significant crossover and non-crossover interactions, as well as significant differences between genotypes. In this study high TOP values (proportion of environments in which a genotype ranked in the top third) and low rank-sum values (sum of ranks of mean yield and Shukla’s stability variance) were associated with high mean yield. Nonetheless, results of the other nonparametric tests were negatively correlated with mean yield. In the simultaneous selection for high yield and stability, only the rank-sum and TOP methods were useful in terms of the principal component analysis (PCA) results, and correlation analysis of nonparametric stability statistics and yield. According to these stability parameters (TOP and rank-sum) G7 (VONA//KS75210/TAM101), G9 (JUP/4/CLLF/3/II14.53/ODIN//CI13431/WA 00477), G20 (Sakin), and G21 (VORONA/KAUZ//1D13.1/MLT) were the most stable genotypes for grain yield. The results also revealed that based on nonparametric test results stability could be classified into 3 groups, according to agronomic and biological concepts of stability.

Kitamura and Stoye (2018) recently proposed a nonparametric statistical test for random utility models of consumer behavior. The test is formulated in terms of linear inequality constraints and a quadratic objective function. While the nonparametric test is conceptually appealing, its practical implementation is computationally challenging. In this paper, we develop a column generation approach to operationalize the test. These novel computational tools generate considerable computational gains in practice, which substantially increases the empirical usefulness of Kitamura and Stoye's statistical test.

We prove a conjecture dating back to a 1978 paper of D.R. Musser [11], namely that four random permutations in the symmetric group Sn generate a transitive subgroup with probability for some independent of n, even when an adversary is allowed to conjugate each of the four by a possibly different element of . In other words, the cycle types already guarantee generation of a transitive subgroup; by a well known argument, this implies generation of An or except for probability as . The analysis is closely related to the following random set model. A random set is generated by including each independently with probability . The sumset is formed. Then at most four independent copies of are needed before their mutual intersection is no longer infinite. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 409–428, 2016