Random Permutation Research Articles

Research on Tennis Score Based on Momentum Quantization Model

The paper goal is to investigate the significance, logic, and effects of momentum in tennis matches in order to offer logical advice to athletes and tennis coaches. the paper created a quantitative momentum model and mostly finished three tasks based on the research objectives. Firstly, the paper develop a momentum quantification model and set four essential indicators to quantify the momentum. By adding together these indicators, the paper may determine a player's momentum score at a given scoring point. And a shift curve of momentum scores is shown using the momentum scores acquired at each scoring point, providing the groundwork for determining the competition's turning point in the future. Secondly, the randomness of momentum is assessed by us. To evaluate coaches' skepticism about the impact of momentum, the paper conducted two randomized experiments to investigate this issue. The first test is a randomness test based on outcome correlation. The paper calculate the correlation between the winner and the player with the highest score on the final score. The second test is the permutation test, which involves generating a large number of random permutations by redistributing the labels of the data points. It then calculates the consistent distribution between the highest momentum scorer and the competition winner under these random permutations. The test results all indicate that the player with the highest momentum score is the winner of the game. The paper proves that momentum plays a crucial role in the matches, and predicting momentum can be utilized to anticipate pivotal moments in the game.

Enhancing Network Threat Detection with Random Forest-Based NIDS and Permutation Feature Importance

Network Intrusion Detection Systems (NIDS) are critical for protecting computer networks from unauthorized activities. Traditional NIDS rely on rule-based signatures, which can be limiting in detecting emerging threats. This study investigates the effectiveness of the random forest classifier in advancing NIDS capabilities through machine learning. Using the CICIDS-2017 dataset, the data are preprocessed to enhance their quality by removing redundancies. feature selection and permutation importance were employed to identify the most relevant features. The methodology involves rigorous testing and analysis of the random forest classifier’s performance, focusing on f1-score rates compared to other machine learning models. Results demonstrate that by optimizing class weights, applying a custom prediction function and leveraging 26 key features, the random forest classifier achieves an outstanding 99.8% in the weighted f1-score and 93.31% in the macro f1-score in various attack types. This research highlights the potential of machine learning to significantly enhance NIDS effectiveness, offering a robust defense mechanism against evolving cybersecurity threats in modern networks.

Markov chains generating random permutations and set partitions

The Chinese Restaurant Process may be considered to be a Markov chain which generates permutations on n elements proportionally to absorption probabilities θ|π|, θ>0, where |π| is the number of cycles of permutation π. We prove a theorem which provides a way of finding Markov chains, restricted to directed graphs called arborescences, and with given absorption probabilities. We find transition probabilities for the Chinese Restaurant Process arborescence with variable absorption probabilities. The method is applied to an arborescence constructing set partitions, resulting in an analogue of the Chinese Restaurant Process for set partitions. We also apply our method to an arborescence for the Feller Coupling Process. We show how to modify the Chinese Restaurant Process, its set partition analogue, and the Feller Coupling Process to generate derangements and set partitions having no blocks of size one.

Random walk in random permutation set theory.

Random walk is an explainable approach for modeling natural processes at the molecular level. The random permutation set theory (RPST) serves as a framework for uncertainty reasoning, extending the applicability of Dempster-Shafer theory. Recent explorations indicate a promising link between RPST and random walk. In this study, we conduct an analysis and construct a random walk model based on the properties of RPST, with Monte Carlo simulations of such random walk. Our findings reveal that the random walk generated through RPST exhibits characteristics similar to those of a Gaussian random walk and can be transformed into a Wiener process through a specific limiting scaling procedure. This investigation establishes a novel connection between RPST and random walk theory, thereby not only expanding the applicability of RPST but also demonstrating the potential for combining the strengths of both approaches to improve problem-solving abilities.

Improving convolutional neural networks for cosmological fields with random permutation

Improving convolutional neural networks for cosmological fields with random permutation

A novel energy-focused slow-time MIMO radar and signal processing scheme

In this paper, we propose a novel energy-focused slow-time MIMO radar and signal processing scheme, aimed at addressing key challenges in slow-time coding and signal processing technology. Conventional slow-time MIMO radar faces issues such as energy waste due to the omnidirectional transmit beampattern of orthogonal coding and the velocity ambiguity problem. To overcome these limitations, the proposed radar system utilizes a method based on Doppler frequency offset diversity (DFOD) for slow-time coding design. This method enables the adjustment of Doppler offset parameters to achieve a rectangular transmit beampattern with any mainlobe width within a single coherent processing interval (CPI), offering the advantage of low computational complexity. Through an analysis of the ambiguity function for DFOD-based coding, we evaluate both Doppler and angular resolution. To further improve Doppler frequency resolution, a slow-time coding design is introduced based on Pulse Random Permutation (PRP). Subsequently, a signal processing scheme based on matched filtering is presented. To tackle the high Doppler sidelobe issue associated with PRP-based coding, we propose a mismatch filter (MMF) design method utilizing convex optimization. Ultimately, the performance enhancement of the proposed slow-time MIMO radar is verified through simulation analysis in comparison to existing technologies.

Comparison of Fisher-Yates Shuffle and Linear Congruent Algorithms for Question Randomization

This research aims to compare Fisher-Yates algorithm and Linear Congruent algorithm in generating random numbers or permutations. The test is conducted using the Chi-Square method to evaluate the quality of randomness generated by both algorithms. The Chi-Square value of the shuffling results is calculated and compared with the critical value of Chi-Square at a significance level of 0.05 with a degree of freedom (df) of 4, which is 9.488. The results show that the Chi-Square value for the Fisher-Yates algorithm is 3.8 and for the Linear Congruent algorithm is 4.3, both of which are below the critical value. This indicates that there is not enough evidence to reject the Null Hypothesis (H₀), implying that the difference in randomness quality between the two algorithms is not statistically significant. Therefore, both algorithms are considered to have equivalent performance. The decision to choose one of the algorithms can be based on other considerations such as complexity and efficiency. Further research is recommended to explore the performance of the algorithms under different conditions.

An advanced cryptographic scheme for DICOM medical image encryption using a novel spatiotemporal chaotic lattice

Ensuring the privacy and security of medical imaging data is crucial. Most existing methods focus on encrypting images in common formats, neglecting the unique challenges posed by medical formats. In this paper an advanced encryption algorithm based on a multi-dimensional and multi-parametric spatiotemporal coupled chaotic lattice (Double-MsCcL) system is proposed. The Double-MsCcL system enhances the complexity of chaotic sequences, and provides a robust foundation for medical image encryption. The encryption method proposed combines double-layer random DNA diffusion and fractal random permutation. Medical information is embedded in the image pixels, serving as the cryptosystem key. The designed double-layer DNA random diffusion algorithm alters pixel values, while the fractal random permutation algorithm disrupts pixel positions. This proposed approach effectively hides sensitive information, strengthens resistance to plaintext attacks, and address the often-overlooked Uint16 format of DICOM images. Comprehensive performance evaluations have shown that the proposed encryption scheme offers significant advantages in protecting sensitive medical imaging data. Our algorithm exhibits better performance in resisting plaintext analysis, statistical, and differential attacks, demonstrating significant security compared to existing state-of-the-art methods, which can be widely applied in the field of modern medical data protection.

On the number of cycles in commutators of random permutations

On the number of cycles in commutators of random permutations

The Yellow River is the key corridor for Tamarix austromongolica to disperse from Asia inlands to east seashores.

Plants of the Tamarix L. genus (Tamaricaceae) mainly occur in arid inlands of Asia, but a few species occur in the coastal areas of China, and the Yellow River may account for this. This study was conducted to elucidate whether and how the Yellow River affects the pattern and development of the Tamarix genus, involving two critical species of Tamarix austromongolica Nakai and Tamarix chinensis Lour. With geographical distribution data, relationships of T. austromongolica with the Yellow River and the pertaining watershed were examined using the method of random permutation. The base-diameter structures of T. austromongolica populations were investigated and compared between different riparian lands that suffer discriminative water inundation. The nearest distances from T. austromongolica locations to the Yellow River and the pertaining watershed were significantly lower than the theoretical expectations in the condition of random distribution (p < .05). In many riparian lands along the Yellow River, wild T. austromongolica populations occurred with vigorous juveniles, despite frequent human disturbances. In coastal areas near the present estuary of the river, wild T. austromongolica plants were still found. In T. austromongolica populations near the Yellow River and sea, the rates of juvenile plants were significantly higher than in other populations situated farther from the river or sea. These findings suggest that the Yellow River can facilitate the eastward dispersal of Tamarix plants that reasonably caused the evolution from T. austromongolica to T. chinensis in ancient coasts in the China east.

Negation of permutation mass function in random permutation sets theory for uncertain information modeling

Negation provides a novel perspective for the representation of information. However, current research seldom addresses the issue of negation within the random permutation set theory. Based on the concept of belief reassignment, this paper proposes a method for obtaining the negation of permutation mass function in the of random set theory. The convergence of proposed negation is verified, the trends of uncertainty and dissimilarity after each negation operation are investigated. Furthermore, this paper introduces a negation-based uncertainty measure, and designs a multi-source information fusion approach based on the proposed measure. Numerical examples are used to verify the rationality of proposed method.

Complexity of Digital Quantum Simulation in the Low-Energy Subspace: Applications and a Lower Bound

Digital quantum simulation has broad applications in approximating unitary evolution of Hamiltonians. In practice, many simulation tasks for quantum systems focus on quantum states in the low-energy subspace instead of the entire Hilbert space. In this paper, we systematically investigate the complexity of digital quantum simulation based on product formulas in the low-energy subspace. We show that the simulation error depends on the effective low-energy norm of the Hamiltonian for a variety of digital quantum simulation algorithms and quantum systems, allowing improvements over the previous complexities for full unitary simulations even for imperfect state preparations due to thermalization. In particular, for simulating spin models in the low-energy subspace, we prove that randomized product formulas such as qDRIFT and random permutation require smaller Trotter numbers. Such improvement also persists in symmetry-protected digital quantum simulations. We prove a similar improvement in simulating the dynamics of power-law quantum interactions. We also provide a query lower bound for general digital quantum simulations in the low-energy subspace.

Variance of a strongly additive function defined on random permutations

Variance of a strongly additive function defined on random permutations

A new orthogonal sum in Random Permutation Set

Random Permutation Set is a newly proposed method for handling uncertainty, which considers the order of the elements in evidence. However, how to fuse RPSs efficiently is still an open issue. To solve this problem, the space composed of order code is defined. Then the mutual mappings between this space and permutation event space are also presented. Finally, the new orthogonal sum is proposed. Compared with the existing left orthogonal sum, the new orthogonal sum can obtain more accurate results with lower entropy and deal with extreme situations. Numerical examples are used to illustrate the superiority of the new orthogonal sum.

Short cycles of random permutations with cycle weights: Point processes approach

We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on {1,…,n}. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as n→∞ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.

Privacy-preserving outsourcing scheme of face recognition based on locally linear embedding

Face recognition is a biometric-based technology that identifies or verifies a person’s identity by analyzing and comparing patterns in their facial features. Among kinds of face recognition algorithms, Locally Linear Embedding (LLE) algorithm has the advantage of preserving the local structure of face data. However, the LLE algorithm requires some computationally intensive matrix calculations, which might introduce latency to the device. In this paper, we propose a privacy-preserving outsourcing scheme of face recognition based on LLE algorithm for the first time. By outsourcing matrix multiplication, linear equations, and eigenvalue decomposition of the LLE algorithm to a cloud server, we reduce the computational load on the client device. To ensure data security, we generate a novel type of sparse orthogonal matrix through square root application and random permutation of diagonal elements, used as secret keys to blind the client’s data. Meanwhile, due to the sparsity of key matrices, the complexity of client-side computation complexity is reduced from O(n3) to O(n2). The experimental results show that our proposed scheme reduces computational costs while maintaining almost identical face recognition accuracy as the original LLE-based algorithm.

The Ulam–Hammersley problem for multiset permutations

Abstract We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in $\{1,\dots,n\}$ occurs k times, where k may depend on n. This generalises the famous Ulam–Hammersley problem of the case $k=1$ . The proof relies on poissonisation and on a careful non-asymptotic analysis of variants of the Hammersley–Aldous–Diaconis particle system.

Increasing subsequences of linear size in random permutations and the Robinson–Schensted tableaux of permutons

AbstractThe study of longest increasing subsequences (LIS) in permutations led to that of Young diagrams via Robinson–Schensted's (RS) correspondence. In a celebrated paper, Vershik and Kerov obtained a limit theorem for such diagrams and found that the LIS of a uniform permutation of size behaves as . Independently and much later, Hoppen et al. introduced the theory of permutons as a scaling limit of permutations. In this paper, we extend in some sense the RS correspondence of permutations to the space of permutons. When the “RS‐tableaux” of a permuton are non‐trivial, we show that the RS‐tableaux of random permutations sampled from this permuton exhibit a linear behavior, in the sense that their first rows and columns have lengths of linear order. In particular, the LIS of such permutations behaves as a multiple of . We also prove some large deviation results for these convergences. Finally, by studying asymptotic properties of Fomin's algorithm for permutations, we show that the RS‐tableaux of a permuton satisfy a partial differential equation.

An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, the authors of a recent paper showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$. This exhibited the fact that random permutations pack consecutive patterns near-perfectly. We use entirely different methods, namely the Stein-Chen method of Poisson approximation, to reprove and slightly improve their result.

Multiplicative arithmetic functions and the generalized Ewens measure

Random integers, sampled uniformly from [1, x], share similarities with random permutations, sampled uniformly from Sn. These similarities include the Erdős–Kac theorem on the distribution of the number of prime factors of a random integer, and Billingsley’s theorem on the largest prime factors of a random integer. In this paper we extend this analogy to non-uniform distributions.Given a multiplicative function α: ℕ → ℝ≥0, one may associate with it a measure on the integers in [1, x], where n is sampled with probability proportional to the value α(n). Analogously, given a sequence {θi}i≥1 of non-negative reals, one may associate with it a measure on Sn that assigns to a permutation a probability proportional to a product of weights over the cycles of the permutation. This measure is known as the generalized Ewens measure.We study the case where the mean value of α over primes tends to some positive θ, as well as the weights α(p) ≈ (log p)γ. In both cases, we obtain results in the integer setting which are in agreement with those in the permutation setting.