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Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.

For exploring the meaning of the power set in evidence theory, a possible explanation of power set is proposed from the view of Pascal’s triangle and combinatorial number. Here comes the question: what would happen if the combinatorial number is replaced by permutation number? To address this issue, a new kind of set, named as random permutation set (RPS), is proposed in this paper, which consists of permutation event space (PES) and permutation mass function (PMF). The PES of a certain set considers all the permutation of that set. The elements of PES are called the permutation events. PMF describes the chance of a certain permutation event that would happen. Based on PES and PMF, RPS can be viewed as a permutation-based generalization of random finite set. Besides, the right intersection (RI) and left intersection (LI) of permutation events are presented. Based on RI and LI, the right orthogonal sum (ROS) and left orthogonal sum (LOS) of PMFs are proposed. In addition, numerical examples are shown to illustrate the proposed conceptions. The comparisons of probability theory, evidence theory, and RPS are discussed and summarized. Moreover, an RPS-based data fusion algorithm is proposed and applied in threat assessment. The experimental results show that the proposed RPS-based algorithm can reasonably and efficiently deal with uncertainty in threat assessment with respect to threat ranking and reliability ranking.

The field containing exactly q elements is referred to as q, where q is a power of a prime. The class of permutations polynomials (PP) with the formula f(x) = b(xq+ax+δ)i+c(xq+ax+δ)j+uxq+vx and its compositional inverse over Fq is examined in this work, where b, c ∈ q, a, δ, u and v ∈ q2 with a1+q = 1, (avq + u)(au − v) ≠ 0 and uq + v = a(u + avq).

Permutation entropy has been used as a powerful nonlinear dynamic tool for randomness measurement of time series and has been used in the area of condition monitoring and early failure fault detection of rolling bearing. However, the detail size relationship between adjacent amplitudes of signal is ignored in the calculation process of the original permutation entropy algorithms. The reverse permutation entropy was developed as a new nonlinear dynamic parameter through introducing distance information to time series with different lengths to improve the performance and stability of permutation entropy. Since the single-scale permutation entropy or reverse permutation entropy cannot completely reflect the complexity features of time series, in this paper, the phase reverse permutation entropy is proposed by introducing phase information into reverse permutation entropy to improve the detection ability of signal dynamic changes as much as possible. Based on phase reverse permutation entropy, the composite multi-scale phase reverse permutation entropy is proposed to extract the complexity information hidden in different time scales and overcome the defects of traditional coarse-grained multi-scale. Also, phase reverse permutation entropy is compared with reverse permutation entropy through simulation data and the result shows that the introduced phase information can increase the sensitivity of phase reverse permutation entropy in mutation characteristics detection of signal. After that, a new fault diagnosis method of rolling bearing was proposed based on composite multi-scale phase reverse permutation entropy for fault feature extraction and the whale optimization algorithm support vector machine for failure mode identification. Finally, the proposed fault diagnosis method was applied to the experimental data analysis of rolling bearing by comparing it with the composite multi-scale permutation entropy, the multi-scale permutation entropy, as well as multi-scale phase reverse permutation entropy based fault diagnosis approaches. The comparison results shows that the proposed method can effectively the fault location and severity of rolling bearings and reaches the highest fault recognition rate among the mentioned methods above.

Permutation entropy (PE), as one of the powerful complexity measures for analyzing time series, has advantages of easy implementation and high efficiency. In order to improve the performance of PE, some improved PE methods have been proposed through introducing amplitude information and distance information in recent years. Weighted-permutation entropy (W-PE) weight each arrangement pattern by using variance information, which has good robustness and stability in the case of high noise level and can extract complexity information from data with spike feature or abrupt amplitude change. Dispersion entropy (DE) introduces amplitude information by using the normal cumulative distribution function (NCDF); it not only can detect the change of simultaneous frequency and amplitude, but also is superior to the PE method in distinguishing different data sets. Reverse permutation entropy (RPE) is defined as the distance to white noise in the opposite trend with PE and W-PE, which has high stability for time series with varying lengths. To further improve the performance of PE, we propose a new complexity measure for analyzing time series, and term it as reverse dispersion entropy (RDE). RDE takes PE as its theoretical basis and combines the advantages of DE and RPE by introducing amplitude information and distance information. Simulation experiments were carried out on simulated and sensor signals, including mutation signal detection under different parameters, noise robustness testing, stability testing under different signal-to-noise ratios (SNRs), and distinguishing real data for different kinds of ships and faults. The experimental results show, compared with PE, W-PE, RPE, and DE, that RDE has better performance in detecting abrupt signal and noise robustness testing, and has better stability for simulated and sensor signal. Moreover, it also shows higher distinguishing ability than the other four kinds of PE for sensor signals.

Given independent samples from P and Q, two-sample permutation tests allow one to construct exact level tests when the null hypothesis is P = Q. On the other hand, when comparing or testing particular parameters θ of P and Q, such as their means or medians, permutation tests need not be level α, or even approximately level α in large samples. Under very weak assumptions for comparing estimators, we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability α in finite samples when the underlying distributions are identical. The ideas are broadly applicable and special attention is given to the k-sample problem of comparing general parameters, whereby a permutation test is constructed which is exact level α under the hypothesis of identical distributions, but has asymptotic rejection probability α under the more general null hypothesis of equality of parameters. A Monte Carlo simulation study is performed as well. A quite general theory is possible based on a coupling construction, as well as a key contiguity argument for the multinomial and multivariate hypergeometric distributions.

As a nonlinear dynamic index, hierarchical permutation entropy (HPE) can effectively represent the complexity change of time series. However, HPE only focuses on extracting time domain information and ignoring the rich information in frequency domain. Meanwhile, HPE is greatly influenced by the length of time series and has poor stability. To address these limitations, a two-dimensional hierarchical time–frequency permutation entropy (HTFPE 2D ) is proposed based on the definition of two-dimensional permutation entropy, and its purpose is to combine time-domain and frequency-domain information. To consider the time–frequency information of the multi-scale low-frequency sequences, the two-dimensional multi-scale hierarchical time–frequency permutation entropy (MHTFPE 2D ) is further established. MHTFPE 2D allows for the synthesis of multidimensional effective information and leads to better feature extraction. Based on the advantages of the MHTFPE 2D , a new fault diagnosis method of rolling bearing is developed by combining the MHTFPE 2D and GOA-SVM. The proposed fault diagnosis method is validated by using the public rolling bearing datasets of CRWU and our rolling bearing datasets of Anhui University of Technology. The comparison results demonstrate that the proposed method achieves high fault identification accuracy, stability and robustness.

In general factorial designs where no homoscedasticity or a particular error distribution is assumed, the well‐known Wald‐type statistic is a simple asymptotically valid procedure. However, it is well known that it suffers from a poor finite sample approximation since the convergence to its χ²limit distribution is quite slow. This becomes even worse with an increasing number of factor levels. The aim of the paper is to improve the small sample behaviour of the Wald‐type statistic, maintaining its applicability to general settings as crossed or hierarchically nested designs by applying a modified permutation approach. In particular, it is shown that this approach approximates the null distribution of the Wald‐type statistic not only under the null hypothesis but also under the alternative yielding an asymptotically valid permutation test which is even finitely exact under exchangeability. Finally, its small sample behaviour is compared with competing procedures in an extensive simulation study.