Random Sequences Of Length Research Articles

Extremal properties of evolving networks: local dependence and heavy tails

A network evolution with predicted tail and extremal indices of PageRank and the Max-Linear Model used as node influence indices in random graphs is considered. The tail index shows a heaviness of the distribution tail. The extremal index is a measure of clustering (or local dependence) of the stochastic process. The cluster implies a set of consecutive exceedances of the process over a sufficiently high threshold. Our recent results concerning sums and maxima of non-stationary random length sequences of regularly varying random variables are extended to random graphs. Starting with a set of connected stationary seed communities as a hot spot and ranking them with regard to their tail indices, the tail and extremal indices of new nodes that are appended to the network may be determined. This procedure allows us to predict a temporal network evolution in terms of tail and extremal indices. The extremal index determines limiting distributions of a maximum of the PageRank and the Max-Linear Model of newly attached nodes. The exposition is provided by algorithms and examples. To validate our theoretical results, our simulation and real data study concerning a linear preferential attachment as a tool for network growth are provided.

Tail Index Estimation of PageRanks in Evolving Random Graphs

Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graphs.

Multi-Image Encryption Algorithm for 2D and 3D Images Based on Chaotic System

In order to accommodate multiple types of image encryption, a multi-image encryption algorithm for 2D and 3D images is designed in this paper. After recording the type and number of images, the pixels/coordinates of multiple images are stored in a cube block and are subjected to confusion and diffusion operations. The confusion step uses the random length sequence position swapping method to swap a row (column) vector of variable length with another row (column) vector of the same length at a random position. The diffusion is done by Exclusive OR to combine pixels/coordinates at different locations with different chaotic matrices. Finally, the cipher images are output one by one. Experimental simulations and security analysis verify the effectiveness and security of the algorithm.

Wind and photovoltaic power time series data aggregation method based on ensemble clustering and Markov chain

Reducing the input wind and photovoltaic power time series data can improve the efficiency of time sequential simulations. In this paper, a wind and photovoltaic power time series data aggregation method based on ensemble clustering and Markov chain (ECMC) is proposed. The ECMC method can effectively reduce redundant information in the data. Firstly, the wind and photovoltaic power time series data were divided into scenarios, and ensemble clustering was used to cluster the divided scenarios. At the same time, the Davies-Bouldin Index (DBI) is adopted to select the optimal number of clusters. Then, according to the temporal correlation between wind and photovoltaic scenarios, the wind and photovoltaic clustering results are merged and reduced to form a set of combined typical day scenarios that can reflect the characteristics of historical data within the calculation period. Finally, based on the Markov Chain, the state transition probability matrix of various combined typical day scenarios is constructed, and the aggregation state sequence of random length is generated, and then, the combined typical day scenarios of wind and photovoltaic were sampled in a sequential one-way sequence according to the state sequence and build into a representative wind and photovoltaic power time series aggregation sequence. A provincial power grid was chosen as an example to compare the multiple evaluation indexes of different aggregation methods. The results show that the ECMC aggregation method improves the accuracy and efficiency of time sequential simulations.

Maxima and sums of non-stationary random length sequences

We study non-stationary random length sequences of random variables with regularly varying tails. Tail and extremal indexes of their maxima and linear combinations are found. We obtain conditions when both sums and maxima of these sequences have the same tail and extremal indexes. Their extremal index corresponds to the tail index of the most heavy-tailed random variable in the sequence.

Microscopic path structure of optimally aligned random sequences

Considering optimal alignments of two i.i.d. random sequences of length $n$, we show that for Lebesgue-almost all scoring functions, almost surely the empirical distribution of aligned letter pairs in all optimal alignments converges to a unique limiting distribution as $n$ tends to infinity. This result helps understanding the microscopic path structure of a special type of last-passage percolation problem with correlated weights, an area of long-standing open problems. Characterizing the microscopic path structure also yields robust alternatives to the use of optimal alignment scores alone for testing the homology of genetic sequences.

A Note on Hall’s Sextic Residue Sequence: Correlation Measure of Order $k$ and Related Measures of Pseudorandomness

It is known that Hall’s sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period $p$ . Here we study its correlation measure of order $k$ and show that it is, up to a constant depending on $k$ and some logarithmic factor, of order of magnitude $p^{1/2}$ , which is close to the expected value for a random sequence of length $p$ . Moreover, we derive from this bound a lower bound on the $N$ th maximum order complexity of order of magnitude $\log p$ , which is the expected order of magnitude for a random sequence of length $p$ .

Response to discussants of "survival models and health sequences".

Survival studies often generate not only a survival time for each patient but also a sequence of health measurements at annual or semi-annual check-ups while the patient remains alive. Such a sequence of random length accompanied by a survival time is called a survival process. Robust health is ordinarily associated with longer survival, so the two parts of a survival process cannot be assumed independent. This paper is concerned with a general technique-reverse alignment-for constructing statistical models for survival processes, here termed revival models. A revival model is a regression model in the sense that it incorporates covariate and treatment effects into both the distribution of survival times and the joint distribution of health outcomes. The revival model also determines a conditional survival distribution given the observed history, which describes how the subsequent survival distribution is determined by the observed progression of health outcomes.

Survival models and health sequences.

Survival studies often generate not only a survival time for each patient but also a sequence of health measurements at annual or semi-annual check-ups while the patient remains alive. Such a sequence of random length accompanied by a survival time is called a survival process. Robust health is ordinarily associated with longer survival, so the two parts of a survival process cannot be assumed independent. This paper is concerned with a general technique-reverse alignment-for constructing statistical models for survival processes, here termed revival models. A revival model is a regression model in the sense that it incorporates covariate and treatment effects into both the distribution of survival times and the joint distribution of health outcomes. The revival model also determines a conditional survival distribution given the observed history, which describes how the subsequent survival distribution is determined by the observed progression of health outcomes.

Regular variation of a random length sequence of random variables and application to risk assessment

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence C(N) = (C 1,…,C N ) of random length N, where C(N) comes from the product of a matrix A(N) of random size N × N and a random sequence X(N) of random length N. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the extremal behavior of ∥C(N)∥, for some norm ∥⋅∥. We propose a generalization of Breiman’s Lemma that gives way to a tail estimate of ∥C(N)∥ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our main result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.

Characterizations in a Random Record Model with a Non-identically Distributed Initial Record

We consider a sequence of random length M of independent absolutely continuous observations Xi, 1 = 2, have cdf F. Let N be the number of upper records and Rn, n >= 1, be the nth record value. We show that N is free of F if and only if G(x) = G0(F (x)) for some cdf G0 and that if E(|X2|) is finite so is E(|Rn|) for n >= 2 whenever N >= n or N = n. We prove that the distribution of N along with appropriately chosen subsequences of E(Rn) characterize F and G, and along with subsequences of E(Rn - Rn - 1) characterize F and G up to a common location shift. We discuss some applications to the identification of the wage offer distribution in job search models.

The algorithm of random length sequences synthesis for frame synchronization of digital television systems

The algorithm of random length sequences synthesis for frame synchronization of digital television systems

A Randomness Test Based on T-Complexity

We propose a randomness test based on the T-complexity of a sequence, which can be calculated using a parsing algorithm called T-decomposition. Recently, the Lempel-Ziv (LZ) randomness test based on LZ-complexity using the LZ78 incremental parsing was officially excluded from the NIST test suite in NIST SP 800-22. This is caused from the problem that the distribution of P-values for random sequences of length 10 6 is strictly discrete for the LZ-complexity. Our proposed test can overcome this problem because T-complexity has almost ideal continuous distribution of P-values for random sequences of length 10 6 . We also devise a new sequential T-decomposition algorithm using forward parsing, while the original T-decomposition is an off-line algorithm using backward parsing. Our proposed test can become a supplement to NIST SP 800-22 because it can detect several undesirable pseudo-random numbers that the NIST test suite almost fails to detect.

Macroscopic non-uniqueness and transversal fluctuation in optimal random sequence alignment

We investigate the optimal alignment of two independent random sequences of length n . We provide a polynomial lower bound for the probability of the optimal alignment to be macroscopically non-unique. We furthermore establish a connection between the transversal fluctuation and macroscopic non-uniqueness.

Phase Transition in Sequence Unique Reconstruction

In this paper, sequence unique reconstruction refers to the property that a sequence is uniquely reconstructable from all its K-tuples. We propose and study the phase transition behavior of the probability P(K) of unique reconstruction with regard to tuple size K in random sequences (iid model). Based on Monte Carlo experiments, artificial proteins generated from iid model exhibit a phase transition when P(K) abruptly jumps from a low value phase (e.g. 0.9). With a generalization to any alphabet, we prove that for a random sequence of length L, as L is large enough, P(K) undergoes a sharp phase transition when p ≤ 0.1015 where p = P (two random letters match). Besides, formulas are derived to estimate the transition points, which may be of practical use in sequencing DNA by hybridization. Concluded from our study, most proteins do not deviate greatly from random sequences in the sense of sequence unique reconstruction, while there are some “stubborn” proteins which only become uniquely reconstructable at a very large K and probably have biological implications.

Sequencing by hybridization with errors: handling longer sequences

Sequencing by hybridization (SBH) is a method for reconstructing a DNA sequence given the set of all subsequences of length k of the target sequence. This set, called the spectrum of the sequence, can be obtained from hybridization with a universal DNA chip. However, the hybridization experiments are error prone, so this leads to the computational problem of reconstructing a sequence from a noisy spectrum. Halperin et al. gave an algorithm for this problem with provable performance in the presence of both false positive and false negative errors. Assuming, for example, that the false positive rate is small, and the probability of false negative is 0.1 , the algorithm can reconstruct a random sequence of length O ( 2 0.7 k ) with an arbitrary small probability of failure. In this paper, we give an algorithm that can reconstruct longer sequences: under the assumptions above, our algorithm can reconstruct sequences of length O ( 2 0.942 k ) . This bound is almost optimal as the bound for the errorless case is Θ ( 2 k ) .

Multi-item working memory--a behavioral study.

Macaque monkeys were trained to recognize the repetition of one of the images already seen in a sequence of random length. On average, performance decreased with sequence length. However, this was due to a complex combination of factors, as follows: performance was found to decrease with the separation in the sequence of the test (repetition image) from the cue (its first appearance in the sequence), for trials with sequences of fixed length. In contrast, performance improved as a function of sequence length, for equal cue-test separations. Reaction times followed a complementary trend: they increased with cue-test separation and decreased with sequence length. The frequency of false positives (FPs) indicates that images are not always removed from working memory between successive trials, and that the monkeys rarely confuse different images. The probability of miss errors depends on number of intervening stimulus presentations, while FPs depend on elapsed time. A simple two-state stochastic model of multi-item working memory is proposed that guides the account for the main effects of performance and false positives, as well as their interaction. In the model, images enter WM when they are presented, or by spontaneous jump-in. Misses are due to spontaneous jump-out of images previously seen.

Performance limits of Reed-Solomon coded CDMA with orthogonal signaling in a Rayleigh-fading channel

The asymptotic performance of Reed-Solomon (RS)-coded M-ary orthogonal signaling with ratio-threshold test (RTT) type demodulation in a Rayleigh-fading channel is considered. We show that the minimum E~/sub b//N/sub 0/ needed for error-free communication is eln2 (2.75 dB) with RTT, and 4.79 (6.8 dB) with hard decisions. The optimum code rate that minimizes the required E~/sub b//N/sub 0/ is e/sup -1/ with RTT and 0.46 with hard decision, and the optimum ratio threshold approaches 1 for large M. Next, we investigate the fundamental limit in a direct-sequence spread-spectrum multiple-access (DS/SSMA) system employing an M-ary orthogonal code of length N=Mm, which is obtained by spreading every row of an M/spl times/M Hadamard matrix with a user-specific random sequence of length N. We derive the minimum E~/sub b//N/sub 0/ for error-free communication as a function of the number of users, the optimum code rate that minimizes E~/sub B//N/sub 0/, and the maximum limit on the total information transmission rate. Then, we consider a multirate DS/SSMA system, where a population of users simultaneously transmit at different power levels a variety of traffic types of different information rates. We derive the minimum required E~/sub B//n/sub 0/ and the optimum code rate for each traffic type.

The maximum and mean of a random length sequence

We obtain a limit theorem for the joint distribution of the maximum value and sample mean of a random length sequence of independent and identically distributed random variables. This extends a previous bivariate convergence result for fixed length sequences and incidentally yields a new proof of Berman's classical limit theorem for the maximum value of a random number of random variables. Our approach uses a property of record time sequences and leads to probabilistically intuitive proofs. We also consider the partition of a finite interval into a random number of subintervals by the points of a non-delayed renewal process. Using the bivariate convergence result for random length sequences, we establish a limit theorem for the joint distribution of the number and maximum length of the subintervals as the interval length becomes large. This leads to limiting results for the ratio of the maximum to the mean subinterval length. Such results are of interest in connection with a simple model of parallel processing.

The maximum and mean of a random length sequence

We obtain a limit theorem for the joint distribution of the maximum value and sample mean of a random length sequence of independent and identically distributed random variables. This extends a previous bivariate convergence result for fixed length sequences and incidentally yields a new proof of Berman's classical limit theorem for the maximum value of a random number of random variables. Our approach uses a property of record time sequences and leads to probabilistically intuitive proofs. We also consider the partition of a finite interval into a random number of subintervals by the points of a non-delayed renewal process. Using the bivariate convergence result for random length sequences, we establish a limit theorem for the joint distribution of the number and maximum length of the subintervals as the interval length becomes large. This leads to limiting results for the ratio of the maximum to the mean subinterval length. Such results are of interest in connection with a simple model of parallel processing.