Stationary Sequence Research Articles

Computer modeling of cardiac rhythm based on vector of stationary random sequences

The article is devoted to a computer modeling method of electrocardiogram rhythm based on a mathematically justified model in the form of a vector of stationary random sequences. The developed computer modeling method allows for generating realizations of vector electrocardiogram rhythm signal (vector components of stationary random sequences) for different types of electrocardiogram signals, both normal and with various types of rhythm pathologies. The modeling of electrocardiogram rhythms was carried out based on the obtained statistical information in the form of estimates of the mathematical expectation and variance of the components of the vector of stationary random sequences. It has been shown that the obtained estimates of statistical characteristics of the modeled vector components (components that describe the electrocardiogram rhythm) are within confidence intervals, which is an indication of the correctness of the experiments conducted using the developed computer simulation method. The accuracy of the computer simulation method for generating realizations of the vector components of stationary random sequences has been investigated, and the error of the computer simulation does not exceed 13% for the investigated vector components.

Using ARIMA and BP neural network to analyse incidence rate of AIDS in China

To analyse the characteristics of AIDS transmission from incidence, we used ARIMA and BP neural networks to model the incidence of AIDS and predict them based on modelling. When the sequence is a small sample sequence and instability, the input of the BP neural network can use raw data or stationary sequence in the ARIMA. When using the stationary sequences of incidence as the input of the BP neural network, we can obtain the output corresponding to raw data by matrix operations. Results show that raw data combined with the stationary sequences as the input of the BP neural network can get better modelling results. Moreover, all the predicted values fall within the 95% CI of the ARIMA model. Although there was also a study (reference 14) using BP to predict the incidence of AIDS, it is the original used stationary series as the input of BP in this study.

Learned Factor Graphs for Inference From Stationary Time Sequences

The design of methods for inference from time sequences has traditionally relied on statistical models that describe the relation between a latent desired sequence and the observed one. A broad family of model-based algorithms have been derived to carry out inference at controllable complexity using recursive computations over the factor graph representing the underlying distribution. An alternative model-agnostic approach utilizes machine learning (ML) methods. Here we propose a framework that combines model-based algorithms and data-driven ML tools for stationary time sequences. In the proposed approach, neural networks are developed to separately learn specific components of a factor graph describing the distribution of the time sequence, rather than the complete inference task. By exploiting stationary properties of this distribution, the resulting approach can be applied to sequences of varying temporal duration. Learned factor graphs can be realized using compact neural networks that are trainable using small training sets, or alternatively, be used to improve upon existing deep inference systems. We present an inference algorithm based on learned stationary factor graphs, which learns to implement the sum-product scheme from labeled data, and can be applied to sequences of different lengths. Our experimental results demonstrate the ability of the proposed learned factor graphs to learn from small training sets to carry out accurate inference for sleep stage detection using the Sleep-EDF dataset, as well as for symbol detection in digital communications with unknown channels.

Asymptotic Extremal Distribution for Non-Stationary, Strongly-Dependent Data

Fisher-Tippet-Gnedenko classical theory shows that the normalized maximum of n iid random variables with distribution F belonging to a very wide class of functions, converges in law to an extremal distribution H, that is determined by the tail of F. Extensions of this theory from the iid case to stationary and weak dependent sequences are well known from the work of Leadbetter, Lindgreen and Rootzén. In this paper, we present a very simple class of random processes that runs from iid sequences to non-stationary and strongly dependent processes, and we study the asymptotic behavior of its normalized maximum. More interesting, we show that when the process is strongly dependent, the asymptotic distribution is no longer an extremal one, but a mixture of extremal distributions. We present very simple theoretical and simulated examples of this result. This provides a simple framework to asymptotic approximations of extremes values not covered by classical extremal theory and its well-known extensions.

Asymptotic behavior of ordered random variables in mixture of two Gaussian sequences with random index

<abstract><p>When the random sample size is assumed to converge weakly and to be independent of the basic variables, the asymptotic distributions of extreme, intermediate, and central order statistics, as well as record values, for a mixture of two stationary Gaussian sequences under an equi-correlated setup are derived. Furthermore, sufficient conditions for convergence are derived in each case. An interesting fact is revealed that in several cases, the limit distributions of the aforementioned statistics are the same when the sample size is random and non-random. e.g., when one mixture component has a correlation that converges to a non-zero value.</p></abstract>

Limit Distributions of Ordered Random Variables in Mixture of Two Gaussian Sequences

In this paper, we study the limit distributions of extreme, intermediate, and central order statistics, as well as record values, of the mixture of two stationary Gaussian sequences under equicorrelated setup.

Large deviation principle for linear processes generated by real stationary sequences under the sub-linear expectation

Let be the linear processes, where is an absolutely summable sequence of real numbers and is a sequence of real stationary independent random variables under the sub-linear expectation space In the framework of sub-linear expectation space, the Beveridge–Nelson decomposition of linear processes and the sub-additivity of capacity are used to establish the large deviation principle for linear processes in the sense of upper probability.

Extremal clustering under moderate long range dependence and moderately heavy tails

We study clustering of the extremes in a stationary sequence with subexponential tails in the maximum domain of attraction of the Gumbel distribution. We obtain functional limit theorems in the space D[0,∞) and in the space of random sup-measures. The limits have the Gumbel distribution if the memory is only moderately long. However, as our results demonstrate rather strikingly, the “heuristic of a single big jump” could fail even in a moderately long range dependence setting. As the tails become lighter, the extremal behavior of a stationary process may depend on multiple large values of the driving noise.

Randomized multivariate Central Limit Theorems for ergodic homogeneous random fields

We present new versions of the CLT which are valid for each ergodic homogeneous multivariate random field (X1(⋅),…,Xd(⋅)) on Rm or Zm(m≥1) (in particular, for each ergodic stationary random process and for each ergodic stationary random sequence) such that for all l E[|Xl(0)|2+δ]<∞ for some δ>0(l=1,...,d); in some statements ergodicity is not assumed. Randomization made it possible to significantly weaken the strong mixing conditions and other restrictions of dependence, that are imposed in the conventional CLTs. These results pave the way to consistent statistical inference for homogeneous random fields and stationary processes with strong dependence.

Algorithm and Results of Short-Term Forecast of Changes in the COVID-19 Spread Coefcient in the Constituent Entities of the Russian Federation

There was a decrease in the number of COVID-19 cases across many entities of the Russian Federation towards the end of summer season-2020. However, the disease remains a relevant threat to the public health and economy and the possibility of a second epidemic wave is not excluded. Rate of infection transmission (Rt) is one of the most important indicators to justify the transition to next stage of removing/introducing restrictive measures on COVID-19.Objective of the work was to describe the algorithm of analysis and short-term forecast of coronavirus spread rate, to assess correspondence between theoretically expected and actual values of this indicator.Materials and methods. Procedure for making a short-term extrapolation forecast of Rt in 10 RF constituent entities, depending on the presence or absence of indicator trends with calculation of a 95 % confdence interval of possible changes in its value is provided.Results and discussion. It is proposed to carry out Rt forecast based on averaged values over a week, combining regression analysis and expert assessment of time series dynamics nature for prompt transition from trend forecasting to extrapolation of stationary observation sequences and vice versa. It has been demonstrated that predicted Rt values are not statistically different from actual values. When making managerial decisions on COVID-19 prevention, special attention should be paid to cases when actual value of Rt exceeds the upper limit of confdence interval. Six (20.0 %) such cases were detected in surveyed entities on calendar weeks 33–35. Three of them were registered in Trans-Baikal Territory, where upward trend was reported during that period of time. The cause of this phenomenon should be analyzed. The put forward algorithm of analysis and forecasting of the Rt value changes, which was tested in 10 entities of Russia, provides a reliable basis for making management decisions on removing/introducing restrictive measures for COVID-19 prevention.

The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation.

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

Robust Filtering of Sequences with Periodically Stationary Multiplicative Seasonal Increments

We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the filtering problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a periodically stationary noise sequence. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal filtering of the functionals. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal linear filtering of the functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.

Stabilization of semi-Markovian jump systems via a quantity limited controller

This paper addresses the stabilization problem of continuous-time semi-Markovian jump systems (s-MJSs), where the quantity of a controller to be designed has a limitation. A principle designing such a controller is proposed based on its stationary probability sequence. Sufficient conditions for the existence of the designed controller are obtained by applying a Lyapunov function approach and presented with LMIs. The controller mentioned above is proved to be a minimal variance approximation of mode-dependent controllers having total modes. More special cases about Markovian switching and without minimal variance approximation are further studied. A practical example is used to demonstrate the effectiveness and superiority of the proposed methods.

Estimation of the limiting availability of a repairable series system with stationary dependent sequences

The independence assumption would lead to poor estimation of the availability and even misleading conclusions when dependence actually exists. In this paper, we provide the estimation of the limiting availability of a repairable series system when the failure times and repair times are two independent sequences of stationary dependent random variables. Furthermore, consistency and asymptotic normality of the estimator of the limiting availability are derived under some regular conditions. Finally, we investigate the limiting availability for a first-order exponential moving average (EMA1) process and a first-order exponential autoregressive (EAR(1)) process. Simulations are presented to study the impact of dependence on the estimation of the limiting availability.

Application of Error Correction model with monetary function factors in the return of bitcoin

Bitcoin is a kind of virtual currency based on block chain technology. In many scenarios, it shows the function of currency to a certain extent. However, many believe that the value of Bitcoin is hollow, speculative and unable to undertake monetary functions, which prevents it to circulate as a real currency. This paper assumes that Bitcoin can undertake monetary functions, which can verify the relationship between monetary functions and Bitcoin earnings. According to the availability of the data and the previous research results, the monthly value of the indexes to measure the monetary function from 2010 to 2020 are selected to form a regression equation with the monthly return data of Bitcoin in the corresponding period. ADF test shows that the explained variable is a first-order simple integer, and the explanatory variables have both first-order simple integer and stationary sequence, and there is a cointegration relationship between them. The result of error correction model shows that the relationship between Bitcoin rate of return and the relevant indicators to measure monetary function is not significant, which rejects the original assumption that Bitcoin can assume monetary function, indicating that Bitcoin does not have the ability and potential to assume monetary function. This may be due to the speculative nature of Bitcoin and the lack of powerful credit support. Therefore, investment in Bitcoin needs to be cautiously thought, but the blockchain technology used in Bitcoin needs to be vigorously developed, which could lead to a technological revolution; in addition, it is momentous for countries to develop digital currencies of central bank to meet the challenge.

Extremal clustering in non-stationary random sequences

It is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but identically distributed sequences of random variables subject to suitable long range dependence restrictions. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence. Based on this new representation we derive the asymptotic distribution for the time between consecutive extreme observations and construct moment and likelihood based estimators for measures of extremal clustering. We specialize our results to random sequences with periodic dependence structure.

Boundedness Condition for Anisotropic Norm of Linear Discrete Time-invariant Systems with Multiplicative Noise

In this paper, the anisotropy-based analysis problem for the linear discrete time-invariant system with state-multiplicative noises is considered. The mean anisotropy of stationary input sequence is assumed to be bounded by given nonnegative value. For this setting, the boundedness condition for anisotropic norm of the system is obtained in terms of Riccati equations.

Evaluating the interplay among stationary phases/ion-pairing reagents/sequences for liquid chromatography mass spectrometry analysis of oligonucleotides

The correlation among stationary phases, ion-pairing reagents (IPR) and sequences for ion-pair reversed-phase liquid chromatography mass spectrometry (IP-RP LC-MS) analysis of oligonucleotide (ODN) remains unclear. The present study aimed to evaluate such correlation using particle-packed C18 columns in order to search for the optimal combination among them. Five C18 columns packed with core-shell silica, polymer, porous silica and hybrid particles, respectively, were evaluated for the analysis of synthetic and chemically modified ODNs with six different IPRs. Our results showed that silica-based porous particles, compared to other particles, retained ODN the strongest no matter which IPR was used. Meanwhile, among the six IPRs hexylamine (HA) produced the longest retention for all ODNs, regardless of the types of C18 particles. For the separation of ODNs, C18 columns performed similarly under identical LC conditions. However, the separation ability of C18 columns is highly dependent on the type of IPR and ODN sequences. Moreover, the type of particles has little impact on the signals of ODNs for the majority of synthetic sequences, but such impact could be dramatic for chemically modified sequences. On the other hand, both the type of IPR and ODN sequence have a significant effect on MS signals for synthetic and chemically modified ODNs.

Analysis and Economic Prediction of Stock Index Based on Index Tracking and ARIMA Model

Investors buy assets in the target index, if all, the cost is too high, generally not feasible. This paper carries on the stationarity test and the unit root test to the time series, establishes the ARIMA model of the stationary sequence obtained after the difference of the non-stationary sequence, determines the constant of the ARIMA model according to the ACF diagram and the PACF diagram. By analyzing the current exponential fluctuation, the time series model is used to predict the exponential fluctuation in the next year. Through the evaluation and analysis of the prediction results, reasonable investment suggestions and strategies can be obtained.

Limit Theorems for Conservative Flows on Multiple Stochastic Integrals

We consider a stationary sequence \((X_n)\) constructed by a multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the multiple integral is non-Gaussian and infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter \(\beta \in (0,1)\) quantifying the conservativity of the system. This parameter \(\beta \) together with the order of the integral determines the decay rate of the covariance of \((X_n)\). The goal of the paper is to establish limit theorems for the partial sum process of \((X_n)\). We obtain a central limit theorem with Brownian motion as limit when the covariance decays fast enough, as well as a non-central limit theorem with fractional Brownian motion or Rosenblatt process as limit when the covariance decays slowly enough.