Special Issue "Stochastic Processes with Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 September 2018)

Special Issue Editors

Guest Editor
Prof. Dr. Antonio Di Crescenzo

Department of Mathematics, University of Salerno, I-84100 Salerno, Italy
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Interests: stochastic processes; applied probability; probability theory
Guest Editor
Prof. Dr. Claudio Macci

Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy
Website | E-Mail
Guest Editor
Dr. Barbara Martinucci

Department of Mathematics, University of Salerno, I-84100 Salerno, Italy
Website | E-Mail
Interests: stochastic processes; large deviations; probability theory

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to publish original research articles that cover recent advances in the theory and applications of stochastic processes. The focus will especially be on applications of stochastic processes as models of dynamic phenomena in various research areas, such as biology, economics, medicine, queuing theory, reliability theory, and statistical physics.

Potential topics include, but are not limited to:

•       Markov processes
•       Markov renewal processes, semi-Markov processes
•       Markov chains
•       Large deviations and limit theorems
•       Random motions
•       Fractional processes
•       Stochastic biological models
•       Reliability, availability, maintenance, inspection
•       Queueing models
•       Queueing network models
•       Computational methods for stochastic models
•       Measures of information, entropy, stochastic orderings
•       Applications to risk theory, insurance and mathematical finance

Prof. Dr. Antonio Di Crescenzo
Prof. Dr. Claudio Macci
Dr. Barbara Martinucci
Guest Editors

Manuscript Submission Information

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Keywords

  • probability theory and stochastic processes

  • applications of stochastic processes

  • computational problems in probability

Published Papers (17 papers)

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Research

Open AccessFeature PaperArticle Stochastic Comparisons and Dynamic Information of Random Lifetimes in a Replacement Model
Mathematics 2018, 6(10), 204; https://doi.org/10.3390/math6100204
Received: 27 September 2018 / Revised: 9 October 2018 / Accepted: 14 October 2018 / Published: 16 October 2018
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Abstract
We consider a suitable replacement model for random lifetimes, in which at a fixed time an item is replaced by another one having the same age but different lifetime distribution. We focus first on stochastic comparisons between the involved random lifetimes, in order [...] Read more.
We consider a suitable replacement model for random lifetimes, in which at a fixed time an item is replaced by another one having the same age but different lifetime distribution. We focus first on stochastic comparisons between the involved random lifetimes, in order to assess conditions leading to an improvement of the system. Attention is also given to the relative ratio of improvement, which is proposed as a suitable index finalized to measure the goodness of the replacement procedure. Finally, we provide various results on the dynamic differential entropy of the lifetime of the improved system. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessFeature PaperArticle Network Reliability Modeling Based on a Geometric Counting Process
Mathematics 2018, 6(10), 197; https://doi.org/10.3390/math6100197
Received: 4 August 2018 / Revised: 28 September 2018 / Accepted: 8 October 2018 / Published: 11 October 2018
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Abstract
In this paper, we investigate the reliability and stochastic properties of an n-component network under the assumption that the components of the network fail according to a counting process called a geometric counting process (GCP). The paper has two parts. In the [...] Read more.
In this paper, we investigate the reliability and stochastic properties of an n-component network under the assumption that the components of the network fail according to a counting process called a geometric counting process (GCP). The paper has two parts. In the first part, we consider a two-state network (with states up and down) and we assume that its components are subjected to failure based on a GCP. Some mixture representations for the network reliability are obtained in terms of signature of the network and the reliability function of the arrival times of the GCP. Several aging and stochastic properties of the network are investigated. The reliabilities of two different networks subjected to the same or different GCPs are compared based on the stochastic order between their signature vectors. The residual lifetime of the network is also assessed where the components fail based on a GCP. The second part of the paper is concerned with three-state networks. We consider a network made up of n components which starts operating at time t = 0 . It is assumed that, at any time t > 0 , the network can be in one of three states up, partial performance or down. The components of the network are subjected to failure on the basis of a GCP, which leads to change of network states. Under these scenarios, we obtain several stochastic and dependency characteristics of the network lifetime. Some illustrative examples and plots are also provided throughout the article. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessArticle Fractional Queues with Catastrophes and Their Transient Behaviour
Mathematics 2018, 6(9), 159; https://doi.org/10.3390/math6090159
Received: 27 July 2018 / Revised: 30 August 2018 / Accepted: 31 August 2018 / Published: 6 September 2018
Cited by 4 | PDF Full-text (321 KB) | HTML Full-text | XML Full-text
Abstract
Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al. in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al. in 2003, we define and study a fractional M/M/1 queue with [...] Read more.
Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al. in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al. in 2003, we define and study a fractional M/M/1 queue with catastrophes. In particular, we focus our attention on the transient behaviour, in which the time-change plays a key role. We first specify the conditions for the global uniqueness of solutions of the corresponding linear fractional differential problem. Then, we provide an alternative expression for the transient distribution of the fractional M/M/1 model, the state probabilities for the fractional queue with catastrophes, the distributions of the busy period for fractional queues without and with catastrophes and, finally, the distribution of the time of the first occurrence of a catastrophe. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
Open AccessArticle Stability Analysis of Cohen–Grossberg Neural Networks with Random Impulses
Mathematics 2018, 6(9), 144; https://doi.org/10.3390/math6090144
Received: 27 July 2018 / Revised: 15 August 2018 / Accepted: 17 August 2018 / Published: 21 August 2018
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Abstract
The Cohen and Grossberg neural networks model is studied in the case when the neurons are subject to a certain impulsive state displacement at random exponentially-distributed moments. These types of impulses significantly change the behavior of the solutions from a deterministic one to [...] Read more.
The Cohen and Grossberg neural networks model is studied in the case when the neurons are subject to a certain impulsive state displacement at random exponentially-distributed moments. These types of impulses significantly change the behavior of the solutions from a deterministic one to a stochastic process. We examine the stability of the equilibrium of the model. Some sufficient conditions for the mean-square exponential stability and mean exponential stability of the equilibrium of general neural networks are obtained in the case of the time-varying potential (or voltage) of the cells, with time-dependent amplification functions and behaved functions, as well as time-varying strengths of connectivity between cells and variable external bias or input from outside the network to the units. These sufficient conditions are explicitly expressed in terms of the parameters of the system, and hence, they are easily verifiable. The theory relies on a modification of the direct Lyapunov method. We illustrate our theory on a particular nonlinear neural network. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessArticle A Within-Host Stochastic Model for Nematode Infection
Mathematics 2018, 6(9), 143; https://doi.org/10.3390/math6090143
Received: 25 June 2018 / Revised: 6 August 2018 / Accepted: 11 August 2018 / Published: 21 August 2018
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Abstract
We propose a stochastic model for the development of gastrointestinal nematode infection in growing lambs under the assumption that nonhomogeneous Poisson processes govern the acquisition of parasites, the parasite-induced host mortality, the natural (no parasite-induced) host mortality and the death of parasites within [...] Read more.
We propose a stochastic model for the development of gastrointestinal nematode infection in growing lambs under the assumption that nonhomogeneous Poisson processes govern the acquisition of parasites, the parasite-induced host mortality, the natural (no parasite-induced) host mortality and the death of parasites within the host. By means of considering a number of age-dependent birth and death processes with killing, we analyse the impact of grazing strategies that are defined in terms of an intervention instant t 0 , which might imply a move of the host to safe pasture and/or anthelmintic treatment. The efficacy and cost of each grazing strategy are defined in terms of the transient probabilities of the underlying stochastic processes, which are computed by means of Strang–Marchuk splitting techniques. Our model, calibrated with empirical data from Uriarte et al and Nasreen et al., regarding the seasonal presence of nematodes on pasture in temperate zones and anthelmintic efficacy, supports the use of dose-and-move strategies in temperate zones during summer and provides stochastic criteria for selecting the exact optimum time instant t 0 when these strategies should be applied. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessArticle On the Number of Periodic Inspections During Outbreaks of Discrete-Time Stochastic SIS Epidemic Models
Mathematics 2018, 6(8), 128; https://doi.org/10.3390/math6080128
Received: 4 July 2018 / Revised: 19 July 2018 / Accepted: 20 July 2018 / Published: 24 July 2018
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Abstract
This paper deals with an infective process of type SIS, taking place in a closed population of moderate size that is inspected periodically. Our aim is to study the number of inspections that find the epidemic process still in progress. As the underlying [...] Read more.
This paper deals with an infective process of type SIS, taking place in a closed population of moderate size that is inspected periodically. Our aim is to study the number of inspections that find the epidemic process still in progress. As the underlying mathematical model involves a discrete time Markov chain (DTMC) with a single absorbing state, the number of inspections in an outbreak is a first-passage time into this absorbing state. Cumulative probabilities are numerically determined from a recursive algorithm and expected values came from explicit expressions. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessArticle An M[X]/G(a,b)/1 Queueing System with Breakdown and Repair, Stand-By Server, Multiple Vacation and Control Policy on Request for Re-Service
Mathematics 2018, 6(6), 101; https://doi.org/10.3390/math6060101
Received: 29 March 2018 / Revised: 28 May 2018 / Accepted: 29 May 2018 / Published: 14 June 2018
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Abstract
In this paper, we discuss a non-Markovian batch arrival general bulk service single-server queueing system with server breakdown and repair, a stand-by server, multiple vacation and re-service. The main server’s regular service time, re-service time, vacation time and stand-by server’s service time are [...] Read more.
In this paper, we discuss a non-Markovian batch arrival general bulk service single-server queueing system with server breakdown and repair, a stand-by server, multiple vacation and re-service. The main server’s regular service time, re-service time, vacation time and stand-by server’s service time are followed by general distributions and breakdown and repair times of the main server with exponential distributions. There is a stand-by server which is employed during the period in which the regular server remains under repair. The probability generating function of the queue size at an arbitrary time and some performance measures of the system are derived. Extensive numerical results are also illustrated. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessFeature PaperArticle Convergence in Total Variation to a Mixture of Gaussian Laws
Mathematics 2018, 6(6), 99; https://doi.org/10.3390/math6060099
Received: 29 April 2018 / Revised: 1 June 2018 / Accepted: 5 June 2018 / Published: 11 June 2018
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Abstract
It is not unusual that XndistVZ where Xn, V, Z are real random variables, V is independent of Z and ZN(0,1). An intriguing feature [...] Read more.
It is not unusual that XndistVZ where Xn, V, Z are real random variables, V is independent of Z and ZN(0,1). An intriguing feature is that PVZA=EN(0,V2)(A) for each Borel set AR, namely, the probability distribution of the limit VZ is a mixture of centered Gaussian laws with (random) variance V2. In this paper, conditions for dTV(Xn,VZ)0 are given, where dTV(Xn,VZ) is the total variation distance between the probability distributions of Xn and VZ. To estimate the rate of convergence, a few upper bounds for dTV(Xn,VZ) are given as well. Special attention is paid to the following two cases: (i) Xn is a linear combination of the squares of Gaussian random variables; and (ii) Xn is related to the weighted quadratic variations of two independent Brownian motions. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
Open AccessArticle The Randomized First-Hitting Problem of Continuously Time-Changed Brownian Motion
Mathematics 2018, 6(6), 91; https://doi.org/10.3390/math6060091
Received: 4 April 2018 / Revised: 24 May 2018 / Accepted: 25 May 2018 / Published: 28 May 2018
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Abstract
Let X(t) be a continuously time-changed Brownian motion starting from a random position η,S(t) a given continuous, increasing boundary, with S(0)0,P(ηS(0) [...] Read more.
Let X ( t ) be a continuously time-changed Brownian motion starting from a random position η , S ( t ) a given continuous, increasing boundary, with S ( 0 ) 0 , P ( η S ( 0 ) ) = 1 , and F an assigned distribution function. We study the inverse first-passage time problem for X ( t ) , which consists in finding the distribution of η such that the first-passage time of X ( t ) below S ( t ) has distribution F , generalizing the results, valid in the case when S ( t ) is a straight line. Some explicit examples are reported. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
Open AccessFeature PaperArticle Some Notes about Inference for the Lognormal Diffusion Process with Exogenous Factors
Mathematics 2018, 6(5), 85; https://doi.org/10.3390/math6050085
Received: 16 April 2018 / Revised: 14 May 2018 / Accepted: 15 May 2018 / Published: 21 May 2018
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Abstract
Different versions of the lognormal diffusion process with exogenous factors have been used in recent years to model and study the behavior of phenomena following a given growth curve. In each case considered, the estimation of the model has been addressed, generally by [...] Read more.
Different versions of the lognormal diffusion process with exogenous factors have been used in recent years to model and study the behavior of phenomena following a given growth curve. In each case considered, the estimation of the model has been addressed, generally by maximum likelihood (ML), as has been the study of several characteristics associated with the type of curve considered. For this process, a unified version of the ML estimation problem is presented, including how to obtain estimation errors and asymptotic confidence intervals for parametric functions when no explicit expression is available for the estimators of the parameters of the model. The Gompertz-type diffusion process is used here to illustrate the application of the methodology. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessArticle A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation
Mathematics 2018, 6(5), 81; https://doi.org/10.3390/math6050081
Received: 6 April 2018 / Revised: 8 May 2018 / Accepted: 9 May 2018 / Published: 11 May 2018
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Abstract
We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by [...] Read more.
We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by a time-varying intensity function. We analyze the transient and the asymptotic behavior of the queueing system. Moreover, we derive a heavy-traffic approximation that allows approximating the state of the systems by a time-non-homogeneous Wiener process subject to jumps to a spurious state (due to catastrophes) and random returns to the zero state (due to repairs). Special attention is devoted to the case of periodic catastrophe and repair intensity functions. The first-passage-time problem through constant levels is also treated both for the queueing model and the approximating diffusion process. Finally, the goodness of the diffusive approximating procedure is discussed. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessFeature PaperArticle On the Bounds for a Two-Dimensional Birth-Death Process with Catastrophes
Mathematics 2018, 6(5), 80; https://doi.org/10.3390/math6050080
Received: 19 April 2018 / Revised: 8 May 2018 / Accepted: 8 May 2018 / Published: 11 May 2018
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Abstract
The model of a two-dimensional birth-death process with possible catastrophes is studied. The upper bounds on the rate of convergence in some weighted norms and the corresponding perturbation bounds are obtained. In addition, we consider the detailed description of two examples with 1-periodic [...] Read more.
The model of a two-dimensional birth-death process with possible catastrophes is studied. The upper bounds on the rate of convergence in some weighted norms and the corresponding perturbation bounds are obtained. In addition, we consider the detailed description of two examples with 1-periodic intensities and various types of death (service) rates. The bounds on the rate of convergence and the behavior of the corresponding mathematical expectations are obtained for each example. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessFeature PaperArticle On Short-Term Loan Interest Rate Models: A First Passage Time Approach
Mathematics 2018, 6(5), 70; https://doi.org/10.3390/math6050070
Received: 2 February 2018 / Revised: 23 April 2018 / Accepted: 25 April 2018 / Published: 3 May 2018
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Abstract
In this paper, we consider a stochastic diffusion process able to model the interest rate evolving with respect to time and propose a first passage time (FPT) approach through a boundary, defined as the “alert threshold”, in order to evaluate the risk of [...] Read more.
In this paper, we consider a stochastic diffusion process able to model the interest rate evolving with respect to time and propose a first passage time (FPT) approach through a boundary, defined as the “alert threshold”, in order to evaluate the risk of a proposed loan. Above this alert threshold, the rate is considered at the risk of usury, so new monetary policies have been adopted. Moreover, the mean FPT can be used as an indicator of the “goodness” of a loan; i.e., when an applicant is to choose between two loan offers, s/he will choose the one with a higher mean exit time from the alert boundary. An application to real data is considered by analyzing the Italian average effect global rate by means of two widely used models in finance, the Ornstein-Uhlenbeck (Vasicek) and Feller (Cox-Ingersoll-Ross) models. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessArticle On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes
Mathematics 2018, 6(4), 55; https://doi.org/10.3390/math6040055
Received: 23 March 2018 / Revised: 2 April 2018 / Accepted: 3 April 2018 / Published: 5 April 2018
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Abstract
We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case [...] Read more.
We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen–Loève expansion together with spectral asymptotics of differential operators and complex analysis methods. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
Open AccessArticle Large Deviation Results and Applications to the Generalized Cramér Model
Mathematics 2018, 6(4), 49; https://doi.org/10.3390/math6040049
Received: 2 March 2018 / Revised: 25 March 2018 / Accepted: 27 March 2018 / Published: 2 April 2018
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Abstract
In this paper, we prove large deviation results for some sequences of weighted sums of random variables. These sequences have applications to the probabilistic generalized Cramér model for products of primes in arithmetic progressions; they could lead to new conjectures concerning the (non-random) [...] Read more.
In this paper, we prove large deviation results for some sequences of weighted sums of random variables. These sequences have applications to the probabilistic generalized Cramér model for products of primes in arithmetic progressions; they could lead to new conjectures concerning the (non-random) set of products of primes in arithmetic progressions, a relevant topic in number theory. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
Open AccessFeature PaperArticle Forecast Combinations in the Presence of Structural Breaks: Evidence from U.S. Equity Markets
Mathematics 2018, 6(3), 34; https://doi.org/10.3390/math6030034
Received: 24 January 2018 / Revised: 13 February 2018 / Accepted: 20 February 2018 / Published: 1 March 2018
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Abstract
Realized volatility, building on the theory of a simple continuous time process, has recently received attention as a nonparametric ex-post estimate of the return variation. This paper addresses the problem of parameter instability due to the presence of structural breaks in realized volatility [...] Read more.
Realized volatility, building on the theory of a simple continuous time process, has recently received attention as a nonparametric ex-post estimate of the return variation. This paper addresses the problem of parameter instability due to the presence of structural breaks in realized volatility in the context of three HAR-type models. The analysis is conducted on four major U.S. equity indices. More specifically, a recursive testing methodology is performed to evaluate the null hypothesis of constant parameters, and then, the performance of several forecast combinations based on different weighting schemes is compared in an out-of-sample variance forecasting exercise. The main findings are the following: (i) the hypothesis of constant model parameters is rejected for all markets under consideration; (ii) in all cases, the recursive forecasting approach, which is appropriate in the absence of structural changes, is outperformed by forecast combination schemes; and (iii) weighting schemes that assign more weight in most recent observations are superior in the majority of cases. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Open AccessArticle Fusion Estimation from Multisensor Observations with Multiplicative Noises and Correlated Random Delays in Transmission
Mathematics 2017, 5(3), 45; https://doi.org/10.3390/math5030045
Received: 20 July 2017 / Revised: 23 August 2017 / Accepted: 29 August 2017 / Published: 4 September 2017
Cited by 1 | PDF Full-text (361 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the information fusion estimation problem is investigated for a class of multisensor linear systems affected by different kinds of stochastic uncertainties, using both the distributed and the centralized fusion methodologies. It is assumed that the measured outputs are perturbed by [...] Read more.
In this paper, the information fusion estimation problem is investigated for a class of multisensor linear systems affected by different kinds of stochastic uncertainties, using both the distributed and the centralized fusion methodologies. It is assumed that the measured outputs are perturbed by one-step autocorrelated and cross-correlated additive noises, and also stochastic uncertainties caused by multiplicative noises and randomly missing measurements in the sensor outputs are considered. At each sampling time, every sensor output is sent to a local processor and, due to some kind of transmission failures, one-step correlated random delays may occur. Using only covariance information, without requiring the evolution model of the signal process, a local least-squares (LS) filter based on the measurements received from each sensor is designed by an innovation approach. All these local filters are then fused to generate an optimal distributed fusion filter by a matrix-weighted linear combination, using the LS optimality criterion. Moreover, a recursive algorithm for the centralized fusion filter is also proposed and the accuracy of the proposed estimators, which is measured by the estimation error covariances, is analyzed by a simulation example. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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