semi-Markov Chain Research Articles

Copula based multivariate semi-Markov models with applications in high-frequency finance

We introduce a new multivariate model of multiple asset returns. Our model is based on weighted indexed semi-Markov chains to describe the single (marginals) asset returns, whereas the dependence structure among the considered assets is described by introducing copula functions. A real application of the proposed multivariate model is presented based on the evolution of 6 stocks from the Italian Stock Exchange. We provide empirical evidence that the model is able to correctly reproduce statistical regularities of multivariate real data such as the cross-correlation function, value-at-risk, marginal value-at-risk and conditional value-at-risk. The model is also used for volatility forecasting of each stock.

Comparative Study of Power Saving and Delay in LTE DRX, Directional-DRX and Hybrid-Directional DRX

With integration of advanced long term evolution networks (LTE networks) in mobile technology, 5G networks are introduced due to broadband spectrum extension which requires higher power consumption. Discontinuous reception (DRX) sleep mode mechanism is provided to accommodate lower power consumption of user equipment terminals (UEs). For mmWave directional communication air interface in LTE between UE and LTE eNB, UE performs beam searching and alignment with LTE eNB (4G base station) after every short/long DRX cycle during ON duration. Dual connectivity of UE to both LTE eNB and NR NodeB (5G New RAN base station) is a hybrid-directional system. This allows us to propose a sleep mode mechanism which is called hybrid directional-DRX (HD-DRX). In HD-DRX, UE performs beam searching and alignment with NR NodeB during ON duration only if data packet intimation. A semi-Markov chain process is introduced to describe UE state transition. Depending on this chain, power saving factor and average delay are calculated. HD-DRX power saving as compared to LTE DRX and Directional-DRX (D-DRX) is conducted. Due to different beam searching procedure, another comparison of average delay is assigned between D-DRX and HD-DRX. Data traffic model parameters are considered and numerical analysis is validated using MATLAB program simulations.

Integrative models for joint analysis of shoot growth and branching patterns.

Plants exhibit dependences between shoot growth and branching that generate highly structured patterns. The characterization of the patterning mechanism is still an open issue because of the developmental processes involved with both succession of events (e.g. internode elongation, axillary shoot initiation and elongation) and complex dependences among neighbouring positions along the parent shoot. Statistical models called semi-Markov switching partitioned conditional generalized linear models were built on the basis of apple and pear tree datasets. In these models, the semi-Markov chain represents both the succession and lengths of branching zones, whereas the partitioned conditional generalized linear models represent the influence of parent shoot growth variables on axillary productions within each branching zone. Parent shoot growth variables were shown to influence specific developmental events. On this basis, the growth and branching patterns of two apple tree (Malus domestica) cultivars, as well as of pear trees (Pyrus spinosa) between two successive growing cycles, were compared. The proposed integrative statistical models were able to decipher the roles of successive developmental events in the growth and branching patterning mechanisms. These models could incorporate other parent shoot explanatory variables, such as the local curvature or the maximum growth rate of the leaf.

MODELING THE SEISMICITY OF CENTRAL IONIAN ISLANDS WITH SEMI-MARKOV MODELS

Earthquakes with M ≥ 5.2 that occurred in the area of central Ionian Islands (1911- 2014) are assumed to form a semi-Markov chain, aiming to contribute in the field of seismic hazard assessment. The sojourn times are considered to be geometric or approximated by Pareto distributions. Destination probabilities are examined and the results demonstrate that in many cases these probabilities become higher adequately forecasting the magnitude class of an anticipated earthquake. The geometrically distributed model can also reveal the more probable occurrence time of the next earthquake since for this model the destination probabilities were found to obtain many times their maximum values for the real occurrence time. The successful forecasting as for the occurrence time is 63.75% for all earthquakes and becomes 71.42% for the larger magnitude events (M ≥ 6.0).

A semi-Markov chain approach to modeling respiratory patterns prior to extubation in preterm infants.

After birth, extremely preterm infants often require specialized respiratory management in the form of invasive mechanical ventilation (IMV). Protracted IMV is associated with detrimental outcomes and morbidities. Premature extubation, on the other hand, would necessitate reintubation which is risky, technically challenging and could further lead to lung injury or disease. We present an approach to modeling respiratory patterns of infants who succeeded extubation and those who required reintubation which relies on Markov models. We compare the use of traditional Markov chains to semi-Markov models which emphasize cross-pattern transitions and timing information, and to multi-chain Markov models which can concisely represent non-stationarity in respiratory behavior over time. The models we developed expose specific, unique similarities as well as vital differences between the two populations.

Modeling Temporal Dominance of Sensations with semi-Markov chains

Temporal Dominance of Sensations (TDS) data are usually represented by TDS curves of dominance rates and analyzed by linear models of dominance durations. Such approaches do not properly take into account the fact that the selection of a new dominant attribute likely depends on the current dominant attribute.Thus, modeling TDS data with a stochastic process seems natural, as recently proposed by Franczak et al. (2015) who used discrete time Markov chains. This approach gives the probabilities of transition from one dominant attribute to another. However Markov chains present some limitations when applied to TDS data. As an alternative, this paper considers semi-Markov chains (SMC), a generalization of Markov chains, which allow the duration of the dominant attribute to be distributed arbitrarily. Because probabilities of transition from one attribute to another one can also depend on time, SMC are applied on sequences split into time periods with specific durations, with one model per time period.Graphs built upon this stochastic pattern can be plotted to represent chronological main transitions between attributes. Contrarily to the TDS curves which summarize a mean panel overview, these graphs can be interpreted as individual’s most probable paths and contribute to a better understanding of consumer perception.

Multi-state models for evaluating conversion options in life insurance

In this paper we propose a multi-state model for the evaluation of the conversion option contract. The multi-state model is based on age-indexed semi-Markov chains that are able to reproduce many important aspects that influence the valuation of the option such as the duration problem, the time non-homogeneity and the ageing effect. The value of the conversion option is evaluated after the formal description of this contract.

Analysis of bearing and branching behavior of two apricot cultivars

The nature of axillary production along shoots is one of the main factors influencing both vegetative and reproductive growth in apricot. This study aimed at analyzing the morphology and branching pattern of one-year-old shoots of 'Velkopavlovicka' cultivar in comparison with 'Bergeron'. 'Velkopavlovicka' is a classical, medium vigor, Czech cultivar which has naturally a spherical crown. In twelve-year-old trees grafted on apricot seedlings and pruned to an open vase, lateral branches were described from the second to the fourth orders. Each shoot was described as a sequence of metamers, and for each metamer the fate (latent, vegetative or floral) of the central bud as well as the number of associated flowers were noted. The presence or absence of development of axillary shoots from vegetative buds was also considered. A hidden semi-Markov chain was built for each cultivar on the basis of the sample of bivariate sequences corresponding to the two observed categorical variables, pooling all one-year-old annual shoots whatever the year. Both cultivars could be characterized by a similar organization in zones, but with different probabilities of occurrence of the median flower bud zone as well as by different mean number of central and associated flowers within this zone. Cultivar 'Bergeron' proved to be more floribund than 'Velkopavlovicka'. Moreover, they revealed biennial bearing in Central European conditions, but different reasons could be responsible for this behavior depending on the cultivar.

Combining 3D digitizing data and architectural modelling to simulate shoot growth and geometry in apple

Various models have been developed to predict tree growth and architecture over years, but few of them are capable of simulating 3D geometry development and shoot typology from a pre-existing crown geometry. In this study, we developed an approach for simulating 3D apple tree structure by combining 3D digitizing data with topological rules adapted from MappleT, an existing architectural model. In this modelling approach, bud fates were simulated with a first and second order Markov chain for terminal successions and using a hidden semi-Markov chain for the branching pattern on the one-year-old shoots. In these Markov models, five bud/shoot types were considered (small, medium, long, floral and dormant). Parametric laws and allometric relationships for each shoot type were used to simulate shoot length and leaf area. Shoot spatial position was set according to branching angle and phyllotaxis of the simulated cultivar. One nine-year-old apple tree (Malus ×domestica 'Jubile®') grown in 110L pots was digitized in 2013 and 2014. The model was calibrated on data recorded in 2012 and 2013. The calibration allowed (i) deriving distribution parameters of shoot length according to their types; (ii) establishing allometric relationships between the internode number, leaf area and the total shoot length; and (iii) evaluating the parameters of the Markov and semi-hidden Markov chains. To validate the model, simulations were performed using the tree digitized in 2013 as inputs of the model. The results were compared to the same tree digitized in 2014, comparing within tree shoot demography and the shoot leaf areas according to their type and length. Using this approach we were able to simulate tree growth on year n+1 from observed architectures at the adult stage. This opens the way for the optimization of management practices on observed trees in orchards conditions.

Study of the seismic activity in central Ionian Islands via semi-Markov modelling

The seismicity of the central Ionian Islands (M ≥ 5.2, 1911–2014) is studied via a semi-Markov chain which is investigated in terms of the destination probabilities (occurrence probabilities). The interevent times are considered to follow geometric (in which case the semi-Markov model reduces to a Markov model) or Pareto distributions. The study of the destination probabilities is useful for forecasting purposes because they can provide the more probable earthquake magnitude and occurrence time. Using the first half of the data sample for the estimation procedure and the other half for forecasting purposes it is found that the time windows obtained by the destination probabilities include 72.9% of the observed earthquake occurrence times (for all magnitudes) and 71.4% for the larger (M ≥ 6.0) ones.

Opportunistic Capacity-Based Resource Allocation for Chunk-Based Multi-Carrier Cognitive Radio Sensor Networks.

The spectrum allocation for cognitive radio sensor networks (CRSNs) has received considerable research attention under the assumption that the spectrum environment is static. However, in practice, the spectrum environment varies over time due to primary user/secondary user (PU/SU) activity and mobility, resulting in time-varied spectrum resources. This paper studies resource allocation for chunk-based multi-carrier CRSNs with time-varied spectrum resources. We present a novel opportunistic capacity model through a continuous time semi-Markov chain (CTSMC) to describe the time-varied spectrum resources of chunks and, based on this, a joint power and chunk allocation model by considering the opportunistically available capacity of chunks is proposed. To reduce the computational complexity, we split this model into two sub-problems and solve them via the Lagrangian dual method. Simulation results illustrate that the proposed opportunistic capacity-based resource allocation algorithm can achieve better performance compared with traditional algorithms when the spectrum environment is time-varied.

Distribution and availability for aggregated second-order semi-Markov ternary system with working time omission

Discrete repairable degradation systems can be modeled by a homogeneous discrete-time second-order semi-Markov chain with finite state space for theoretical and practical sake. In the present paper, the state space can be divided into three subclasses, namely excellent states, good states (middle states) and failure states. The transitions among states may be driven by degradations and inside shocks of the system, repair actions, self-healings, energy supplement and other recovery ways. The Z-transform is used to give distributions for some interesting problems such as distributions of an I-period (oscillation & working time) and a gap between I-periods, and instantaneous and steady availabilities of the system. Finally, some numerical examples are given to illustrate the results obtained in the paper. The work may be used in reliability and maintenance analysis of discrete time repairable systems.

Enhanced uplink multi-users scheduling for future 802.11ax networks: wait-to-pick-as-available enhanced

One of the key enablers of the upcoming IEEE 802.11ax standard is the inclusion of uplink multi-users (UL MU) transmission model, which is performed using multiple-input multiple-output (MIMO) and beamforming techniques. So far, UL MU-MIMO has not been standardized in any of IEEE 802.11 amendments, because of technical issues facing its definition. One of these significant issues is the scheduling of UL MU-MIMO transmissions, which refers to selection rules and resource allocation procedures performed before simultaneous data transmissions. Indeed, simultaneous transmitters and receiver should exchange some information to correctly transmit the parallel data frames. However, this information exchange adds overheads, leading to reduce network performance. In this regard, we have proposed, in a previous work, a novel 802.11ax medium access control protocol aiming at reducing elapsed time in managing the establishment of an UL -MU communication called Wait-to-Pick-As-Available (W2PAA). Our contribution in this paper is twofold. First, we introduce an analytical model based on semi-Markov Chains to evaluate the performance of W2PAA; taking into a detailed behavior of the backoff counter (i.e. freezing period). Second, we propose an enhanced version of W2PAA aiming at improving both system and user-oriented performances. Aiming at validating the analytical model and comparing the performance of both versions of W2PAA, we used computer simulation. Obtained results, validate the analytical model in one hand, and clearly indicated the gain of both W2PAA versions by report to the basic UL-Single User on the other hand. Copyright © 2016 John Wiley & Sons, Ltd.

The Quality of Life Via Semi Markov Reward Modelling

The quality of life is recognized as an important element which is generally treated at least as primary or secondary criterion in most clinical trials. So, its measurement and statistical analysis remain an issue. In the present, a non-homogeneous semi Markov model is defined for the description of the evolution of a patient’s health status. Also, a stochastic process where the state space elements are the quality of life states of a patient and the transition probabilities are dependent on the health status is considered. A methodology via Markov modelling in order to combine the health and the quality of life status of a patient is provided. Next, we extend the above model by attaching rewards of making transitions between the states of the two kinds for indexing through rewards the quality of life for patients of all health states. The mean, the second non central moment, the variance and the high order moments of a patients interval reward are calculated by means of the basic parameters of the semi Markov chain and the quality status process. The above results provide us with a tool which could operate as a measure for indicating the quality of life status. Finally, the theoretical results are illustrated numerically with synthesized data.

On the limit distribution of a second-order semi-Markov chain in state and duration

ABSTRACTIn this paper, we derive conditions under which second-order semi-Markov chains in state and duration admits a limit distribution.

Reliability of non-repairable systems with cyclic-mission switching and multimode failure components

This paper considers a mission-based non-repairable system which has multiple and cyclic mission sequences, and whose components have multimode failures. For example, the valve system with two failure modes cannot only fail to close when required to close but can also fail to open when required to open. A discrete-time semi-Markov chain within the time evolution framework is constructed, which is more properly to model the realistic problems because the sojourn times between the successive missions may be not satisfied Markov property. Using the aggregated stochastic theory and Z-transform, the explicit form of the solution of system reliability is obtained, which can simplify calculations and get some distributions of some sojourn times that the readers relish, including perfect and imperfect operations. In addition, two special valve systems with two-mission and three-mission switching are provided respectively. Finally, numerical examples are presented to illustrate the proposed model and the method.

A Novel Replication Model with Enhanced Data Availability in P2P Platforms

With the development of P2P technology, more and more distributed systems are applying it for deploying large-scale storage infrastructure. However, maintaining desirable data availability in P2P environments is still an opening issue because of the vulnerability of participated peers. In this paper, a novel replication model is proposed to improving the data availability of P2P storage platforms. In the proposed replication model, the probability of peer’s failure is estimated by semi-Markov chain, which enables us to significantly improving the prediction accuracy of peer’s failure in a given period. Massive experiments are conducted in a real-world P2P platform to examining the performance of the proposed replication model, and the results indicate that it can achieve better tradeoffs between performance and data availability comparing with other replication schemes.

First hitting probabilities for semi-Markov chains and estimation

ABSTRACTWe first consider a stochastic system described by an absorbing semi-Markov chain (SMC) with finite state space, and we introduce the absorption probability to a class of recurrent states. Afterwards, we study the first hitting probability to a subset of states for an irreducible SMC. In the latter case, a non-parametric estimator for the first hitting probability is proposed and the asymptotic properties of strong consistency and asymptotic normality are proven. Finally, a numerical application on a five-state system is presented to illustrate the performance of this estimator.

Controlled Semi-Markov Chains with Risk-Sensitive Average Cost Criterion

This work concerns with semi-Markov decision chains on a finite state space. Assuming that the controller has a constant and positive risk-sensitive coefficient, an optimality equation for the corresponding (long-run) risk-sensitive average cost index is formulated and, under suitable continuity-compactness conditions, it is shown that a solution of such an equation determines the optimal average cost, as well as an optimal stationary policy. Additionally, if the underlying Markov chain is communicating, then it is proved that the optimality equation has a solution.

Reducing the energy consumption at the user equipment through multi-stream cross-carrier-aware discontinuous reception (DRX) in LTE-advanced systems

In order to minimize the amount of energy consumption at the user equipment (UE) level, the 3rd Generation Partnership Project (3GPP) presented in the Long Term Evolution (LTE) an approach called discontinuous reception (DRX). Nevertheless, existing models for the LTE DRX and their extension to scenarios that support carrier aggregation (CA) and multi-stream carrier aggregation (MSCA) have several drawbacks. In this paper, we utilize a semi-Markov Chain to model the operation of the LTE DRX and characterize its performance metrics. Then, we exploit the new features introduced in LTE Advanced (LTE-A) to develop a novel cross-carrier-aware DRX for scenarios that support CA and MSCA since the energy consumption in such scenarios can be significantly higher and existing techniques simply reapplied the traditional DRX scheme. We present a detailed examination of our DRX solution along with the analytical expressions of its performance metrics. The accuracy of our modeling approach for both the classical and our novel LTE DRX is validated through extensive simulations across a wide range of parameters. We evaluate the performance of our cross-carrier-aware DRX solution and show that it can significantly outperform the classical one, especially under a low tolerable delay. In addition, we also show the effects of implementation-dependent power levels on the performance of our cross-carrier-aware DRX.