Probability Density Function Of Time Research Articles

Substorm Occurrence and Intensity Associated With Three Types of Solar Wind Structure

AbstractThis paper presents the results of a study of the characteristics of substorms that occurred during three distinct types of solar wind: coronal mass ejection (CME) associated, high‐speed streams (HSS), and slow solar wind (SSW). A total number of 53,468 geomagnetic substorm onsets from 1983 to 2009 is used and sorted by the three solar wind types. It is found that the probability density function (PDF) of the intersubstorm time can be fitted by the combination of a dominant power law with an exponential cutoff component and a minor lognormal component, implying that substorms are associated with two distinctly different dynamical processes corresponding, perhaps, to the “externally driven” and “internally driven” processes, respectively. We compare substorm frequency and intensity associated with the three types of solar wind. It is found that the intersubstorm time is the longest during SSW and shortest during CME intervals. The averaged intersubstorm time for the internally driven substorms is 3.13, 3.15, and 7.96 h for CME, HSS, and SSW, respectively. The substorm intensity PDFs, as represented by the peak value of |SML| (the generalization of AL), can be fitted by two lognormal distribution functions. The averaged substorm intensity for either component is largest for CME (292 and 674 nT) and smallest for SSW (265 and 434 nT). We argue that the externally driven substorms are more intense than those driven internally. We conclude that the dynamical process of substorms is controlled mainly by the direct solar wind‐magnetosphere coupling, whereas the internally driven process only plays a very modest minor role.

2014 Mainshock-Aftershock Activity Versus Earthquake Swarms in West Bohemia, Czech Republic

A singular sequence of three episodes of $$M_{L}$$ 3.5, 4.4 and 3.6 mainshock-aftershock occurred in the West Bohemia/Vogtland earthquake-swarm region during 2014. We analysed this activity using the WEBNET data and compared it with the swarms of 1997, 2000, 2008 and 2011 from the perspective of cumulative seismic moment, statistical characteristics, space-time distribution of events, and prevailing focal mechanisms. For this purpose, we improved the scaling relation between seismic moment $$M_{0}$$ and local magnitude $$M_{L}$$ by WEBNET. The total seismic moment released during 2014 episodes ( $$M_{0\mathrm{tot}}\approx 1.58\times 10^{15}$$ Nm) corresponded to a single $$M_{L}4.6+$$ event and was comparable to $$M_{0\mathrm{tot}}$$ of the swarms of 2000, 2008 and 2011. We inferred that the $$M_{L}4.8$$ earthquake is the maximum expected event in Nový Kostel (NK), the main focal zone. Despite the different character of the 2014 sequence and the earthquake swarms, the magnitude-frequency distributions (MFDs) show the b-values $$\approx$$ 1 and probability density functions (PDFs) of the interevent times indicate the similar event rate of the individual swarms and 2014 activity. Only the a-value (event-productivity) in the MFD of the 2014 sequence is significantly lower than those of the swarms. A notable finding is a significant acceleration of the seismic moment release in each subsequent activity starting from the 2000 swarm to the 2014 sequence, which may indicate an alteration from the swarm-like to the mainshocks-aftershock character of the seismicity. The three mainshocks are located on a newly activated fault segment/asperity (D in out notation) of the NK zone situated in the transition area among fault segments A, B, C, which hosted the 2000, 2008 and 2011 swarms. The segment D appears to be predisposed to an oblique-thrust faulting while strike-slip faulting is typical of segments A, B and C. In conclusion, we propose a basic segment scheme of the NK zone which should be improved gradually.

Predicting population extinction from early observations of the Lotka–Volterra system

Population extinction is one of the central themes in population biology. We propose a statistical algorithm for long-term prediction of an extinction event in the paradigmatic predator–prey model. The algorithm is based on noisy and sporadic observations of the Lotka–Volterra (LV) system at the early stages of its evolution, when the system is still very far from extinction. There are two stages in the algorithm: first, the unknown parameters (reaction rates) of the LV system are estimated using the Approximate Bayesian Computation method; then an analytical expression for the time-scale of extinction (which involves the estimated parameters) is applied to compute the probability density function of extinction time. The proposed algorithm is validated by numerical simulations for the case of a stochastic LV system specified by the birth–death rate equations. The algorithm can be seen as an initial step in the quest for long-term prediction of rare “catastrophic” events in complex stochastic dynamic systems (epidemics, host-parasite dynamics, enzyme kinetics, dynamic trading, etc.).

Micromagnetic Analysis of Statistical Switching in Perpendicular STT-MRAM With Interfacial Dzyaloshinskii–Moriya Interaction

The technological implementation of STT-magnetic random-access memory (MRAM) in real devices needs a complete description of the statistical switching behavior at room temperature. In this paper, we investigate, by means of a full micromagnetic model, the effect of the interfacial Dzyaloshinskii-Moriya interaction (IDMI) on the critical current density and probability density function (PDF) of the switching time in perpendicular STT-MRAMs. We show that for large enough values of the symmetric exchange interaction, the negative effect of the IDMI on the critical current is strongly reduced. In addition, a comprehensive analysis of the four main statistical moments (mean, standard deviation, skewness, and kurtosis) points out that to fit the switching time PDF in order to maintain a quadratic error at least one order of magnitude smaller than skew normal PDF, a Pearson type IV PDF has to be used also in presence of the IDMI.

Non-Poisson renewal events and memory.

We study two different forms of fluctuation-dissipation processes generating anomalous relaxations to equilibrium of an initial out-of-equilibrium condition, the former being based on a stationary although very slow correlation function and the latter characterized by the occurrence of crucial events, namely, non-Poisson renewal events, incompatible with the stationary condition. Both forms of regression to equilibrium have the same nonexponential Mittag-Leffler structure. We analyze the single trajectories of the two processes by recording the time distances between two consecutive origin recrossings and establishing the corresponding waiting time probability density function (PDF), ψ(t). In the former case, with no crucial events, ψ(t) is an exponential, and in the latter case, with crucial events, ψ(t) is an inverse power law PDF with a diverging first moment. We discuss the consequences that this result is expected to have for the correct interpretation of some anomalous relaxation processes.

Analysis of the forward and backward in time pair-separation probability density functions for inertial particles in isotropic turbulence

In this paper we investigate, using theory and direct numerical simulations (DNS), the forward in time (FIT) and backward in time (BIT) probability density functions (PDFs) of the separation of inertial particle pairs in isotropic turbulence. In agreement with our earlier study (Bragg et al., Phys. Fluids, vol. 28, 2016, 013305), where we compared the FIT and BIT mean-square separations, we find that inertial particles separate much faster BIT than FIT, with the strength of the irreversibility depending upon the final/initial separation of the particle pair and their Stokes number $St$. However, we also find that the irreversibility shows up in subtle ways in the behaviour of the full PDF that it does not in the mean-square separation. In the theory, we derive new predictions, including a prediction for the BIT/FIT PDF for $St\geqslant O(1)$, and for final/initial separations in the dissipation regime. The prediction shows how caustics in the particle relative velocities in the dissipation range affect the scaling of the pair-separation PDF, leading to a PDF with an algebraically decaying tail. The predicted functional behaviour of the PDFs is universal, in that it does not depend upon the level of intermittency in the underlying turbulence. We also analyse the pair-separation PDFs for fluid particles at short times, and construct theoretical predictions using the multifractal formalism to describe the fluid relative velocity distributions. The theoretical and numerical results both suggest that the extreme events in the inertial particle-pair dispersion at the small scales are dominated by their non-local interaction with the turbulent velocity field, rather than due to the strong dissipation range intermittency of the turbulence itself. In fact, our theoretical results predict that for final/initial separations in the dissipation range, when $St\gtrsim 1$, the tails of the pair-separation PDFs decay faster as the Taylor Reynolds number $Re_{\unicode[STIX]{x1D706}}$ is increased, the opposite of what would be expected for fluid particles.

Modeling service time reliability in urban ferry system

The urban ferry system can carry a large number of travelers, which may alleviate the pressure on road traffic. As an indicator of its service quality, service time reliability (STR) plays an essential part in attracting travelers to the ferry system. A wide array of studies have been conducted to analyze the STR of land transportation. However, the STR of ferry systems has received little attention in the transportation literature. In this study, a model was established to obtain the STR in urban ferry systems. First, the probability density function (PDF) of the service time provided by ferry systems was constructed. Considering the deficiency of the queuing theory, this PDF was determined by Bayes’ theorem. Then, to validate the function, the results of the proposed model were compared with those of the Monte Carlo simulation. With the PDF, the reliability could be determined mathematically by integration. Results showed how the factors including the frequency, capacity, time schedule and ferry waiting time affected the STR under different degrees of congestion in ferry systems. Based on these results, some strategies for improving the STR were proposed. These findings are of great significance to increasing the share of ferries among various urban transport modes.

Continuous-time random-walk model for anomalous diffusion in expanding media

Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "big crunch" effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium. From this hierarchy, the full time evolution of the second-order moment is obtained for some specific types of expansion. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained, whence the long-time behavior of moments of arbitrary order is subsequently inferred. Our analytical and numerical results for both Lévy flights and subdiffusive CTRWs confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called "Lévy horizon."

Load variation analysis by bus of distribution systems based on PEVs charging modeling

This paper presents an algorithm that evaluated the load variation analysis influence by bus upon the distribution system by calculating the daily load curve of PEVs charging by bus based on the daily charging patterns of PEVs according to PEVs penetration scenarios. Up to now, PEVs clustering methodology was simply considered to calculate number of PEVs per household by applying the Stochastic model, thereby failing to reflect charging characteristics by battery type, charging time and number of PEVs by bus of the distribution systems. To overcome these problems, the proposed algorithm calculates the number of PEVs to estimate the number of households by bus; the probability density function of the charging start time of PEVs, considering driving characteristics of PEVs; and the daily load curve of PEVs charging by bus considering battery characteristics according to PEVs penetration scenarios. The algorithm then adds the calculated daily load curve of PEVs charging and the conventional daily load curve to evaluate the load variation influence on the target bus according to PEVs penetration scenarios. To verify the evaluation of the load variation influence by bus on the distribution system in terms of the proposed algorithm, the cases were reviewed on the target bus (apartments, detached houses #1 and detached houses #3) among the feeders of the distribution systems at Dongtan new town in South Korea.

Stability and Stabilization of Semi-Markov Jump Linear Systems With Exponentially Modulated Periodic Distributions of Sojourn Time

This paper is concerned with a class of discrete-time semi-Markov jump linear systems (S-MJLSs) subject to exponentially modulated periodic (EMP) probability density function (PDF) of sojourn time, and the problems of stability and stabilization are addressed. Setting a relatively large period, the considered systems are capable of approximating the general S-MJLSs (without any requirements on sojourn-time PDFs) for which numerically testable stability and stabilization conditions are rather difficult to obtain. Necessary and sufficient criterion for mean square stability of the general S-MJLSs is first derived, which involves an infinite number of conditions and as such not checkable. However, the developments lay a foundation to further establish the numerically testable conditions for the systems when the PDF of sojourn time is EMP albeit the sojourn time can tend to infinity. The derivations explicitly depend on the PDF of sojourn time, which circumvents the difficulty in obtaining the memory transition probabilities of S-MJLSs. The adopted Lyapunov function is sojourn-time-dependent (STD), by which the existence conditions of STD controller are developed as well using certain techniques that can eliminate the terms of power of matrices in the stability conditions. Two illustrative examples including a class of population ecological systems are presented to show the validity and applicability of the developed theoretical results.

Identify source location and release time for pollutants undergoing super-diffusion and decay: Parameter analysis and model evaluation

Backward models have been applied for four decades by hydrologists to identify the source of pollutants undergoing Fickian diffusion, while analytical tools are not available for source identification of super-diffusive pollutants undergoing decay. This technical note evaluates analytical solutions for the source location and release time of a decaying contaminant undergoing super-diffusion using backward probability density functions (PDFs), where the forward model is the space fractional advection-dispersion equation with decay. Revisit of the well-known MADE-2 tracer test using parameter analysis shows that the peak backward location PDF can predict the tritium source location, while the peak backward travel time PDF underestimates the tracer release time due to the early arrival of tracer particles at the detection well in the maximally skewed, super-diffusive transport. In addition, the first-order decay adds additional skewness toward earlier arrival times in backward travel time PDFs, resulting in a younger release time, although this impact is minimized at the MADE-2 site due to tritium’s half-life being relatively longer than the monitoring period. The main conclusion is that, while non-trivial backward techniques are required to identify pollutant source location, the pollutant release time can and should be directly estimated given the speed of the peak resident concentration for super-diffusive pollutants with or without decay.

Theory of earthquakes interevent times applied to financial markets

We analyze the probability density function (PDF) of waiting times between financial loss exceedances. The empirical PDFs are fitted with the self-excited Hawkes conditional Poisson process with a long power law memory kernel. The Hawkes process is the simplest extension of the Poisson process that takes into account how past events influence the occurrence of future events. By analyzing the empirical data for 15 different financial assets, we show that the formalism of the Hawkes process used for earthquakes can successfully model the PDF of interevent times between successive market losses.

H∞ fuzzy control of semi-Markov jump nonlinear systems under σ-error mean square stability

ABSTRACTThis paper is concerned with the problem of time-varying H∞ fuzzy control for a class of semi-Markov jump nonlinear systems in the sense of σ-error mean square stability. The nonlinear plant is described via the Takagi–Sugeno fuzzy model. By defining a time-varying mode-dependent Lyapunov function, a set of sufficient stability and stabilisation criteria for non-disturbance case is first derived and then applied to the investigation of H∞ performance analysis and H∞ fuzzy controller design problems of semi-Markov jump nonlinear systems. Different from the traditional stochastic switching system framework, the probability density function of sojourn time is exploited to circumvent the complex computation of transition probabilities. The derived conditions can cover the time-invariant mode-dependent and time-invariant mode-independent H∞ fuzzy control schemes as special cases. A classic cart-pendulum system is presented to demonstrate the effectiveness and advantages of the proposed theoretical results.

Quantum resonant activation.

Quantum resonant activation is investigated for the archetype setup of an externally driven two-state (spin-boson) system subjected to strong dissipation by means of both analytical and extensive numerical calculations. The phenomenon of resonant activation emerges in the presence of either randomly fluctuating or deterministic periodically varying driving fields. Addressing the incoherent regime, a characteristic minimum emerges in the mean first passage time to reach an absorbing neighboring state whenever the intrinsic time scale of the modulation matches the characteristic time scale of the system dynamics. For the case of deterministic periodic driving, the first passage time probability density function (pdf) displays a complex, multipeaked behavior, which depends crucially on the details of initial phase, frequency, and strength of the driving. As an interesting feature we find that the mean first passage time enters the resonant activation regime at a critical frequency ν^{*} which depends very weakly on the strength of the driving. Moreover, we provide the relation between the first passage time pdf and the statistics of residence times.

The residence time of water in the atmosphere revisited

Abstract. This paper revisits the knowledge on the residence time of water in the atmosphere. Based on state-of-the-art data of the hydrological cycle we derive a global average residence time of 8.9 ± 0.4 days (uncertainty given as 1 standard deviation). We use two different atmospheric moisture tracking models (WAM-2layers and 3D-T) to obtain atmospheric residence time characteristics in time and space. The tracking models estimate the global average residence time to be around 8.5 days based on ERA-Interim data. We conclude that the statement of a recent study that the global average residence time of water in the atmosphere is 4–5 days, is not correct. We derive spatial maps of residence time, attributed to evaporation and precipitation, and age of atmospheric water, showing that there are different ways of looking at temporal characteristics of atmospheric water. Longer evaporation residence times often indicate larger distances towards areas of high precipitation. From our analysis we find that the residence time over the ocean is about 2 days less than over land. It can be seen that in winter, the age of atmospheric moisture tends to be much lower than in summer. In the Northern Hemisphere, due to the contrast in ocean-to-land temperature and associated evaporation rates, the age of atmospheric moisture increases following atmospheric moisture flow inland in winter, and decreases in summer. Looking at the probability density functions of atmospheric residence time for precipitation and evaporation, we find long-tailed distributions with the median around 5 days. Overall, our research confirms the 8–10-day traditional estimate for the global mean residence time of atmospheric water, and our research contributes to a more complete view of the characteristics of the turnover of water in the atmosphere in time and space.

Exact distributions of cover times for N independent random walkers in one dimension.

We study the probability density function (PDF) of the cover time t_{c} of a finite interval of size L by N independent one-dimensional Brownian motions, each with diffusion constant D. The cover time t_{c} is the minimum time needed such that each point of the entire interval is visited by at least one of the N walkers. We derive exact results for the full PDF of t_{c} for arbitrary N≥1 for both reflecting and periodic boundary conditions. The PDFs depend explicitly on N and on the boundary conditions. In the limit of large N, we show that t_{c} approaches its average value of 〈t_{c}〉≈L^{2}/(16DlnN) with fluctuations vanishing as 1/(lnN)^{2}. We also compute the centered and scaled limiting distributions for large N for both boundary conditions and show that they are given by nontrivial N independent scaling functions.

Stability and Stabilization of a Class of Discrete-Time Fuzzy Systems With Semi-Markov Stochastic Uncertainties

This paper addresses the problems of stability and stabilization for a class of discrete-time Takagi–Sugeno fuzzy systems with semi-Markov stochastic uncertainties. By means of the semi-Markov kernel, the probability density function of the sojourn time for different modes in describing the underlying semi-Markov stochastic uncertainties can be mode-dependent in contrast with the previous studies. Both the sojourn-time-independent and sojourn-time-dependent Lyapunov functions are proposed, by which the stability criteria are obtained in terms of a new $\boldsymbol {\sigma }$ -error mean square stability concept, and the latter is demonstrated to be less conservative and more practical than some existing results. Then, control synthesis problem is investigated and the sufficient conditions on the existence of admissible mode-dependent state-feedback stabilizing controller are developed. A numerical example and a cart-pendulum system are given to show the effectiveness and potential of the new design techniques.

Two-boundary first exit time of Gauss-Markov processes for stochastic modeling of acto-myosin dynamics.

We consider a stochastic differential equation in a strip, with coefficients suitably chosen to describe the acto-myosin interaction subject to time-varying forces. By simulating trajectories of the stochastic dynamics via an Euler discretization-based algorithm, we fit experimental data and determine the values of involved parameters. The steps of the myosin are represented by the exit events from the strip. Motivated by these results, we propose a specific stochastic model based on the corresponding time-inhomogeneous Gauss-Markov and diffusion process evolving between two absorbing boundaries. We specify the mean and covariance functions of the stochastic modeling process taking into account time-dependent forces including the effect of an external load. We accurately determine the probability density function (pdf) of the first exit time (FET) from the strip by solving a system of two non singular second-type Volterra integral equations via a numerical quadrature. We provide numerical estimations of the mean of FET as approximations of the dwell-time of the proteins dynamics. The percentage of backward steps is given in agreement to experimental data. Numerical and simulation results are compared and discussed.

Sediment sound speed inversion with time-frequency analysis and modal arrival time probability density functions.

The dispersion pattern of a received signal is critical for understanding physical properties of the propagation medium. The objective of this work is to estimate accurately sediment sound speed using modal arrival times obtained from dispersion curves extracted via time-frequency analysis of acoustic signals. A particle filter is used that estimates probability density functions of modal frequencies arriving at specific times. Employing this information, probability density functions of arrival times for modal frequencies are constructed. Samples of arrival time differences are then obtained and are propagated backwards through an inverse acoustic model. As a result, probability density functions of sediment sound speed are estimated. Maximum a posteriori estimates indicate that inversion is successful. It is also demonstrated that multiple frequency processing offers an advantage over inversion at a single frequency, producing results with reduced variance.

Lightweight TRNG Based on Multiphase Timing of Bistables

The paper presents a concept of a True Random Number Generator (TRNG) that utilizes phase noise of a pair of ring oscillators (ROs) to increase the variance of the initial condition of a bistable. For this purpose a special TRNG D-latch architecture (TDL) has been proposed, which can either operate in the oscillatory ring-oscillator mode or the nearly-metastable mode. The RO mode increases the probability of the nearly metastable operation of the TDL, which in turn increases the mean value and variance of the resolve time. Moreover, due to the oscillatory nature of the TDL metastability, the resolve time can be easily measured and used for further randomness harvesting in the TRNG. The proposed TRNG uses a pair of TDLs to reduce sensitivity to process or temperature variation, whereas TDLs' individual resolve time, their resolve time difference and logical state contribute to randomness. In the article, the impact of the devices' imbalance (tolerance) resulting from process variation on the resultant entropy is also explored. The proposed TRNG based on TDLs is scalable and has been implemented in modern CPLD and FPGA devices whereas the bit rate reached up to 1 Mbit. The article also discusses theoretical issues related to a transformation of phase and resolve time probability density functions and their influence on TRNG parameters.