Semi-Markov Reliability Models with Recurrence Times and Credit Rating Applications

AbstractThis study proposes a non-homogeneous continuous-time Markov regenerative process with recurrence times, in particular, forward and backward recurrence processes. We obtain the transient solution of the process in the form of a generalized Markov renewal equation. A distinguishing feature is that Markov and semi-Markov processes result as special cases of the proposed model. To model the credit rating dynamics to demonstrate its applicability, we apply the proposed stochastic process to Standard and Poor’s rating agency’s data. Further, statistical tests confirm that the proposed model captures the rating dynamics better than the existing models, and the inclusion of recurrence times significantly impacts the transition probabilities.

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Recurrence times of draft-patterns from reservoirs

Journal of Applied Probability ◽

10.1017/s002190020004852x ◽

1975 ◽

Vol 12 (03) ◽

pp. 647-652 ◽

Cited By ~ 1

Author(s):

G. G. S. Pegram

Keyword(s):

Markov Chain ◽

Discrete Time ◽

Recurrence Time ◽

Reservoir Model ◽

Discrete State ◽

Discrete Time Markov Chain ◽

Recurrence Times ◽

Mean And Variance ◽

The Mean ◽

Run Theory

Expressions for the mean and variance of the recurrence time of non-overlapping draft-patterns of draft from a Moran Reservoir Model (discrete-state and discrete-time Markov chain) are derived using Feller's Renewal argument. In addition an expression for the mean recurrence time for self-overlapping patterns of draft is derived using run-theory.

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Standard deviation of recurrence times for piecewise linear transformations

Asian-European Journal of Mathematics ◽

10.1142/s1793557117500097 ◽

2017 ◽

Vol 10 (01) ◽

pp. 1750009

Author(s):

Mimoon Ismael ◽

Rodney Nillsen ◽

Graham Williams

Keyword(s):

Standard Deviation ◽

Dynamical Systems ◽

Piecewise Linear ◽

Recurrence Formula ◽

Recurrence Time ◽

Linear Transformations ◽

Recurrence Times ◽

Borel Set ◽

Bounded Interval ◽

The Mean

This paper is concerned with dynamical systems of the form [Formula: see text], where [Formula: see text] is a bounded interval and [Formula: see text] comes from a class of measure-preserving, piecewise linear transformations on [Formula: see text]. If [Formula: see text] is a Borel set and [Formula: see text], the Poincaré recurrence time of [Formula: see text] relative to [Formula: see text] is defined to be the minimum of [Formula: see text], if the minimum exists, and [Formula: see text] otherwise. The mean of the recurrence time is finite and is given by Kac’s recurrence formula. In general, the standard deviation of the recurrence times need not be finite but, for the systems considered here, a bound for the standard deviation is derived.

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STATISTICS OF EARTHQUAKE RECURRENCE TIME IN NORTH AEGEAN TROUGH

Bulletin of the Geological Society of Greece ◽

10.12681/bgsg.11841 ◽

2017 ◽

Vol 50 (3) ◽

pp. 1349

Author(s):

Ch. Kourouklas ◽

E. Papadimitriou ◽

G. Tsaklidis ◽

V. Karakostas

Keyword(s):

Stochastic Processes ◽

Source Parameters ◽

Information Criteria ◽

Recurrence Time ◽

Time Interval ◽

Data Sets ◽

Inverse Gaussian ◽

Strong Earthquakes ◽

Recurrence Times ◽

North Aegean

The determination of recurrence time of strong earthquakes of certain magnitude on a specific fault or fault segment is an important component of seismic hazard assessment. The occurrence of these earthquakes is neither periodic nor completely random but often clustered in time. This fact in connection with their limited number inhibits a deterministic approach for recurrence times calculation and thus application of stochastic processes is required. For recurrence times determination in the area of North Aegean, all the available information on strong earthquakes (historical and instrumental) with M6.0 is collected. Given that source parameters of historical events contain larger uncertainties, reassessment of their focal parameters before the application of stochastic processes is necessary, which was performed by applying the method of Bakun and Wentworth (1997). The reasses sed catalogue was divided into three data sets, according to the strong events spatial distribution and their association with distinctive fault segments. Three statistical distributions (Weibull, inverse Gaussian, lognormal) were applied and evaluated with the Anderson–Darling test and the Akaike and Bayesian Information Criteria. The Weibull distribution exhibited better performance in two out of three data sets and the Inverse Gaussian distribution in the third. With given distributions the occurrence probabilities were calculated for strong events above a certain magnitude and for certain time interval.

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Slab Avalanche Release: Data and Field Experiments

Snow Avalanches ◽

10.1093/oso/9780198866930.003.0004 ◽

2020 ◽

pp. 25-35

Author(s):

Francois Louchet

Keyword(s):

Large Scale ◽

Field Experiments ◽

Power Laws ◽

Recurrence Time ◽

Recurrence Times ◽

Elastic Bending ◽

Release Data ◽

Statistical Laws

Starting zone sizes are shown to obey statistical laws, named “power laws”, stating that the recurrence time of an event of a given size increases in a precise proportion with its size. Extrapolation of such laws fitted on small-sized events allows a determination of recurrence times for big and uncommon events. The key role of the weak layer (WL) failure is illustrated by “Propagation Saw Tests” (PST), showing that the collapse of a WL zone of a few decimeters may act as a switch, triggering a very large scale spontaneous WL failure. However, the consequences of such a collapse may be damped down by sintering of broken WL grains. We analyze bridging indexes, often used to estimate WL resistance to collapse under loading. We define a new bridging index, extending the usual one to the case of elastic bending, and we discuss the validity domains of both of them.

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Mean recurrence times for k successes within m trials

Journal of Applied Probability ◽

10.1017/s0021900200104176 ◽

1976 ◽

Vol 13 (03) ◽

pp. 604-607 ◽

Cited By ~ 3

Author(s):

Raymond J. Huntington

Keyword(s):

Recurrent Event ◽

Recurrence Time ◽

Bernoulli Trials ◽

Closed Expression ◽

Recurrence Times ◽

The Mean

Given an unlimited sequence of Bernoulli trials, let E be the recurrent event that k or more successes occur within m consecutive trials in the sequence. Let μk, m denote the mean recurrence time for E. Previous work finds expressions for μk, m for limited values of k and m. The present paper derives a closed expression for μk, m for all k, m. Tables for μk, m are also presented.

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Dynamics of a seismogenic fault subject to variable strain rate

Nonlinear Processes in Geophysics ◽

10.5194/npg-18-431-2011 ◽

2011 ◽

Vol 18 (3) ◽

pp. 431-439 ◽

Cited By ~ 4

Author(s):

M. Dragoni ◽

A. Piombo

Keyword(s):

Strain Rate ◽

Constant Strain Rate ◽

Recurrence Time ◽

Seismogenic Fault ◽

Recurrence Times ◽

Monotonic Change ◽

Average Value ◽

First Case ◽

Rate Oscillations ◽

Time Changes

Abstract. The behaviour of seismogenic faults is generally investigated under the assumption that they are subject to a constant strain rate. We consider the effect of a slowly variable strain rate on the recurrence times of earthquakes generated by a single fault. To this aim a spring-block system is employed as a low-order analog of the fault. Two cases are considered: a sinusoidal oscillation in the driver velocity and a monotonic change from one velocity value to another. In the first case, a study of the orbit of the system in the state space suggests that the seismic activity of the equivalent fault is organized into cycles that include several earthquakes and repeat periodically. Within each cycle the recurrence times oscillate about an average value equal to the recurrence period for constant strain rate. In the second case, the recurrence time changes gradually from the value before the transition to the value following it. Asymptotic solutions are also given, approximating the case when the amplitude of the oscillation or of the monotonic change is much smaller than the average driver velocity and the period of oscillation or the duration of the transition is much longer than the recurrence times of block motions. If the system is not isolated but is subject to perturbations in stress, the perturbation anticipates or delays the subsequent earthquake. The effects of stress perturbations in the two cases of strain rate oscillations and monotonic change are considered.

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New Observational Frontiers of ER UMa (= PG 0943+521) Type Dwarf Novae

International Astronomical Union Colloquium ◽

10.1017/s0252921100038203 ◽

1996 ◽

Vol 158 ◽

pp. 59-60

Author(s):

D. Nogami ◽

T. Kato ◽

S. Masuda ◽

R. Hirata ◽

K. Matsumoto ◽

...

Keyword(s):

Mass Transfer ◽

Transfer Rate ◽

Mass Transfer Rate ◽

Natural Extension ◽

Light Curves ◽

Recurrence Time ◽

Dwarf Novae ◽

Recurrence Times ◽

Tidal Torque ◽

Duty Cycles

The ER UMa-type, including ER UMa, V1159 Ori and RZ LMi, is a subgroup of SU UMa-type dwarf novae. Outbursts of these stars are characterized by: (i) the extremely short recurrence time of the superoutburst (~ 40 d in ER UMa and V1159 Ori, ~ 20 d in RZ LMi), (ii) short outbursts with short recurrence times (~ 4 d) between the bright states, and (iii) extremely long duty cycles (~ 0.5). Assuming that the mass transfer rate from the secondary is ten times higher than that of ordinary SU UMa stars, which was invoked by Kato & Kunjaya (1995), and a weak tidal torque in the case of RZ LMi, Osaki (1996) showed that the light curves of these stars can be reproduced by the disk instability model, which does not require mass transfer bursts. These indicate that ER UMa, V1159 Ori and RZ LMi are not on the natural extension of SU UMa-type dwarf novae.

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Criticism of some forecasts of the national earthquake prediction evaluation council

Bulletin of the Seismological Society of America ◽

10.1785/bssa0810030862 ◽

1991 ◽

Vol 81 (3) ◽

pp. 862-881 ◽

Cited By ~ 1

Author(s):

J. C. Savage

Keyword(s):

Ad Hoc ◽

Recurrence Time ◽

North Coast ◽

Recurrence Times ◽

The North ◽

San Andreas ◽

Parkfield Earthquake ◽

Characteristic Earthquakes ◽

Earthquake Sequences ◽

Near Future

Abstract The Working Group on California Earthquake Probabilities has assigned probabilities for rupture in the interval from 1988 to 2018 to various segments of the San Andreas fault on the basis of the lognormal distribution of recurrence times of characteristic earthquakes postulated by Nishenko and Buland (1987). I question the validity of those probabilities on the basis of three separate arguments: (1) The distributions of recurrence times of the four, best-observed, characteristic-earthquake sequences are each only marginally consistent with the Nishenko - Buland Iognormal distribution. (2) The range of possible 30-year conditional probabilities for many of the fault segments is so great due to uncertainty in the average recurrence time for that segment that the assigned probability is virtually meaningless. (3) The 1988 forecasts not subject to the foregoing objection are those in which there is a low probability of an earthquake in the near future (e.g., only a 5 per cent chance of rupture of the North Coast segment before the year 2049 and of the Carrizo segment before the year 2018). The same reasoning would assign only a 5 per cent chance of rupture before mid-1993 to the southern Santa Cruz Mountains segment, the segment that failed in October 1989. Finally, the forecast of the next Parkfield earthquake (95 per cent probability before 1993.0) by Bakun and Lindh (1985) depends upon an ad hoc explanation of the out-of-sequence 1934 earthquake. A less-contrived forecast would have assigned a conditional probability of about 60 ± 20 per cent to the 1985.0 to 1993.0 interval and 30 ± 15 per cent to the 1990.0 to 1993.0 interval.

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Recurrence times of draft-patterns from reservoirs

Journal of Applied Probability ◽

10.2307/3212886 ◽

1975 ◽

Vol 12 (3) ◽

pp. 647-652 ◽

Cited By ~ 4

Author(s):

G. G. S. Pegram

Keyword(s):

Markov Chain ◽

Discrete Time ◽

Recurrence Time ◽

Reservoir Model ◽

Discrete State ◽

Discrete Time Markov Chain ◽

Recurrence Times ◽

Mean And Variance ◽

The Mean ◽

Run Theory

Expressions for the mean and variance of the recurrence time of non-overlapping draft-patterns of draft from a Moran Reservoir Model (discrete-state and discrete-time Markov chain) are derived using Feller's Renewal argument. In addition an expression for the mean recurrence time for self-overlapping patterns of draft is derived using run-theory.

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