Extreme value theory for processes with periodic variances

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equationThe eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce toaN = bN = N2when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namelySuppose first that N is an odd integer. Then there is an expansionwhereThese functions π satisfyandOn Substituting (3) in (1), one obtains the algebraic equationwhereExplicitly,{11} = q{lm} = 0 otherwise.

<abstract> <bold><sc>Abstract.</sc></bold> Methods are needed to estimate R-factor values and their seasonal distribution from daily rainfall totals in regions with little or no sub-daily rainfall intensity data. This study used calculated R-factor values for a 29-year period (1956-1984) for 22 locations in mainland China to evaluate two estimators of rainfall erosivity for large-scale application of the Universal Soil Loss Equation (USLE) or the Revised USLE (RUSLE), one for an R-factor estimated from mean annual precipitation and another for event erosivity from daily rainfall. This study shows that as a first approximation, the mean annual precipitation is non-linearly related to the R-factor, and that the relationship can be used to predict the R-factor for mainland China locations (Nash-Sutcliffe model efficiency E_c = 0.88, RMSE = 28% of the mean). The nonlinear relationship for China is quite similar to those reported in other studies for Australia and the U.S. The study also shows that a daily rainfall erosivity model using average parameter values previously applied for Australia can satisfactorily estimate the seasonal variation in rainfall erosivity in addition to the R-factor for the mainland China locations (E_c = 0.90, RMSE = 25% of the mean). Once calibrated, model estimates of the R-factor and its monthly distribution from daily rainfall for the mainland China locations improved noticeably (E_c = 0.99, RMSE < 10% of the mean). The daily rainfall erosivity model and a gridded daily precipitation dataset (the China Gridded Daily Precipitation Product) with a 0.25° (~25 km) resolution were then used to produce a concurrent daily rainfall erosivity map for the mainland China locations (E_c = 0.96, RMSE = ~23% of the mean) for a 5-year period from April 2008 to March 2013. This study shows that: (1) mean annual precipitation can be used to estimate the R-factor for mainland China, Australia, and the U.S.; and (2) a calibrated daily rainfall erosivity model performed well in estimating the seasonal and interannual variations in rainfall erosivity, in addition to the R-factor, for large-scale erosion monitoring and assessments in China.

Extreme short-duration rainfall is intensifying with climate warming, and growing evidence suggests that subhourly rainfall extremes are increasing faster than more widely studied durations at hourly and daily timescales. In this case study, we used 55 years (1968–2022) of 5-min precipitation data from Mahantango Creek, a long-term experimental agricultural watershed in east-central Pennsylvania, United States, to examine annual and seasonal changes in subhourly (15-min), hourly, and daily rainfall extremes. Specifically, we evaluated temporal trends in the magnitude and frequency of subhourly, hourly, and daily rainfall extremes. We then estimated apparent scaling rates between rainfall extremes and dew point temperature (Td) and compared these rates to the Clausius-Clapeyron (CC) rate (∼ 7% per °C). We also determined the coincidence of extreme rainfall trends with indicators of atmospheric instability and convective-type precipitation. Overall, we found the most significant changes in rainfall extremes at 15-min durations during the spring, with magnitudes of these subhourly extremes increasing by 0.6 to 0.9% per year, and frequencies rising by 3.4% per year. Apparent scaling rates in the spring showed that 15-min rainfall extremes transitioned from sub-CC scaling to greater than 2CC scaling when Td reached 11° C, implying a possible shift from stratiform rains to more intense convective rains above this Td threshold. Notably, trends in maximum hourly convective available potential energy (CAPE) increased during spring, as did the ratio of 15-min rainfall extremes to their corresponding daily rainfall totals. Findings indicate that convective-type precipitation may be playing an increasing role in the intensification of springtime 15-min rainfall extremes in Mahantango Creek.

There are limitations in using monthly rainfall totals in studies of rainfall climatology as well as in hydrological and agricultural investigations. Variations in rainfall may be considered to result from frequency changes in the daily rainfall of the respective regime. In the present study, daily rainfall data of the stations inside the Koyna catchment has been analysed for the period of 1961–2005 to understand the relationship between the rain and rainy days, mean daily intensity (MDI) and seasonal rainfall over the catchment on monthly as well as seasonal scale. Considering the topographical location of the catchment, analysis of seasonal rainfall data of 8 stations suggests that a linear relationship fits better than the logarithmic relationship in the case of seasonal rainfall versus mean daily intensity. So far as seasonal rainfall versus number of rainy days is considered, the logarithmic relationship is found to be better.

Daily precipitation totals at 55 sites were used to investigate geographic variability in winter (DJF) rainfall over Cumbria, NW England, over an 11-year period. Winter is the wettest season (>800 mm in the mountainous Lake District), with rainfall mechanisms closely linked to North Atlantic forcing. The Lamb weather type catalogue was used to identify rainfall distributions under different wind directions. Precipitation magnitude over Cumbria is much more sensitive to a change in wind direction than the geographic pattern in rainfall, with southwesterly (easterly) winds producing the highest (lowest) spatially averaged daily rainfall totals of 8.2 mm (0.6 mm). S-mode principal components analysis was used to identify the main patterns of precipitation variability. Three principal components (PCs) were retained as being statistically significant (cumulative explained variance for unrotated PCs = 84.3%), with a correlated PC structure (direct oblimin rotation) best describing the spatial variance in rainfall. PC 1 has a very high index of strength (variance measure = 40.9), indicating that there is one dominant rainfall pattern. PC 1 shows a gradient between wetter conditions in southwest Cumbria and over the central Lake District and drier conditions in NE Cumbria, and is usually caused by active zonal west to southwest flows. Almost of equal importance to PC 1 is PC 3 (variance measure = 39.3), which has a more uniform rainfall distribution than PC 1 and is usually caused by fronts stalling over the region. PC 2, which shows an east to west decline in rainfall totals, is much less important than PCs 1 and 3 (variance measure = 18.6). PC 2’s rainfall pattern can be caused by easterly flows with high pressure over Scandinavia and low pressure over the Continent, or by strong southwesterly flows, with depressions often centred over Scotland. Finally, cluster analysis was carried out to identify precipitation regions for all days and for each wind direction. Clusters were found to be largely stable to changes in wind direction, with stations in the central Lake District often clustered together, thus highlighting the importance of orographic enhancement of rainfall in this region.

ABSTRACTReference analysis is an objective Bayesian approach to finding noninformative prior distributions. For various models involving nuisance parameters the reference prior has been shown to be superior to the multiparameter Jeffreys prior. In this article, the performance of the reference prior is evaluated for models that are defined at the extremes of parameter ranges, in which case reference and Jeffreys priors are shown to be potentially informative. Two quantities of interest are analyzed: the ratio of two multinomial parameters and the ratio of two independent binomial parameters, where the latter is just one example of inference based on the common 2 × 2 contingency table. For these two specific examples, we show that reference and/or Jeffreys priors lead to overinformative inference in extreme data situations that would seem common in, for example, many medical contexts when interest is in rare events. It is recommended that in the context of noninformative priors and binary data with extreme...

The real negative zeros equation and equation — a finite positive integer) in the first parameter k, resp. â of the real Kummer confluent hypergeometric function equation with â, ĉ and x — all real, â 0 and x > 0, â = ĉ/2 + k (k = â – ĉ/2 — real, negative, −∞ < k < −ĉ/2 ), resp., â — arbitrary real negative, are investigated numerically for selected values of n, assuming x and ĉ as parameters. The results are presented in the form of Tables and graphs. Analyzing them, certain properties of the zeros are revealed and summarized in three theorems which are substantiated numerically and analytically. Theorem 1 states that when x → +∞, then equation and equation tend to the real positive numbers equation, relevantly equation, and in case x → 0, both quantities of interest go to −∞. Theorem 2 determines the number n of zeros studied. It ascertains that it is finite and depends on their values. According to it n = 1; 2; …, p, where equation, correspondingly equation, (equation is the greatest integer less than or equal to equation). Theorem 3 specifies the value of the first zero at x = ĉ to equation, accordingly to equation. An application of the quantities considered for ĉ = 3 is given in the theory of azimuthally magnetized circular ferrite waveguide under slow equation modes excitation.

Context. Photonuclear interactions between ultrahigh-energy cosmic ray (UHECR) nuclei and photon fields are key to connecting the compositions observed at Earth to those emitted from the sources as well as understanding their evolution during propagation. The stochastic nature of these interactions implies that any deterministic description is an approximation, but an exact probabilistic formulation is not yet available. Currently, Monte Carlo simulations are the only means to probe the probability space of these events. Aims. We aim to obtain a rigorous probabilistic description of UHECR interactions based on the theory of stochastic processes and to derive closed-form expressions for the probability distributions of the distance, with the inclusion of all energy losses and cosmological effects. Additionally, we aim to demonstrate the application of this description in astrophysical scenarios where UHECR-photon interactions are relevant, such as radiation models of UHECR sources and extragalactic propagation of UHECRs. Methods. We applied the theory of Markov jump processes to the nuclear cascades produced by photonuclear interactions of UHECRs, providing definitions to characterize the cascades within this new framework. By comparing different choices of input originating from nuclear physics experiments and electromagnetic observations, we improved our understanding of the underlying physics. Results. We developed the sought-after probabilistic description and derived analytical expressions for various types of cascades, which are defined in terms of the number of nuclei and disintegration channels involved. We discuss the fundamental properties of these cascades and their relation to physical parameters, such as the UHECR composition, the Lorentz boost, and the target photon thickness. We discuss the role of continuous energy losses in this description and show their impact in the propagation horizons of UHECRs. We computed several quantities of interest, such as the UHECR disintegration distance as a function of the nuclear mass, the composition evolution in a gamma-ray burst (GRB) source scenario, and the angular diffusion of disintegration products produced during extragalactic propagation, while accounting for interactions.

From a mathematical viewpoint, the frequency domain analysis of vessel motion responses due to wave actions is based on integration of system dynamics idealized in terms of response amplitude operators for six degrees-of-freedom (DOFs) rigid body motions and an input wave spectrum in order to obtain the response spectrum. Various quantities of interest can be deduced from the response spectrum, which are then used for deriving response-based operational limits for marine operations, also including extreme value and fatigue analysis. The variation of such quantities, owing to the uncertainties associated with the vessel system parameters, can be quantified by performing uncertainty propagation and consequent sensitivity analysis. This study emphasizes and proposes a computational-efficient way of assessing the sensitivity of the system model output with respect to the uncertainties residing in the input parameters by operating on a surrogate model representation. In this respect, the global sensitivity analysis was effectively carried out by deploying an efficient nonintrusive polynomial chaos expansion surrogate model built using a point collocation strategy. Successively, Sobol’ indices were obtained from the analytical decomposition of the polynomial coefficients. The indices, eventually, are employed to quantitatively measure the effects of input uncertainties on the output 6DOF vessel root-mean-square responses.

Modeling uncertainties in complex systems

Importance sampling is a variance reduction technique for efficient simulation via a change of measure. In particular, it can be applied to rare event simulation of Markov chains. An optimal change of measure yielding a zero variance estimator always exists, but it explicitly depends on the unknown quantity of interest. Thus, it is typically neither known in advance nor available in simulations. In this paper, we investigate the form of optimal importance sampling for estimating state probabilities in discrete-time Markov chains, both over finite horizon and in steady state. We derive the optimal change of measure by utilizing only a general property, without using a priori knowledge of the quantity of interest. Our results show that optima! importance sampling for Markov chains cannot be performed by simulating an alternative Markov chain, since transition probabilities in each step must depend on the history of the already simulated process. This particularly indicates that combined dynamic/adaptive techniques should be applied, and that some traditional approaches are not really promising, which holds not only for Markov chains but also for higher level models such as Petri nets and Markovian queueing models.

Spatial interpolation of daily totals of rainfall

A record of daily rainfall totals from 1827 at the Radcliffe Observatory, Oxford, United Kingdom, is presented. Changing instrumentation over the period and homogenization of the record are described. Brief analysis of the record is provided, including reference to the monthly record from 1767. Most heavy falls of rain occur in summer. The twentieth century tendency toward wetter winters was not accompanied by any significant increase in heavy falls of rain. There were very few heavy falls in summer during the 1990s but a large number in autumn. Preliminary analysis suggests that there are more heavy downpours in summers when there is reduced storminess in the North Atlantic region.

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour asn →∞ of the process conditioned on non-absorption until timen.By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.

Recent studies have noted a worldwide increase in the occurrence of extreme-precipitation events. Here we use daily rainfall data from 1951 to 2010 of 659 meteorological stations in China and on the basis of duration days defined long duration heavy rainfall. Results indicate that: on the spatial distribution, short duration heavy rainfall shows gradually decreasing phenomenon from the southeast coastal to northwest inland in China from 1951 to 2010. And long duration heavy rainfall is concentrated in the southeast coastal areas, such as Guangdong, Guangxi and Hainan. On the temporal change, the interannual and interdecadal short and long duration heavy rainfall both show increasing trend. In precipitation contribution ratio, the proportion of total heavy rainfall amounts to total rainfall amounts and total heavy rainfall days to total rainfall days are 6.1%∼27.7% and 0.6%∼27.7% respectively from 1951 to 2010 in China. Short duration storm to occupy the dominant position that the proportion of short duration heavy rainfall amounts to total heavy rainfall amounts and short duration heavy rainfall days to total heavy rainfall days are 75.9%∼89.4% and 75.6%∼89.4% respectively in the same period. Long duration heavy rainfall occupy a secondary position that the proportion of long duration heavy rainfall amounts to total heavy rainfall amounts and short duration heavy rainfall days to total heavy rainfall days are only 10.6%∼24.1% and 10.8%∼24.4% respectively. On the trend of precipitation contribution, China’s the contribution rate of total heavy rainfall to the total rainfall show increasing trend with contribution of rainfall amounts and rainfall days trend are 2.1%/10a and 0.2%/10a respectively. The contribution rate of short duration heavy rainfall to the total heavy rainfall show increasing trend with contribution of rainfall amounts and rainfall days trend are 0.5%/10a and 0.4%/10a respectively. On the contrary, the contribution rate of long duration heavy rainfall to the total heavy rainfall show decreasing trend with contribution of rainfall amounts and rainfall days trend are -0.5%/10a and -0.4%/10a respectively. The results suggest that the precipitation in China are changing to extremely accompanied by short duration storm increased significantly.