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In this paper we build a methodology for pricing of insurance-linked securities which are tied to multiple natural catastrophe perils. As a representative example, we construct a multi-peril catastrophe (CAT) bond which can be linked to the industry loss indices or actual losses incurred by an insurer. We provide pricing formulas for such CAT bonds. We illustrate the introduced methodology on the US natural catastrophe data obtained from Property Claim Services (PCS). Within this dataset, we specifically examine two types of risks: losses associated with wind and thunderstorm events, and those linked to winter storm events. Then, we fit and validate the underlying compound non-homogeneous Poisson processes taking into account the fact that the data are left-truncated. The best fitted loss distributions appear to be Burr and Generalised Extreme Value and for the first peril and log-normal for the second. Finally, we visualise the zero-coupon CAT bond prices for the selected best-fitted models. [ABSTRACT FROM AUTHOR]

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically. [ABSTRACT FROM AUTHOR]

We point out a minor mistake in Fig. 10 in the published version of our paper [M. Balcerek et al., Chaos 32, 093114 (2022)]. The conclusions drawn from the illustration remain the same. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two tempered linear and non-linear time series models, namely, an autoregressive tempered fractionally integrated moving average (ARTFIMA) with α -stable noise and ARTFIMA with generalized autoregressive conditional heteroskedasticity (GARCH) noise (ARTFIMA-GARCH). We provide estimation procedures for the processes and explain the connection between ARTFIMA and their tempered continuous-time counterparts. Next, we demonstrate an application of the processes to modeling of heavy-tailed data from solar flare soft x-ray emissions. To this end, we study the solar flare data during a period of solar minimum, which occurred most recently in July, August, and September 2017. We use a two-state hidden Markov model to classify the data into two states (lower and higher activity) and to extract stationary trajectories. We do an end-to-end analysis and modeling of the solar flare data using both ARTFIMA and ARTFIMA-GARCH models and their non-tempered counterparts. We show through visual inspection and statistical tests that the ARTFIMA and ARTFIMA-GARCH models describe the data better than the ARFIMA and ARFIMA-GARCH, especially in the second state, which justifies that tempered processes can serve as the state-of-the-art approach to model signals originating from a power-law source with long memory effects. [ABSTRACT FROM AUTHOR]

Gaussian processes are powerful tools for modeling and predicting various numerical data. Hence, checking their quality of fit becomes a vital issue. In this article, we introduce a testing methodology for general Gaussian processes based on a quadratic form statistic. We illustrate the methodology on three statistical tests recently introduced in the literature, which are based on the sample autocovariance function, time average mean-squared displacement, and detrended moving average statistics. We compare the usefulness of the tests by taking into consideration three very important Gaussian processes: the fractional Brownian motion, which is self-similar with stationary increments (SSSIs), scaled Brownian motion, which is self-similar with independent increments (SSIIs), and the Ornstein–Uhlenbeck (OU) process, which is stationary. We show that the considered statistics' ability to distinguish between these Gaussian processes is high, and we identify the best performing tests for different scenarios. We also find that there is no omnibus quadratic form test; however, the detrended moving average test seems to be the first choice in distinguishing between same processes with different parameters. We also show that the detrended moving average method outperforms the Cholesky method. Based on the previous findings, we introduce a novel procedure of discriminating between Gaussian SSSI, SSII, and stationary processes. Finally, we illustrate the proposed procedure by applying it to real-world data, namely, the daily EURUSD currency exchange rates, and show that the data can be modeled by the OU process. [ABSTRACT FROM AUTHOR]

In this article, we concentrate on modelling heavy-tailed data which can be subjected to left-truncation. We modify an existing procedure for modelling left-truncated data via a compound non-homogeneous Poisson process to make it systematically applicable in the context heavy-tailed data. The introduced procedure can be applied when the underlying severities of the process follow Burr type XII, Generalised Pareto and Generalised Extreme Value distributions by using the Maximum Product of Spacings (MPS) parameter estimation technique. As a natural consequence of the MPS technique, we consider how Moran's log spacings statistic for testing goodness-of-fit of the severity distributions can be adapted to suit left-truncated data. Thereafter, we compare the performance of this new fitting procedure against traditional maximum likelihood estimation in the context of natural catastrophe loss data, and evidence in favour of MPS is found. Within the context of these data, we also compare our procedure to a one that does not account for left-truncation. We end our contribution by proposing, for our modelling procedure, a Monte Carlo importance sampling algorithm which ensures that large losses are satisfactorily simulated. In closing, we illustrate the potential usage of both the new fitting and simulation procedures by presenting catastrophe bond prices with a trigger based on the analysed heavy-tailed data. • A rigorous methodology for modelling heavy-tailed left-truncated data is introduced. • A modification of the maximum product of spacings estimation technique is studied. • An importance-sampling algorithm for heavy-tailed distributions is proposed. • Catastrophe bond prices with the trigger based on the PCS loss index are presented. [ABSTRACT FROM AUTHOR]

The Sun is variable in activity with changes on time-scales as short as minutes to as long as a solar cycle. Although the most accurate measurements are limited to the satellite era, the past four decades, looking at the solar variability over this period provides a possible link between complex dynamics of the Sun and the accompanying radiation. Measurements of the latter and their analysis by sophisticated time series methods encourage forecasting future values of the time series. Our data analysis work focuses on the soft X-ray emission observed at the current solar minimum, in 2017 September July. We have found two different (active and inactive) states of the solar activity using a Hidden Markov Model, and we show that in the periods of high-solar activity the energy distribution of soft X-ray solar flares is well described by an ARFIMA-GARCH model, whereas in the case of low-solar activity an ARFIMA model is best fitted. Switching from the inactive state to the active one is caused by explosive phenomena in the Sun. The model describes three effects detected in our empirical studies. One of them is a long-term dependence, the second is variance changing in time, and the third corresponds to heavy-tailed distributions of the X-ray data. Moreover, the model takes into account memory effects in soft X-ray emission due to the Sun's magnetic field evolution. All this together allows us to suggest a statistically justified model for explaining the solar activity variability at the current solar minimum. [ABSTRACT FROM AUTHOR]

In this paper we propose an algorithm to distinguish between light- and heavy-tailed probability laws underlying random datasets. The idea of the algorithm, which is visual and easy to implement, is to check whether the underlying law belongs to the domain of attraction of the Gaussian or non-Gaussian stable distribution by examining its rate of convergence. The method allows to discriminate between stable and various non-stable distributions. The test allows to differentiate between distributions, which appear the same according to standard Kolmogorov–Smirnov test. In particular, it helps to distinguish between stable and Student’s t probability laws as well as between the stable and tempered stable, the cases which are considered in the literature as very cumbersome. Finally, we illustrate the procedure on plasma data to identify cases with so-called L-H transition. [ABSTRACT FROM AUTHOR]

In this paper, we extend a method of estimation of parameters of the fractional autoregressive integrated moving average (FARIMA) process with stable noise to the case of negative memory parameter d. We construct an estimator that is a modification of that of Kokoszka and Taqqu and prove its consistency for -1/2

We show how to use a fractional autoregressive integrated moving average (FARIMA) model to a statistical analysis of the subdiffusive dynamics. The discrete time FARIMA(l,

We address the problem of recognizing &agr;-stable Lévy distribution with Lévy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and larque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Lévy to Gaussian fluctuation statistics. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we show that the logarithmic returns of the Hang Seng index from January 2, 1987 to November 14, 2005 statistically resemble a sequence of independent identically distributed Lévy stable random variables. This is in stark contrast to Xiu and Jin (2007) , where long-memory FARIMA processes with Gaussian noise were suggested as well fitted to the data. The lack of memory is checked by using Lo’s modified statistic and a new method of estimation of the memory parameter which applies the notion of empirical mean-squared displacement. In order to test stability of the data we employ several statistical tests based on the empirical distribution function. Finally, we also show that the returns possess no conditional heteroscedasticity property thus excluding the ARCH/GARCH family of processes as possible underlying models. [Copyright &y& Elsevier]

We analyze the time series of soft X-ray emission registered in the years 1974-2007 by the GOES spacecrafts. We show that in the periods of high solar activity, namely 1977-1981, 1988-1992, 1999-2003 the maximum energy of soft X-ray solar flares exhibits both heavy tails and long-range dependence. We investigate the presence of long-range dependence by means of different self-similarity estimators. This analysis gives a promising start to model the appearance of such solar events during solar maxima. [ABSTRACT FROM AUTHOR]

Competition between subdiffusion and Lévy flights is conveniently described by the fractional Fokker--Planck equation with temporal and spatial fractional derivatives. The equivalent approach is based on the subordinated Langevin equation with stable noise. In this paper we examine the properties of such Langevin equation with the heavy-tailed noise belonging to the class of geometric stable distributions. In particular, we consider two physically relevant examples of geometric stable noises, namely Linnik and Mittag--Leffler. We describe in detail a numerical algorithm for visualization of subdiffusion coexisting with Lévy flights. Using Monte Carlo simulations we demonstrate the realizations as well as the probability density functions of the considered anomalous diffusion process. [ABSTRACT FROM AUTHOR]

Abstract: We demonstrate that continuous-time FARIMA processes with -stable noise provide a new stochastic tool for studying the solar flare phenomenon in the framework of fractional Langevin equation. Simple computer tests to check the origins of -stability and self-similarity are implemented for empirical time series describing the energy of solar flares. Based on observed physical time series we solve the challenging problem of how to detect long-range dependence from real data and how to model it via fractional dynamics (Langevin or Fokker–Planck). We employ here codifference as a proper measure for long-range dependence. It is applicable to empirical data from the distribution lacking the second moment. [Copyright &y& Elsevier]

Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index 0 < H < 1 . In nature one often observes changes in the dynamics of a system over time. For example, this is true in single-particle tracking experiments where a transient behavior is revealed. The stationarity of increments of FBM restricts substantially its applicability to model such phenomena. Several generalizations of FBM have been proposed in the literature. One of these is called multifractional Brownian motion (MFBM) where the Hurst index becomes a function of time. In this paper, we introduce a rigorous statistical test on MFBM based on its covariance function. We consider three examples of the functions of the Hurst parameter: linear, logistic, and periodic. We study the power of the test for alternatives being MFBMs with different linear, logistic, and periodic Hurst exponent functions by utilizing Monte Carlo simulations. We also analyze mean-squared displacement (MSD) for the three cases of MFBM by comparing the ensemble average MSD and ensemble average time average MSD, which is related to the notion of ergodicity breaking. We believe that the presented results will be helpful in the analysis of various anomalous diffusion phenomena. [ABSTRACT FROM AUTHOR]

We present here the autoregressive tempered fractionally integrated moving average (ARTFIMA) process obtained by taking the tempered fractional difference operator of the non-Gaussian stable noise. The tempering parameter makes the ARTFIMA process stationary for a wider range of the memory parameter values than for the classical autoregressive fractionally integrated moving average, and leads to semi-long range dependence and transient anomalous behavior. We investigate ARTFIMA dependence structure with stable noise and construct Whittle estimators. We also introduce the stable Yaglom noise as a continuous version of the ARTFIMA model with stable noise. Finally, we illustrate the usefulness of the ARTFIMA process on a trajectory from the Golding and Cox experiment. [ABSTRACT FROM AUTHOR]

In this paper we analyze the relationship between the sunspot numbers and the average monthly precipitation measured at meteorological stations in the US. The results indicate that there is a significant correlation between the solar activity and the monthly average precipitation data for selected months and time delays. In order to confirm the relationship, for the precipitation data we check the forecasting abilities of a popular time series model with and without additional information on the sun activity. Namely, first we fit the autoregressive moving average (ARMA) time series model to the US precipitation data and calculate one-step ahead forecasts. Next, we repeat the same steps with the ARMA model with exogenous input (ARMAX) which is the sunspot number data. We show that the forecast errors are essentially lower for the ARMAX model for months and time delays found in the correlation analysis. • Influence of solar activity on precipitation in the US is addressed. • The impact is studied by means of ARMA and ARMAX models. • Information on the number of sunspots helps to improve precipitation forecasts. • The presented scheme can be also applied to other meteorological parameters. [ABSTRACT FROM AUTHOR]

In this paper we show that an autoregressive fractionally integrated moving average time-series model can identify two types of motion of membrane proteins on the surface of mammalian cells. Specifically we analyze the motion of the voltage-gated sodium channel Nav1.6 and beta-2 adrenergic receptors. We find that the autoregressive (AR) part models well the confined dynamics whereas the fractionally integrated moving average (FIMA) model describes the nonconfined periods of the trajectories. Since the Ornstein-Uhlenbeck process is a continuous counterpart of the AR model, we are also able to calculate its physical parameters and show their biological relevance. The fitted FIMA and AR parameters show marked differences in the dynamics of the two studied molecules. [ABSTRACT FROM AUTHOR]

Diffusion of nanoparticles in the cytoplasm of live cells has frequently been reported to exhibit an anomalous and even heterogeneous character, i.e. particles seem to switch gears during their journey. Here we show by means of a hidden Markov model that individual trajectories of quantum dots in the cytoplasm of living cultured cells feature a dichotomous switching between two distinct mobility states with an overall subdiffusive mode of motion of the fractional Brownian motion (FBM) type. Using the extracted features of experimental trajectories as input for simulations of different variants of a two-state FBM model, we show that the trajectory-intrinsic and the ensemble-wise heterogeneity in the experimental data is mostly due to variations in the (local) transport coefficients, with only minor contributions due to locally varying anomaly exponents. Altogether, our approach shows that diffusion heterogeneities can be faithfully extracted and quantified from fairly short trajectories obtained by single-particle tracking in highly complex media. [ABSTRACT FROM AUTHOR]

Statistical inference in the form of hypothesis tests and confidence intervals often assumes that the underlying distribution is normal. Similarly, many signal processing techniques rely on the assumption that a stationary time series is normal. As a result, a number of tests have been proposed in the literature for detecting departures from normality. In this article we develop a novel approach to the problem of testing normality by constructing a statistical test based on the Edgeworth expansion, which approximates a probability distribution in terms of its cumulants. By modifying one term of the expansion, we define a test statistic which includes information on the first four moments. We perform a comparison of the proposed test with existing tests for normality by analyzing different platykurtic and leptokurtic distributions including generalized Gaussian, mixed Gaussian, α-stable and Student's t distributions. We show for some considered sample sizes that the proposed test is superior in terms of power for the platykurtic distributions whereas for the leptokurtic ones it is close to the best tests like those of D'Agostino-Pearson, Jarque-Bera and Shapiro-Wilk. Finally, we study two real data examples which illustrate the efficacy of the proposed test. [ABSTRACT FROM AUTHOR]

Anomalous diffusion in crowded fluids, e.g., in cytoplasm of living cells, is a frequent phenomenon. A common tool by which the anomalous diffusion of a single particle can be classified is the time-averaged mean square displacement (TAMSD). A classical mechanism leading to the anomalous diffusion is the fractional Brownian motion (FBM). A validation of such process for single-particle tracking data is of great interest for experimentalists. In this paper we propose a rigorous statistical test for FBM based on TAMSD. To this end we analyze the distribution of the TAMSD statistic, which is given by the generalized chi-squared distribution. Next, we study the power of the test by means of Monte Carlo simulations. We show that the test is very sensitive for changes of the Hurst parameter. Moreover, it can easily distinguish between two models of subdiffusion: FBM and continuous-time random walk. [ABSTRACT FROM AUTHOR]

Abstract This paper addresses an important problem of modeling and prediction of phenomena with antipersistent behavior and variance changing in time. As a proper stochastic model we propose an autoregressive fractionally integrated moving average (ARFIMA) process with generalized autoregressive conditional heteroskedasticity (GARCH) noise. First, we introduce a simple identification and validation algorithm for such model. Second, we apply the algorithm to weekday data of hot water usage at urban residential blocks. We extract the deterministic sinusoidal component from the data and fit successfully the ARFIMA–GARCH model to the stochastic part. The goodness of fit is checked by examining model errors and prediction performance. All analyses are performed by the rigorous statistical procedure. The proposed model allows for real-time accurate predictions and when implemented at a hot water supply level will lead to a better optimization of the control system and energy efficiency use. Highlights • A problem of modeling and prediction of hot water usage is addressed. • Hot water usage data depicts antipersistent behavior and time-dependent variance. • A simple identification and validation algorithm for ARFIMA–GARCH is introduced. • The presented scheme can be also applied to other phenomena. [ABSTRACT FROM AUTHOR]

Accurately characterizing the anomalous diffusion of a tracer particle has become a central issue in biophysics. However, measurement errors raise difficulty in the characterization of single trajectories, which is usually performed through the time-averaged mean square displacement (TAMSD). In this paper, we study a fractionally integrated moving average (FIMA) process as an appropriate model for anomalous diffusion data with measurement errors. We compare FIMA and traditional TAMSD estimators for the anomalous diffusion exponent. The ability of the FIMA framework to characterize dynamics in a wide range of anomalous exponents and noise levels through the simulation of a toy model (fractional Brownian motion disturbed by Gaussian white noise) is discussed. Comparison to the TAMSD technique, shows that FIMA estimation is superior in many scenarios. This is expected to enable new measurement regimes for single particle tracking (SPT) experiments even in the presence of high measurement errors. [ABSTRACT FROM AUTHOR]

Single particle tracking is an essential tool in the study of complex systems and biophysics and it is commonly analyzed by the time-averaged mean square displacement (MSD) of the diffusive trajectories. However, past work has shown that MSDs are susceptible to significant errors and biases, preventing the comparison and assessment of experimental studies. Here, we attempt to extract practical guidelines for the estimation of anomalous time averaged MSDs through the simulation of multiple scenarios with fractional Brownian motion as a representative of a large class of fractional ergodic processes. We extract the precision and accuracy of the fitted MSD for various anomalous exponents and measurement errors with respect to measurement length and maximum time lags. Based on the calculated precision maps, we present guidelines to improve accuracy in single particle studies. Importantly, we find that in some experimental conditions, the time averaged MSD should not be used as an estimator. [ABSTRACT FROM AUTHOR]