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Abstract The molecular movement in single particle tracking (SPT) experiments shows a crucial role of diffusion in many biological processes such as signaling, cellular organization, transport mechanisms, and more. The SPT analysis detects not only classical Brownian motion but diffusion with other features. These include directed diffusion and confined motion. The behavior remains a challenging problem for several reasons. Due to the action of many physical processes, random trajectories of cellular molecules are segmented in different diffusive modes. Often their study requires sophisticated algorithms for the analysis of statistical properties. In this paper we consider the segment analysis for trajectories of G proteins and coupled-receptors in living cells. Their movement is often transient and switches among free diffusion, confined diffusion, directed diffusion, and immobility. Moreover, the confined segments can have both Gaussian and non-Gaussian statistics. The types of alternation of diffusive modes along the trajectories of G proteins and coupled-receptors are analyzed.

Entropic Value-at-Risk (EVaR) measure is a convenient coherent risk measure. Due to certain difficulties in finding its analytical representation, it was previously calculated explicitly only for the normal distribution. We succeeded to overcome these difficulties and to calculate Entropic Value-at-Risk (EVaR) measure for Poisson, compound Poisson, Gamma, Laplace, exponential, chi-squared, inverse Gaussian distribution and normal inverse Gaussian distribution with the help of Lambert function that is a special function, generally speaking, with two branches.

This manuscript presents a new two-parameter unit stochastic distribution, obtained by transforming the Laplace distribution, using a generalized logistic map, into a unit interval. The distribution thus obtained is named the Laplace-logistic unit (abbreviated LLU) distribution, and its basic stochastic properties are examined in detail. Also, the procedure for estimating parameters based on quantiles is provided, along with the asymptotic properties of the obtained estimates and the appropriate numerical simulation study. Finally, the application of the LLU distribution in dynamic and regression analysis of real-world data with accentuated “peaks” and “fat” tails is also discussed.

Traditional convolutional neural networks (CNNs) lack equivariance for transformations such as rotation and scaling. Consequently, they typically exhibit weak robustness when an input image undergoes generic transformations. Moreover, the complex model structure complicates the interpretation of learned low- and mid-level features. To address these issues, we introduce a Lie group equivariant convolutional neural network predicated on the Laplace distribution. This model’s Lie group characteristics blend multiple mid- and low-level features in image representation, unveiling the Lie group geometry and spatial structure of the Laplace distribution function space. It efficiently computes and resists noise while capturing pertinent information between image regions and features. Additionally, it refines and formulates an equivariant convolutional network appropriate for the Lie group feature map, maximizing the utilization of the equivariant feature at each level and boosting data efficiency. Experimental validation of our methodology using three remote sensing datasets confirms its feasibility and superiority. By ensuring a high accuracy rate, it enhances data utility and interpretability, proving to be an innovative and effective approach.

Circadian systems are composed of multiple oscillatory elements that contain both circadian and ultradian oscillations. The relationships between these components maintain a stable temporal function in organisms. They provide a suitable phase to recurrent environmental changes and ensure a suitable temporal sequence of their own functions. Therefore, it is necessary to identify these interactions. Because a circadian rhythm of activity can be recorded in each crayfish cheliped, this paired organ system was used to address the possibility that two quasi-autonomous oscillators exhibiting both circadian and ultradian oscillations underlie these rhythms. The presence of both oscillations was found, both under entrainment and under freerunning. The following features of interactions between these circadian and ultradian oscillations were also observed: (a) circadian modal periods could be a feature of circadian oscillations under entrainment and freerunning; (b) the average period of the rhythm is a function of the proportions between the circadian and ultradian oscillations; (c) the release of both populations of oscillations of Zeitgeber effect results in the maintenance or an increase in their number and frequency under freerunning conditions. These circadian rhythms of activity can be described as mixed probability distributions containing circadian oscillations, individual ultradian oscillations, and ultradian oscillations of Gaussian components. Relationships among these elements can be structured in one of the following six probability distributions: Inverse Gaussian, gamma, Birnbaum–Saunders, Weibull, smallest extreme value, or Laplace. It should be noted that at one end of this order, the inverse Gaussian distribution most often fits the freerunning rhythm segments and at the other end, the Laplace distribution fits only the segments under entrainment. The possible relationships between the circadian and ultradian oscillations of crayfish motor activity rhythms and between the probability distributions of their periodograms are discussed. Also listed are some oscillators that could interact with cheliped rhythms.

This manuscript deals with a novel, nonlinear, and non-stationary stochastic model with symmetric, Laplacian distributed innovations. The obtained model, named Laplacian Split-BREAK (LSB) process, is intended for dynamic analysis of time series with pronounced and permanent fluctuations. By using the method of characteristic functions (CFs), the basic stochastic properties of the LSB process are proven, with a special emphasis on its asymptotic behaviour. The different procedures for estimating its parameters are also given, along with numerical simulations of the obtained estimators. Finally, it has been shown that the LSB process, as an adequate stochastic model, can be applied in the analysis of dynamics in the world market of crude oil and natural gas.

Genomic prediction (GP) has become a valuable tool for predicting the performance of selection candidates for the next breeding cycle. A vast majority of statistical linear models on which GP is based rely on the assumption of normality of the residuals and therefore on the response variable itself. In this study, we propose to use Bayesian regularized quantile regression (BRQR) in the context of GP; the model has been successfully used in other research areas. We evaluated the prediction ability of the proposed model and compared it with the Bayesian ridge regression (BRR; equivalent to genomic best linear unbiased predictor, GBLUP). In addition, BLUP can be used with pedigree information obtained from the coefficient of coancestry (ABLUP). We have found that the prediction ability of BRQR is comparable to that of BRR and, in some cases, better; it also has the potential to efficiently deal with outliers. A program written in the R statistical package is available as Supplementary material.

This paper is concerned with the robust parameter estimation for linear parameter varying (LPV) finite impulse response (FIR) model. The practical process data are typically polluted by outliers and conventional parameter estimation methods may fail to derive an unbiased estimate. In order to deal with outliers, the Laplace noise model is adopted and the robust system model for the described parameter estimation problem is established. The robust parameter estimation for LPV FIR model is formulated and solved in the EM algorithm scheme and the equations to estimate all the unknown parameters are derived. The efficacy of the proposed method is verified through a numerical simulation and a chemical unit.

This paper presents a robust method for dealing with switching regression problems. Regression models with switch-points are broadly employed in diverse areas. Many traditional methods for switching regressions can falter in the presence of outliers or heavy-tailed distributions because of the modeling assumptions of Gaussian errors. The outlier corruption of datasets is often unavoidable. When misapplied, the Gaussian assumption can lead to incorrect inference making. The Laplace distribution is known as a longer-tailed alternative to the normal distributions and connected with the robust least absolute deviation regression criterion. We propose a robust switching regression model of Laplace distributed errors. To advance robustness, we extend the Laplace switching model to a fuzzy class model and create a robust algorithm named FCL through the fuzzy classification maximum likelihood procedure. The robustness properties and the advance of resistance against high-leverage outliers are discussed. Simulations and sensitivity analyses illustrate the effectiveness and superiority of the proposed algorithm. The experimental results indicate that FCL is much more robust than the EM-based algorithm. Furthermore, the Laplace-based algorithm is more time-saving than the t-based procedure. Diverse real-world applications demonstrate the practicality of the proposed approach.

In this article, the Laplace distribution is employed in lieu of the well-known normal distribution for finding better scalar values of risk. Explicit formulas for value-at-risk (VaR) and conditional value-at-risk (CVaR) are studied and used to manage the risk involved in a stock movement by using the GARCH model. Numerical simulations are given for a variety of stocks in equity markets to uphold the findings.

Grain production is one of the Ukrainian agro-industrial complex main branches. An indicator of the grain production efficiency is its profitability. It is characterized by significant annual fluctuations that induce risk. Redistribution of existing grain crops areas taking into consideration their profitability can bring to increase of production efficiency. The paper observes Markowitz’s optimal portfolio theory appliance to grain branch. The central aim of this work is the development and justification of a new technique of sown areas portfolio risk evaluating. In this study, the annual profitability of four cereal crops cultivated in the Rivne region: wheat, barley, corn, and oats, have been analyzed. It is shown that the profitability of the cereals is not normally distributed. Under these conditions, the portfolio variance loses part of its informativity and can not serve as a good risk measure. It has been determined that the profitability of crops with good precision follows Laplace distribution (double exponential distribution). The analytical expression for Value-at-Risk measures has been obtained using the Laplace distribution function. Numerical risk assessments performed. Using a modified Markowitz model and obtained risk estimates the efficient frontiers of cereal sown areas portfolios in the Rivne region were constructed. Obtained results allow indicating ways for optimization the region grain industry structure.

This paper introduces a robust identification solution for the linear parameter varying Autoregressive Exogenous systems with outlier-contaminated outputs. The Laplace distribution with heavy tails and the expectation maximization algorithm are combined to build the robust system identification framework. To overcome the obstacles brought by the outliers, the Laplace distribution which can be decomposed into infinite Gaussian components, is applied to mathematically model the system noise. The problem of parameter estimation is solved using the expectation maximization algorithm, and the equations to infer the system model and noise parameters are simultaneously provided in the developed identification method. Finally, the verification tests performed on a numerical example and a mechanical unit are used to prove the validity of the developed identification method.

Pairs trading under the copula approach is revisited in this paper. It is well known that financial returns arising from indices in markets may not follow the features of normal distribution and may exhibit asymmetry or fatter tails, in particular. Due to this, the Laplace distribution is employed in this work to fit the marginal distribution function, which will then be employed in a copula function. In fact, a multivariate copula function is constructed on two indices (based on the Laplace marginal distribution), enabling us to obtain the associated probabilities required for the process of pairs trade and creating an efficient tool for trading.

We analyse the time evolution of the empirical cross-sectional distribution of firms’ profit and growth rates. In particular, we analyse the conditional properties of the empirical distributions depending on the size of the firms and the business cycle phase. In order to do so, we employ the Laplace distribution as a benchmark, further considering the Subbotin and Asymmetric Exponential Power (AEP hereafter) distributions, to capture the potential asymmetry and leptokurtosis of the empirical distribution. Our results show that the profit rates of large firms are characterised by an asymmetric Laplace distribution with parameters largely independent of the business cycle phase. Small firms, instead, are characterised by the AEP distribution, which accounts for the conditional dependence of distribution on the phase of the business cycle. We observe that the largest firms are more robust to downturns compared to the small firms, given their invariant distributional characteristics during crisis periods.

The Global Navigation Satellite System (GNSS) allows for the cost-effective estimation of the ionospheric total electron content (TEC). However, research on error characteristics of the derived TEC is scarce, which provides insights into the quality of the GNSS ionospheric observation. We investigate characteristics of errors in the derived TEC with data from ~260 GNSS dual-frequency receivers of the Crustal Movement Observation Network of China (CMONOC). The slant TEC is calculated from carrier phase measurements and the vertical TEC over China is fitted with a spatial resolution of 1° by 1° in latitude and longitude in four seasons of 2014. It is found that the errors of both the slant TEC and the derived TEC follow Laplace distribution rather than Gaussian distribution in all seasons. The errors of the slant TEC have sharper peaks than those of the derived TEC. The Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) of the slant TEC are typically 0.04 TECU and 0.2 TECU, while the MAE and RMSE of the fitting residuals for the derived TEC are typically 1 TECU and under 2 TECU, respectively. Both MAEs and RMSEs of the derived TEC have the largest value in spring and the smallest value in summer, while the seasonal dependence is only observed in RMSE of the slant TEC.

This paper examines the return distributions of 7 markets in the Asian region, namely Hongkong, Indonesia, Malaysia, Korea, Japan, Shanghai, and Singapore, to find out whether the return distributions in those markets follow a specific distribution. Using data from January 2000 to September 2009, the return distributions of each market were constructed and was first fitted to the normal distribution to find out whether or not each market behaves according to the standard theory of finance and investment – which stated that the financial time series follow a random walk – and thus would fit the normal distribution. The result of fitting the return distributions of the 7 markets to normal distribution shows that none of the return distributions follows the normal distribution as evident from the leptokurtic phenomena marked by the excess kurtosis compared to the normal distribution curve and also from the fatter than normal distribution tails and the existence of returns that lie outside the area predictedby the normal distribution.The return distributions were then fitted to a series of theoretical probability distribution. Each of the distribution was fitted to the theoretical. The results are: the Hongkong and Shanghai markets follow the Laplace distribution while the other five markets: Indonesia, Malaysia, Korea, Japan, and Singapore follow the Johnson SU distribution.

Many environmental and physiological factors affect plant temperature. The objective of this study was to use thermal imagery to investigate robust features for early diagnosis of Botrytis cinerea infection in cut rose flowers under the postharvest application of Methyl Jasmonate (MeJA). Three cases treated with different concentrations of MeJA (0.1, 0.2, and 0.3 µl.l−1), a control (0 µl.l−1 MeJA) and an ethanol-treated case (20 µl.l−1 ethanol) were considered as five treatments in this study. Infrared images of MeJA-treated and non-treated flowers were captured during five consecutive days. Eight days after inoculation, disease severity in all concentrations of MeJA was significantly lower than that of control and ethanol treatments. Maximum temperature difference (MTD) index and median temperature could be used to diagnose the existence and growth of fungal pathogen, at least a day before any significant visual symptoms appear. To identify some robust features for classifying the infected and non-infected flowers, analysis of temperature frequency distribution was implemented. Laplace and normal distributions were considered as the best fitted probability distributions based on the shape of thermal histograms. Parameters of normal and Laplace probability density functions were estimated and the most effective attributes were selected. A radial-basis-function neural network with 60 neurons in the hidden layer was designed to classify and distinguish the infected flowers from the healthy ones. Results showed that the network can classify the infected and non-infected flowers with a 96.4% correct estimation rate.

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.

The representations of record ranges via sums of independent identically distributed exponential random variables are obtained for asymmetrical Laplace distributions. This result generalizes the corresponding relations for record values in the cases of exponential and negative exponential distributions

The conception of direct and inverse problem of random variable modelling is introduced. The direct problem is a problem for getting value of continuous random variable, which is contributed according to the given distribution law, which parameters are known. The inverse problem is a problem for defining the distribution law parameters, which are necessary for modelling of continuous one-dimensional random variable, for which the distribution law, mathematical expectation and dispersion are known. For its solution by known type of distribution it is necessary to find the parameter dependence of simulated distribution on set initial characteristics – ensemble average and standard deviation. The assigned problem is solved in explicit form for the following cases: normal distribution, exponential distribution, Laplace distribution, extreme value minimum distribution, extreme value maximum distribution, double exponential distribution, logistic distribution, gamma distribution, Erlang distribution of n-th order, Rayleigh distribution, Maxwellian distribution, parabolic distribution, Simpson distribution, arc sine distribution, inverse Gaussian distribution , Cauchy distribution, one-parameter distribution of n-dimansional random value, hyperexponential distribution, beta distribution, common- beta distribution, Birnbaum-Sanders distribution. For random variables, which are distributed according to the laws: Erlang second order, beta-distribution of second order, logarithmic normal distribution, it is described the interactive procedure to solve the modelling inverse problem, which realizes theNewton's method for solution of linear equation system. The expressions for elements of matrix solution are received. The solution procedure of assigned task for Weibull and Nakagami distribution is set, which is based on construction of regressive equations, which interpolate the table values to determine links of distribution law parameters and initial characteristics of random variable, which is distributed according to the given law.

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.

Multiresolution models such as the wavelet-domain hidden Markov tree (HMT) model provide a powerful approach for image modeling and processing because it captures the key features of the wavelet coefficients of real-world data. It is observed that the Laplace distribution is peakier in the center and has heavier tails compared with the Gaussian distribution. Thus we propose a new HMT model based on the two-state, zero-mean Laplace mixture model (LMM), the LMM-HMT, which provides significantly potential for characterizing real-world textures. By using the HMT segmentation framework, we develop LMM-HMT based segmentation methods for image textures and dynamic textures. The experimental results demonstrate the effectiveness of the introduced model and segmentation methods.

The Riemannian geometry of the space Pm, of m × m symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision and radar signal processing. Still, an open challenge remains, which consists of extending these tools to correctly handle the presence of outliers (or abnormal data), arising from excessive noise or faulty measurements. The present paper tackles this challenge by introducing new probability distributions, called Riemannian Laplace distributions on the space Pm. First, it shows that these distributions provide a statistical foundation for the concept of the Riemannian median, which offers improved robustness in dealing with outliers (in comparison to the more popular concept of the Riemannian center of mass). Second, it describes an original expectation-maximization algorithm, for estimating mixtures of Riemannian Laplace distributions. This algorithm is applied to the problem of texture classification, in computer vision, which is considered in the presence of outliers. It is shown to give significantly better performance with respect to other recently-proposed approaches.