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This paper introduces a novel non-homogeneous stochastic diffusion process, useful for modeling both decreasing and increasing trend data. The model is based on a generalized Gamma-like curve. We derive the probabilistic characteristics of the proposed process, including a closed-form unique solution to the stochastic differential equation, the transition probability density function, and both conditional and unconditional trend functions. The process parameters are estimated using the maximum likelihood (ML) method with discrete sampling paths. A small Monte Carlo experiment is conducted to evaluate the finite sample behavior of the trend function. The practical utility of the proposed process is demonstrated by fitting it to two real-world data sets, one exhibiting a decreasing trend and the other an increasing trend. [ABSTRACT FROM AUTHOR]

Spatial interpolation of meteorological data can have immense implications on risk management and climate change planning. Kriging with external drift (KED) is a spatial interpolation variant that uses auxiliary information in the estimation of target variables at unobserved locations. However, traditional KED methods with linear trend functions may not be able to capture the complex and non-linear interdependence between target and auxiliary variables, which can lead to an inaccurate estimation. In this work, a novel KED method using least squares support vector regression (LSSVR) is proposed. This machine learning algorithm is employed to construct trend functions regardless of the type of variable interrelations being considered. To evaluate the efficiency of the proposed method (KED with LSSVR) relative to the traditional method (KED with a linear trend function), a systematic simulation study for estimating the monthly mean temperature and pressure in Thailand in 2017 was conducted. The KED with LSSVR is shown to have superior performance over the KED with the linear trend function. [ABSTRACT FROM AUTHOR]

Manufactures always face a big challenge to obtain the sufficient reliability information for high‐quality products when only a relatively short time period is available for an internal life testing. Fortunately, if QCs exists, whose degradation over time can be related to reliability, an approach is suggested to collect sufficient degradation data for estimating the product's lifetime distribution more accurately. In numerical applications, the gamma process (GP) is commonly used when the degradation path is strictly increasing. However, in some circumstances, the GP is not able to successfully capture the degradation path. One resolution to tackle this problem is to consider the random effect into the GP model. In this paper, we propose a trend gamma process (TGP), which integrated the merits of trend function into a GP model, as an alternative approach to overcome this obstacle. The proposed TGP model is a generalized formulation of GP model, and it is attempted to transform a monotonic stochastic process into a GP. The results based on simulation studies and real data examples illustrate that our proposed TGP model is suitable for widely use when the degradation path is monotonically increasing. [ABSTRACT FROM AUTHOR]

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Objective: The aim of this publication is to analyze the value of the number of new cases and deaths from COVID-19 in selected European Union countries: Poland, France and Belgium. Research Design & Methods: Data were collected from the on-line database: https://covid.ourworldindata.org/data/owid-covid-data.xlsx, which demonstrate the daily number of new cases and deaths due to the COVID-19 pandemic. The forecast was based on a linear trend function and a 7-period moving average, using Statistica 13 software. Findings: The test results facilitated an evaluation of the diversity in the number of cases and the number of deaths in the assessed countries. Implications & Recommendations: From the obtained results, it can be concluded that the pandemic caused by the SARS-CoV-2 virus will end in 2021, about a year after the first case that appeared in Europe, provided that the vaccines are also effective against the mutated form of the virus. Implications & Recommendations: Based on the results obtained by China, where the pandemic ended after a year, it can be assumed that EU countries will also win the fight against Covid-19 at a similar time provided that the vaccines are also effective against the mutated forms of the virus. This is indicated by the results of research obtained in this paper. However, it should be remembered that the pandemic is unpredictable and it is difficult to predict the values of variables for a longer period of time. Contribution & Value Added: The article indicates the methods of combating Covid-19 in selected countries of the European Union. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of nonparametric estimation of trend function for stochastic differential equations driven by a weighted fractional Brownian motion (weighted-fBm). Under some general conditions, the consistent uniform, the rate of convergence as well as the asymptotic normality of our estimator are established. In addition, a numerical example is provided to illustrate the validity of the considered estimator. [ABSTRACT FROM AUTHOR]

Discrete groundwater level datasets are interpolated often using kriging group of models to produce a spatially continuous groundwater level map. There is always some level of uncertainty associated with different interpolation methods. Therefore, we developed a new trend function with the mean groundwater level as a drift variable in the regression kriging approach to predict the groundwater levels at the unvisited locations. Groundwater level data for 29 observation wells in Adyar River Basin were used to assess the performance of the developed regression kriging models. The cross-validation results shows that the proposed regression kriging method in the spatial domain outperforms other physical and kriging-based methods with R2 values of 0.96 and 0.98 during pre-monsoon and post-monsoon seasons, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study a new one-dimensional homogeneous stochastic process, termed the Square of the Brennan-Schwartz model, which is used in various contexts. We first establish the probabilistic characteristics of the model, such as the analytical expression solution to Itô's stochastic differential equation, after which we determine the trend functions (conditional and non-conditional) and the likelihood approach in order to estimate the parameters in the drift. Then, in the diffusion coefficient, we consider the problem of parameter estimation, doing so by a numerical approximation. Finally, we present an application to population growth by the use of real data, namely the growth of the total population aged 65 and over, resident in the Arab Maghreb, to illustrate the research methodology presented. [ABSTRACT FROM AUTHOR]

Surrogate modeling is commonly used to replace expensive simulations of engineering problems. Kriging is a popular surrogate for deterministic approximation due to its good nonlinear fitting ability. Previous researches demonstrate that constructing an appropriate trend function or a better stochastic process can improve the prediction accuracy of Kriging. However, they are not improved simultaneously to estimate the model parameters, thus limiting the further improvement on the prediction capability. In this paper, a novel penalized blind likelihood Kriging (PBLK) method is proposed to obtain better model parameters and improve the prediction accuracy. It improves the trend function and stochastic process with regularization techniques simultaneously. First, the formulation of the penalized blind likelihood function is introduced, which penalizes the regression coefficients and correlation parameters at the same time. It is a general expression and therefore can incorporate any type of penalty functions easily. To maximize the penalized blind likelihood function effectively and efficiently, a nested optimization algorithm is proposed to estimate the model parameters sequentially with gradient and Hessian information. As different regularization parameters can lead to different optimal model parameters and influence the prediction accuracy, a cross-validation-based grid search method is proposed to select good regularization parameters. The proposed PBLK method is tested on several analytical functions and two engineering examples, and the experimental results confirm the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Kriging is a popular surrogate for approximating computationally expensive computer experiments. When sample points are limited, it is difficult to identify the overall trend of the problem at hand properly. Thanks to the interpolating characteristic of the Kriging model, a constant is widely used as the trend function, which neglects the overall trend presented by data. However, previous researches prove that an appropriate trend function considering high-order terms is able to enhance the approximation ability of the Kriging model. In this paper, a regularization approach is proposed to construct the trend function in the Kriging model to improve the prediction accuracy. First, a new weighting scheme, which is formulated as an optimization problem with regularization terms, is used to solve the regression coefficients. Then, the other model parameters are estimated by maximizing the likelihood function, which leads to a nested optimization problem. It is solved iteratively to obtain the final estimation of the model parameters. From a Bayesian point of view, the proposed regularization method can adaptively tune the parameter of the prior distribution on the regression coefficients in the iterative algorithm. To select good regularization parameters, a cross-validation method is used. The implementation is tested on several analytical examples and physical examples, and the experimental results confirm the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Trend renewal process (TRP) is a generalized stochastic point process with two elements of a nonhomogeneous Poisson process (NHPP) and a renewal process (RP), and plays a significant role to represent a sub-class of general repair models. Nonparametric estimation of TRPs receives considerable concern in estimating the statistical properties of repairable systems under uncertainty, when the parametric forms of NHPP and/or RP cannot be known in advance. In this paper, we provide an alternative nonparametric kernel-based approach of the TRP for a given trend function which is equivalent to the intensity function of NHPP. In details, we estimate the TRPs with the kernel-based estimator and apply two bandwidth estimation methods which are based on the cross-validation technique. We evaluate effectiveness of our proposed kernel estimation algorithms throughout simulation experiments and, show application examples for a preventive maintenance problem of repairable systems. [ABSTRACT FROM AUTHOR]

This paper considers testing procedures for the null hypothesis of a unit root process against the alternative of a fractional process, called a fractional unit root test. We extend the Lagrange Multiplier (LM) tests of Robinson (1994) and Tanaka (1999), which are locally best invariant and uniformly most powerful, to allow for a slope change in trend with or without a concurrent level shift under both the null and alternative hypotheses. We show that the limit distribution of the proposed LM tests is standard normal. Finite sample simulation experiments show that the tests have good size and power. As an empirical analysis, we apply the tests to the Consumer Price Indices of the G7 countries. [ABSTRACT FROM AUTHOR]

Copyright of Research Papers of the Wroclaw University of Economics / Prace Naukowe Uniwersytetu Ekonomicznego we Wroclawiu is the property of Uniwersytet Ekonomiczny we Wroclawiu and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Adulthood trajectories of outcomes such as depression and the sense of control measure aspects of the human condition that Americans may view as objects of change. Social science should provide information on that progress, or its absence. Whether these trajectories change their shape, and how and why if they do, is important theoretically too. A range of birth cohorts coexist in time, place, and social relationship. Each cohort, as it goes through adulthood, follows in aggregate a path left by older ones, reshaping that path as it goes. The shapes of the trajectories, and the trends reshaping them, represent two inseparable aspects of the same phenomenon. This report describes methods for mapping aging trajectories and intercohort trends, using linear latent-growth models of relatively brief follow-up data (six years in the examples). The author reviews shared research ideals that led to the model: put theory into modeling, go where the data lead, use what you have, go beyond where you have been, and risk being precisely wrong. [ABSTRACT FROM PUBLISHER]

Spatiotemporal models have been applied in several scientific disciplines. A crucial problem is estimation of the trend function. Although nonparametric regression for spatial data has been studied in many papers, it is not the case for spatio-temporal data. In this article, we propose a local linear fitting method for spatio-temporal data and investigate the problem under what conditions the proposed method can work well. To guarantee the uniformly weakly consistent and asymptotically normal properties, it is just required that at a fixed location i0, {R(i0, t), t∈Tn} is strictly stationary, at a fixed moment t0, {R(i, t0), i∈Λn} is strictly stationary which is weaker than {R(i, t), i∈Λn, t∈Tn} is strictly stationary both in time and space locations. This assumption can be met often in practice, and the proposed estimation method can be applied widely. The simulation results and case study show that the estimator performs well. [ABSTRACT FROM AUTHOR]

In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data. [ABSTRACT FROM AUTHOR]