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Methodological and Applied Contributions on Stochastic Modelling and Forecasting
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Methodological and Applied Contributions on Stochastic Modelling and Forecasting

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D1: Probability and Statistics".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 45519

Special Issue Editors

Prof. Dr. Ana M. Aguilera

E-Mail Website
Guest Editor
Department of Statistics and O.R. and IEMath-GR, University of Granada, 18071-Granada, Spain
Interests: functional data analysis; categorical data analysis; applied stochastic modelling and forecasting; data driven statistics in econometrics chemometrics, biomechanics, engineering, and survival analysis
Dr. M. Carmen Aguilera-Morillo

E-Mail Website
Guest Editor
Department of Applied Statistics and Operational Research, and Quality, Universitat Politècnica de València, 46022-València, Spain, and UC3M-BS Santander Big Data Institute, Madrid, Spain
Interests: functional regression models; smoothing techniques; variable selection methods and penalized regression models for high-dimensional data and applications in chemometrics, spectroscopy, biomedical science, and engineering

Special Issue Information

Dear Colleagues,

We are pleased to invite you to contribute to this Special Issue on “Methodological and Applied Contributions on Stochastic Modeling and Forecasting” with an original research article focused on theoretical or data-driven contributions for solving real problems in challenging research areas, such as biosciences or engineering.

With the advance of modern technology, more and more data are being recorded continuously on a discrete or continuous domain. Techniques that address these problems are framed in the theory of stochastic processes. The focus of this Issue is mainly on stochastic models under classical (Markov models and stochastic differential equations) and non-classical (functional data analysis and non-parametric regression) assumptions. Contributions with algorithms and computational tools that facilitate the application of the proposed methodologies will be especially appreciated.

Prof. Dr. Ana M. Aguilera
Dr. M. Carmen Aguilera-Morillo
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Stochastic processes
  • Functional data analysis
  • Smoothing techniques
  • Dynamic prediction
  • Data-driven stochastic models
  • Applications in biosciences, engineering. and other areas of interest

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Published Papers (15 papers)

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21 pages, 497 KiB  
Article
Causal Discovery of Stochastic Dynamical Systems: A Markov Chain Approach
by Marcell Stippinger, Attila Bencze, Ádám Zlatniczki, Zoltán Somogyvári and András Telcs
Mathematics 2023, 11(4), 852; https://doi.org/10.3390/math11040852 - 7 Feb 2023
Cited by 2 | Viewed by 2104
Abstract
Our proposed method for exploring the causal discovery of stochastic dynamic systems is designed to overcome the limitations of existing methods in detecting hidden and common drivers. The method is based on a simple principle and is presented in a nonparametric structural vector [...] Read more.
Our proposed method for exploring the causal discovery of stochastic dynamic systems is designed to overcome the limitations of existing methods in detecting hidden and common drivers. The method is based on a simple principle and is presented in a nonparametric structural vector autoregressive modeling framework. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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29 pages, 1067 KiB  
Article
Multisensor Fusion Estimation for Systems with Uncertain Measurements, Based on Reduced Dimension Hypercomplex Techniques
by Rosa M. Fernández-Alcalá, José D. Jiménez-López, Jesús Navarro-Moreno and Juan C. Ruiz-Molina
Mathematics 2022, 10(14), 2495; https://doi.org/10.3390/math10142495 - 18 Jul 2022
Cited by 2 | Viewed by 1840
Abstract
The prediction and smoothing fusion problems in multisensor systems with mixed uncertainties and correlated noises are addressed in the tessarine domain, under Tk-properness conditions. Bernoulli distributed random tessarine processes are introduced to describe one-step randomly delayed and missing measurements. Centralized and [...] Read more.
The prediction and smoothing fusion problems in multisensor systems with mixed uncertainties and correlated noises are addressed in the tessarine domain, under Tk-properness conditions. Bernoulli distributed random tessarine processes are introduced to describe one-step randomly delayed and missing measurements. Centralized and distributed fusion methods are applied in a Tk-proper setting, k=1,2, which considerably reduce the dimension of the processes involved. As a consequence, efficient centralized and distributed fusion prediction and smoothing algorithms are devised with a lower computational cost than that derived from a real formalism. The performance of these algorithms is analyzed by using numerical simulations where different uncertainty situations are considered: updated/delayed and missing measurements. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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29 pages, 778 KiB  
Article
Two Multi-Sigmoidal Diffusion Models for the Study of the Evolution of the COVID-19 Pandemic
by Antonio Barrera, Patricia Román-Román, Juan José Serrano-Pérez and Francisco Torres-Ruiz
Mathematics 2021, 9(19), 2409; https://doi.org/10.3390/math9192409 - 28 Sep 2021
Cited by 2 | Viewed by 2063
Abstract
A proposal is made to employ stochastic models, based on diffusion processes, to represent the evolution of the SARS-CoV-2 virus pandemic. Specifically, two diffusion processes are proposed whose mean functions obey multi-sigmoidal Gompertz and Weibull-type patterns. Both are constructed by introducing polynomial functions [...] Read more.
A proposal is made to employ stochastic models, based on diffusion processes, to represent the evolution of the SARS-CoV-2 virus pandemic. Specifically, two diffusion processes are proposed whose mean functions obey multi-sigmoidal Gompertz and Weibull-type patterns. Both are constructed by introducing polynomial functions in the ordinary differential equations that originate the classical Gompertz and Weibull curves. The estimation of the parameters is approached by maximum likelihood. Various associated problems are analyzed, such as the determination of initial solutions for the necessary numerical methods in practical cases, as well as Bayesian methods to determine the degree of the polynomial. Additionally, strategies are suggested to determine the best model to fit specific data. A practical case is developed from data originating from several Spanish regions during the first two waves of the COVID-19 pandemic. The determination of the inflection time instants, which correspond to the peaks of infection and deaths, is given special attention. To deal with this particular issue, point estimation as well as first-passage times have been considered. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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17 pages, 501 KiB  
Article
Calendar Effect and In-Sample Forecasting Applied to Mesothelioma Mortality Data
by Alex Isakson, Simone Krummaker, María Dolores Martínez-Miranda and Ben Rickayzen
Mathematics 2021, 9(18), 2260; https://doi.org/10.3390/math9182260 - 14 Sep 2021
Viewed by 2917
Abstract
In this paper, we apply and further illustrate a recently developed extended continuous chain ladder model to forecast mesothelioma deaths. Making such a forecast has always been a challenge for insurance companies as exposure is difficult or impossible to measure, and the latency [...] Read more.
In this paper, we apply and further illustrate a recently developed extended continuous chain ladder model to forecast mesothelioma deaths. Making such a forecast has always been a challenge for insurance companies as exposure is difficult or impossible to measure, and the latency of the disease usually lasts several decades. While we compare three approaches to this problem, we show that the extended continuous chain ladder model is a promising benchmark candidate for asbestosis mortality forecasting due to its flexible and simple forecasting strategy. Furthermore, we demonstrate how the model can be used to provide an update for the forecast of the number of deaths due to mesothelioma in Great Britain using in recent Health and Safety Executive (HSE) data. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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23 pages, 753 KiB  
Article
Out-of-Sample Prediction in Multidimensional P-Spline Models
by Alba Carballo, María Durbán and Dae-Jin Lee
Mathematics 2021, 9(15), 1761; https://doi.org/10.3390/math9151761 - 26 Jul 2021
Viewed by 1873
Abstract
The prediction of out-of-sample values is an interesting problem in any regression model. In the context of penalized smoothing using a mixed-model reparameterization, a general framework has been proposed for predicting in additive models but without interaction terms. The aim of this paper [...] Read more.
The prediction of out-of-sample values is an interesting problem in any regression model. In the context of penalized smoothing using a mixed-model reparameterization, a general framework has been proposed for predicting in additive models but without interaction terms. The aim of this paper is to generalize this work, extending the methodology proposed in the multidimensional case, to models that include interaction terms, i.e., when prediction is carried out in a multidimensional setting. Our method fits the data, predicts new observations at the same time, and uses constraints to ensure a consistent fit or impose further restrictions on predictions. We have also developed this method for the so-called smooth-ANOVA model, which allows us to include interaction terms that can be decomposed into the sum of several smooth functions. We also develop this methodology for the so-called smooth-ANOVA models, which allow us to include interaction terms that can be decomposed as a sum of several smooth functions. To illustrate the method, two real data sets were used, one for predicting the mortality of the U.S. population in a logarithmic scale, and the other for predicting the aboveground biomass of Populus trees as a smooth function of height and diameter. We examine the performance of interaction and the smooth-ANOVA model through simulation studies. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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19 pages, 3162 KiB  
Article
A Distance Correlation Approach for Optimum Multiscale Selection in 3D Point Cloud Classification
by Manuel Oviedo-de la Fuente, Carlos Cabo, Celestino Ordóñez and Javier Roca-Pardiñas
Mathematics 2021, 9(12), 1328; https://doi.org/10.3390/math9121328 - 9 Jun 2021
Cited by 7 | Viewed by 2518
Abstract
Supervised classification of 3D point clouds using machine learning algorithms and handcrafted local features as covariates frequently depends on the size of the neighborhood (scale) around each point used to determine those features. It is therefore crucial to estimate the scale or scales [...] Read more.
Supervised classification of 3D point clouds using machine learning algorithms and handcrafted local features as covariates frequently depends on the size of the neighborhood (scale) around each point used to determine those features. It is therefore crucial to estimate the scale or scales providing the best classification results. In this work, we propose three methods to estimate said scales, all of them based on calculating the maximum values of the distance correlation (DC) functions between the features and the label assigned to each point. The performance of the methods was tested using simulated data, and the method presenting the best results was applied to a benchmark data set for point cloud classification. This method consists of detecting the local maximums of DC functions previously smoothed to avoid choosing scales that are very close to each other. Five different classifiers were used: linear discriminant analysis, support vector machines, random forest, multinomial logistic regression and multilayer perceptron neural network. The results obtained were compared with those from other strategies available in the literature, being favorable to our approach. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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22 pages, 15662 KiB  
Article
Wildfires Vegetation Recovery through Satellite Remote Sensing and Functional Data Analysis
by Feliu Serra-Burriel, Pedro Delicado and Fernando M. Cucchietti
Mathematics 2021, 9(11), 1305; https://doi.org/10.3390/math9111305 - 7 Jun 2021
Cited by 4 | Viewed by 3951
Abstract
In recent years, wildfires have caused havoc across the world, which are especially aggravated in certain regions due to climate change. Remote sensing has become a powerful tool for monitoring fires, as well as for measuring their effects on vegetation over the following [...] Read more.
In recent years, wildfires have caused havoc across the world, which are especially aggravated in certain regions due to climate change. Remote sensing has become a powerful tool for monitoring fires, as well as for measuring their effects on vegetation over the following years. We aim to explain the dynamics of wildfires’ effects on a vegetation index (previously estimated by causal inference through synthetic controls) from pre-wildfire available information (mainly proceeding from satellites). For this purpose, we use regression models from Functional Data Analysis, where wildfire effects are considered functional responses, depending on elapsed time after each wildfire, while pre-wildfire information acts as scalar covariates. Our main findings show that vegetation recovery after wildfires is a slow process, affected by many pre-wildfire conditions, among which the richness and diversity of vegetation is one of the best predictors for the recovery. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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35 pages, 38288 KiB  
Article
Development of a Backward–Forward Stochastic Particle Tracking Model for Identification of Probable Sedimentation Sources in Open Channel Flow
by Chelsie Chia-Hsin Liu, Christina W. Tsai and Yu-Ying Huang
Mathematics 2021, 9(11), 1263; https://doi.org/10.3390/math9111263 - 31 May 2021
Cited by 3 | Viewed by 2201
Abstract
As reservoirs subject to sedimentation, the dam gradually loses its ability to store water. The identification of the sources of deposited sediments is an effective and efficient means of tackling sedimentation problems. A state-of-the-art Lagrangian stochastic particle tracking model with backward–forward tracking methods [...] Read more.
As reservoirs subject to sedimentation, the dam gradually loses its ability to store water. The identification of the sources of deposited sediments is an effective and efficient means of tackling sedimentation problems. A state-of-the-art Lagrangian stochastic particle tracking model with backward–forward tracking methods is applied to identify the probable source regions of deposited sediments. An influence function is introduced into the models to represent the influence of a particular upstream area on the sediment deposition area. One can then verify if a specific area might be a probable source by cross-checking the values of influence functions calculated backward and forward, respectively. In these models, the probable sources of the deposited sediments are considered to be in a grid instead of at a point for derivation of the values of influence functions. The sediment concentrations in upstream regions must be known a priori to determine the influence functions. In addition, the accuracy of the different types of diffusivity at the water surface is discussed in the study. According to the results of the case study of source identification, the regions with higher sediment concentrations computed by only backward simulations do not necessarily imply a higher likelihood of sources. It is also shown that from the ensemble results when the ensemble mean of the concentration is higher, the ensemble standard deviation of the concentration is also increased. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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17 pages, 1597 KiB  
Article
Bi-Smoothed Functional Independent Component Analysis for EEG Artifact Removal
by Marc Vidal, Mattia Rosso and Ana M. Aguilera 
Mathematics 2021, 9(11), 1243; https://doi.org/10.3390/math9111243 - 28 May 2021
Cited by 8 | Viewed by 4800
Abstract
Motivated by mapping adverse artifactual events caused by body movements in electroencephalographic (EEG) signals, we present a functional independent component analysis based on the spectral decomposition of the kurtosis operator of a smoothed principal component expansion. A discrete roughness penalty is introduced in [...] Read more.
Motivated by mapping adverse artifactual events caused by body movements in electroencephalographic (EEG) signals, we present a functional independent component analysis based on the spectral decomposition of the kurtosis operator of a smoothed principal component expansion. A discrete roughness penalty is introduced in the orthonormality constraint of the covariance eigenfunctions in order to obtain the smoothed basis for the proposed independent component model. To select the tuning parameters, a cross-validation method that incorporates shrinkage is used to enhance the performance on functional representations with a large basis dimension. This method provides an estimation strategy to determine the penalty parameter and the optimal number of components. Our independent component approach is applied to real EEG data to estimate genuine brain potentials from a contaminated signal. As a result, it is possible to control high-frequency remnants of neural origin overlapping artifactual sources to optimize their removal from the signal. An R package implementing our methods is available at CRAN. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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18 pages, 1784 KiB  
Article
Volatility Modeling: An Overview of Equity Markets in the Euro Area during COVID-19 Pandemic
by Pierdomenico Duttilo, Stefano Antonio Gattone and Tonio Di Battista
Mathematics 2021, 9(11), 1212; https://doi.org/10.3390/math9111212 - 27 May 2021
Cited by 22 | Viewed by 5851
Abstract
Volatility is the most widespread measure of risk. Volatility modeling allows investors to capture potential losses and investment opportunities. This work aims to examine the impact of the two waves of COVID-19 infections on the return and volatility of the stock market indices [...] Read more.
Volatility is the most widespread measure of risk. Volatility modeling allows investors to capture potential losses and investment opportunities. This work aims to examine the impact of the two waves of COVID-19 infections on the return and volatility of the stock market indices of the euro area countries. The study also focuses on other important aspects such as time-varying risk premium and leverage effect. This investigation employed the Threshold GARCH(1,1)-in-Mean model with exogenous dummy variables. Daily returns of the euro area stock markets indices from 4 January 2016 to 31 December 2020 has been used for the analysis. The results reveal that euro area stock markets respond differently to the COVID-19 pandemic. Specifically, the first wave of COVID-19 infections had a notable impact on stock market volatility of euro area countries with middle-large financial centres while the second wave had a significant impact only on stock market volatility of Belgium. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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21 pages, 658 KiB  
Article
New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques
by Alfonso García-Pérez
Mathematics 2021, 9(7), 762; https://doi.org/10.3390/math9070762 - 1 Apr 2021
Cited by 4 | Viewed by 2308
Abstract
Let Z(s)=(Z1(s),…,Zp(s))t be an isotropic second-order stationary multivariate spatial process. We measure the statistical association between the p random components of Z with [...] Read more.
Let Z(s)=(Z1(s),…,Zp(s))t be an isotropic second-order stationary multivariate spatial process. We measure the statistical association between the p random components of Z with the correlation coefficients and measure the spatial dependence with variograms. If two of the Z components are correlated, the spatial information provided by one of them can improve the information of the other. To capture this association, both within components of Z(s) and across s, we use a cross-variogram. Only two robust cross-variogram estimators have been proposed in the literature, both by Lark, and their sample distributions were not obtained. In this paper, we propose new robust cross-variogram estimators, following the location estimation method instead of the scale estimation one considered by Lark, thus extending the results obtained by García-Pérez to the multivariate case. We also obtain accurate approximations for their sample distributions using saddlepoint techniques and assuming a multivariate-scale contaminated normal model. The question of the independence of the transformed variables to avoid the usual dependence of spatial observations is also considered in the paper, linking it with the acceptance of linear variograms and cross-variograms. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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22 pages, 1901 KiB  
Article
Functional Modeling of High-Dimensional Data: A Manifold Learning Approach
by Harold A. Hernández-Roig, M. Carmen Aguilera-Morillo and Rosa E. Lillo
Mathematics 2021, 9(4), 406; https://doi.org/10.3390/math9040406 - 19 Feb 2021
Cited by 3 | Viewed by 2926
Abstract
This paper introduces stringing via Manifold Learning (ML-stringing), an alternative to the original stringing based on Unidimensional Scaling (UDS). Our proposal is framed within a wider class of methods that map high-dimensional observations to the infinite space of functions, allowing the use of [...] Read more.
This paper introduces stringing via Manifold Learning (ML-stringing), an alternative to the original stringing based on Unidimensional Scaling (UDS). Our proposal is framed within a wider class of methods that map high-dimensional observations to the infinite space of functions, allowing the use of Functional Data Analysis (FDA). Stringing handles general high-dimensional data as scrambled realizations of an unknown stochastic process. Therefore, the essential feature of the method is a rearrangement of the observed values. Motivated by the linear nature of UDS and the increasing number of applications to biosciences (e.g., functional modeling of gene expression arrays and single nucleotide polymorphisms, or the classification of neuroimages) we aim to recover more complex relations between predictors through ML. In simulation studies, it is shown that ML-stringing achieves higher-quality orderings and that, in general, this leads to improvements in the functional representation and modeling of the data. The versatility of our method is also illustrated with an application to a colon cancer study that deals with high-dimensional gene expression arrays. This paper shows that ML-stringing is a feasible alternative to the UDS-based version. Also, it opens a window to new contributions to the field of FDA and the study of high-dimensional data. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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16 pages, 1167 KiB  
Article
A Complex Model via Phase-Type Distributions to Study Random Telegraph Noise in Resistive Memories
by Juan E. Ruiz-Castro, Christian Acal, Ana M. Aguilera and Juan B. Roldán
Mathematics 2021, 9(4), 390; https://doi.org/10.3390/math9040390 - 16 Feb 2021
Cited by 10 | Viewed by 2353
Abstract
A new stochastic process was developed by considering the internal performance of macro-states in which the sojourn time in each one is phase-type distributed depending on time. The stationary distribution was calculated through matrix-algorithmic methods and multiple interesting measures were worked out. The [...] Read more.
A new stochastic process was developed by considering the internal performance of macro-states in which the sojourn time in each one is phase-type distributed depending on time. The stationary distribution was calculated through matrix-algorithmic methods and multiple interesting measures were worked out. The number of visits distribution to a determine macro-state were analyzed from the respective differential equations and the Laplace transform. The mean number of visits to a macro-state between any two times was given. The results were implemented computationally and were successfully applied to study random telegraph noise (RTN) in resistive memories. RTN is an important concern in resistive random access memory (RRAM) operation. On one hand, it could limit some of the technological applications of these devices; on the other hand, RTN can be used for the physical characterization. Therefore, an in-depth statistical analysis to model the behavior of these devices is of essential importance. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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11 pages, 365 KiB  
Article
Compression-Based Methods of Time Series Forecasting
by Konstantin Chirikhin and Boris Ryabko
Mathematics 2021, 9(3), 284; https://doi.org/10.3390/math9030284 - 31 Jan 2021
Cited by 2 | Viewed by 3413
Abstract
Time series forecasting is an important research topic with many practical applications. As shown earlier, the problems of lossless data compression and prediction are very similar mathematically. In this article, we propose several forecasting methods based on real-world data compressors. We consider predicting [...] Read more.
Time series forecasting is an important research topic with many practical applications. As shown earlier, the problems of lossless data compression and prediction are very similar mathematically. In this article, we propose several forecasting methods based on real-world data compressors. We consider predicting univariate and multivariate data, describe how multiple data compressors can be combined into one forecasting method with automatic selection of the best algorithm for the input data. The developed forecasting techniques are not inferior to the known ones. We also propose a way to reduce the computation time of the combined method by using the so-called time-universal codes. To test the proposed techniques, we make predictions for real-world data such as sunspot numbers and some social indicators of Novosibirsk region, Russia. The results of our computations show that the described methods find non-trivial regularities in data, and time universal codes can reduce the computation time without losing accuracy. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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19 pages, 874 KiB  
Article
Limiting Genotype Frequencies of Y-Linked Genes with a Mutant Allele in a Two-Sex Population
by Miguel González, Cristina Gutiérrez and Rodrigo Martínez
Mathematics 2021, 9(2), 131; https://doi.org/10.3390/math9020131 - 9 Jan 2021
Viewed by 1933
Abstract
A two-type two-sex branching process is considered to model the evolution of the number of carriers of an allele and its mutations of a Y-linked gene. The limiting growth rates of the different types of couples and males (depending on the allele, [...] Read more.
A two-type two-sex branching process is considered to model the evolution of the number of carriers of an allele and its mutations of a Y-linked gene. The limiting growth rates of the different types of couples and males (depending on the allele, mutated or not, that they carry on) on the set of coexistence of both alleles and on the fixation set of the mutant allele are obtained. In addition, the limiting genotype of the Y-linked gene and the limiting sex frequencies on those sets are established. Finally, the main results have been illustrated with simulated studies contextualized in problems of population genetics. Full article
(This article belongs to the Special Issue Methodological and Applied Contributions on Stochastic Modelling and Forecasting)
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