Multivariate Stochastic Model Research Articles

A multivariate stochastic model for the generation of synthetic time series at multiple time scales reproducing long-term persistence

A time series generator is presented, employing a robust three-level multivariate scheme for stochastic simulation of correlated processes. It preserves the essential statistical characteristics of historical data at three time scales (annual, monthly, daily), using a disaggregation approach. It also reproduces key properties of hydrometeorological and geophysical processes, namely the long-term persistence (Hurst–Kolmogorov behaviour), the periodicity and intermittency. Its efficiency is illustrated through two case studies in Greece. The first aims to generate monthly runoff and rainfall data at three reservoirs of the hydrosystem of Athens. The second involves the generation of daily rainfall for flood simulation at five rain gauges. In the first emphasis is given to long-term persistence – a dominant characteristic in the management of large-scale hydrosystems, comprising reservoirs with carry-over storage capacity. In the second we highlight to the consistent representation of intermittency and asymmetry of daily rainfall, and the distribution of annual daily maxima.

Studious by association? Effects of teacher’s attunement to students’ peer relations

We study the role of teacher-perceived friendship on teachers’ assessments of their students’ learning efforts. Based on earlier studies, we expect that teacher-perceived friendship differs from students’ actual, self-reported friendship, and can account for teachers’ judgement of students’ learning efforts, above and beyond effects of students’ self-reported study effort and friendship. Data were collected as part of the Networks and Actor Attributes in Early Adolescence study, which traces the development of individual behaviour, norms, and classroom social structure in 126 school classes (3332 students, 14 schools) during the first year at secondary school in the Netherlands. Students’ individual study effort was assessed based on students’ self-reports of four effort-related behaviours and as an overall judgement by the class teacher. Students’ networks were assessed with the classroom as boundary, on the one hand as students’ self-reported friendships, on the other hand as friendships perceived by the class teachers. Results were obtained with multivariate stochastic actor-based modelling of the feedback processes between study effort, study norms, and friendship networks.

From Amazon to Apple: Modeling Online Retail Sales, Purchase Incidence, and Visit Behavior

In this study, we propose a multivariate stochastic model for Web site visit duration, page views, purchase incidence, and the sale amount for online retailers. The model is constructed by composition from carefully selected distributions and involves copula components. It allows for the strong nonlinear relationships between the sales and visit variables to be explored in detail, and can be used to construct sales predictions. The model is readily estimated using maximum likelihood, making it an attractive choice in practice given the large sample sizes that are commonplace in online retail studies. We examine a number of top-ranked U.S. online retailers, and find that the visit duration and the number of pages viewed are both related to sales, but in very different ways for different products. Using Bayesian methodology, we show how the model can be extended to a finite mixture model to account for consumer heterogeneity via latent household segmentation. The model can also be adjusted to accommodate a more accurate analysis of online retailers like apple.com that sell products at a very limited number of price points. In a validation study across a range of different Web sites, we find that the purchase incidence and sales amount are both forecast more accurately using our model, when compared to regression, probit regression, a popular data-mining method, and a survival model employed previously in an online retail study. Supplementary materials for this article are available online.

Forecasting bunker prices; A nonstationary, multivariate methodology

This paper suggests a methodological approach for the forecasting of marine fuel prices. The prediction of the bunker prices is of outmost importance for operators, as bunker prices affect heavily the economic planning and financial viability of ventures and determine decisions related to compliance with regulations. A multivariate nonstationary stochastic model available in the literature is being retrieved, after appropriate adjustment and testing. The model belongs to the class of periodically correlated stochastic processes with annual periodic components. The time series are appropriately transformed to become Gaussian, and then are decomposed to deterministic seasonal characteristics (mean value and standard deviation) and a residual time series. The residual part is proved to be stationary and then is modeled as a Vector AutoRegressive Mooving Average (VARMA) process. Finally, using the methodology presented, forecasts of a tetra-variate and an octa-variate time series of bunker prices are produced and are in good agreement with actual values. The obtained results encourages further research and deeper investigation of the driving characters of the multivariate time series of bunker prices.

Optimal placement and sizing of plug in electric vehicles charging stations within distribution networks with high penetration of photovoltaic panels

This paper proposes a scenario optimization based algorithm in order to allocate charging station of plug in electric vehicles (PEVs) within a commercial area with the aim of increasing penetration level of photovoltaic (PV) panels as well as decreasing side effects of vehicular loads. A multivariate stochastic modeling methodology based on the notion of copula is provided in order to develop a probabilistic model of the load demand due to PEVs. The suggested method considers the correlation issue among the related random variables of the PEVs. This model, in addition to the models provided for PV generation and commercial loads, is utilized to construct synthetic data required for the optimal allocation of the station. Considering the prepared scenarios, particle swarm optimization (PSO) algorithm is utilized to minimize energy loss as well as voltage deviation in the distribution system. The proposed methodology has been applied on a test distribution network in order to demonstrate its efficiency in simultaneous integration of vehicular loads and PV generations.

From Amazon to Apple: Modeling Online Retail Sales, Purchase Incidence and Visit Behavior

In this study we propose a multivariate stochastic model for website visit duration, page views, purchase incidence and the sale amount for online retailers. The model is constructed by composition from carefully selected distributions, and involves copula components. It allows for the strong nonlinear relationships between the sales and visit variables to be explored in detail, and can be used to construct sales predictions. The model is readily estimated using maximum likelihood, making it an attractive choice in practice given the large sample sizes that are commonplace in online retail studies. We examine a number of top-ranked U.S. online retailers, and find that the visit duration and the number of pages viewed are both related to sales, but in very different ways for different products. Using Bayesian methodology we show how the model can be extended to a finite mixture model to account for consumer heterogeneity via latent household segmentation. The model can also be adjusted to accommodate a more accurate analysis of online retailers like apple.com that sell products at a very limited number of price points. In a validation study across a range of different websites, we find that the purchase incidence and sales amount are both forecast more accurately using our model, when compared to regression, probit regression, a popular data-mining method and a survival model employed previously in an online retail study.

Generalised polynomial chaos-based uncertainty quantification for planning MRgLITT procedures

Purpose: A generalised polynomial chaos (gPC) method is used to incorporate constitutive parameter uncertainties within the Pennes representation of bioheat transfer phenomena. The stochastic temperature predictions of the mathematical model are critically evaluated against MR thermometry data for planning MR-guided laser-induced thermal therapies (MRgLITT). Methods: The Pennes bioheat transfer model coupled with a diffusion theory approximation of laser tissue interaction was implemented as the underlying deterministic kernel. A probabilistic sensitivity study was used to identify parameters that provide the most variance in temperature output. Confidence intervals of the temperature predictions are compared to MR temperature imaging (MRTI) obtained during phantom and in vivo canine (n = 4) MRgLITT experiments. The gPC predictions were quantitatively compared to MRTI data using probabilistic linear and temporal profiles as well as 2-D 60 °C isotherms. Results: Optical parameters provided the highest variance in the model output (peak standard deviation: anisotropy 3.51 °C, absorption 2.94 °C, scattering 1.84 °C, conductivity 1.43 °C, and perfusion 0.94 °C). Further, within the statistical sense considered, a non-linear model of the temperature and damage-dependent perfusion, absorption, and scattering is captured within the confidence intervals of the linear gPC method. Multivariate stochastic model predictions using parameters with the dominant sensitivities show good agreement with experimental MRTI data. Conclusions: Given parameter uncertainties and mathematical modelling approximations of the Pennes bioheat model, the statistical framework demonstrates conservative estimates of the therapeutic heating and has potential for use as a computational prediction tool for thermal therapy planning.

Multivariate stochastic downscaling models for generating precipitation and temperature scenarios of climate change based on atmospheric circulation

The spatial resolution of General Circulation Models (GCMs) is too coarse to represent regional climate changes at the scales required for environmental impact assessment. Therefore, downscaling of precipitation and temperature has to be carried out from the GCM grids to smaller scales of a few square kilometres. Daily precipitation and temperature are modelled as stochastic processes coupled to atmospheric circulation. Precipitation is linked to circulation patterns (CPs) using conditional model parameters. Temperature is modelled using a simple autoregressive model conditioned on atmospheric circulation and local daily precipitation. The models use an automated objective classification of daily atmospheric circulation patterns based on optimized fuzzy rules. Both temperature and precipitation are downscaled to several locations taking into account the CP dependent spatial correlation. The models were applied to the Mesochora medium-sized mountainous catchment in Central Greece for validation using observed precipitation and temperature and observed classified geo-potential heights (at 700 hPa). GCM scenarios of the ECHAM4 model for 1xCO2 and 2xCO2 cases were used to make climate change predictions (by using classified GCM geopotential heights). Simulated values agree fairly well with historical data. Most of the GCM results (incl. mean daily values, renewal process probabilities, spell lengths) under the 2xCO2 case reflect a somewhat wetter and a more variable precipitation regime over the Mesochora catchment with significantly increased daily mean temperatures.

Asymptotically Optimal Methods of Early Change-Point Detection

In this article the problem of early change-point detection is considered. The main difference from traditional sequential change-point detection problems consists in the nature of changes: we consider both gradual and abrupt changes and wish to detect instants of the beginning of these changes as soon as possible on condition that false alarms are few. We prove the theoretical informational inequalities for the main performance characteristics of early change-point detection methods that help us to consider the asymptotically optimal methods. Our criterion of asymptotic optimality does not coincide with traditional criteria of Shiryaev (1963), Lorden (1971), and Pollak (1985) but allows us to consider both univariate and multivariate nonstationary stochastic models. Besides the theoretical analysis of univariate and multivariate models with changes, we present results of the Monte Carlo study of the proposed methods.

Pricing and Hedging Variable Annuity Guarantees with Multiasset Stochastic Investment Models

Variable annuities are often sold with guarantees to protect investors from downside investment risk. The majority of variable annuity guarantees are written on more than one asset, but in practice, single-asset (univariate) stochastic investment models are mostly used for pricing and hedging these guarantees. This practical shortcut may lead to problems such as basis risk. In this article, we contribute a multivariate framework for pricing and hedging variable annuity guarantees. We explain how to transform multivariate stochastic investment models into their risk-neutral counterparts, which can then be used for pricing purposes. We also demonstrate how dynamic hedging can be implemented in a multivariate framework and how the potential hedging error can be quantified by stochastic simulations.

The Estimation of the Firm Hydroelectric Potential of the Çoruh Basin, Turkey

The main aim of this study is to estimate firm hydroelectric potential of Çoruh Basin, which is one of the most picturesque, yet significantly underdeveloped, regions of Turkey. For this purpose, the most suitable multivariate stochastic model for daily streamflow observations of nine stations located in the basin must first be obtained. Then, using the best model, a total of 100 synthetic streamflow series, each having the same length with the historical time series, are generated. Finally, upper and lower limits of firm hydroelectric potential of Çoruh Basin are calculated within 95% confidence intervals.

Nonparametric multivariate weather generator and an extreme value theory for bandwidth selection

Summary A multivariate stochastic generation model for daily weather variables is proposed that extends the multivariate k -nearest neighbor resampling approach (MKNN). Major drawbacks of the MKNN approach include repetitive historical multivariate patterns, underestimating variance and serial correlation, and reshuffling of historical data. These drawbacks cause under-generation of events that are extreme in their frequency and magnitude. In this study, these drawbacks are addressed by applying a stochastic optimization technique (i.e., a genetic algorithm (GA)) and a perturbation using a gamma kernel density estimate (GKDE). The competitive selection operator in the GA was used to better preserve the historical variance and serial correlation as well as to produce unprecedented multivariate patterns. By employing the GKDE, the resampled precipitation data are perturbed, and thus new precipitation values are generated. To preserve the distribution of the annual maximum events fitted to a general extreme value (GEV), the GKDE bandwidth was selected by employing the statistics of the historical annual maximum. The proposed method was applied to generate six daily weather variables (maximum temperature, minimum temperature, dew point temperature, solar radiation, wind speed, and precipitation) of the summer season (June–September) for a station in Seoul, South Korea. The presented results indicate that the suggested weather generator is an appropriate alternative for generating daily weather variables while reproducing the historical extreme distribution.

Multivariate Exponential Smoothing for Forecasting Tourist Arrivals

In this article, we propose a new set of multivariate stochastic models that capture time-varying seasonality within the vector innovations structural time-series (VISTS) framework. These models encapsulate exponential smoothing methods in a multivariate setting. The models considered are the local level, local trend, and damped trend VISTS models with an additive multivariate seasonal component. We evaluate the forecasting accuracy of these models against the forecasting accuracy of univariate alternatives using international tourist arrivals from 11 source countries to Australia and New Zealand. In general, the newly proposed multivariate models improve on forecast accuracy over the univariate alternatives.

Ordering of multivariate risk models with respect to extreme portfolio losses

Abstract The notion of asymptotic portfolio loss order is introduced to compare multivariate stochastic risk models with respect to extreme portfolio losses. In the framework of multivariate regular variation comparison criteria are derived in terms of spectral measures. This allows for analytical and numerical verification in applications. Worst and best case dependence structures with respect to the asymptotic portfolio loss order are determined. Comparison criteria in terms of further stochastic ordering notions are derived. The examples include elliptical distributions and multivariate regularly varying models with Gumbel, Archimedean, and Galambos copulas. Particular interest is paid to the inverse influence of dependence on the diversification of risks with infinite expectations.

Reduced Rank Stochastic Regression with a Sparse Singular value Decomposition

SummaryFor a reduced rank multivariate stochastic regression model of rank r*, the regression coefficient matrix can be expressed as a sum of r* unit rank matrices each of which is proportional to the outer product of the left and right singular vectors. For improving predictive accuracy and facilitating interpretation, it is often desirable that these left and right singular vectors be sparse or enjoy some smoothness property. We propose a regularized reduced rank regression approach for solving this problem. Computation algorithms and regularization parameter selection methods are developed, and the properties of the new method are explored both theoretically and by simulation. In particular, the regularization method proposed is shown to be selection consistent and asymptotically normal and to enjoy the oracle property. We apply the proposed model to perform biclustering analysis with microarray gene expression data.

Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo.

Computational systems biology is concerned with the development of detailed mechanistic models of biological processes. Such models are often stochastic and analytically intractable, containing uncertain parameters that must be estimated from time course data. In this article, we consider the task of inferring the parameters of a stochastic kinetic model defined as a Markov (jump) process. Inference for the parameters of complex nonlinear multivariate stochastic process models is a challenging problem, but we find here that algorithms based on particle Markov chain Monte Carlo turn out to be a very effective computationally intensive approach to the problem. Approximations to the inferential model based on stochastic differential equations (SDEs) are considered, as well as improvements to the inference scheme that exploit the SDE structure. We apply the methodology to a Lotka-Volterra system and a prokaryotic auto-regulatory network.

A multivariate semi-logistic autoregressive process and its characterization

An autoregressive multivariate stochastic model is constructed which yields a stationary Markov process with a marginal invariant distribution as a multivariate semi-logistic distribution. This model is denoted as an MSL-AR(1) process. Some properties of the MSL-AR(1) process are studied and its characterization is also derived.

A Multivariate Stochastic Hybrid Model with Switching Coefficients and Jumps: Solution and Distribution

In this work, a class of multidimensional stochastic hybrid dynamic models is studied. The system under investigation is a first‐order linear nonhomogeneous system of Itô‐Doob type stochastic differential equations with switching coefficients. The switching of the system is governed by a discrete dynamic which is monitored by a non‐homogeneous Poisson process. Closed‐form solutions of the systems are obtained. Furthermore, the major part of the work is devoted to finding closed‐form probability density functions of the solution processes of linear homogeneous and Ornstein‐Uhlenbeck type systems with jumps.

Penalized maximum likelihood estimation of a stochastic multivariate regression model

We study the large-sample properties of the penalized maximum likelihood estimator of a multivariate stochastic regression model with contemporaneously correlated data. The penalty is in terms of the square norm of some (vector) linear function of the regression coefficients. The model subsumes the so-called common transfer function model useful for extracting common signals in a panel of short time series. We show that, under mild regularity conditions, the penalized maximum likelihood estimator is consistent and asymptotically normal. The asymptotic bias of the regression coefficient estimator is also derived.

Photothermal Quotient Specifications to Improve Wheat Cultivar Yield Component Models

Previous research has demonstrated the importance and statistical significance of the photothermal quotient (PTQ) to predict and explain wheat (Triticum aestivum L.) yields. The objective of this study is to respecify PTQ to enhance the explanatory power of statistical models used to explain grains per square meter (GM−2), and increase understanding of weather's impact on yields. The primary objective is to identify and quantify potential gains from including separate components of solar radiation (Solar) and temperature (Temp) in place of PTQ (Solar/Temp) to improve wheat yield model explanatory ability. The study also determines the optimal time interval that defines PTQ by varying the number of days before and after 50% anthesis. Using wheat test plot data from Centro Internacional de Mejoramiento de Maíz y Trigo (CIMMYT) in Mexico's Yaqui Valley, a multivariate regression‐based stochastic model of wheat yields was used to estimate the impact of altering PTQ definition and specification. Results support previous research: the maximum F‐test value of 66.02 and adjusted R2 value of 0.446 were obtained for 31 d before to 1 d after 50% anthesis. Interpretation and analysis were also enhanced by disaggregating PTQ into separate variables Solar and Temp. A 1 MJ m−2 d−1 increase in Solar increased GM−2 by 1.25%, whereas a 1°C increase in Temp decreased GM−2 by 2.8%. This difference in yield responsiveness to weather components results in greater statistical significance, explanatory power, and interpretation of GM−2 models. Future research that builds on these results will better explain, predict, and forecast crop yields.