Stochastic Realization Research Articles

Probabilistic delineation of subsurface connected pathways in alluvial aquifers under geological uncertainty

The present work aims to address the issue of the delineation of subsurface pathways carved in sedimentary aquifers from a stochastic perspective. A probabilistic approach to recognize preferential alluvial pathways is proposed relying only upon borehole data. These data can be employed to identify locations where the surficial aquifer is characterized by highly permeable (alluvial) hydrofacies. A Monte Carlo (MC) approach is set up to simulate stochastic realizations of local alluvial soil ratios in a bi-dimensional regularly gridded domain. Alluvial pathways are defined as the collection of cells whose simulated alluvial ratios are greater or equal to a first threshold, imposed on the simulated sediments ratio, and sharing a minimum occurrence probability, equal to a second probabilistic threshold. Then, probable highly permeable subsurface pathways throughout the domain for a real case-study application, within the province of Lecco (Lombardy, Northern Italy), are appraised. Subsurface alluvial pathways delineated upon geological data show channel-like patterns, like their surface counterparts. Subsurface discharge peaks in the central portion of these pathways, where permeable sediment probability is higher than elsewhere. Moreover, subsurface alluvial pathways features are consistent with available hydrogeological and piezometric information. The present approach has been also compared with a Multiple-Points Statistics approach, proving better at capturing the spatial patterns of connected subsurface pathways. Therefore, the present approach seems promising to be employed in the preliminarily understanding of large-scale groundwater circulation patterns, in particular for poorly monitored areas or serving as further analysis background.

Stochastic Merging of Soil Hydraulic Properties for Vadose Zone Hydrological Modeling

Soil hydraulic properties (SHPs) are commonly determined in soil samples with replicas. Whether these replicas are taken at a same location to represent a specific point or at several locations to represent a larger area, the results should be merged into a final data set to be used in modeling. For this data set to be representative, standard errors and a correlation matrix must be considered in the merging process. We present a method to perform this merging and give an example using stochastic realizations of van Genuchten-Mualem (VGM) parameters generated by Cholesky decomposition to merge the SHP and associated statistics into a merged parameter set. To do so, we used VGM parameters obtained at sample scale in three replicas from a Brazilian savanna soil through inverse modeling of laboratory evaporation experiments. The effectiveness and representativeness of the proposed methodology were evaluated by observing the frequency distribution of different levels of output, comparing individual and merged sample properties. The outputs include VGM parameters, retention and conductivity characteristics, and water balance components stochastically predicted by a hydrological model. The performed stochastic merging correctly represented the variability of the combined replicas, especially with respect to hydrological model outputs of soil water balance components. Using the mean hydraulic property parameter values to deterministically predict water balance components may yield values that are substantially different from the mean values of stochastic realizations. This suggests that the deterministic prediction using mean parameter values in vadose zone hydrological modeling may result in unrepresentative outputs.

Robust stock assortment and cutting under defects in automotive glass production

We address an assortment‐and‐cutting problem arising in the glass industry. The objective is to provide minimum waste solutions that are robust against such raw material imperfections as those possibly occurring with float glass production technology. The stochastic realization of defects is modeled as a spatial Poisson point process. A mixed integer program in the classical vein of robust optimization is presented and tested on data taken from a real plant application. Defective final products must in any case be discarded as waste but, if a recourse strategy is adopted, faults in glass sheets can sometimes be recovered. Closed forms for the computation of faulty item probabilities are provided in simple cases, and obtained via Monte Carlo simulation in more complex ones. The computational results demonstrate the benefits of the robust approach in terms of the reduction of back‐orders and overproduction, thereby showing that recourse strategies can enable nonnegligible improvements. Encouraged by this result, the management is presently evaluating the possibility of adopting the proposed model in plant operation.

Spontaneous symmetry breaking in frustrated triangular atom arrays due to cooperative light scattering

We demonstrate the presence of an optical phase transition with frustration-induced spontaneous symmetry breaking in a triangular planar atomic array due to cooperative light-mediated interactions. We show how the array geometry of triangle unit cells at low light intensities leads to degenerate collective radiative excitations forming nearly flat bands. We drive degenerate pairs of collective excitations to be equally populated in both cases of the atomic polarization in the lattice plane and perpendicular to it. At higher intensities, above specific threshold values, this symmetry in the populations is spontaneously broken. We also develop an effective few-mode model that provides semianalytic descriptions of the symmetry-breaking threshold and infinite-lattice limit phase transition. Surprisingly, we find how excitations due to dipolar interactions correspond to optical analogs of those found in frustrated magnets and superfluids, with closely related symmetry-breaking mechanisms despite the significant physical differences between these systems, opening potential for simulating even quantum magnetism. Transmitted light through the array conveys information about symmetry breaking in the hysteresis behavior of the spectrum. Moreover, in a Mott-insulator state, the atomic positions are subject to zero-point quantum fluctuations. Interpreting each stochastic realization as a light-induced quantum measurement of the atomic position configuration, we find how strong nonlinearities and even weak position uncertainties lead to considerable measurement-induced symmetry breaking, while ensemble-averaging over many realizations restores the original symmetry and the unbroken state. Larger position uncertainty results in the formation of domains of different broken symmetries.

A Realization Approach to Lossy Network Compression of a Tuple of Correlated Multivariate Gaussian RVs.

Examined in this paper is the Gray and Wyner source coding for a simple network of correlated multivariate Gaussian random variables, and . The network consists of an encoder that produces two private rates and , and a common rate , and two decoders, where decoder 1 receives rates and reproduces by , and decoder 2 receives rates and reproduces by , with mean-square error distortions . Use is made of the weak stochastic realization and the geometric approach of such random variables to derive test channel distributions, which characterize the rates that lie on the Gray and Wyner rate region. Specific new results include: (1) A proof that, among all continuous or finite-valued random variables, , Wyner’s common information, , is achieved by a Gaussian random variable, of minimum dimension n, which makes the two components of the tuple conditionally independent according to the weak stochastic realization of , and a the formula where are the canonical correlation coefficients of the correlated parts of and , and a realization of which achieves this. (2) The parameterization of rates that lie on the Gray and Wyner rate region, and several of its subsets. The discussion is largely self-contained and proceeds from first principles, while connections to prior literature is discussed.

A Lubrication‐Based Solver for Shear‐Thinning Flow in Rough Fractures

AbstractThe depth‐averaged (2D) lubrication theory is often adopted to simulate Newtonian flow in rough fractures. This approach, which is computationally much less expensive than using 3D CFD solvers, allows addressing large ensembles of stochastic fracture realizations. For creeping flow, the degree of approximation introduced is limited as long as the apertures vary relatively smoothly. We propose the first generalization of this approach addressing the flow of fluids whose rheology, described by the Ellis model, is shear‐thinning (ST) above a crossover shear stress and Newtonian (of viscosity μ0) below. The resulting nonlinear Reynolds equation for pressures is solved for a vast range of realistic rheological parameter values using a novel and specifically designed finite volume‐based numerical model. The spatial discretization takes inspiration from the graph p‐Laplacian to yield a symmetric Newton Jacobian, allowing for a highly efficient inexact implementation of the preconditioned conjugate gradient‐based Newton‐Krylov method. This is combined with a parameter continuation strategy to increase code robustness and ensure global convergence for flow indices as low as 0.1 with an excellent efficiency. This original solver is used to investigate realistic synthetic rough fracture geometries, which exhibits both self‐affinity and a correlation length. The results show that the ST rheology mitigates the effects of aperture heterogeneities, increasing fracture transmissivity by several orders of magnitudes as compared to the Newtonian flow of viscosity μ0 if the imposed macroscopic gradient is sufficiently large, and even rendering the rough fracture up to 10 times more permeable than a smooth fracture of identical mean aperture.

On the doubly stochastic realization of spectra

An n-list $$\lambda = \left( r; \lambda _2, \ldots , \lambda _n\right)$$ of complex numbers with $$r>\max _{2\le j\le n}|\lambda _j|,$$ is said to be realizable if $$\lambda$$ is the spectrum of $$n\times n$$ nonnegative matrix A and in this case A is said to be a nonnegative realization of $$\lambda .$$ If, in addition, each row and column sum of A is equal to r, then $$\lambda$$ is said to be doubly stochastically realizable and in such case A is said to be a doubly stochastic realization for $$\lambda .$$ In 1997, Guo proved that if $$\left( \lambda _2,\ldots , \lambda _n\right)$$ is any list of complex numbers which is closed under complex conjugation then there exists a least real number $$\lambda _0$$ with $$\max _{2\le j\le n}|\lambda _j|\le \lambda _0\le 2n\max _{2\le j\le n}|\lambda _j|$$ such that the list of complex numbers $$\left( \lambda _1,\lambda _2,\ldots ,\lambda _n\right)$$ is realizable if and only if $$\lambda _1\ge \lambda _0.$$ Many researchers deal with sharpening this upper bound and others are concerned with finding the exact value of $$\lambda _0$$ for particular classes of matrices (Andrade et al. in Linear Algebra Appl 556:301–322, 2018; Andrade et al. in Linear Algebra Appl 551:36–56, 2018; Julio and Soto in Electron J Linear Algebra 36:484–502, 2020; Robbiano in Linear Algebra Appl 564:15–27, 2019). In this paper, we first describe a method for passing from a nonnegative realization to a doubly stochastic realization. As applications, we give a new sufficient condition for a stochastic matrix A to be cospectral to a doubly stochastic matrix B and in this case B is shown to be the unique closest doubly stochastic matrix to A with respect to the Frobenius norm. Then, our next result gives an improvement of Guo’s bound which also sharpens the existing known bound for the case when one of at least one of $$\left\{ \lambda _2, \ldots , \lambda _n\right\}$$ is real. Furthermore, we investigate the case when $$\left\{ \lambda _2, \ldots , \lambda _n\right\}$$ are all non-real which has not been dealt before. Our main results here also sharpen Guo’s bound. Next, for doubly stochastic realizations, we obtain an upper bound that improves Guo’s bound as well. Finally, for certain particular cases, we give a further improvement of our last bound for doubly stochastic realization.

Modelling CO2 plume spreading in highly heterogeneous rocks with anisotropic, rate-dependent saturation functions: A field-data based numeric simulation study of Otway

In this field-data-based simulation study, we apply novel rate-dependent, anisotropic saturation functions curve-fitted to match the flow behaviour of laminated sand- and siltstones from the CO2CRC's Otway International Test Centre (Australia), in well-spot simulations of plume spreading at the site. Scenario analysis is conducted by performing simulations on various stochastic model realisations under different conditions, investigating what controls plume migration. Results indicate that high-permeability streaks and intraformational baffles determine plume spreading in flat-lying stratigraphy. Displacement of CO2 is unstable and focussed by high-permeability streaks, leading to multi-layer plumes. In inclined stratigraphy, buoyancy forces control plume migration channelling it up dip. However, high-permeability streaks still influence plume shape. Comparisons for different injection rates show that heterogeneity-induced fingering dominates at high injection rates, whereas gravity override and vertical movement of the plume are prominent at low injection rates. Continuous build-up of fluid pressure in the storage horizon influences the properties of CO2 when no-flow (closed) lateral boundary conditions are applied. By contrast, a marked pressure drop occurs in the near-well region for hydrostatic (open) lateral boundary conditions. These also foster uneven heterogeneity-fingering of the plume. Since flow velocity in the highly heterogeneous Paaratte formation varies over many orders of magnitude, force balances range between the capillary and the viscous limit even at some distance to the injector. A comparison with simulations based on standard Brooks-Corey saturation functions shows that taking flow-rate dependence and anisotropy into account leads to significant differences in the shape of the evolving CO2 plume and the saturation within.

Exploring the role of Northeast Atlantic cod in the Barents Sea food web using a multi‐model approach

AbstractIt is commonly accepted that no ecosystem model is the ‘best’, but rather that ecosystem models should be used in ensembles. This is also the case for the Barents Sea ecosystem, where we have used two different ecosystem models to explore the role of the top‐predator Northeast Arctic (NEA) stock of Atlantic cod (Gadus morhua, Gadidae) in the food web. The two models differ in complexity; Gompertz being less complex in terms of food web (7 components) and processes compared to the complex Nordic and Barents Seas Atlantis model (53 components). On the other hand, Gompertz provides thousands of stochastic realizations for each scenario, whereas Atlantis provides only one deterministic simulation. To compare the response to changes in NEA cod on two key prey species, capelin (Mallotus villosus, Osmeridae) and polar cod (Boreogadus saida, Gadidae), we perturbed the historical fishing pressure by ±50% and used the same NEA cod biomass in both models. Even though the links between NEA cod and the prey species are similar in the two models, the results from the study reveal that indirect effects through other food‐web components might be as important as direct predator–prey interactions. Differences in spatial structure and overlap between species also influence the species response to the perturbations. In this study, we focus on the mechanisms that drives the changes in the models, and advise on potential consequences for fisheries management. The two models can complement each other, and the differences between them point to areas where more knowledge is needed.

How does the quantity, resolution, and scaling of turbulence boxes affect aeroelastic simulation convergence?

Aeroelastic wind turbine simulations are extensively used in the turbine design process to estimate structural fatigue loading. While it is assumed that more stochastic realisations (seeds) and higher resolution simulations will converge to the ‘true’ result faster, a detailed study has not been published to support how many seeds and what resolution is required to achieve an arbitrary level of convergence. In this study, we perform a comprehensive study on the convergence of turbine fatigue statistics as a function of turbulence grid resolution, number of seeds, and the type of turbulence scaling. The study is performed using the HAWC2 aeroelastic code in conjunction with Mann turbulence fields of varying resolution. Key findings are that over 21 seeds are required to ensure a standard error of the mean of less than 5% for fatigues statistics, and a turbulence resolution of at least 2048×64×64 (in the longitudinal, lateral and vertical, respectively) is required to ensure statistical convergence of a 10 minute simulation on one turbine. The presented study provides a valuable insight into the uncertainties involved in turbine fatigue load calculations, as well as recommendations on how to mitigate them.

Analytical representations of inherent structural irregularities in turnout crossing with moveable point

The inherent structural irregularities in turnouts are an impact source during train passing. The position of the wheel–rail contact point on the rails and the wheel trajectory can help reveal the characteristics of such structural irregularities. In this work, for a given nominal turnout, several parameterised single harmonic-shaped functions are selected as analytical representations to describe the inherent structural irregularities in a crossing with moveable point using multiobjective optimisation. The objective of the optimisation is achieved by ensuring that the responses of the analytical representation applied to a plain track are close to those of the turnout model under the constraint of maximum and minimum wheel–rail contact forces as well as a correlation coefficient. Thus, a plain track with irregularity functions can be considered an equivalent model to enable a fast modelling procedure and efficient assessment of vehicles. For a robust design in different traffic situations, two stochastic variables, namely the rail pad stiffness and axle load, are selected and are found to be significant. The influence of the random distributions of the input parameters on the statistical outputs of the wheel–rail contact forces is assessed using a descriptive sampling technique for several stochastic realizations.

Complete Characterization of Gorbunov and Pinsker Nonanticipatory Epsilon Entropy of Multivariate Gaussian Sources: Structural Properties

This paper derives the optimal test channel distribution and the complete characterization of the classical Gorbunov and Pinsker (1973), Gorbunov and Pinsker (1974) nonanticipatory epsilon entropy of multivariate Gaussian Markov sources with square-error fidelity, which remained an open problem since 1974. The paper also formulates a state dependent nonanticipatory epsilon entropy, in which past reproductions are available to the decoder and not to the encoder, the test channel is specified with respect to an auxiliary (state) random process, and the reproduction process is a causal function of past reproduction and the auxiliary random process. This variation is analogous to the Wyner and Ziv (1976) and Wyner (1978) rate distortion function (RDF), of memoryless sources. It is shown that the operational rate of zero-delay codes, with past reproductions available to the decoder but not to the encoder is bounded below by the state dependent nonanticipatory epsilon entropy rate. For the case of multivariate Gaussian Markov sources with square-error fidelity, the optimal test channel distribution and the complete characterization of the state dependent of nonanticipatory epsilon entropy are derived, and also shown that that the two nonanticipatory epsilon entropies coincide. The derivations are new; they are based on structural properties of the stochastic realizations of the reproduction process that induce the optimal test channel distributions. They are derived using, achievable lower bounds on information theoretic measures, properties of mean-square estimation theory, Hadamard’s inequality, and canonical correlation coefficients of a tuple of multivariate jointly Gaussian random processes. Applications of the nonanticipatory epsilon entropy and its state dependent variation are discussed to the areas of control of unstable Gaussian systems over limited memory channels, design of causal estimators for Gaussian Markov sources with a fidelity criterion, computation of the rate loss of causal and zero-delay codes of Gaussian Markov sources with respect to non-causal codes.

Exponential Growth Model and Stochastic Population Models: A Comparison via Population Data

A population dynamic model explains the changes of a population in the near future, given its current status and the environmental conditions that the population is exposed to. In modelling a population dynamic, deterministic model and stochastic models are used to describe and predict the observed population. For modelling population size deterministic model may provide sufficient biological understanding about the system, but if the population numbers do become small, then a stochastic model is necessary with certain conditions. In this study, both types of models such exponential, discrete-time Markov chain (DTMC), continuous-time Markov chain (CTMC) and stochastic differential equation (SDE) are applied to goat population data. Results from the simulations of stochastic realisations as well as deterministic counterparts are shown and tested by root mean square error (RMSE). The SDE model gives the smallest RMSE value which indicate the best model in fitting the data.

Spectral element numerical simulation of the 2009 L’Aquila earthquake on a detailed reconstructed domain

SUMMARY In this paper, we simulate the earthquake that hit the city of L’Aquila on 2009 April 6 using SPEED (SPectral Elements in Elastodynamics with Discontinuous Galerkin), an open-source code able to simulate the propagation of seismic waves in complex 3-D domains. Our model includes an accurate 3-D reconstruction of the Quaternary deposits, according to the most up-to-date data obtained from the Microzonation studies in Central Italy and a detailed model of the topography incorporated using a newly developed tool. The sensitivity of our results with respect to different kinematic seismic sources is investigated. The results obtained are in good agreement with the recordings at the available seismic stations at epicentral distances within a range of 20 km. Finally, a blind source prediction scenario application shows that a reasonably good agreement between simulations and recordings can be obtained by simulating stochastic rupture realizations with basic input data. These results, although limited to nine simulated scenarios, demonstrate that it is possible to obtain a satisfactory reconstruction of a ground shaking scenario employing a stochastic source constrained on a limited amount of ex-ante information. A similar approach can be used to model future and past earthquakes for which little or no information is typically available, with potential relevant implications for seismic risk assessment.

Joint Spatial Modeling of Nutrients and Their Ratio in the Sediments of Lake Balaton (Hungary): A Multivariate Geostatistical Approach

Eutrophication, water quality, and environmental status of lakes is a global issue that depends not only on external loadings from industrial, agricultural, and municipal sources but often also on internal loadings from lake sediments. In the latter case, in addition to the quality and quantity of nutrients stored in sediments, their relative content may be an important factor. In the example of Lake Balaton, we jointly modeled the spatial distribution of the nutrients nitrogen (N) and phosphorus (P) and their ratio (i.e., N:P) in the sediments of the lake and then provided spatial predictions at different scales (i.e., point, basin, and entire lake) with the associated uncertainty. Our aim was to illustrate the merits of applying multivariate geostatistics when spatial modeling of more than one variable is targeted at various scales in water ecosystems. Variography confirmed that there is a spatial interdependence between the nutrients. The results revealed that multivariate geostatistics allows this interdependence to be taken into account and exploited to provide coherent and accurate spatial models. Additionally, stochastic realizations, reproducing the joint spatial variability, can be generated that allow providing spatially aggregated predictions with the associated uncertainty at various scales. Our study highlighted that it is worthy of applying multivariate geostatistics in case spatial modeling of two or more variables, which jointly vary in space, is targeted in water ecosystems.

Efficient Stochastic Computing FIR Filtering Using Sigma-Delta Modulated Signals

This work presents a soft-filtering digital signal processing architecture based on sigma-delta modulators and stochastic computing. A sigma-delta modulator converts the input high-resolution signal to a single-bit stream enabling filtering structures to be realized using stochastic computing’s negligible-area multipliers. Simulation in the spectral domain demonstrates the filter’s proper operation and its roll-off behavior, as well as the signal-to-noise ratio improvement using the sigma-delta modulator, compared to typical stochastic computing filter realizations. The proposed architecture’s hardware advantages are showcased with synthesis results for two FIR filters using FPGA and synopsys tools, while comparisons with standard stochastic computing-based hardware realizations, as well as with conventional binary ones, demonstrate its efficacy.

Deterministic and Stochastic Seismic Inversion techniques towards a better prediction for the reservoir distribution in NEAG-2 Field, north Western Desert, Egypt

NEAG 2 field is one of the key economic oil fields in the company that was contributing to 35% of total liquid production in 2016. Despite that, the full field potentiality is still not estimated precisely. The field lies within the Northeast part of Abu El-Gharadig (NEAG Extension concession). Albian-Cenomanian Kharita-Bahariya is the targeted reservoir section. The reservoir distribution trends are not clear and cannot be predicted easily. Thus, the main uncertainty is to trace the reservoir extension. Since inversions of seismic into absolute impedance rely on well data for their parameterization, they are typically conducted in brownfields like NEAG 2. In addition, the reservoir quality and thickness in NEAG 2 encouraged the application of the seismic inversion approach, to gain a large-scale insight into the reservoir distribution qualitatively and quantitatively, if possible. Also, to mitigate the reservoir trends uncertainty, and assess its spatial continuity in the uncalibrated areas. There were some concerns about the seismic quality, although, several trials were implemented to get the best results and overcome the seismic vagueness. The post-stack seismic inversion workflow was applied in Hampson-Russel Software Suite. A Deterministic approach was followed by a Stochastic approach in an adopted workflow. The workflow was accomplished by generating several Stochastic Impedance realizations. To cover the uncertainty range, P10, P50 and, P90 realizations were generated and analyzed. Additionally, a connectivity analysis was done, to compute the probability of communication between the wells. The inverted volume helped to distinguish between different reservoir bodies' geometries and dimensions, which cannot be revealed from the conventional seismic interpretation. The reservoir bodies were resolved and quantified as channels and bars. Followed by the Stochastic workflow which produced multiple realizations, allowing the seismic uncertainty to be assessed. This research work will be the first of its kind for the Kharita-Bahariya section in NE Abu-Gharadig Basin.

Bayesian Inversion for Surface Magnetic Resonance Tomography Based on Geostatistics

Magnetic resonance tomography (MRT) has the advantages of direct, quantitative and unique interpretation in the field of groundwater detection. Currently, the inversion of MRT data primarily uses the QT (Q-time) inversion method based on Tikhonov regularization. However, when the heterogeneity of an aquifer is high, and the water content distribution is markedly uneven, this method removes many details in the model and cannot perform uncertainty analysis on the results. To solve these problems, we propose a Bayesian inversion for MRT data based on geostatistics. The method uses previously known geological data, such as drilling, to determine prior information model containing variograms and mixture Gaussian probability distributions for generating many stochastic realizations. Under the Bayesian framework, a modified Markov chain Monte Carlo strategy (MCMC) is used to obtain the posterior probability distributions of subsurface aquifers and hydraulic conductivity, and the results of quantitative uncertainty analysis. By comparing the inversion performance for simulated models, the imaging result of the Bayesian method is found to be markedly more accurate than that of the QT method for subsurface two-dimensional aquifers, particularly in explaining the stochastic model (i.e., the water-bearing model with an uneven distribution of water content). This method can also intuitively quantify the uncertainty of the imaging results, which mitigates the shortcomings of existing inversion methods. This paper also discusses the effects of prior information, number of chains and noise levels on the results, and also validates the effectiveness and practicability of the proposed method using field-measured data.

Two parameter optimization methods of multi-point geostatistics

Multi-point geostatistics has been widely used, but it involves many parameters when applying the method. These modeling parameters are usually sensitive to the quality of the model, such as the template size of the simpat algorithm, the number of nodes in the neighborhood searched of the Snesim algorithm, and the multigrid in common use. What kind of evaluation criteria for selecting the value and ranges of the parameters has become an urgent problem to be solved. Some methods to optimize these modeling parameters have been proposed. These methods either statistics the pattern feature distribution of training image, or use some certain spatial structure as the index to quantitatively evaluate the quality of training image reconstruction of stochastic realizations, or use cross-validation to compare the sample matching degree between the realization and the training image. However, these methods have their limitations in optimizing modeling parameters. In this paper, two methods to optimize the modeling parameters of multi-point geostatistics were proposed based on gray level co-occurrence matrix and convolutional neural network. Both methods can optimize the appropriate parameters by analyzing the influence of the parameters on modeling quality. Taking the frequently used parameters of Snesim and simpat as an example, the parameter optimization results of the GLCM-based method and the Deep Learning-based method are very close, which indicates that both methods can effectively extract and distinguish the spatial features of the model. In addition, the relationship between modeling parameters and evaluation indexes accords with the visual features of stochastic models. Therefore, the method proposed in this paper will be helpful to solve the difficulty of parameter setting in multi-point geostatistical modeling.

The Geometric Brownian Motion of Indosat Telecommunications Daily Stock Price During the Covid-19 Pandemic in Indonesia

One of the major telecommunication and network service providers in Indonesia is PT Indosat Tbk. During the coronavirus (COVID-19) pandemic, the daily stock price of that company was influenced by government policies. This study addresses stock data movement from February 5, 2020 to February 5, 2021, resulted in 243 data, using the Geometric Brownian motion (GBM). The stochastic process realization of this stock price fluctuates and increases exponentially, especially in the 40 latest data. Because of this situation, the realization is transformed into log 10 and calculated its return. As a result, weak stationary in variance is obtained. Furthermore, only data from December 7, 2020 to February 5, 2021 fulfill the GBM assumption of stock price return, as . The main idea of this study is adding datum one by one as much as 10% – 15% of the total data , starting from December 4, 2020 backwards. Following this procedure, and based on the 3% < p-value < 10%, the study shows that its datum can be included in , so and form five other data groups, . Considering Mean Absolute Percentage Error (MAPE) and amount of data from each group, is selected for modelling. Thus, GBM succeeded in representing the stock price movement of the second most popular Indonesian telecommunication company during COVID-19 pandemic.