Direct Sampling Algorithm Research Articles

A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting

We present a faster direct sampling algorithm for random equilateral closed polygons in three-dimensional space. This method improves on the moment polytope sampling algorithm of Cantarella et al (2016 J. Phys. A: Math. Theor. 49 275202) and has (expected) time per sample quadratic in the number of edges in the polygon. We use our new sampling method and a new code for computing invariants based on the Alexander polynomial to investigate the probability of finding unknots among equilateral closed polygons.

Direct Sampling for Spatially Variable Extreme Event Generation in Resampling‐Based Stochastic Weather Generators

AbstractResampling‐based weather generators simulate new time series of weather variables by reordering the observed values such that the statistics of the simulated data are consistent with the observed ones. These generators are fully data‐driven, easy to implement, and capable of reproducing the dynamics among weather variables. However, although the simulated time series is new, the weather fields produced at arbitrary time steps are replicas of those found in observations, limiting the spatial variability of simulations and preventing the generation of extreme weather fields beyond the range of observed values. To address these limitations, we propose the integration of the Direct Sampling algorithm—a data‐driven method for producing simulations—into resampling‐based weather generators. By incorporating Direct Sampling as a post‐processing step on the outputs of the weather generator, we enhance the spatial variability of the generated weather fields and enable the generation of extreme weather fields. We introduce an approach for generating out‐of‐sample extreme weather fields using Direct Sampling. This method involves utilizing a set of control points in conjunction with Direct Sampling, where the values of these control points are informed by return period analysis. The proposed approach is validated using precipitation, temperature, and cloud cover weather fields in a region of northwest India. The experimental results confirm that Direct Sampling enhances the spatial variability of the weather fields and facilitates the generation of out‐of‐sample precipitation fields that accurately adhere to the spatial statistics provided by return precipitation level maps, as well as the observed precipitation weather field employed in the analysis.

Integrating Multi-Point Geostatistics, Machine Learning, and Image Correlation for Characterizing Positional Errors in Remote-Sensing Images of High Spatial Resolution

Remote-sensing images of high spatial resolution (HSR) are valuable sources of fine-grained spatial information for various applications, such as urban surveys and governance. There is continuing research on positional errors in remote-sensing images and their impacts in geoprocessing and applications. This paper explores the combined use of multi-point geostatistics (MPS), machine learning—in particular, generalized additive modeling (GAM)—and computer-image correlation for characterizing positional errors in images—in particular, HSR images. These methods are employed because of the merits of MPS in being flexible for non-parametric and joint simulation of positional errors in X and Y coordinates, the merits of GAM in being capable of handling non-stationarity in-positional errors through error de-trending, and the merits of computer-image correlation in being cost-effective in furnishing the training data (TD) required in MPS. Procedurally, image correlation is applied to identify homologous image points in reference-test image pairs to extract image displacements automatically in constructing TD. To cope with the complexity of urban scenes and the unavailability of truly orthorectified images, visual screening is performed to clean the raw displacement data to create quality-enhanced TD, while manual digitization is used to obtain reference sample data, including conditioning data (CD), for MPS and test data for performance evaluation. GAM is used to decompose CD and TD into trends and residuals. With CD and TD both de-trended, the direct sampling (DS) algorithm for MPS is applied to simulate residuals over a simulation grid (SG) at 80 m spatial resolution. With the realizations of residuals and, hence, positional errors generated in this way, the means, standard deviation, and cross correlation in bivariate positional errors at SG nodes are computed. The simulated error fields are also used to generate equal-probable realizations of vertices that define some road centerlines (RCLs), selected for this research through interpolation over the aforementioned simulated error fields, leading to error metrics for the RCLs and for the lengths of some RCL segments. The enhanced georectification of the RCLs is facilitated through error correction. A case study based in Shanghai municipality, China, was carried out, using HSR images as part of generalized point clouds that were developed. The experiment results confirmed that by using the proposed methods, spatially explicit positional-error metrics, including means, standard deviation, and cross correlation, can be quantified flexibly, with those in the selected RCLs and the lengths of some RCL segments derived easily through error propagation. The reference positions of these RCLs were obtained through error correction. The positional accuracy gains achieved by the proposed methods were found to be comparable with those achieved by conventional image georectification, in which the CD were used as image-georectification control data. The proposed methods are valuable not only for uncertainty-informed image geolocation and analysis, but also for integrated geoinformation processing.

A parsimonious parametrization of the Direct Sampling algorithm for multiple-point statistical simulations

Multiple-point statistics algorithms allow modeling spatial variability from training images. Among these techniques, the Direct Sampling (DS) algorithm has advanced capabilities, such as multivariate simulations, treatment of non-stationarity, multi-resolution capabilities, conditioning by inequality or connectivity data. However, finding the right trade-off between computing time and simulation quality requires tuning three main parameters, which can be complicated since simulation time and quality are affected by these parameters in a complex manner. To facilitate the parameter selection, we propose the Direct Sampling Best Candidate (DSBC) parametrization approach. It consists in setting the distance threshold to 0. The two other parameters are kept (the number of neighbors and the scan fraction) as well as all the advantages of DS. We present three test cases that prove that the DSBC approach allows to identify efficiently parameters leading to comparable or better quality and computational time than the standard DS parametrization. We conclude that the DSBC approach could be used as a default mode when using DS, and that the standard parametrization should only be used when the DSBC approach is not sufficient.

Markov chains improve the significance computation of overlapping genome annotations.

MotivationGenome annotations are a common way to represent genomic features such as genes, regulatory elements or epigenetic modifications. The amount of overlap between two annotations is often used to ascertain if there is an underlying biological connection between them. In order to distinguish between true biological association and overlap by pure chance, a robust measure of significance is required. One common way to do this is to determine if the number of intervals in the reference annotation that intersect the query annotation is statistically significant. However, currently employed statistical frameworks are often either inefficient or inaccurate when computing P-values on the scale of the whole human genome.ResultsWe show that finding the P-values under the typically used ‘gold’ null hypothesis is -hard. This motivates us to reformulate the null hypothesis using Markov chains. To be able to measure the fidelity of our Markovian null hypothesis, we develop a fast direct sampling algorithm to estimate the P-value under the gold null hypothesis. We then present an open-source software tool MCDP that computes the P-values under the Markovian null hypothesis in time and memory, where m and n are the numbers of intervals in the reference and query annotations, respectively. Notably, MCDP runtime and memory usage are independent from the genome length, allowing it to outperform previous approaches in runtime and memory usage by orders of magnitude on human genome annotations, while maintaining the same level of accuracy.Availability and implementationThe software is available at https://github.com/fmfi-compbio/mc-overlaps. All data for reproducibility are available at https://github.com/fmfi-compbio/mc-overlaps-reproducibility.Supplementary information Supplementary data are available at Bioinformatics online.

Local Ranking of Geological Conceptual Models in Non-stationary Settings Using Multi-point Geostatistics

In geomodeling, it is commonly accepted that the distribution of physical properties is controlled by the architecture of geological objects. However, insufficient data and the complexity of earth processes create an ill-posed problem where many architectures are plausible. Consequently, several geologists will produce different geological models for the same location. This contribution proposes a way to objectivize the ranking of those conceptual models by comparing them with hard data, both globally for the whole study region and locally for certain of its sectors. The idea is to extend the multi-point geostatistics direct sampling algorithm to be able to extract data events from different training images, representing several competing geological models, and to record the training image origin of values pasted on simulation grid cells. By tracking the frequency with which every training image is visited, we can rank the likelihood of each geological model. Histograms of the frequency of usage of each training image will provide a global ranking of the several conceptual models, while maps of these frequencies can be used to produce the local rankings. We demonstrate this method in two synthetic fluvial depositional environments where three distinct geological concepts are being proposed, with different abundances of hard data. Results indicate that the proposed method could be a useful tool in defining which geological concept dominates at a particular region and which is the frequency ranking for each training image on that region.

Direct sampling of projected entangled-pair states

Variational Monte Carlo studies employing projected entangled-pair states (PEPS) have recently shown that they can provide answers to long-standing questions such as the nature of the phases in the two-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. The sampling in these Monte Carlo algorithms is typically performed with Markov chain Monte Carlo algorithms employing local update rules, which often suffer from long autocorrelation times and interdependent samples. We propose a sampling algorithm that generates independent samples from a PEPS, bypassing all problems related to finite autocorrelation times. This algorithm is a generalization of an existing direct sampling algorithm for unitary tensor networks. We introduce an auxiliary probability distribution from which independent samples can be drawn, and combine it with importance sampling in order to evaluate expectation values accurately. We benchmark our algorithm on the classical Ising model and on variational optimization of two-dimensional quantum spin models.

An enhanced direct sampling (DS) approach to model the geological domain with locally varying proportions: Application to Golgohar iron ore mine, Iran

A critical challenge in evaluating mineral resources is to model the uncertainty of geological boundaries and to classify the block model into domains of similar rock or ore types. The aim of this study is to deal with the problem of simulating geological boundaries in the Golgohar iron deposit based on two datasets extracted from exploration boreholes and blast holes. In the first step, contact analysis tools are used to identify the nature of geological boundaries. Next, a modified version of the direct sampling algorithm is proposed to simulate geological domains based on a training image and local probabilities derived from a vertical proportion matrix. The purpose of using the modified direct sampling is to control the local proportion of each geological domain in the distance-based multiple-point geostatistical approaches. A set of 100 conditional realizations of two ore types are generated. The reconciliation between the realizations produced with the proposed algorithm and actual data values is provided by checking the reproduction of single-, two- and multiple-point statistical parameters and the connectivity of geological boundaries. The proposed method proves to be a useful variant of the direct sampling algorithm for simulating the spatial layout of geological domains when there is a strong difference in the domain proportions between the hard data and the TI.

Conditioning Multiple‐Point Statistics Simulation to Inequality Data

AbstractStochastic modeling is often employed in environmental sciences for the analysis and understanding of complex systems. For example, random fields are key components in uncertainty analysis or Bayesian inverse modeling. Multiple‐point statistics (MPS) provides efficient simulation tools for simulating fields reproducing the spatial statistics depicted in a training image (TI), while accounting for local or block conditioning data. Among MPS methods, the direct sampling algorithm is a flexible pixel‐based technique that consists in first assigning the conditioning data values (so‐called hard data) in the simulation grid, and then in populating the rest of the simulation domain in a random order by successively pasting a value from a TI cell sharing a similar pattern. In this study, an extension of the direct sampling method is proposed to account for inequality data, that is, constraints in given cells consisting of lower and/or upper bounds for the simulated values. Indeed, inequality data are often available in practice. The new approach involves the adaptation of the distance used to compare and evaluate the match between two patterns to account for such constraints. The proposed method, implemented in the DeeSse code, allows generating random fields both reflecting the spatial statistics of the TI and honoring the inequality constraints. Finally examples of topography simulations illustrate and show the capabilities of the proposed method.

A Python Based Multi-Point Geostatistics by using Direct Sampling Algorithm

Multi-Point Geostatistics (MPS) is a type of geostatistical method used to estimate the value of an unsampled location by utilizing several data points around it simultaneously. The MPS method estimates it by defining a model based on initial data in the form of a training image, which is a collection of data in the form of a geological conceptual model in the research area with the integration of geological and geophysical knowledge. The MPS method is currently starting to develop because it differs from conventional covariance-based geostatistical methods such as simple kriging and ordinary kriging, which only use a variogram based on the relationship between two points rapidly. In this study, we evaluated the use of the MPS method by using a direct sampling algorithm with Python that will directly sample the training image and then retrieve the data based on the sample data. A braided channel training image is used as the initial model to estimate the distribution of reservoir properties in lithology with sand and shale types. This study shows that MPS could reconstruct geological features better than kriging.

Using geostatistics and maximum entropy model to identify geochemical anomalies: A case study in Mila Mountain region, southern Tibet

Separating geochemical anomalies from background values is crucial for the processing of geochemical data. In the present study, a workflow for identifying geochemical anomalies was constructed by using the direct sampling algorithm of multi-point geostatistics, the maximum entropy model, and local singularity analysis. The smoothing effect and the uncertainty of the unsampled point value in the traditional interpolation method were taken into consideration in this workflow. Based on the statistic of singular exponential distribution of each element with equal probability, the geochemical anomaly probability distribution of each element was obtained (Ag、Cd、Cu、Pb、Zn). Based on the five anomaly probability distributions, the maximum entropy model was used to establish a comprehensive perspective of geochemical anomaly uncertainty evaluation. The validity of the method was verified by analyzing geochemical data of stream sediment samples from the Mila Mountain region in Tibet. The results showed that the prospectivity map of copper deposits generated by the maximum entropy model can effectively link the probability of multivariate geochemical anomalies with the known positions of copper deposits and greatly increase the precision of the potential exploration areas for copper deposits.

3D multiple-point statistics simulations of the Roussillon Continental Pliocene aquifer using DeeSse

Abstract. This study introduces a novel workflow to model the heterogeneity of complex aquifers using the multiple-point statistics algorithm DeeSse. We illustrate the approach by modeling the Continental Pliocene layer of the Roussillon aquifer in the region of Perpignan (southern France). When few direct observations are available, statistical inference from field data is difficult if not impossible and traditional geostatistical approaches cannot be applied directly. By contrast, multiple-point statistics simulations can rely on one or several alternative conceptual geological models provided using training images (TIs). But since the spatial arrangement of geological structures is often non-stationary and complex, there is a need for methods that allow to describe and account for the non-stationarity in a simple but efficient manner. The main aim of this paper is therefore to propose a workflow, based on the direct sampling algorithm DeeSse, for these situations. The conceptual model is provided by the geologist as a 2D non-stationary training image in map view displaying the possible organization of the geological structures and their spatial evolution. To control the non-stationarity, a 3D trend map is obtained by solving numerically the diffusivity equation as a proxy to describe the spatial evolution of the sedimentary patterns, from the sources of the sediments to the outlet of the system. A 3D continuous rotation map is estimated from inferred paleo-orientations of the fluvial system. Both trend and orientation maps are derived from geological insights gathered from outcrops and general knowledge of processes occurring in these types of sedimentary environments. Finally, the 3D model is obtained by stacking 2D simulations following the paleo-topography of the aquifer. The vertical facies transition between successive 2D simulations is controlled partly by the borehole data used for conditioning and by a sampling strategy. This strategy accounts for vertical probability of transitions, which are derived from the borehole observations, and works by simulating a set of conditional data points from one layer to the next. This process allows us to bypass the creation of a 3D training image, which may be cumbersome, while honoring the observed vertical continuity.

Convergence acceleration of Monte Carlo many-body perturbation methods by direct sampling.

In the Monte Carlo many-body perturbation (MC-MP) method, the conventional correlation-correction formula, which is a long sum of products of low-dimensional integrals, is first recast into a short sum of high-dimensional integrals over electron-pair and imaginary-time coordinates. These high-dimensional integrals are then evaluated by the Monte Carlo method with random coordinates generated by the Metropolis-Hasting algorithm according to a suitable distribution. The latter algorithm, while advantageous in its ability to sample nearly any distribution, introduces autocorrelation in sampled coordinates, which, in turn, increases the statistical uncertainty of the integrals and thus the computational cost. It also involves wasteful rejected moves and an initial "burn-in" step as well as displays hysteresis. Here, an algorithm is proposed that directly produces a random sequence of electron-pair coordinates for the same distribution used in the MC-MP method, which is free from autocorrelation, rejected moves, a burn-in step, or hysteresis. This direct-sampling algorithm is shown to accelerate second- and third-order Monte Carlo many-body perturbation calculations by up to 222% and 38%, respectively.

Downscaling Multispectral Satellite Images Without Colocated High-Resolution Data: A Stochastic Approach Based on Training Images

Very high-resolution satellite imagery from the latest generation commercial platforms provides an unprecedented capacity for imaging the Earth with very high spatial detail. However, these data are generally expensive, particularly if large areas or temporal sequences are required. In recent years, lower quality imagery has been enabled through the launch of constellations of small satellites with short revisit time. In this article, we apply for the first time a statistical approach to downscale and bias-correct these multispectral satellite data using the information contained in a limited training set of very high-resolution images. The technique, based on the direct sampling algorithm, aims at extending the coverage of high-resolution images by sampling data from a training data set, where similar lower resolution data patterns are found. Unlike the majority of the current downscaling techniques, the approach does not require colocated fine-resolution data, but it is based on the use of training images similar to the target zone. A novel specific setup is proposed, which is adaptive to different types of landscapes with no additional user effort. The results show that the proposed technique can generate more realistic images than the traditional approaches based on the parametric bias correction and bicubic interpolation. In particular, properties such as the intensity histogram, spatial correlation, and connectivity are accurately preserved. The proposed approach can be used to extend the footprint of the high-resolution images to generate new time frames or to downscale the remote sensing imagery based on a distant but structurally similar training image.

Multiple-point statistics using multi-resolution images

Multiple-point statistics (MPS) is a simulation technique allowing to generate images that reproduce the spatial features present in a training image (TI). MPS algorithms consist in sequentially filling a simulation grid such that patterns around the simulated values come from the TI. Following this principle, joint simulations of multiple variables can be handled and complex heterogeneous fields can be generated. However, inconsistent patterns are often found in the results and some spatial features can be difficult to reproduce. In this paper, a new MPS algorithm based on a multi-resolution representation of the TI is proposed to enhance the quality of the realizations. The method consists in first building a pyramid of images from the TI by successive convolution using Gaussian-like kernels. Secondly, a MPS simulation is done at the lowest resolution level. Then, the result is expanded to the next level of resolution (one rank higher) and used as a conditioning variable for a joint MPS simulation at that level. This last step is repeated up to the initial resolution, where the final simulation is retrieved. The method is implemented in the DeeSse code based on the direct sampling algorithm. Most of the features provided by the direct sampling (conditioning to hard data, uni- or multi-variate simulation of categorical and continuous variables, scaling and rotation of the training structures) are compatible with the proposed method and the usability is maintained. Finally, various examples show that in most of the situations, combining Gaussian pyramids with MPS allows to get results of better quality and in less time compared to direct MPS simulations.

Joint simulation of compositional and categorical data via direct sampling technique – Application to improve mineral resource confidence

Ore deposits usually consist of ore materials with different discrete (e.g. rock and alteration types) and continuous (e.g. geochemical and mineral composition) features. Financial feasibility studies are highly dependent on the modelling of these features and their associated joint uncertainties. Few geostatistical techniques have been developed for the joint modelling of high-dimensional mixed data (continuous and categorical) or constrained data, such as compositional data. The compositional nature of the mineral and geochemical data induces several challenges for multivariate geostatistical techniques, because such data carry relative information and are known for spurious statistical and spatial correlation effects. This paper investigates the application of the direct sampling algorithm for joint modelling of compositional and categorical data. In some mining projects the amount of available data may be enormous in some parts of the deposit and if the density of measurements is sufficient, multivariate geospatial patterns can be derived from that data and be simulated (without model inference) at other undersampled areas of the deposit with similar characteristics. In this context, the direct sampling multiple-point simulation method can be implemented for this reconstruction process. The compositional nature of the data is addressed via implementing an isometric log-ratio transformation. The approach is illustrated through two case studies, one synthetic and one real. The accuracy of the results is checked against a set of validation data, revealing the potential of the proposed methodology for joint modelling of compositional and categorical information. The direct sampling technique can be considered as a smart move to assess the future risk and uncertainty of a resource by making use of all the information hidden within the early data.

A fast direct sampling algorithm for equilateral closed polygons

Sampling equilateral closed polygons is of interest in the statistical study of ring polymers. Over the past 30 years, previous authors have proposed a variety of simple Markov chain algorithms (but have not been able to show that they converge to the correct probability distribution) and complicated direct samplers (which require extended-precision arithmetic to evaluate numerically unstable polynomials). We present a simple direct sampler which is fast and numerically stable, and analyze its runtime using a new formula for the volume of equilateral polygon space as a Dirichlet-type integral.

Conditioning multiple-point statistics simulations to block data

Multiple-points statistics (MPS) allows to generate random fields reproducing spatial statistics derived from a training image. MPS methods consist in borrowing patterns from the training set. Therefore, the simulation domain is assumed to be at the same resolution as the conceptual model, although geometrical deformations can be handled by such techniques. Whereas punctual conditioning data corresponding to the scale of the grid node can be easily integrated, accounting for data available at larger scales is challenging. In this paper, we propose an extension of MPS able to deal with block data, i.e. target mean values over subsets of the simulation domain. Our extension is based on the direct sampling algorithm and consists to add a criterion for the acceptance of the candidate node scanned in the training image to constrain the simulation to block data. Likelihood ratios are used to compare the averages of the simulated variable taken on the informed nodes in the blocks and the target mean values. Moreover, the block data may overlap and their support can be of any shape and size. Illustrative examples show the potential of the presented algorithm for practical applications.

Tomogram-based comparison of geostatistical models: Application to the Macrodispersion Experiment (MADE) site

We propose a new framework to compare alternative geostatistical descriptions of a given site. Multiple realizations of each of the considered geostatistical models and their corresponding tomograms (based on inversion of noise-contaminated simulated data) are used as a multivariate training image. The training image is scanned with a direct sampling algorithm to obtain conditional realizations of hydraulic conductivity that are not only in agreement with the geostatistical model, but also honor the spatially varying resolution of the site-specific tomogram. Model comparison is based on the quality of the simulated geophysical data from the ensemble of conditional realizations. The tomogram in this study is obtained by inversion of cross-hole ground-penetrating radar (GPR) first-arrival travel time data acquired at the MAcro-Dispersion Experiment (MADE) site in Mississippi (USA). Various heterogeneity descriptions ranging from multi-Gaussian fields to fields with complex multiple-point statistics inferred from outcrops are considered. Under the assumption that the relationship between porosity and hydraulic conductivity inferred from local measurements is valid, we find that conditioned multi-Gaussian realizations and derivatives thereof can explain the crosshole geophysical data. A training image based on an aquifer analog from Germany was found to be in better agreement with the geophysical data than the one based on the local outcrop, which appears to under-represent high hydraulic conductivity zones. These findings are only based on the information content in a single resolution-limited tomogram and extending the analysis to tracer or higher resolution surface GPR data might lead to different conclusions (e.g., that discrete facies boundaries are necessary).

Conditioning of Multiple-Point Statistics Facies Simulations to Tomographic Images

Geophysical tomography captures the spatial distribution of the underlying geophysical property at a relatively high resolution, but the tomographic images tend to be blurred representations of reality and generally fail to reproduce sharp interfaces. Such models may cause significant bias when taken as a basis for predictive flow and transport modeling and are unsuitable for uncertainty assessment. We present a methodology in which tomograms are used to condition multiple-point statistics (MPS) simulations. A large set of geologically reasonable facies realizations and their corresponding synthetically calculated cross-hole radar tomograms are used as a training image. The training image is scanned with a direct sampling algorithm for patterns in the conditioning tomogram, while accounting for the spatially varying resolution of the tomograms. In a post-processing step, only those conditional simulations that predicted the radar traveltimes within the expected data error levels are accepted. The methodology is demonstrated on a two-facies example featuring channels and an aquifer analog of alluvial sedimentary structures with five facies. For both cases, MPS simulations exhibit the sharp interfaces and the geological patterns found in the training image. Compared to unconditioned MPS simulations, the uncertainty in transport predictions is markedly decreased for simulations conditioned to tomograms. As an improvement to other approaches relying on classical smoothness-constrained geophysical tomography, the proposed method allows for: (1) reproduction of sharp interfaces, (2) incorporation of realistic geological constraints and (3) generation of multiple realizations that enables uncertainty assessment.