Multidimensional Probability Distribution Research Articles

Characteristic properties of equivalent structures in compositional models

Compositional model theory serves as an alternative approach to multidimensional probability distribution representation and processing. Every compositional model over a finite non-empty set of variables N is uniquely defined by its generating sequence – an ordered set of low-dimensional probability distributions. A generating sequence structure induces a system of conditional independence statements over N valid for every multidimensional distribution represented by a compositional model with this structure. The equivalence problem is how to characterise whether all independence statements induced by structure P are induced by a second structure P ′ and vice versa. This problem can be solved in several ways. A partial solution of the so-called direct characterisation of an equivalence problem is represented here. We deduce and describe three properties of equivalent structures necessary for equivalence of the respective structures. We call them characteristic properties of classes of equivalent structures.

No-arbitrage conditions, scenario trees, and multi-asset financial optimization

Many numerical optimization methods use scenario trees as a discrete approximation for the true (multi-dimensional) probability distributions of the problem’s random variables. Realistic specifications in financial optimization models can lead to tree sizes that quickly become computationally intractable. In this paper we focus on the two main approaches proposed in the literature to deal with this problem: scenario reduction and state aggregation. We first state necessary conditions for the node structure of a tree to rule out arbitrage. However, currently available scenario reduction algorithms do not take these conditions explicitly into account. State aggregation excludes arbitrage opportunities by relying on the risk-neutral measure. This is, however, only appropriate for pricing purposes but not for optimization. Both limitations are illustrated by numerical examples. We conclude that neither of these methods is suitable to solve financial optimization models in asset–liability or portfolio management.

Gaussian concentration for a class of spherically invariant measures

Concentration and logarithmic Sobolev inequalities are derived for a class of multidimensional probability distributions, including spherically invariant log-concave measures. Bibliography: 17 titles.

Agglomeration detection based on attractor comparison in horizontal stirred bed reactors by acoustic emission sensors

AbstractIn industrial propylene polymerization technology of Innovene and Chisso, agglomeration of polymer particles in horizontal stirred bed reactors (HSBR) has significant negative impacts on the reactor efficiency. If the hydrodynamic change could be detected early enough, then large chunks may not be formed nor cause an unscheduled shutdown of the plant. A novel non‐invasive acoustic emission (AE) technique combined with an attractor comparison method is presented. It is based on the comparison of the time series of AE signals acquired from a normal state with the time series measured during operation of the bed. The nature of the method is to determine whether or not two time series are generated by the same mechanism. A statistical characteristic S is used to test the null hypothesis that the two multidimensional probability distributions are identical. Experiments were carried out at room temperature in a 475 mm i.d. laboratory HSBR. The results demonstrated that the method is sensitive to small changes in the particle size distribution. It is, therefore, indicated that the proposed method not only can offer “early and accurate warning,” but also has the potential to locate the agglomeration in HSBR with multiple AE sensors. © 2009 American Institute of Chemical Engineers AIChE J, 2009

Stochastic Kinetic Analysis of the Frank Model. Stochastic Approach to Flow-Through Reactors

The continuous time discrete state stochastic kinetic approach and its extension to flow-through reactors was used to study a straightforward modification of the Frank model to interpret absolute asymmetric synthesis, which is impossible using deterministic approaches. Computational methods for calculating multidimensional probability distributions and expectations for enantiomeric excess were developed. The results showed that narrow focus on the conventionally defined enantiomeric excess could lead to misleading conclusions and the yield-adjusted enantiomeric excess is often more useful. Closed systems proved to be more favorable for the formation of high enantiomeric excesses than flow-through reactors and the importance of mutual antagonism can also be questioned in the original Frank model. It was also shown that a flow-through reactor with a relatively small number of molecules predicts the behavior of much larger systems well.

Water demand and flows in the São Francisco River Basin (Brazil) with increased irrigation

Most activities that support economic growth in the São Francisco River Basin (Brazil) need water. Allocation of the water resources to each competing use needs quantification in order to develop an integrated water management plan. Irrigation agriculture is the largest water consuming activity in the basin. It has produced large economic and social advancements in the region and has potential for further development. The local development agency in the São Francisco River has projected an increase of more than 500,000 ha in irrigation developments distributed within the basin. Water requirements of the projected irrigation expansions and their effects on river flow were quantified. A semi-distributed model was constructed to simulate the water balance in 16 watersheds within the basin. The watersheds were hydrologically characterized by the average precipitation, atmospheric demand and runoff as well as their variability. Water requirements for increased irrigated agriculture were calculated using an agronomic mass balance. A Monte Carlo procedure generated the variability of irrigation requirements and resulting decreased river flows from the multidimensional probability distribution of the hydrologic variables of each watershed. Irrigation requirements were found to be more variable during the wet season because of weather variability. In contrast to what might be expected, in drier years, irrigation requirements were often larger during the wet season than in the dry season because the cropped area is largest in the wet months and variability of precipitation is greater. Increased irrigation shifted downward the distribution of river flows but not enough to affect other strategic water uses such as hydropower. Further irrigation expansion may be limited by wet season flows.

Computational methods for parameter estimation in climate models

Intensive computational methods have been used by Earth scientists in a wide range of problems in data inversion and uncertainty quantification such as earthquake epicenter location and climate projections. To quantify the uncertainties resulting from a range of plausible model configurations it is necessary to estimate a multidimensional probability distribution. The computational cost of estimating these distributions for geoscience applications is impractical using traditional methods such as Metropolis/Gibbs algorithms as simulation costs limit the number of experiments that can be obtained reasonably. Several alternate sampling strategies have been proposed that could improve on the sampling efficiency including Multiple Very Fast Simulated Annealing (MVFSA) and Adaptive Metropolis algorithms. The performance of these proposed sampling strategies are evaluated with a surrogate climate model that is able to approximate the noise and response behavior of a realistic atmospheric general circulation model (AGCM). The surrogate model is fast enough that its evaluation can be embedded in these Monte Carlo algorithms. We show that adaptive methods can be superior to MVFSA to approximate the known posterior distribution with fewer forward evaluations. However the adaptive methods can also be limited by inadequate sample mixing. The Single Component and Delayed Rejection Adaptive Metropolis algorithms were found to resolve these limitations, although challenges remain to approximating multi-modal distributions. The results show that these advanced methods of statistical inference can provide practical solutions to the climate model calibration problem and challenges in quantifying climate projection uncertainties. The computational methods would also be useful to problems outside climate prediction, particularly those where sampling is limited by availability of computational resources.

Color textural analysis of the blood preparation images

A technology of the color textural features formation is presented. The method is based on estimation of the statistical characteristics of multidimensional probability distribution of the color image components. Full scale experiments of the method are presented in application to the blood preparation images classification. The results of the developed method evaluation are discussed in compare with the existing methods.

Statistical Signal Processing by Using the Higher‐Order Correlation between Sound and Vibration and Its Application to Fault Detection of Rotational Machine

In this study, a stochastic diagnosis method based on the changing information of not only a linear correlation but also a higher‐order nonlinear correlation is proposed in a form suitable for online signal processing in time domain by using a personal computer, especially in order to find minutely the mutual relationship between sound and vibration emitted from rotational machines. More specifically, a conditional probability hierarchically reflecting various types of correlation information is theoretically derived by introducing an expression on the multidimensional probability distribution in orthogonal expansion series form. The effectiveness of the proposed theory is experimentally confirmed by applying it to the observed data emitted from a rotational machine driven by an electric motor.

Compositional Belief Function Models

Analogously to Graphical Markov models, also Compositional models serve as an efficient tool for multidimensional models representation. The main idea of the latter models resembles a jig saw puzzle: Multidimensional models are assembled (composed) from a large number of small pieces, from a large number of low-dimensional models. Originally they were designed to represent multidimensional probability distributions. In this paper they will be used to represent multidimensional belief functions (or more precisely, multidimensional basic belief assignments) with the help of a system of low-dimensional ones. In addition to a number of basic properties of such models, in the paper it will be shown that these models can serve as a real enrichment of probabilistic models. They can relieve a drawback of probabilistic models that can be, in case that the initial building blocks of the model are inconsistent, undefined. As a side result of the paper we propose a new way how to define the concept of conditional independence for belief functions.

Analysis of the Multiplane Method for Stochastic Simulations of Reaction Networks with Fluctuations

Chemical reaction networks are commonly modeled by rate equations, which are systems of ordinary differential equations describing the evolution of species concentrations. Such models break down at low concentrations, where stochastic effects become dominant. Instead, one has to solve the master equation that governs the multidimensional probability distribution of particle populations. For large networks such an approach is often computationally prohibitive due to the exponential dependence of the number of states on the number of components. The multiplane method is a dimension reduction technique that exploits the structure of the network to derive approximate dynamics for the marginal distributions of pairs of coreacting species. This method was introduced in [A. Lipshtat and O. Biham, Phys. Rev. Lett., 93 (2004), 170601] as an uncontrolled approximation for specific examples in the context of interstellar chemical reactions. In this paper we formalize the method and prove that it is asymptotically exact in the two extreme limits of small and large population sizes. Our analysis concentrates on steady-state conditions, although numerical simulations indicate that the method is equally well applicable to time-dependent solutions. This analysis partially explains the surprisingly high accuracy of the method.

Validation of statistical estimation of transmission loss in the presence of geoacoustic inversion uncertainty

Often the ocean acoustic environment is not well known and sonar performance prediction will be affected by this uncertainty. Here, a method for estimating transmission loss (TL) is proposed which incorporates these environmental uncertainties. Specifically, we derive an approach for the statistical estimation of TL based on the posterior probability density of environmental parameters obtained from the geoacoustic inversion process. First, a Markov chain Monte Carlo procedure is employed in the inversion process to sample the posterior probability density of the geoacoustic parameters. Then, these sampled parameters are mapped to the transmission loss domain where a full multidimensional probability distribution of TL as a function of range and depth is obtained. In addition, TL is also characterized by its summary statistics including the median, percentiles, and correlation coefficients. The approach is illustrated using a data set obtained from the ASIAEX 2001 East China Sea experiment. Based on the geoacoustic inversion results, the predicted TL and its variability are estimated and then compared with the measured TL. In general, there is a good agreement with the percentage of observed number of data points inside the credibility interval.

Validation of statistical estimation of transmission loss in the presence of geoacoustic inversion uncertainty

Often the ocean acoustic environment is not well known and sonar performance prediction will be affected by this uncertainty. Here, a method for characterizing transmission loss (TL) is proposed which incorporates these environmental uncertainties. The statistical estimation of TL based on the posterior probability density of environmental parameters obtained from the geoacoustic inversion process is the main theme of this study. First, a Markov chain Monte Carlo procedure is employed to sample the posterior probability density of the geoacoustic parameters. Then, these parameter uncertainties are mapped to the transmission loss domain, where a full multidimensional probability distribution of TL as a function of range and depth is obtained. In addition, TL is also characterized by its summary statistics including the median, percentiles, and correlation coefficients. The approach is illustrated using a data set obtained from the ASIAEX 2001 East China Sea experiment. Based on the geoacoustic inversion results, the predicted TL and its variability are estimated and then compared with the measured TL. In general, the results show that approximately 80% of the observed TL data fall within 90% of the TL probability distribution.

Estimation of Higher Order Correlation between Electromagnetic and Sound Waves Leaked from VDT Environment Based on Fuzzy Probability and the Prediction of Probability Distribution

In this paper, a signal processing method considering not only linear correlation but also the higher order nonlinear correlation information is proposed on the basis of fuzzy observation data, in order to find the mutual relationship between sound and electromagnetic waves leaked from electronic information equipment. More specifically, by applying the well-known fuzzy probability to an expression on the multi-dimensional probability distribution in an orthogonal expansion series form reflecting systematically various types of correlation information, a method to estimate precisely the correlation information between the variables from the conditional moment statistics of fuzzy variables is proposed. The effectiveness of the proposed theory is experimentally confirmed by applying it to the observation data leaked from VDT in the actual work environment

Estimation of Higher Order Correlation Between Electromagnetic and Sound Waves Leaked from VDT Environment Based on Fuzzy Probability and the Prediction of Probability Distribution

In this paper, a signal processing method considering not only linear correlation but also the higher order nonlinear correlation information is proposed on the basis of fuzzy observation data, in order to find the mutual relationship between sound and electromagnetic waves leaked from electronic information equipment. More specifically, by applying the well-known fuzzy probability to an expression on the multi-dimensional probability distribution in an orthogonal expansion series form reflecting systematically various types of correlation information, a method to estimate precisely the correlation information between the variables from the conditional moment statistics of fuzzy variables is proposed. The effectiveness of the proposed theory is experimentally confirmed by applying it to the observation data leaked from VDT in the actual work environment

Assessing the effects of foregrounds and sky removal in WMAP

Many recent analyses have indicated that large scale Wilkinson Microwave Anisotropy Probe (WMAP) data display anomalies that appear inconsistent with the standard cosmological paradigm. However, the effects of foreground contamination, which require elimination of some fraction of the data, have not been fully investigated due to the complexity in the analysis. Here we develop a general formalism of how to incorporate these effects into any analysis of this type. Our approach is to compute the full multidimensional probability distribution function of all possible sky realizations that are consistent with the data and with the allowed level of contamination. Any statistic can be integrated over this probability distribution to assess its significance. As an example we apply this method to compute the joint probability distribution function for the possible realizations of quadrupole and octopole using the WMAP data. This 12 dimensional distribution function is explored using the Markov Chain Monte Carlo technique. The resulting chains are used to assess the statistical significance of the low quadrupole using frequentist methods, which we find to be 3-4%. Octopole is normal and the probability of it being anomalously low or as low as WMAP reported value is very small. We address the quadrupole-octopole alignmentmore » using several methods that have been recently used to argue for anomalies, such as angular momentum dispersion, multipole vectors and a new method based on feature matching. While we confirm that the full-sky map Internal Linear Combination Map (ILC) suggest an alignment, we find that removing the most contaminated part of the data also removes any evidence of alignment: the probability distributions strongly disfavor the alignment. This suggests that most of the evidence for it comes from non-Gaussian features in the part of the data most contaminated by the foregrounds. We also present an example, that of octopole alignment with the ecliptic, where the statistical significance can be enhanced by removing the contamination.« less

Optimum Bid Markup Calculation Using Neurofuzzy Systems and Multidimensional Risk Analysis Algorithm

The paper presents a methodology for arriving at optimum bid markups in static competitive bidding environments by use of neurofuzzy systems and integrated multidimensional risk analysis algorithms. The neurofuzzy framework enables integration of the objective and subjective factors found in the underlying decision-making process, and serves as the stepping stone for the generation of the multidimensional probability distribution function that governs competitive bidding. Subsequent bid optimization is achieved by employing a multidimensional risk analysis algorithm.

Fully automated biomedical image segmentation by self-organized model adaptation

In this paper, we present a fully automated image segmentation method based on an algorithm that provides adaptive plasticity in function approximation problems: the deformable (feature) map (DM) algorithm. The DM approach reduces a class of similar function approximation problems to the explicit supervised one-shot training of a single data set. This is followed by a subsequent, appropriate similarity transformation, which is based on a self-organized deformation of the underlying multidimensional probability distributions. We apply this algorithm to the real-world problem of fully automated voxel-based multispectral image segmentation, employing magnetic resonance data sets of the human brain. In contrast to previous segmentation approaches, the knowledge obtained within the segmentation procedure of a single prototypical reference data set can be re-utilized for the segmentation of new, ‘similar’ data employing a strategy of incremental adaptive learning based on the DM algorithm. Thus, we obtain a fully automatic segmentation method that does neither require manual contour tracing of training regions, visual classification of voxel clusters, nor any other kind of human intervention. Our application demonstrates that flexible learning by a strategy of self-organized incremental model adaptation can contribute to increase the efficiency and practicability of biomedical image processing systems.

Use of probability distribution functions for discriminating between cloud and aerosol in lidar backscatter data

In this paper, we describe the algorithm that will be used during the upcoming Cloud‐Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission for discriminating between clouds and aerosols detected in two‐wavelength backscatter lidar profiles. We first analyze single‐test and multiple‐test classification approaches based on one‐dimensional (1‐D) and multidimensional probability distribution functions (PDFs) in the context of a two‐class feature identification scheme. From these studies we derive an operational algorithm. This algorithm is a 3‐D approach utilizing the layer mean attenuated backscatter at 532 nm, the layer‐integrated 1064‐nm to 532‐nm volume color ratio, and the midlayer altitude. A data set acquired by the Cloud Physics Lidar (CPL) is used to test the algorithm. Comparisons are conducted between the 3‐D CALIPSO algorithm results and those derived from an existing 2‐D algorithm. The results obtained show generally good agreement between the two methods. However, of a total of 228,264 layers analyzed, ∼5.7% are classified as different types by the two algorithms. This disparity is shown to be due largely to the misclassification of optically thin clouds as aerosols by the 2‐D algorithm. The use of 3‐D PDFs in the CALIPSO algorithm is found to significantly reduce this type of error because the separation between cloud and aerosol clusters is more complete in this 3‐D space. Dust presents a special case. Because the intrinsic scattering properties of dust layers can be very similar to those of clouds, additional algorithm testing was performed using an optically dense layer of Saharan dust measured during the Lidar In‐space Technology Experiment (LITE). In general, the method is shown to distinguish reliably between dust layers and clouds. The relatively few erroneous classifications that occurred most often in the analysis of the LITE data occurred, in those regions of the Saharan dust layer where the optical thickness was the highest.

An Efficient Stochastic Bayesian Approach to Optimal Parameter and Uncertainty Estimation for Climate Model Predictions

Abstract One source of uncertainty for climate model predictions arises from the fact that climate models have been optimized to reproduce observational means. To quantify the uncertainty resulting from a realistic range of model configurations, it is necessary to estimate a multidimensional probability distribution that quantifies how likely different model parameter combinations are, given knowledge of the uncertainties in the observations. The computational cost of mapping a multidimensional probability distribution for a climate model using traditional means (e.g., Monte Carlo or Metropolis/Gibbs sampling) is impractical, requiring 104–106 model evaluations for problems involving less than 10 parameters. This paper examines whether such a calculation is more feasible using a particularly efficient but approximate algorithm called Bayesian stochastic inversion, based on multiple very fast simulated annealing (VFSA). Investigated here is how the number of model parameters, natural variability, and the d...