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Analysis of equation for conditional probability distribution of state of nonlinear system and application to nonlinear filtering
The lack of well-developed techniques for modeling changing statistical moments in our observations has stymied the application of stochastic process theory for many scientific and engineering applications. Non linear effects of the observation methodology is one of the most perplexing aspects to modeling non stationary processes. This perplexing problem was encountered when modeling the effect of non stationary receiver fluctuations on the performance of radiometer calibration architectures. Existing modeling approaches were found not applicable; particularly problematic is modeling processes across scales over which they begin to exhibit non stationary behavior within the time interval of the calibration algorithm. Alternatively, the radiometer output is modeled as samples from a sequence random variables; the random variables are treated using a conditional probability distribution function conditioned on the use of the variable in the calibration algorithm. This approach of treating a process as a sequence of random variables with non stationary stochastic moments produce sensible predictions of temporal effects of calibration algorithms. To test these model predictions, an experiment using the Millimeter wave Imaging Radiometer (MIR) was conducted. The MIR with its two black body calibration references was configured in a laboratory setting to observe a third ultra-stable reference (CryoTarget). The MIR was programmed to sequentially sample each of the three references in approximately a 1 second cycle. Data were collected over a six-hour interval. The sequence of reference measurements form an ensemble sample set comprised of a series of three reference measurements. Two references are required to estimate the receiver response. A third reference is used to estimate the uncertainty in the estimate. Typically, calibration algorithms are designed to suppress the non stationary effects of receiver fluctuations. By treating the data sequence as an ensemble collection, it is possible to apply temporal algorithms which exacerbate the non stationary effects. By varying the algorithm, information about the properties of the non stationary receiver fluctuations is obtained. Comparisons of analytical calculations and statistical analysis of data demonstrate impressive agreement.
The James Webb Space Telescope (JWST) and Atacama Large Millimeter/submillimeter Array have detected emission lines from the ionized interstellar medium (ISM) in some of the first galaxies at z ≳ 6. These measurements present an opportunity to better understand galaxy assembly histories and may allow important tests of state-of-the-art galaxy formation simulations. It is challenging, however, to model these lines in their proper cosmological context. In order to meet this challenge, we introduce a novel subgrid line emission modeling framework. The framework uses the high-z zoom-in simulation suite from the Feedback in Realistic Environments (FIRE) collaboration. The line emission signals from H II regions within each simulated FIRE galaxy are modeled using the semianalytic HIIL INES code. A machine learning approach is then used to determine the conditional probability distribution for the line luminosity to stellar-mass ratio from the H II regions around each simulated stellar particle. This conditional probability distribution can then be applied to predict the line luminosities around stellar particles in lower-resolution, yet larger volume cosmological simulations. As an example, we apply this approach to the IllustrisTNG simulations at z = 6. The resulting predictions for the [O II ], [O III ], and Balmer line luminosities as a function of star formation rate agree well with current observations. Our predictions differ, however, from related works in the literature, which lack detailed subgrid ISM models. This highlights the importance of our multiscale simulation modeling framework. Finally, we provide forecasts for future line luminosity function measurements from the JWST and quantify the cosmic variance in such surveys.
We present a study of the distribution of ozone in the lowermost stratosphere with the goal of understanding the relative contribution to the observations of air of either distinctly tropospheric or stratospheric origin. The air in the lowermost stratosphere is divided into two population groups based on Ertel's potential vorticity at 300 hPa. High [low] potential vorticity at 300 hPa suggests that the tropopause is low [high], and the identification of the two groups helps to account for dynamic variability. Conditional probability distribution functions are used to define the statistics of the mix from both observations and model simulations. Two data sources are chosen. First, several years of ozonesonde observations are used to exploit the high vertical resolution. Second, observations made by the Halogen Occultation Experiment [HALOE] on the Upper Atmosphere Research Satellite [UARS] are used to understand the impact on the results of the spatial limitations of the ozonesonde network. The conditional probability distribution functions are calculated at a series of potential temperature surfaces spanning the domain from the midlatitude tropopause to surfaces higher than the mean tropical tropopause [about 380K]. Despite the differences in spatial and temporal sampling, the probability distribution functions are similar for the two data sources. Comparisons with the model demonstrate that the model maintains a mix of air in the lowermost stratosphere similar to the observations. The model also simulates a realistic annual cycle. By using the model, possible mechanisms for the maintenance of mix of air in the lowermost stratosphere are revealed. The relevance of the results to the assessment of the environmental impact of aircraft effluence is discussed.
We present a study of the distribution of ozone in the lowermost stratosphere with the goal of understanding the relative contribution to the observations of air of either distinctly tropospheric or stratospheric origin. The air in the lowermost stratosphere is divided into two population groups based on Ertel's potential vorticity at 300 hPa. High [low] potential vorticity at 300 hPa suggests that the tropopause is low [high], and the identification of the two groups helps to account for dynamic variability. Conditional probability distribution functions are used to define the statistics of the mix from both observations and model simulations. Two data sources are chosen. First, several years of ozonesonde observations are used to exploit the high vertical resolution. Second, observations made by the Halogen Occultation Experiment [HALOE) on the Upper Atmosphere Research Satellite [UARS] are used to understand the impact on the results of the spatial limitations of the ozonesonde network. The conditional probability distribution functions are calculated at a series of potential temperature surfaces spanning the domain from the midlatitude tropopause to surfaces higher than the mean tropical tropopause [approximately 380K]. Despite the differences in spatial and temporal sampling, the probability distribution functions are similar for the two data sources. Comparisons with the model demonstrate that the model maintains a mix of air in the lowermost stratosphere similar to the observations. The model also simulates a realistic annual cycle. By using the model, possible mechanisms for the maintenance of mix of air in the lowermost stratosphere are revealed. The relevance of the results to the assessment of the environmental impact of aircraft effluence is discussed.
We present a study of the distribution of ozone in the lowermost stratosphere with the goal of characterizing the observed variability. The air in the lowermost stratosphere is divided into two population groups based on Ertel's potential vorticity at 300 hPa. High (low) potential vorticity at 300 hPa indicates that the tropopause is low (high), and the identification of these two groups is made to account for the dynamic variability. Conditional probability distribution functions are used to define the statistics of the ozone distribution from both observations and a three-dimensional model simulation using winds from the Goddard Earth Observing System Data Assimilation System for transport. Ozone data sets include ozonesonde observations from northern midlatitude stations (1991-96) and midlatitude observations made by the Halogen Occultation Experiment (HALOE) on the Upper Atmosphere Research Satellite (UARS) (1994- 1998). The conditional probability distribution functions are calculated at a series of potential temperature surfaces spanning the domain from the midlatitude tropopause to surfaces higher than the mean tropical tropopause (approximately 380K). The probability distribution functions are similar for the two data sources, despite differences in horizontal and vertical resolution and spatial and temporal sampling. Comparisons with the model demonstrate that the model maintains a mix of air in the lowermost stratosphere similar to the observations. The model also simulates a realistic annual cycle. Results show that during summer, much of the observed variability is explained by the height of the tropopause. During the winter and spring, when the tropopause fluctuations are larger, less of the variability is explained by tropopause height. This suggests that more mixing occurs during these seasons. During all seasons, there is a transition zone near the tropopause that contains air characteristic of both the troposphere and the stratosphere. The relevance of the results to the assessment of the environmental impact of aircraft effluence is also discussed.
Abstract Measures of inconsistency and tension between datasets have become an essential part of cosmological analyses. It is important to accurately evaluate the significance of such tensions when present. We propose here a Bayesian interpretation of inconsistency measures that can extract information about physical inconsistencies in the presence of data scatter. This new framework is based on the conditional probability distribution of the level of physical inconsistency given the obtained value of the measure. We use the index of inconsistency as a case study to illustrate the new interpretation framework, but this can be generalized to other metrics. Importantly, there are two aspects in the quantification of inconsistency that behave differently as the number of model parameters increases. The first is the probability for the level of physical inconsistency to reach a threshold which drops with the increase of the number of parameters under consideration. The second is the actual level of physical inconsistency which remains rather insensitive to such an increase in parameters. The difference between these two aspects is often overlooked, which leads to a long-standing ambiguity: when a given inconsistency is found between two constraints, its “significance” seems to be lower when considered in a higher-dimensional parameter space. This ambiguity is resolved by the Bayesian interpretation we introduce in this work because the conditional probability distribution includes all the statistical information of the level of physical inconsistency. Finally, we apply the Bayesian interpretation to examine the (in)consistency between Planck versus the Cepheid-based local measurement, the Dark Energy Survey (DES), the Atacama Cosmology Telescope (ACT) and WMAP. We confirm and revisit the degrees of previous physical inconsistencies and show the stability of the new interpretation with respect to the number of cosmological parameters compared to the commonly used n-σ interpretation when applied to cosmological tensions in multi-parameter spaces.
We introduce a conditional pseudo-reversible normalizing flow (PR-NF) that directly learns conditional probability distributions from noisy physical models to efficiently quantify both forward and inverse uncertainty propagation. Traditional surrogate modeling approaches approximate only the deterministic component of physical models, requiring separate noise characterization and computationally expensive sampling methods for inverse problems. Here, in this work, we develop the conditional PR-NF model to directly learn and efficiently generate samples from the conditional probability density functions (PDFs). The training process utilizes dataset consisting of input-output pairs without requiring prior knowledge about the noise and the function. Once trained, our model efficiently generates samples from conditional PDFs for any input within the training domain. Moreover, the pseudo-reversibility feature allows for the use of fully connected neural network architectures, which simplifies the implementation and enables theoretical analysis. We provide a rigorous convergence analysis of the conditional PR-NF model, showing its ability to converge to the target conditional PDF using the Kullback−Leibler divergence. To demonstrate the effectiveness of our method, we apply it to several benchmark tests and a real-world geologic carbon storage problem.
Report presents conditional-probability interpretation of Kanerva's sparse distributed memory (SDM). SDM is conceptual digital electronic memory in which addresses chosen sparsely in space of many dimensions and content of memory at each address positive or negative integer written (incremented or decremented) or read according to special rules. Contents of SDM constitute Monte Carlo approximation to multidimensional conditional-probability integral.
A financial accounting model that incorporates physical and institutional uncertainties was developed for geothermal projects. Among the uncertainties it can handle are well depth, flow rate, fluid temperature, and permit and construction times. The outputs of the model are cumulative probability distributions of financial measures such as capital cost, levelized cost, and profit. These outputs are well suited for use in an investment decision incorporating risk. The model has the powerful feature that conditional probability distribution can be used to account for correlations among any of the input variables. The model has been applied to a geothermal reservoir at Heber, California, for a 45-MW binary electric plant. Under the assumptions made, the reservoir appears to be economically viable.
Estimation of daily suspended sediment concentration (SSC) is required for water resources and environment management. In this paper, a copula-based stochastic method was proposed for daily SSC simulation. Here, the multivariate copula function, constructed based on a bivariate copula and two bivariate conditional probability distributions, was used to model the temporal and cross dependence structures in daily SSCs. Then, the daily SSCs were generated by sampling from the multivariate conditional distribution. As a result, synthetic long-term SSCs data beyond the limited observation period can be provided for water resources managers, which plays a critical role in accurately estimating frequency and magnitude of extreme SSCs events. The proposed method was under rigorous examination by applying to a case study at Pingshan station in the Jinsha River Basin, China. Results showed that the generated daily SSC sequences not only had a high degree of accuracy in preserving the statistical characteristics of the daily SSC observations, but also captured both the temporal correlation and the cross-correlation between the daily streamflow and daily SSC. Specifically, the average daily relative error values corresponding to mean, standard deviation, skewness, lag-1 temporal correlation, and cross correlation were 0.87%, 4.24%, 7.52%, 0.51% and 2.02%, respectively. The multivariate copula framework proposed here can accurately and efficiently generate long-term daily SSC data for water resources management such as frequency analysis and risk assessment of extreme SSC events.
The ENDF/B-VIII.1 evaluation library has seen a great growth in the thermal neutron scattering sub-library. The SCALE code system has traditionally approached CE transport by assuming that the CE library on disk represented the fully expanded cumulative probability distributions, conditional on exiting angle and marginal on exiting energy. While this is a complete description of the data, it comes at the potential cost of large amounts of on-disk storage. This approach was strained by several TSL files in ENDF/B-VIII.1, such as graphite, which contained data for a large number of Bragg edges. In the fully expanded probability distributions, this was found to be a disproportionately large fraction of the SCALE CE library.
The availability of fine grained time series data is a pre-requisite for research in smart-grids. While data for transmission systems is relatively easily obtainable, issues related to data collection, security and privacy hinder the widespread public availability/accessibility of such datasets at the distribution system level. This has prevented the larger research community from effectively applying sophisticated machine learning algorithms to significantly improve the distribution-level accuracy of predictions and increase the efficiency of grid operations. Synthetic dataset generation has proven to be a promising solution for addressing data availability issues in various domains such as computer vision, natural language processing and medicine. However, its exploration in the smart grid context remains unsatisfactory. Previous works have tried to generate synthetic datasets by modeling the underlying system dynamics: an approach which is difficult, time consuming, error prone and often times infeasible in many problems. In this work, we propose a novel data-driven approach to synthetic dataset generation by utilizing deep generative adversarial networks (GAN) to learn the conditional probability distribution of essential features in the real dataset and generate samples based on the learned distribution. To evaluate our synthetically generated dataset, we measure the maximum mean discrepancy (MMD) between real and synthetic datasets as probability distributions, and show that their sampling distance converges. To further validate our synthetic dataset, we perform common smart grid tasks such as k-means clustering and short-term prediction on both datasets. Experimental results show the efficacy of our synthetic dataset approach: the real and synthetic datasets are indistinguishable by solely examining the output of these tasks.
Efficient outage location is critical to enhancing the resilience of power distribution systems. However, accurate outage location requires combining massive evidence received from diverse data sources, including smart meter (SM) last gasp signals, customer trouble calls, social media messages, weather data, vegetation information, and physical parameters of the network. This is a computationally complex task due to the high dimensionality of data in distribution grids. In this paper, we propose a multi-source data fusion approach to locate outage events in partially observable distribution systems using Bayesian networks (BNs). A novel aspect of the proposed approach is that it takes multi-source evidence and the complex structure of distribution systems into account using a probabilistic graphical method. Our method can radically reduce the computational complexity of outage location inference in high-dimensional spaces. The graphical structure of the proposed BN is established based on the network’s topology and the causal relationship between random variables, such as the states of branches/customers and evidence. Utilizing this graphical model, accurate outage locations are obtained by leveraging a Gibbs sampling (GS) method, to infer the probabilities of de-energization for all branches. Compared with commonly-used exact inference methods that have exponential complexity in the size of the BN, GS quantifies the target conditional probability distributions in a timely manner. As a result, a case study of several real-world distribution systems is presented to validate the proposed method.
Probabilistic aerodynamic databases are a crucial component of the development lifecycle for aerospace vehicles. A key challenge when building aerodynamic databases is that most data used to construct them represent various simplifications of the real flight vehicle. For example, wind tunnel models often simplify the vehicle geometry and surface roughness characteristics, while CFD computations often make simplifications to the physics being modeled, such as fully laminar or turbulent calculations. Multifidelity data fusion models rely on a user being able to define a hierarchy of fidelity levels anchored to some "truth" data. This approach is unsatisfactory when no data can be considered to accurately reflect real flight conditions. In this work, we provide an alternative approach by presenting a consistent mathematical framework for building probabilistic aerodynamic databases in the form of a conditional probability distribution described by an ensemble of multifidelity Gaussian Processes. Instead of relying on a single hierarchy of data fidelity levels, the presented framework identifies a "nominal" configuration and potential corrections to the nominal which represent specific physical phenomena not represented in the nominal data. The nominal and correction functions themselves are constructed as multifidelity Gaussian Processes and linearly combined to form an ensemble model which fuses the uncertainties associated nominal and correction models. Results obtained using the proposed framework on a simplified Orion Crew Module wind tunnel dataset demonstrate the predictive capability of the multihierarchy framework. We further demonstrate the benefits of such a probabilistic aerodynamic database approach through function sampling and computing the conditional distributions of derived quantities, such as the trim angle of attack and aerodynamic coefficients at trim.
Individual quarks and gluons at small x inside an unpolarized hadron can be regarded as Bell states in which qubits in the spin and orbital angular momentum spaces are maximally entangled. Using the machinery of quantum information science, we generalize this observation to all values 0 < x <1 and describe gluons (but not quarks) as maximally entangled states between a qubit and a qudit. We introduce the conditional probability distribution P(l z |s z ) of a gluon’s orbital angular momentum l z given its helicity s z . Restricting to the three states l z =0,±1, which constitute a qutrit, we explicitly compute P as a function of x.
The ResStock™ analysis tool was built on NREL's OpenStudio® platform, and is a project geared at modeling existing residential building stocks at national, regional, or local scales with a high-degree of granularity (e.g., one physics-based simulation model for every 200 dwelling units), using the EnergyPlus® simulation engine. Information about ComStock™, a sister tool for modeling the commercial building stock, can be found here: https://www.nrel.gov/buildings/comstock.html This repository contains: Housing characteristics of the U.S. residential building stock, in the form of conditional probability distributions stored as tab-separated value (.tsv) files. Comments at the bottom of each file document data sources and assumptions for each. A library of housing characteristic "options" that translate high-level characteristic parameters into arguments for OpenStudio measures, and which are referenced by the housing characteristic .tsv files and building energy upgrades defined in project definition files Project definition files: v2.3.0 and later: buildstockbatch YML files openable in any text editor v2.2.5 and prior: Project folder openable in PAT Unit-level OpenStudio Measures for automatically constructing OpenStudio Models of each representative dwelling unit model: v3.0.0 and later: OpenStudio-HPXML Measures v2.5.0 and prior: OpenStudio Measures Higher-level OpenStudio Measures for controlling simulation inputs and outputs This repository does not contain software for running ResStock simulations, which can be found as follows: Versions 2.3.0 and later only support the use of buildstockbatch for deploying simulations on high-performance or cloud computing. Version 2.3.0 also removed separate projects for single-family detached and multifamily buildings, in lieu of a combined project_national representing the U.S. residential building stock. See the changelog for more details. Versions 2.2.5 and prior support the use of the publicly available OpenStudio-PAT software as an interface for deploying simulations on cloud computing. Read the documentation for v2.2.5.
The probability of exceeding interference power levels and the duration of interference at the Deep Space Network (DSN) antenna is calculated parametrically when the state vector of an Earth-orbiting satellite over the DSN station view area is not known. A conditional probability distribution function is derived, transformed, and then convolved with the interference signal uncertainties to yield the probability distribution of interference at any given instant during the orbiter's mission period. The analysis is applicable to orbiting satellites having circular orbits with known altitude and inclination angle.