Conditional Probability Distribution Function Research Articles

A Method for Enhancing the Simulation Continuity of the Snesim Algorithm in 2D Using Multiple Search Trees

Multiple-point geostatistics (MPS) has more advantages than two-point geostatistics in reproducing the continuity of geobodies in subsurface reservoir modeling. For fluvial reservoir modeling, the more continuous a channel, the more consistent it is with geological knowledge in general, and fluvial continuity is also of paramount importance when simulating fluid flow. Based on the pixel-based MPS algorithm Snesim, this study proposes a method that utilizes multiple search trees (MSTs) to enhance simulation continuity in 2D fluvial reservoir modeling. The objective of the MST method is to capture complete data events from a training image (TI), which aims to achieve enhanced continuity in fluvial reservoir sublayer modeling. By resorting to search neighborhoods based on their proximity to the central node of the data template, multiple data templates that correspond to the MSTs will be generated. Here, four data templates were generated by arranging the relative search neighborhood coordinates in ascending and descending order with respect to the central node. Parallel computing was tried for the construction of the search trees. This work calculated the conditional probability distribution function (CPDF) of the simulating nodes by averaging the CPDFs derived from the MSTs, and double retrieval was employed to filter out the search trees that possessed an inaccurate local CPDF for the simulating nodes. In addition, the connected component labeling (CCL) method was introduced to evaluate the simulation continuity in MPS. The results indicated that the MST method can enhance the simulation continuity of the Snesim algorithm by reproducing the fine connectivity of channel facies in 2D fluvial reservoir modeling.

Modeling forecast errors for microgrid operation using Gaussian process regression

Microgrids, denoting small-scale and self-sustaining grids, constitute a pivotal component in future power systems with a high penetration of renewable generators. The inherent uncertainty tied to renewable power generation, typified by photovoltaic and wind turbine systems, necessitates counterbalancing mechanisms. These mechanisms encompass Energy storage systems or conventional thermal fossil-fuel generators imbued with heightened flexibility. Addressing the uncertainty stemming from renewable generators mandates a cost-effective assessment and operational strategy for said compensatory devices. To this end, myriad uncertainty factors warrant scrutiny, conceivably concretized into a unified probability distribution function (PDF) that takes into account their temporal inter-dependencies. Diverse uncertainty factors, characterized by varying marginal distributions and scales, can be assimilated into a multivariate probability distribution through a conversion to normal distributions via rank correlation. However, with the escalation in the number of uncertainty factors embraced within a microgrid context, the endeavour becomes notably intricate when aiming to define conditional probability distributions originating from joint PDFs. This paper presents a method proposing the modelling of net-load forecast error distribution, considering the interplay among uncertainty factors. The approach introduces a data-driven Gaussian process regression technique for training and validating conditional PDFs among these uncertainty factors. Notably, this approach facilitates the transformation of said factors into normal distributions while preserving their inherent marginal characteristics. The resultant conditional density function, as per the proposed methodology, exhibits enhanced suitability for estimating net-load error distribution. Consequently, the conditional density function stemming from this proposed approach demonstrates superior aptitude in approximating the distribution of net load error.

Bayesian calibration and fitting of nuclear thermal–hydraulic models by Markov chain Monte Carlo methods using the Gibbs sampler

This paper applies Bayesian methods to obtain the joint probability distribution function (PDF) of the unknown parameters of some of the correlation types used in thermal hydraulics, which are conditioned to a set of results of separate effect experiments. The problem is solved by Markov chain Monte Carlo methods using the Gibbs sampler. In this paper it is shown how to obtain the conditional PDFs for each one of the parameters, which define two typical correlations that appear in thermal–hydraulic problems, i) the heat transfer in the walls of a pipe and ii) the penetration length of steam discharged in a subcooled pool. The equations for the conditional probabilities of each one of the parameters are deduced and then used to build the Gibbs sampler which is then programmed in R. The PDFs are obtained for each one of the parameters and from these the maximum a posteriori of each one of the parameters is obtained. The new correlations obtained in this way are compared with the experimental data and show satisfactory performance for the case of the penetration of steam in a subcooled pool. Also, we study two cases one with a small number of experimental data and another one with a large number, it is observed that if the number of parameters is small the Markov chain converges fast to the region of high probability of the parameters for both cases with a small number of data and with large number of data, but if the number of parameters increases then the convergence for the case with small number of data becomes poor, however the convergence continuous being very fast for the case with large number of experimental data.

Identification of dynamic drought propagation from a nonstationary perspective and its application to drought warnings

Accurate drought warnings in changing environments remain challenging since the nonstationary behaviors of drought propagation. The novelty of this study is the presentation of a nonstationary framework for drought propagation analysis and the objective determination of drought propagation thresholds. The calculated propagation probabilities and thresholds are responsive to changing environments. The Generalized Addictive Models in Location, Scale and Shape (GAMLSS) and time-varying Clayton copula function were adopted to establish the nonstationary joint distribution between meteorological and hydrological drought indices. Several indices reflecting the climatic and anthropic impacts on hydrological process were selected as covariates. Combined with the conditional probability distribution function, the time-varying drought propagation probabilities can be estimated to describe the watershed drought tolerance variations. The receiver operator characteristic (ROC) curves were utilized to determine the propagation thresholds for drought warnings. The upper Yellow River Basin (UYRB) was selected as the study area. Results indicate that the nonstationary models show better fitting performance for two drought indices series. The Clayton copula function with dynamic parameters offers a more accurate description of the linkages between two types of drought. The analysis of watershed drought tolerance reveals a declining trend from June to October, contrasting with an increasing trend during other months. In comparison to the stationary approach, the hit rates of the proposed method improve by 0.18, reaching 0.88, while the false alarm rates decrease by 0.11, reaching 0.24. The criteria for propagation threshold determination in the UYRB is established as a conditional probability of 20%. These findings can provide a foundation for drought warning management in a changing environment.

Statistical refinement of the North American Multi-Model Ensemble precipitation forecasts over Karoon basin, Iran

Abstract An effective postprocessing approach has been examined to improve the skill of North American Multi-Model Ensemble (NMME) precipitation forecasts in the Karoon basin, Iran. The Copula–Bayesian approach was used along with the Normal Kernel Density marginal distribution and the Kernel Copula function. This process creates more than one postprocessing precipitation value as results candidates (first pass). A similar process is used for a second pass to obtain preprocessed values based on the candidate inputs, which helps identify the most suitable postprocessed value. The application of the technique for order preference by similarity to the ideal solution method based on conditional probability distribution functions of the first and second passes leads to achieving final improved forecast data among the existing candidates. To validate the results, data from 1982–2010 and 2011–2018 were used for the calibration and forecast periods. The results show that while the GFDL and CFS2 models tend to overestimate precipitation, most other NMME models underestimate it. Postprocessing improves the accuracy of forecasts for most models by 20%–40%. Overall, the proposed Copula–Bayesian postprocessing approach could provide more reliable forecasts with higher spatial and temporal consistency, better detection of extreme precipitation values, and a significant reduction in uncertainties.

Validation of uncertainty predictions in digital soil mapping

It is quite common in digital soil mapping (DSM) to quantify the uncertainty of issued predictions, that is to make probabilistic predictions. Yet, little attention has been paid to its validation. Probabilistic predictions are only of value for end users if they are reliable and ideally also sharp. Reliability refers to the consistency between predicted conditional probabilities and observed frequencies of independent test data. Sharpness refers to the concentration of a conditional probability distribution function, i.e. its narrowness. The prediction interval coverage probability (PICP) is currently used in DSM to validate the reliability of prediction intervals but it is ignorant of a potential one-sided bias of its boundaries. Therefore, we propose to extend the current validation procedure with metrics used in the broader probabilistic literature. These metrics not only evaluate probabilistic predictions in prediction interval format but also quantiles or full conditional probability distributions. We suggest the quantile coverage probability (QCP) and probability integral transform (PIT) histogram as alternatives to PICP and proper scoring rules for relative comparisons of competing probabilistic models. As scoring rules, we present the interval score (IS) and the continuous ranked probability score (CRPS), which can be decomposed into a reliability part (RELI). We illustrated the use of these metrics in a case study using soil pH and soil organic carbon from the LUCAS-soil database. Thereby, probabilistic predictions of five different models were compared: a reference null model (NM), quantile regression forest (QRF), quantile regression post-processing of a random forest (QRPP RF), kriging with external drift (KED) and quantile regression neural network (QRNN). For KED and QRNN, one-sided bias was found. This was not apparent from PICP but was shown by use of the PIT histogram and QCP. RELI summarized the trends found in QCP, PICP and PIT histograms to one numerical value. CRPS and IS were especially harsh to outliers and low sharpness. According to CRPS and IS, the best probabilistic predictions were obtained by QRF and QRPP RF and the worst by NM.

Directional wind characteristics analysis in the mountainous area based on field measurement

Wind characteristics are one of the most important factors when designing long-span bridges. In a typical deep-cutting gorge, the wind field is influenced by local terrain and shows dependence on wind direction according to preliminary numerical simulations. Field measurement is then employed to evaluate the impact of wind direction further. Instead of the common method, which splits samples into sectors by referring to the canyon direction, a qualitative computation approach is adopted. Specifically, marginal probability distribution functions (PDFs) of various wind characteristic parameters are explored initially, followed by binary and ternary joint PDFs associated with wind direction using the modified Bernstein copula. The directional wind characteristics are then examined utilizing conditional PDFs and mathematical expectations for all wind directions. The results show that both the distribution and mean value of wind parameters in severe winds exhibit a substantial association with wind direction. The statistical properties change significantly and cannot be captured using the common method when airflows enter from downstream of the canyon and are blocked by the ridge near the bridge. When designing long-span bridges sensitive to wind action, the impact of wind direction should be carefully factored, especially when local terrain blocks the frequent incoming flow.

A Method of Probability Distribution Modeling of Multi-Dimensional Conditions for Wind Power Forecast Error Based on MNSGA-II-Kmeans

How to consider both the influence of weather and wind power in the modeling process of probability distribution of wind power forecast error (WPFE), and to emphasize the application value of conditional modeling, is rarely studied at present. This paper proposes a novel method of conditional probability distribution modeling for WPFE. This method uses a proposed MNSGA-II-Kmeans algorithm to perform multi-objective clustering of multi-dimensional influencing factors (MDIF), including weather and wind power. It can maximize the difference between the probability distributions of each MDIF mode’s WPFE while clustering, thus ensuring the application value of the conditional modeling way. Based on the clustering results, by using the versatile distribution to simulate the probability distribution of WPFE and the support vector machine to realize the recognition of MDIF modes, the specific conditional probability distribution function of WPFE can be provided to stochastic economic dispatch by identifying the forecast MDIF data. A wind plant of north China with historical data is selected for calculation. The results verify the effectiveness of the proposed method, and by comparison with the non-conditional probability distribution of WPFE that does not consider MDIF, it can effectively increase the wind power consumption of the power system.

Comparison between bivariate and trivariate flood frequency analysis using the Archimedean copula functions, a case study of the Karun River in Iran

Historically, severe floods have caused great human and financial losses. Therefore, the flood frequency analysis based on the flood multiple variables including flood peak, volume and duration poses more motivation for hydrologists to study. The main goal of this paper is conducting a tri-variate flood frequency analysis through simultaneously processing the three main variables of any flood event. In addition to this analysis, three bi-variate flood frequency analyses are also performed considering the variables pairwise. Meanwhile, the Archimedean copula functions are employed to conduct such analyses, and finally compared based on their performance in estimating the accurate and reliable flood frequencies. Bivariate and trivariate flood frequency analysis and modeling using Archimedean copula functions is focused. For this purpose, the annual flood data over a 55-year historical period recorded at the Dez Dam hydrometric station were used. According to the goodness-of-fit criteria along with the analytical tail dependence results based on the extreme value theory, the Frank function built upon the couple of the flood peak-volume and the couple of the flood peak-duration as well as the Clayton function built upon the flood volume-duration were identified to be the best copula families to be adopted. The trivariate analysis was conducted and the Clayton family was chosen as the best copula function. Thereafter, the common and conditional cumulative probability distribution functions were built and analyzed to determine the periodic "and", "or" and "conditional" bivariate and trivariate flood return periods. The results suggest that the bivariate conditional return period obtained for short-term periods is more reliable than the trivariate conditional return period. Additionally, the trivariate conditional return period calculated for long-term periods is more reliable than the bivariate conditional return period.

Copula based post-processing for improving the NMME precipitation forecasts

Using reliable and timely precipitation forecasts on a monthly or seasonal scale could be useful in many water resources management planning, especially in countries facing drought challenges. Amongst many, the North American Multi-Model Ensemble (NMME) is one of the most well-known models. In this study, a Bayesian method based on Copula functions has been applied to improve NMME precipitation forecasts. This method is based on the existence of a correlation between the raw forecast and observational data. Two main factors affect the results of rainfall improvement based on the selected method. This research has presented innovative methods in these regards namely; 1) the approach of selecting the appropriate statistical distribution for variables and 2) the selection method of improved data according to the conditional probability distribution functions (CPDF). To evaluate the effectiveness of the statistical distribution, firstly the precipitation forecast improvement model has been developed based on the application of parametric (Exponential, Normal, Gamma, LogNormal and General Exreteme Value (GEV)) and non-parametric distributions (Standard Normal Kernel). Then the novel mixed distribution function based on GEV parametric distribution and Standard Normal Kernel (non-parametric distribution) has been suggested. As the second aim, a new method for selecting improved data based on the center of mass of estimated CPDF is presented. The evaluation of the proposed method for estimating the statistical distribution of data and improving the forecast precipitation by the NMME model has been performed in Sistan and Baluchestan province in Iran. In this regard, the data of 1982–2010 for the calibration period and the data of 2012–2016 for the validation of the results have been used. According to the results, the non-parametric distribution best fitted with the data in the time series and selecting the appropriate bandwidth increased the efficiency of this distribution. Besides, due to the weakness of non-parametric distributions in the boundaries, the use of GEV distribution with a high ability to estimate boundary conditions as semi-parametric distribution, led to improved performance of the proposed distribution. Finally, the selection of the improved data based on the center of the mass method has efficiently provided much improvement compared to the maximum likelihood method commonly used.

Stress Testing and Systemic Risk Measures Using Elliptical Conditional Multivariate Probabilities

Systemic risk, in a complex system with several interrelated variables, such as a financial market, is quantifiable from the multivariate probability distribution describing the reciprocal influence between the system’s variables. The effect of stress on the system is reflected by the change in such a multivariate probability distribution, conditioned to some of the variables being at a given stress’ amplitude. Therefore, the knowledge of the conditional probability distribution function can provide a full quantification of risk and stress propagation in the system. However, multivariate probabilities are hard to estimate from observations. In this paper, I investigate the vast family of multivariate elliptical distributions, discussing their estimation from data and proposing novel measures for stress impact and systemic risk in systems with many interrelated variables. Specific examples are described for the multivariate Student-t and the multivariate normal distributions applied to financial stress testing. An example of the US equity market illustrates the practical potentials of this approach.

Universal Conditional Distribution Function of [O ii] Luminosity of Galaxies, and Prediction for the [O ii] Luminosity Function at Redshift z < 3

Abstract The star-forming emission line galaxies (ELGs) with a strong [O ii] doublet are one of the main spectroscopic targets for the ongoing and upcoming fourth-generation galaxy redshift surveys. In this work, we measure the [O ii] luminosity L [O II] and the absolute magnitude in the near-ultraviolet (NUV) band M NUV for a large sample of galaxies in the redshift range 0.6 ≤ z < 1.45 from the Public Data Release 2 (PDR-2) of the VIMOS Public Extragalactic Redshift Survey (VIPERS). We aim to construct the intrinsic relationship between L [O II] and M NUV through Bayesian analysis. In particular, we develop two different methods to properly correct for the incompleteness effect and observational errors in the [O ii] emission line measurement. Our results indicate that the conditional distribution of L [O II] at a given M NUV can be well described by a universal probability distribution function (PDF), which is independent of M NUV and redshift. Convolving the L [O II] conditional PDF with the NUV luminosity function (LF) available in the literature, we make a prediction for [O ii] LFs at z < 3. The predicted [O ii] LFs are in good agreement with the observational results from the literature. Finally, we utilize the predicted [O ii] LFs to estimate the number counts of [O ii] emitters for the Subaru Prime Focus Spectrograph survey. This universal conditional PDF of L [O II] provides a novel way to optimize the source targeting strategy for [O ii] emitters in future galaxy redshift surveys, and to model [O ii] emitters in theories of galaxy formation.

Investigation the Effects of Climate Variables on Yield and Yield Risk of Dry-land Wheat Using Moment-Based Models

The yield of agricultural products and their risk are highly conditional on climate conditions. Thus, investigating how these changes will affect yields of different crops and their risk of production is of interest to producers and agricultural policy makers. This study tries to investigate consequences of changes in climate parameters on yield and risk of dry-land wheat production in Mashhad city. To this end, annual time series data on dry-land wheat yield and monthly climate information over 1991 to 2017 in Mashhad were used to specify crop yield conditional probability distribution functions, using Moment-based regression models. Unconditional yield distribution then, was derived using the yield conditional probability distributions. Finally, simulation approach was used to simulate the effects of changes in the climate parameters on the mean and variance of yield distribution. Results reveal that the rainfall factor in the vegetative growth phase and in the germination stage, the total annual precipitation, and the numbers of days that go below zero degree of Celsius in the sleep stage are important factors in determining the yield and yield risk of dry-land wheat in Mashhad. The simulation results indicate that the yield’s mean of dry-land wheat will fluctuate between +21.40% and -20.49% relative to the base line yield, given the best or the worst reviewed climate condition is prevailed. In addition, given the upward trend of average annual temperature and downward trend of total annual precipitation during the past 26 years in Mashhad, and according to IPCC climate change prediction, an increase in the risk of production and a decrease in the mean of the yield is expected in the future. Accordingly, developing an appropriate variety which is dry resistance, or searching for an appropriate plant to be substituted for the dry-land wheat in Mashhad can be suggested as a way to face with climate change challenges in this region.

Gaussian mixture model for robust design optimization of planar steel frames

A new method is presented for an application of the Gaussian mixture model (GMM) to a multi-objective robust design optimization (RDO) of planar steel frame structures under aleatory (stochastic) uncertainty in material properties, external loads, and discrete design variables. Uncertainty in the discrete design variables is modeled in the wide range between the smallest and largest values in the catalog of the cross-sectional areas. A weighted sum of Gaussians is statistically trained based on the sampled training data to capture an underlying joint probability distribution function (PDF) of random input variables and the corresponding structural response. A simple regression function for predicting the structural response can be found by extracting the information from a conditional PDF, which is directly derived from the captured joint PDF. A multi-objective RDO problem is formulated with three objective functions, namely, the total mass of the structure, and the mean and variance values of the maximum inter-story drift under some constraints on design strength and serviceability requirements. The optimization problem is solved using a multi-objective genetic algorithm utilizing the trained GMM for calculating the statistical values of objective and constraint functions to obtain Pareto-optimal solutions. Since the three objective functions are highly conflicting, the best trade-off solution is desired and found from the obtained Pareto-optimal solutions by performing fuzzy-based compromise programming. The robustness and feasibility of the proposed method for finding the RDO of planar steel frame structures with discrete variables are demonstrated through two design examples.

Exploiting geographical-temporal awareness attention for next point-of-interest recommendation

With the prosperity of the location-based social networks, next point-of-interest (POI) recommendation has become an increasingly significant requirement since it can benefit both users and business. Obtaining insight into user mobility for the next POI recommendations is a vital yet challenging task. Existing approaches to understanding user mobility mainly gloss over the check-in sequence, making it fail to explicitly capture the subtle POI–POI interactions across the entire user check-in history and distinguish relevant check-ins from the irrelevant. In this paper, we proposed a novel recommendation approach, namely geographical-temporal awareness hierarchical attention network (GT-HAN) to resolve those issues. We first establish a geographical-temporal attention network to simultaneously uncover the overall sequence dependence and the subtle POI–POI relationships. Then, a context-specific co-attention network was designed to learn to change user preferences by adaptively selecting relevant check-in activities from check-in histories, which enabled GT-HAN to distinguish degrees of user preference for different check-ins. Finally, we make a POI recommendation using a conditional probability distribution function. Experimental results on real world datasets (obtained from Foursquare and Gowalla) show that the GT-HAN model significantly outperforms current state-of-the-art approaches, and demonstrating the benefits produced by new technologies incorporated into GT-HAN.

Stress Testing and Systemic Risk Measures Using Multivariate Conditional Probability

Systemic risk, in a complex system with several interrelated variables, such as a financial market, is quantifiable from the multivariate probability distribution describing the reciprocal influence between the system’s variables. The effect of stress on the system is reflected by the change in such a multivariate probability distribution, conditioned to some of the variables being at a given stress’ amplitude. Therefore, the knowledge of the conditional probability distribution function can provide a full quantification of risk and stress propagation in the system. However, multivariate probabilities are hard to estimate from observations. In this paper, I investigate the vast family of multivariate elliptical distributions, discussing their estimation from data and proposing novel measures for stress impact and systemic risk. Specific examples are developed for the multivariate Student-t and the multivariate Normal distributions applied to financial stress testing. An example of the US equity market illustrates the practical potentials of this approach.

Predicting Conditional Expectations For Path-Dependent Events Using TDBP-Learning

Conditional expectations (or probabilities) of events with multiple time steps into the future provide information about the likelihood of future events and, as such, are central to all decision making processes. The knowledge of such formulae is of fundamental importance, and has applications in all sectors of the economy. Hence, we are going to explore the problem of predicting conditional expectations for some path-dependent events driven by stochastic processes. That is, multi-step prediction problems where the correctness of the prediction is not revealed until more than one step after the prediction is made. On one hand, parametric models for computing conditional expectations of path-dependent events are tied to their ability of capturing the dynamics of the underlying stochastic processes. As a result, their misspecification will lead to computation errors, since their formulae depend on the particular form of the dynamics of the stochastic processes chosen. On the other hand, non-parametric models, such as neural networks (NNs), use data to estimate the implicit stochastic processes driving the dynamics of the future events and their relationship with conditional expectations. However, while these methods are good at solving single-step prediction problems, they cannot solve multi-step prediction problems. One solution is to consider methods specifically designed for solving multi-step prediction problems, such as temporal difference (TD) procedures, which are widely used in reinforcement learning (RL). We show that these methods are particularly well suited for capturing the conditional probability distribution function of stochastic processes. To infer the characteristics of conditional expectations for path-dependent events, we propose to combine the error backpropagation algorithm with the TD methods, getting the temporal difference backpropagation (TDBP) model. We propose a framework for learning conditional expectations, both in discrete and continuous space, and perform a thorough analysis of our model by testing it on well known continuous stochastic processes, namely, the Black-Scholes-model, the Merton model, the Heston model, and the Bates model. We show that directly feeding the continuous stochastic processes into Deep Networks improve results, leading to smoother and more accurate conditional expectations. Further, since Deep Networks generalise learning across similar states, they can be used for learning more complex nonlinear functions. Another advantage of using Deep Networks with the TDBP model is that the continuous process can be a vector, allowing for the computation of multiple processes conditional expectations. We will show that, in the continuous space, the TDBP-model recover nearly exactly the forward price, the call option price, and the digital option generated with Monte Carlo simulations. Further, we will extend our analysis to strong path-dependent payoffs such as forward starting contracts, barrier options, and American options. Finally, we will compute multiunderlying contingent claims such as basket options. In conclusion, we can approximately predict conditional expectations for path-dependent events with a single TDBP-model or an ensemble of such models.

Predicting forest fires burned area and rate of spread from pre-fire multispectral satellite measurements

Operational forest fire danger rating systems rely on the recent evolution of meteorological variables to estimate dead fuel condition. Further combining the latter with meteorological and environmental variables, they predict fire occurrence and spread. In this study we retrieved live fuel condition from MODIS multispectral measurements in the near infrared and shortwave infrared. Next, we combined these retrievals with an extensive dataset on actual forest fires in Campania (13,595 km2), Italy, to determine how live fuel condition affects the probability distribution functions of fire characteristics. Accordingly, the specific objective of this study was to develop and evaluate a new approach to estimate the probability distribution functions of fire burned area, duration and rate of spread as a function of the Perpendicular Moisture Index (PMI), whose value decreases with decreasing live fuel moisture content (LFMC). To this purpose, available fire data was intersected with MODIS 8-day composited reflectance data so to associate each fire event with the corresponding pre-fire PMI observation. Fires were then grouped in ten decile bins of PMI, and the conditional probability distribution functions of burned area, fire duration and rate of spread were determined in each bin. Distributions of burned area and rate of spread vary across PMI decile bins, while no significant difference was observed for fire duration. Further testing this result with a likelihood ratio test confirmed that PMI is a covariate of burned area and rate of spread, but not of fire duration. We defined an extreme event as a fire whose burned area (respectively rate of spread) exceeds the 95th percentile of the frequency distribution of all observed fire events. The probability distribution functions in the ten decile bins of PMI were combined to obtain a conditional probability distribution function, which was then used to predict the probability of extreme fires, as defined. It was found that the probability of extreme events steadily increases with decreasing PMI. Overall, at the end of the dry season the probability of extreme events is about the double than at the beginning. These results may be used to produce frequently (e.g. daily) updated maps of the probability of extreme events given a PMI map retrieved from e.g. MODIS reflectance data.

A Bayesian Hydrometeor Classification Algorithm for C-Band Polarimetric Radar

A hydrometeor classification algorithm is developed by applying Bayes’ theorem to C-band polarimetric weather radar measurements. The Bayesian hydrometeor classification algorithm (BHCA) includes eight hydrometeor types: hail, rain, graupel, dry snow, wet snow, crystal, biological scatterers (BS) and ground clutter (GC). The conditional likelihood probability distribution functions (PDFs) for each hydrometeor type are constructed with training data from radar observations. The prior PDFs include not only temperature information but also background information about occurrence frequency of hydrometeor types at each altitude, which is incorporated by a hydrometeor classification algorithm for the first time. The BHCA is evaluated by comparing with the Marzano-Bayesian hydrometeor classification algorithm (MBHC) and NCAR fuzzy logic classifier (NFLC). Results show that wet snow is largely missed in MBHC, while crystals are not adequately identified by NFLC. This may be due to the inappropriate conditional likelihood PDFs or membership functions. The prior PDFs in the MBHC may cause unexpected hail due to unreasonable variation above 0 °C. In addition, the prior PDFs of graupel and dry snow in the MBHC appear below −52 °C, which is not realistic. The BHCA proposed in this study overcomes these shortcomings in the prior PDFs and produces an overall reasonable classification product over the Yangtze-Huaihe River Basin (YHRB), Eastern China.

Theoretical Analysis of Label Distribution Learning

As a novel learning paradigm, label distribution learning (LDL) explicitly models label ambiguity with the definition of label description degree. Although lots of work has been done to deal with real-world applications, theoretical results on LDL remain unexplored. In this paper, we rethink LDL from theoretical aspects, towards analyzing learnability of LDL. Firstly, risk bounds for three representative LDL algorithms (AA-kNN, AA-BP and SA-ME) are provided. For AA-kNN, Lipschitzness of the label distribution function is assumed to bound the risk, and for AA-BP and SA-ME, rademacher complexity is utilized to give data-dependent risk bounds. Secondly, a generalized plug-in decision theorem is proposed to understand the relation between LDL and classification, uncovering that approximation to the conditional probability distribution function in absolute loss guarantees approaching to the optimal classifier, and also data-dependent error probability bounds are presented for the corresponding LDL algorithms to perform classification. As far as we know, this is perhaps the first research on theory of LDL.