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The numerical strength of the evidence of matching DNA profiles is conveyed by a likelihood ratio. This is the probability under one hypothesis, generally from the Prosecution, divided by the probability of the evidence under an alternative hypothesis, generally from the Defense. The probabilities can be calculated to reflect the effects of population structure or of relatedness between individuals listed under the alternative hypotheses.

Probability theory provides the general framework within which assignments of probabilities of past, present, and future events are coherently modified in the light of observed events or, more generally, new information. Forensic scientists, as an illustrative example, routinely face tasks of reasoning under uncertainty when they seek to assist members of the judiciary in evaluating or interpreting the meaning of items of scientific evidence. As a consequence of the laws of probability theory and related concepts, Bayes’ theorem is the key rule according to which to conduct such reasoning in order to comply with the requirement of rationality. This quantification, though, does not represent the end of the matter as the forensic scientist may also be confronted with questions of how to make a rational choice amongst alternative courses of action. This article presents the role of Bayes’ theorem, and its extension to decision analysis, in categorical and continuous data analysis in forensic science applications. It emphasizes the importance of propositional hierarchies, the role of background information, the interpretation of probability as personal degrees of belief and the personal quantification of the consequences of decisions. The discussion also includes a sketch of some common pitfalls of intuition associated with probabilistic reasoning in legal contexts.

Globally, solar photovoltaic (PV) installations on distribution LV feeders have increased significantly. The increased penetration leads to several technical problems on existing networks and impacts utilities’ business models as energy sales drop. For proactive management of these challenges, utilities need to continually monitor the capacity of installed PV. To this end, some utilities typically require PV installations to be registered and sometimes use GIS mapping to approximate the installed PV capacity. However, these GIS PV capacity estimations methods are unreliable. Therefore, to obtain reliable PV estimates at a distribution level, comprehensive modeling is required to accurately represent the generation output and the distribution load. This paper proposes a novel probabilistic PV estimation method that uses time-series historical sets of load and irradiance data to estimate the embedded PV capacity while considering the uncertainty in solar irradiance and the measured net load. Uncertainty characterization is implemented using empirical probability density functions, and simulation is performed stochastically using Monte-Carlo simulation methods. A novel quantile analysis approach is developed and used in the computation of the final capacity of PV. The proposed methodology is tested using measured load data from Ausgrid customers, Australia, and achieves reasonable accuracy (between 86% and 90% for the tested cases) and a 10% mean absolute percentage error. This approach is robust to the effects of input uncertainty and can be used by distribution utilities to estimate and monitor PV capacity installed on the distribution networks without incurring extra advanced metering investment costs.

In order to accurately estimate wind farm output and subsequently optimise it, a method of wind speed distribution approximation is suggested. The method is based on period-by-period accumulation of wind speed measurements, transforming them into empirical probabilities, and observing the moving approximation to the expected power produced by the wind turbine or entire wind farm. A year is a minimal term during which wind statistics are to be accumulated. The sufficient validity and reliability of the wind speed distribution approximation is supported by controlling root -mean-square deviations and maximal absolute deviations with respect to the moving average of the expected power. The approximation quality can be regulated by adjusting constants defining the requirements to the moving deviations.

Domain adaptation algorithms are designed to minimize the misclassification risk of a discriminative model for a target domain with little training data by adapting a model from a source domain with a large amount of training data. Standard approaches measure the adaptation discrepancy based on distance measures between the empirical probability distributions in the source and target domain. In this setting, we address the problem of deriving generalization bounds under practice-oriented general conditions on the underlying probability distributions. As a result, we obtain generalization bounds for domain adaptation based on finitely many moments and smoothness conditions.

In this paper, we discuss physical model based method to generate channel coefficients for Nakagami-m distribution. We consider the phase envelope joint distribution so that the phase of the faded signal is also considered. Mainly, the coefficients are generated by exploring the physical model that relates the Nakagami-m fading distribution with Gaussian and gamma distributions for which generation of coefficients is available in commonly used simulation tools. The empirical probability density function (PDF) of generated coefficients are compared with the theoretical values and they are found in excellent agreements. The empirical PDFs for envelope and phase of the generated coefficients are validated using the Kolmogorov–Smirnov test.

Highlights • A new a deterministic and stochastic epidemic model with varying total population size is proposed. • The thresholds of the deterministic and stochastic epidemic model are established respectively. • Our methods can improve some existing results. • We carry out numerical and find a critical isolation rate 0.0217. • The empirical probability density are presented., In this paper, a stochastic and a deterministic SIS epidemic model with isolation and varying total population size are proposed. For the deterministic model, we establish the threshold R0. When R0 is less than 1, the disease-free equilibrium is globally stable, which means the disease will die out. While R0 is greater than 1, the endemic equilibrium is globally stable, which implies that the disease will spread. Moreover, there is a critical isolation rate δ*, when the isolation rate is greater than it, the disease will be eliminated. For the stochastic model, we also present its threshold R0s. When R0s is less than 1, the disease will disappear with probability one. While R0s is greater than 1, the disease will persist. We find that stochastic perturbation of the transmission rate (or the valid contact coefficient) can help to reduce the spread of the disease. That is, compared with stochastic model, the deterministic epidemic model overestimates the spread capacity of disease. We further find that there exists a critical the stochastic perturbation intensity of the transmission rate σ*, when the stochastic perturbation intensity of the transmission rate is bigger than it, the disease will disappear. At last, we apply our theories to a realistic disease, pneumococcus amongst homosexuals, carry out numerical simulations and obtain the empirical probability density under different parameter values. The critical isolation rate δ* is presented. When the isolation rate δ is greater than δ*, the pneumococcus amongst will be eliminated.

The phenomenon of wave set-up may substantially contribute to the formation of devastating coastal flooding in certain coastal areas. We study the appearance and properties of empirical probability density distributions of the occurrence of different set-up heights on an approximately 80 km long section of coastline near Tallinn in the Gulf of Finland, eastern Baltic Sea. The study area is often attacked by high waves propagating from various directions, and the typical approach angle of high waves varies considerably along the shore. The distributions in question are approximated by an exponential distribution with a quadratic polynomial as the exponent. Even though different segments of the study area have substantially different wave regimes, the leading term of this polynomial is usually small (between −0.005 and 0.005) and varies insignificantly along the study area. Consequently, the distribution of wave set-up heights substantially deviates from a Rayleigh or Weibull distribution (that usually reflect the distribution of different wave heights). In about three-quarters of the occasions, it is fairly well approximated by a standard exponential distribution. In about 25 % of the coastal segments, it qualitatively matches a Wald (inverse Gaussian) distribution. The Kolmogorov–Smirnov test (D value) indicates that the inverse Gaussian distribution systematically better matches the empirical probability distributions of set-up heights than the Weibull, exponential, or Gaussian distributions.

A study has been made of the random motion of inertial particles in a homogeneous isotropic turbulent gas flow. Fluctuations of the gas velocity along the particle path were modeled by the Gaussian random process with a finite time of degeneracy of the autocorrelation function. A closed equation has been obtained for the probability density function of the particle velocity, for which two methods of numerical solution have been proposed: using the finite-difference scheme and using one based on direct numerical modeling of an empirical probability density function. The empirical probability density function was obtained as a result of the averaging of random particle paths, which are a solution of a system of ordinary stochastic differential equations. The results of numerical calculation have been compared with the analytical solution describing the dynamics of the probability density function of the particle-velocity distribution.

The probability distributions of hydrological series are primarily determined according to the best fitting of empirical probability from observation data. The fitted distribution functions are variable in different watersheds, and thus difficult to be applied to estimate probability of any discharge occurrence at other stations. In this study, chaotic characteristics of annual discharge series are analyzed by using correlation dimension analysis at 12 hydrological stations in the six main rivers in China. It is found that the correlation integral of annual discharge series represents the probability distribution of difference between any two annual discharges. The derived probability function can be perfectly fitted by the Pearson III function. Although the six basins have distinctively different climate conditions from the northern cold temperature zone to the southern subtropical zone, the correlation dimension of chaotic characteristics for annual discharges are similar at the 12 hydrological stations. The mean of difference between any two annual discharges, the maximum and minimum of annual discharge are found to be high correlation with the mean of annual discharge. Moreover, the derived probability distributions at 12 hydrological stations can be normalized by a universal distribution function, and the derived probability is high correlation with the probability distribution of annual discharge. This universal probability distribution can be easily applied to obtain probability distributions of annual discharge series over any stations.

The selection of an analytical expression approximating an empirical probability distribution function is considered. For specific examples, the problems that arise in the analysis of data from simulations and tests of aerospace products are identified. These problems cannot be solved by classical statistical methods. A universal approach based on estimation of the distance between the empirical and hypothetical distribution functions permits the selection of the best solution from those available.

A recent method for estimating a lower bound of the population size in capture-recapture samples is studied. Specifically, some asymptotic properties, such as strong consistency and asymptotic normality, are provided. The introduced estimator is based on the empirical probability generating function (pgf) of the observed data, and it is consistent for count distributions having a log-convex pgf ( LC -class). This is a large family that includes mixed and compound Poisson distributions, and their independent sums and finite mixtures as well. The finite-sample performance of the lower bound estimator is assessed via simulation showing a better behavior than some close competitors. Several examples of application are also analyzed and discussed.

Optimal reactive power dispatch (ORPD) problems in power system have been solved by using several variants of traditional nature inspired particle swam optimization (PSO) with aim to achieve a promising solution for a given objective such as line loss, voltage deviation and overall cost minimization. Several schemes have been designed to improve the performance of the optimization technique in tunning the operational variables and analyzed by evaluating the final results. In this article, a different method is designed to solve ORPD problems, by introducing Shannon entropy based diversity in the fractional order PSO dynamics, i.e., FOPSO-EE. The results show that synergy of both, the Shannon entropy and the fractional calculus can be used as the useful tools for enhancing the optimization strength of algorithm while solving the ORPD problems in standard IEEE 30 and 57 bus power systems. The performance of the design FOPSO-EE is further validated through results of statistical interpretations in terms of histogram analysis, box plot illustration, quantile-quantile probability plot and empirical probability distribution function.

The objective of this chapter is to demonstrate how to convert data into probabilities to solve managerial decisions. Real historical (empirical) data does not necessarily fit a known distribution, yet these data frequencies and rankings can be used to estimate the appropriate empirical probability distribution. Subsequently, the empirical distributions are used in decision trees and simulations to make optimum managerial decisions.

The histogram is widely used for visualizing continuous data and calculating empirical probabilities, but it is usually not clear how to choose the number and size of the bins. We propose a new histogram method that optimally deals with the two main tasks of the histogram: detecting features such as mode, increase, spike, gap, outlier and discrete component embedded in the empirical distribution and providing empirical probabilities. We follow Kolmogorov's approach for measuring distance between a histogram distribution function (hdf) and the empirical distribution function (edf), and select a histogram to minimize the distance subject to an upper bound for the number of bins. The idea is that the edf tightly holds all the information in the data and a histogram will serve the main tasks well if the distance between hdf and edf is small. Our comparison study with real-life data demonstrates that the proposed method performs uniformly well in detecting data-features and approximating empirical probabilities. Moreover, the proposed method is appropriate for continuous data with or without discrete component. We provide Rcodes for computing the proposed histogram in the Appendix.

The efficient operation of emergency medical services is critical for any society. Typically, optimisation and simulation models support decisions on emergency ambulance stations’ locations and ambulance management strategies. Essential inputs for such models are the spatiotemporal characteristics of ambulance trips. Access to data on the movements of ambulances is limited, and therefore modelling efforts often rely on assumptions (e.g., the Euclidean distance is used as a surrogate of the ambulance travel time; the closest available ambulance is dispatched to a call; or the travel time estimates, offered by application programming interfaces for ordinary vehicles, are applied to ambulances). These simplifying assumptions are often based on incomplete data or common sense without being fully supported by the evidence. Thus, data-driven research to model ambulance trips is required. We investigated a unique dataset of global positioning system-based measurements collected from seventeen emergency ambulances over three years. We enriched the data by exploring external sources and designed a rule-based procedure to extract ambulance trips for emergency cases. Trips were split into training and test sets. The training set was used to develop a series of statistical models that capture the spatiotemporal characteristics of emergency ambulance trips. The models were used to generate synthetic ambulance trips, and those were compared with the test set to decide which models are the most suitable and to evaluate degrees to which they fit the statistical properties of real-world trips. As confirmed by the low values of the Kullback–Leibler divergence (0.004–0.229) and by the Kolmogorov–Smirnov test at the significance level of 0.05, we found a very good fit between the probability distributions of spatiotemporal properties of synthetic and real trips. A reasonable modelling choice is a model where the exponential dependency on the population density is used to locate emergency cases, emergency cases are allocated to hospitals following empirical probabilities, and ambulances are routed using the fastest paths. The models we developed can be used in optimisations and simulations to improve their validity.

The operation of a mass spectrometry instrument, used in analytical chemistry, for custom applications requires the careful tuning of several instrument settings by an expert. In this work, we developed a model that allows the instrument to tune itself. The approach employs a fast, adaptive evolutionary algorithm, the Covariance Matrix Adaptation evolutionary strategy, to tune a mass spectrometry instrument. By developing a scheme for normalizing the values of the outcome variables (resolution, intensity and peak shape of a calibrant peak signal) based on their experimental probability distributions, we combined the outcomes into a single score that was used as the fitness score for the search algorithm. This approach resulted in a more thorough examination of the search space, and in an economical amount of time by being adaptive, resulting in a more stable tuning, no matter the initial state of the settings involved.

Previous knowledge of the possible spatial relationships between land cover types is one factor that makes remote sensing image classification “smarter”. In recent years, knowledge graphs, which are based on a graph data structure, have been studied in the community of remote sensing for their ability to build extensible relationships between geographic entities. This paper implements a classification scheme considering the neighborhood relationship of land cover by extracting information from a graph. First, a graph representing the spatial relationships of land cover types was built based on an existing land cover map. Empirical probability distributions of the spatial relationships were then extracted using this graph. Second, an image was classified based on an object-based fuzzy classifier. Finally, the membership of objects and the attributes of their neighborhood objects were joined to decide the final classes. Two experiments were implemented. Overall accuracy of the two experiments increased by 5.2% and 0.6%, showing that this method has the ability to correct misclassified patches using the spatial relationship between geo-entities. However, two issues must be considered when applying spatial relationships to image classification. The first is the “siphonic effect” produced by neighborhood patches. Second, the use of global spatial relationships derived from a pre-trained graph loses local spatial relationship in-formation to some degree.

As the growth of interferometric synthetic aperture radar (In-SAR) data continues to accelerate, there is an increasing need to detect surface deformation features from gigantic datasets in an automated fashion. We have developed a computer vision algorithm for automatic detection of InSAR deformation signals using multi-scale Laplacian-of-Gaussian (LoG) filters. We applied the algorithm to InSAR-derived land surface deformation maps in West Texas, where there has been a rise in the level of seismic activity near areas of hydrocarbon production. Our algorithm detected numerous surface deformation features without the need for manual inspection of signals. To control the false detection rate, here we employed a realistic tropospheric turbulence noise model and computed an empirical probability density for the magnitudes of candidate detections under the null hypothesis. We showed that the likelihoods from this PDF can be used to set an acceptable false detection rate for near real-time deformation monitoring applications.

In this paper, we address semantics-empowered communication strategies that simultaneously incorporate semantic aware active sampling and transmission over a shared communication medium. We consider a setting where the goal is to devise a joint sampling and communication scheme over a wireless multiple access channel (MAC) that enables to compute the empirical probability measure of a quantity of interest (QoI) at the destination. We conceptualize a simple application example that illustrates the potential of the approach to provide scalable and resource-efficient operation in massive connectivity settings. We numerically assess the performance of the scheme against an upper bound derived under the assumption that direct access to the noise-free measurements is provided at the destination.

Electromechecanical actuators in the aerospace industry are gradually replacing hydraulic ones. In these circumstances, prognostics and health management are innovative frameworks to ensure better safety on board, especially in flight controls where jamming is dreaded. It allows the user to assess and predict system health in real–time. The first step is to collect temporal data from the monitored actuator and perform a data mining procedure to gain insight into its current health. Clustering encompasses several data-driven methods used to reveal patterns. However, getting a set of classes usually requires providing the algorithm with prior knowledge, such as the number of groups to seek. To avoid this drawback, we have developed a clustering algorithm using a deep neural network, as its core, to get the number of groups in data associated with their likelihood. Temporal sequences are reshaped into pictures to be fed into an artificially trained neural network: U-NET. The latter outputs segmented images from which one-dimensional information is extracted and filtered, without any need for parameter selection. A kernel density estimation finally transforms the signal into a candidate density. This new method provides a robust clustering result coupled with an empirical probability to label the times series. It lays the groundwork for future training of diagnosis and prognosis structures in the PHM framework.

Probability distribution models (PDMs) of variables are cornerstones in the performance-based earthquake engineering framework. For liquefiable ground in earthquake-prone regions, the lique...

In view of that there exist complicated correlations among failure modes of the stochastic structure because of the same uncertain variables that are contained in different failure modes, many reliability analysis methods based on empirical probability model have been developed to calculate and assess the structural reliability. However, the empirical distributions of uncertain variables are not always available or not accurate. Thus, based on the available sample data of random variables of the stochastic structural system to be analyzed, this paper presents a new reliability analysis method in which the mixed Copula is constructed to describe the correlations among multi-failure modes. First, the Gumbel Copula and Clayton Copula, which have the upper and lower tail dependence separately, are utilized to construct a mixed Copula so as to describe the correlations among multi-failure modes. Then, the sample data of the random variables is substituted into the performance functions of failure modes to calculate the sequence values thereof, and the probability density function (PDF) and cumulative distribution function (CDF) of each performance function of the structure are estimated by using the nonparametric kernel density estimation (NKDE) technique. Subsequently, the Canonical Maximum Likelihood (CML) method is applied to estimate the correlation parameters in the mixed Copula. Finally, the main girder structure of an overhead traveling crane is investigated by using the proposed method, and the results indicate that the new reliability analysis method is effective and valid. Compared with traditional reliability analysis methods, the new method, in which the available sample data of random variables is directly used to calculate the values of performance functions so as to avoid the errors caused by empirical distributions, provides some important theories and engineering contributions for the reliability design analyses and safety assessments of actual engineering structures, especially these with complex correlations among multi-failure modes.

We propose three methods for estimating the joint tail probabilities based on a d -variate copula with dimension d ≥ 2 . For the first two methods, we use two different tail expansions of the copula which are valid under mild regularity conditions. We estimate the coefficients of these expansions using the maximum likelihood approach with appropriate data beyond a threshold in the tail. For the third method, we propose a family of tail-weighted measures of multivariate dependence and use these measures to estimate the coefficients of the second tail expansion using regression. This expansion is then used to estimate the joint tail probabilities when the empirical probabilities cannot be used because of lack of data in the tail. The three proposed methods can also be used to estimate tail dependence coefficients of a multivariate copula. Simulation studies are used to indicate when the methods give more accurate estimates of the tail probabilities and tail dependence coefficients. We apply the proposed methods to analyze tail properties of a data set of financial returns.

This paper analyzes the possibilities and constraints in determining the skewness and kurtosis of slopes by radio sounding data at angles at which the backscattered signal is determined by quasi-specular reflection. It is shown that direct recovery of skewness and kurtosis based on the dependence of the normalized backscattering cross section on the incidence angle leads to significant errors. The determination of these parameters by minimizing the discrepancies between the empirical probability density and the truncated Gram—Charlier series given a priori depends significantly on the choice of this model.

The paper proposes an alternative algorithm to implement the current manual choosing mechanism of energy companies whose dedicated department has a library of electric load models and one best model is chosen manually everyday for daily forecast. The proposed algorithm is a combination of an estimation of change point, and a fast ECM algorithm based on the empirical probability function, as well as methods of a hidden markov chain. We train parameters of the proposed algorithm based on a historical dataset consisting of loads, exogenous information such as temperature, and the daily recommended best model which is unavailable sometimes. Simulations and a test on a real-world dataset show that compared with other state-of-art algorithms, the proposed algorithm is fast and efficient for short-term electric load forecasting. An implement to the proposed algorithm written in Matlab is provided in supplement file.

The blackout risks of cascading failures in power systems are notably associated with the failures of transmission lines. Line capacity temporary expansion can reduce blackout risk by decreasing the line failures due to the overloads during the cascading failures. To efficiently quantify the impact of line capacity temporary expansion on blackout risks, we propose a data based state-failure–network method in this paper. The state-failure network, which is formed by the cascading failure data generated by cascading failure simulations, contains the empirical probabilities that correspond to the failure probabilities of lines. Since implementing line capacity temporary expansion to the system can change the failure probabilities of lines and reduce the blackout risk, the empirical probabilities offer the link between line capacity temporary expansion and state-failure network. By updating the values in the state-failure network with changed empirical probabilities, the blackout risk after line capacity temporary expansion is implemented can be efficiently worked out by state-failure network. Thus, the impact of line capacity temporary expansion on blackout risk is quantified by comparing the newly calculated blackout risk with the risk before the line capacity temporary expansion is implemented. The advantage of the proposed method lies in the high accuracy and efficiency of quantifying the impacts of any line capacity temporary expansion schemes once the state-failure network is formed. Case studies verify the accuracy and efficiency of the proposed method.

The trading volume in stock markets is known as an important variable which reflects the liquidity of the financial markets and therefore is regarded to be greatly important for the measurement of market liquidity risk. In this work, a new concept called recurrence duration is introduced for study of daily trading volumes, which is inspired by idea of the volatility duration that was proposed and studied in our previous work. The recurrence duration is thought as the shortest passing time that the following days’ trading volume takes to exceed or go below the current trading volume which is time-varying. Similar to the volatility duration distribution of the price returns, the power-law function could describe the empirical probability distribution of recurrence durations of trading volumes, and their tail distributions can be fitted by two stretched exponential functions. Further, the correlation relationships of trading volumes between Chinese stock indices as well as the correlations of recurrence durations are investigated. One approach employed is a recently proposed method, time-dependent intrinsic correlation (TDIC), which is based on the empirical mode decomposition (EMD) to decompose nonlinear and nonstationary signals into the intrinsic mode functions (IMFs), the instantaneous periods of which are used then in determination of the sizes of sliding windows to compute the running correlation coefficients for the multiscale signals. The empirical results reveal rich patterns of correlations for both trading volumes and recurrence durations at different scales for different modes. Another approach is the widely-used DCCA cross-correlation coefficient, by which the level of cross-correlation is measured for both original series and IMF modes of the stock indices.

Polygenic risk scores (PRSs) for predicting disease risk have become increasingly accurate, leading to rising popularity of PRS tests. Consider an individual whose PRS test has placed him/her at the top q-quantile of genetic risk. Recently, Reid et al. (Circ Genom Precis Med. 2021;14:e003262) have investigated whether such a finding should motivate cascade screening in the proband’s siblings. Specifically, using data from the UK biobank, Reid et al. computed the empirical probability of a sibling of the proband to also have a PRS at the top q-quantile. In this short note, I use the liability threshold model to compute this probability analytically (for either a sibling of the proband or for a more distant relative), showing excellent agreement with the empirical results of Reid et al., including that this probability is disease-independent. Further, I compute the probability of the relative of the proband to be affected, as a function of the quantile threshold q, the proportion of variance explained by the score, and the disease prevalence.

We derive an improvement for probably approximately correct (PAC) sublevel-set bounds for the multinomial distributed discrete random variables. Previous bounds (including Sanov's Theorem) show that the Kullback Leibler (KL) divergence between the empirical probability mass function (pmf) and the true PMF converges with rate O(log(N)/N), where $N$ is the number of independent and identically distributed (i.i.d.) samples used to compute the empirical pmf. We interpret the KL divergence as bounding the probability that a multinomial distributed random variable (RV) deviates into a halfspace and construct improved uniform PAC sublevel-set bounds that converge with rates $O$ (log (log (N)) / N). These results bound the worst case performance for a number of machine learning algorithms. Finally, the ‘halfspace bound’ methodology suggests further improvements are possible for non-uniform bounds. In this paper, we derive an improvement (on the convergence rate) for various Probably Approximately Correct (PAC) bounds (including Sanov's Theorem) for multinomially distributed discrete random variables.

Uncertainty quantification is an important topic for many environmental studies, such as identifying zones where potentially toxic materials exist in the soil. In this work, the nonparametric geostatistical framework of histogram via entropy reduction (HER) is adapted to address local and spatial uncertainty in the context of risk of soil contamination. HER works with empirical probability distributions, coupling information theory and probability aggregation methods to estimate conditional distributions, which gives it the flexibility to be tailored for different data and application purposes. To explore the method adaptation for handling estimations of threshold-exceeding probabilities, it is used to map the risk of soil contamination by lead in the well-known dataset of the region of Swiss Jura. Its results are compared to indicator kriging (IK) and to an ordinary kriging (OK) model available in literature. For the analyzed dataset, IK and HER achieved the best performance and exhibited comparable accuracy and precision of their predictions. When compared to IK, HER has shown to be a unique approach for dealing with uncertainty estimation in a fine resolution, without the need of modeling multiple indicator variograms, correcting order-relation violations, or defining interpolation/extrapolation of distribution. Finally, to avoid the well-known smoothing effect when using point estimations (this is the case with kriging, but also with HER) and to provide maps that reflect the spatial fluctuation of the revealed reality, we demonstrate how HER can be used in combination with sequential simulation to assess spatial uncertainty (uncertainty jointly over several locations).

Quantification of information content and its temporal variation in intracellular calcium spike trains in neurons helps one understand functions such as memory, learning, and cognition. Such quantification could also reveal pathological signaling perturbation that potentially leads to devastating neurodegenerative conditions including Parkinson's, Alzheimer's, and Huntington's diseases. Accordingly, estimation of entropy rate, an information-theoretic measure of information content, assumes primary significance. However, such estimation in the present context is challenging because, while entropy rate is traditionally defined asymptotically for long blocks under the assumption of stationarity, neurons are known to encode information in short intervals and the associated spike trains often exhibit nonstationarity. Against this backdrop, we propose an entropy rate estimator based on empirical probabilities that operates within windows, short enough to ensure approximate stationarity. Specifically, our estimator, parameterized by the length of encoding contexts, attempts to model the underlying memory structures in neuronal spike trains. In an example Markov process, we compared the performance of the proposed method with that of versions of the Lempel-Ziv algorithm as well as with that of a certain stationary distribution method and found the former to exhibit higher accuracy levels and faster convergence. Also, in experimentally recorded calcium responses of four hippocampal neurons, the proposed method showed faster convergence. Significantly, our technique detected structural heterogeneity in the underlying process memory in the responses of the aforementioned neurons. We believe that the proposed method facilitates large-scale studies of such heterogeneity, which could in turn identify signatures of various diseases in terms of entropy rate estimates.

Statistical similarities between neuronal spike trains could reveal significant information on complex underlying processing. In general, the similarity between synchronous spike trains is somewhat easy to identify. However, the similar patterns also potentially appear in an asynchronous manner. However, existing methods for their identification tend to converge slowly, and cannot be applied to short sequences. In response, we propose Hellinger distance measure based on empirical probabilities, which we show to be as accurate as existing techniques, yet faster to converge for synthetic as well as experimental spike trains. Further, we cluster pairs of neuronal spike trains based on statistical similarities and found two non-overlapping classes, which could indicate functional similarities in neurons. Significantly, our technique detected functional heterogeneity in pairs of neuronal responses with the same performance as existing techniques, while exhibiting faster convergence. We expect the proposed method to facilitate large-scale studies of functional clustering, especially involving short sequences, which would in turn identify signatures of various diseases in terms of clustering patterns.

Probability theory provides a mathematical model for the study of randomness and uncertainty. The concept of probability occupies an important role in the decision-making process, whether the problem is one faced in business, in engineering, in government, in sciences, or just in one’s own everyday life. Most decisions are made in the face of uncertainty. The mathematical models of probability theory enable us to make predictions about certain mass phenomena from the necessarily incomplete information derived from sampling techniques. It is the probability theory that enables one to proceed from descriptive statistics to inferential statistics. In fact, probability theory is the most important tool in statistical inference.

Using classifiers for decision making requires well-calibrated probabilities for estimation of expected utility. Furthermore, knowledge of the reliability is needed to quantify uncertainty. Outputs of most classifiers can be calibrated, typically by using isotonic regression that bins classifier outputs together to form empirical probability estimates. However, especially for highly imbalanced problems it produces bins with few samples resulting in probability estimates with very large uncertainty. We provide a formal method for quantifying the reliability of calibration and extend isotonic regression to provide reliable calibration with guarantees for width of credible intervals of the probability estimates. We demonstrate the method in calibrating purchase probabilities in e-commerce and achieve significant reduction in uncertainty without compromising accuracy.

Using a distribution-free "payoffs" CAPM derived under power utility, we examine the parameters of the payoff distribution that have greatest effect on the market equilibrium price, cost of capital and investor welfare. Results are necessarily all numerical, and are obtained by simulating from lognormal and normal payoff distributions. Those distributions are calibrated to approximate the empirical probability distribution of returns and payoffs on the S&P500. The overriding result is that the benefits to investors of better information arise most strongly via better estimates of the mean payoff. Estimation risk surrounding the payoff risk or variance prove surprisingly much less important. That result under our CRRA CAPM replicates similar results in finance under the conventional CARA mean-variance CAPM. We also confirm the inherent disconnection under equilibrium between the market cost of capital and investor welfare.

Recently, many researchers are investigating quantum annealing as a solver for real-world combinatorial optimization problems. However, due to the format of problems that quantum annealing solves and the structure of the physical annealer, these problems often require additional setup prior to solving. We study how these setup steps affect performance and provide insight into the interplay among them using the job-shop scheduling problem for our evaluation. We show that the empirical probability of success is highly sensitive to problem setup and that excess variables and large embeddings reduce performance. We then show that certain problem instances are unable to be solved without the use of additional post-processing methods. Finally, we investigate the effect of pausing during the anneal. Our results show that pausing within a certain time window can improve the probability of success, which is consistent with other work. However, we also show that the performance improvement due to pausing can be masked depending on properties of the embedding, and thus, special care must be taken for embedded problems.

Computer simulation has become the main engine of development in statistical mechanics. In structural biology, computer simulation constitutes the main theoretical tool for structure determination of proteins and for calculation of the free energy of binding, which are important in drug design. Entropy and Free Energy in Structural Biology leads the reader to the simulation technology in a systematic way. The book, which is structured as a course, consists of four parts: Part I is a short course on probability theory emphasizing (1) the distinction between the notions of experimental probability, probability space, and the experimental probability on a computer, and (2) elaborating on the mathematical structure of product spaces. These concepts are essential for solving probability problems and devising simulation methods, in particular for calculating the entropy. Part II starts with a short review of classical thermodynamics from which a non-traditional derivation of statistical mechanics is devised. Theoretical aspects of statistical mechanics are reviewed extensively. Part III covers several topics in non-equilibrium thermodynamics and statistical mechanics close to equilibrium, such as Onsager relations, the two Fick's laws, and the Langevin and master equations. The Monte Carlo and molecular dynamics procedures are discussed as well. Part IV presents advanced simulation methods for polymers and protein systems, including techniques for conformational search and for calculating the potential of mean force and the chemical potential. Thermodynamic integration, methods for calculating the absolute entropy, and methodologies for calculating the absolute free energy of binding are evaluated. Enhanced by a number of solved problems and examples, this volume will be a valuable resource to advanced undergraduate and graduate students in chemistry, chemical engineering, biochemistry biophysics, pharmacology, and computational biology.

Incipient fault detection is a crucial procedure to recognize abnormal behaviors of a complex system. It allows to define necessary actions to be done before further large deteriorations, and helps to prevent the system breakdown abruptly. In these early stage fault detection, amplitude of drift is close to the noise ones. Then it is more complex to distinguish the fault itself. In this paper, a nonparametric method based on empirical probability density is proposed to detect incipient fault in noisy environment. The proposed method not only improves the accuracy of fault detection compared to existing methods but is also suitable for real-time application. Besides, fault position is estimated in the meantime. The performance of the proposed method in different noise levels is discussed either for Gaussian than non-Gaussian type noise. Simulation result validates the effectiveness and noise robustness of the proposed method.

Interpolation of spatial data has been considered in many different forms. This study proposes a stochastic, non-parametric, geostatistical estimator that combines measures of information theory with probability aggregation method. Histogram via entropy reduction (HER) can be used to analyze the data spatial correlation and for predicting distributions at unobserved locations directly based on empirical probability. The method minimizes estimation uncertainty, relaxes normality assumptions and therefore avoids the risk of adding information not available in data (or losing available information). In particular, the applied probability aggregation method provides a proper framework for uncertainty estimation that reflects both the spatial configuration of the data as well as data values, while allowing to infer (or introduce) physical properties (continuous or discontinuous characteristics) from the field under study. Three different aggregation methods were explored in terms of uncertainty, resulting in predictions ranging from conservative to more confident ones. We investigate the performance of the framework using four synthetically generated datasets from known Gaussian processes and demonstrate the efficacy of the method in ascertaining the underlying true field with varying sample sizes. By comparing the method performance to popular benchmark models, namely nearest neighbors (NN), inverse distance weighting (IDW) and ordinary kriging (OK), we were able to obtain competitive results with respect to OK, with the advantage of presenting generalization properties. The novel method brings a new perspective of spatial and uncertainty analysis to geostatistics and statistical learning, using the lens of information theory.

This paper proposes a modulation classification method based on Kullback-Leibler divergence (KLD). The proposed method involves computation of the empirical probability mass functions (PMFs) of the decision statistics and their subsequent comparison with the theoretical PMFs of the decision statistics under each candidate modulation scheme by using KLD. We use quadrature components of the received signal as the decision statistics to compute the PMFs and consider the classification of linear digital modulation schemes such as the phase shift keying and quadrature amplitude modulation schemes in an additive white Gaussian noise channel. Through computer simulations, we show that the proposed KLD-based method outperforms conventional Kolmogorov-Smirnov test-based methods in classification performance.