Choice Of Probability Distribution Research Articles

Dissecting unsupervised learning through hidden Markov modeling in electrophysiological data.

Unsupervised, data-driven methods are commonly used in neuroscience to automatically decompose data into interpretable patterns. These patterns differ from one another depending on the assumptions of the models. How these assumptions affect specific data decompositions in practice, however, is often unclear, which hinders model applicability and interpretability. For instance, the hidden Markov model (HMM) automatically detects characteristic, recurring activity patterns (so-called states) from time series data. States are defined by a certain probability distribution, whose state-specific parameters are estimated from the data. But what specific features, from all of those that the data contain, do the states capture? That depends on the choice of probability distribution and on other model hyperparameters. Using both synthetic and real data, we aim to better characterize the behavior of two HMM types that can be applied to electrophysiological data. Specifically, we study which differences in data features (such as frequency, amplitude, or signal-to-noise ratio) are more salient to the models and therefore more likely to drive the state decomposition. Overall, we aim at providing guidance for the appropriate use of this type of analysis on one- or two-channel neural electrophysiological data and an informed interpretation of its results given the characteristics of the data and the purpose of the analysis.NEW & NOTEWORTHY Compared with classical supervised methods, unsupervised methods of analysis have the advantage to be freer of subjective biases. However, it is not always clear what aspects of the data these methods are most sensitive to, which complicates interpretation. Focusing on the hidden Markov model, commonly used to describe electrophysiological data, we explore in detail the nature of its estimates through simulations and real data examples, providing important insights about what to expect from these models.

Probabilistic characterization of soil properties based on the maximum entropy method from fractional moments: Model development, case study, and application

In probabilistic reliability analysis for geotechnical engineering applications, determining the underlying probability distributions of soil properties and corresponding parameters from observed data is a critical initial step because subsequent risk and reliability analyses depend upon these evaluations. Conventionally, the choice of probability distribution is dictated by subjective familiarity with a classical (e.g., normal or lognormal) distribution. This paper proposes an objective and unbiased method to estimate probability distributions of a soil property using the maximum entropy method from fractional moments of observed data. The probability distribution is based on the concept of maximum entropy and is free from the assumptions of classical distributions. A case study is presented for the undrained shear strength of soil in the Nipigon River landslide area, Ontario, Canada. The maximum entropy distributions of the soil property from fractional moments are compared to the frequency histogram, normal and lognormal distributions, and maximum entropy distributions from integral moments. The maximum entropy distribution with two-order fractional moments, almost equivalent to that with four-order integral moments, is verified with the chi-square goodness-of-fit test. Issues related to overfitting/underfitting, minimum sample size, fractional orders, negative moments, and limitations are discussed. Application of the method to geotechnical reliability analysis is illustrated.

Evaluation of the Areal Reduction Factor in an Urban Area through Rainfall Records of Limited Length: A Case Study

AbstractThis paper addresses the determination of the areal reduction factor (ARF) as a function of area and duration. The analysis is carried out using the data recorded in a 10-year period at the rainfall gauge network of the city of Milan (Italy). Four types of probability distributions [exponential, the extreme value type 1 (EV1), the generalized extreme value (GEV), and generalized Pareto], two parameter estimation methods (probability-weighted moments and partial probability-weighted moments), and four different regression models of ARF on area and duration are considered. A sensitivity analysis is carried out to outline the effect exerted on ARF by the choice of probability distribution and the parameter estimation method and by that of the model. The effect of model choice is more important than the choice of the distribution and estimation method. The models that fit the data best are the newly developed ones. The ARF model that has the best fit presents a root-mean-square error (RMSE) equal to 0...

Quantifying the Effectiveness of Interventions in Workflows

Workflows are common in today’s business world, and are an integral part of current enterprise content management, product lifecycle management, and enterprise resource planning systems. When a task assignee does not complete a task on time, workflow systems are commonly configured to send out reminders. Reminders are a form of intervention in the workflow. It is tacitly assumed that workflow intervention is effective, yet, to date, there has been no quantitative characterization of the benefits of workflow intervention. This study first develops a mathematical model for workflow intervention. The controlling parameters are identified: the choice of probability distribution, the skewness of the probability distribution, the intervention interval, and the effectiveness of individual interventions. To the extent that closed-form solutions are available (e.g., for uniform or triangular probability density functions), they are presented. More generally, results are presented by representing the wait time using the Weibull probability density function. Cases where closed-form solutions are intractable are simulated using the Petri net method. Results indicate that, while interventions always reduce the mean cycle time for a workflow, there are certain circumstances where the cycle time reduction is dramatic (i.e., >50%).

The effect of the different uncertainty models in dose expected volume histogram computation

Dose expected volume histograms are a useful alternative to dose volume histograms in order to take into account computation uncertainty when prescribing, planning and reporting external beam radiation therapy. Due to the type B nature of computation uncertainties, a rectangular probability distribution was assumed in its definition. In the present work, the changes of dose expected volume histograms when using other recommended point dose uncertainty models are investigated. Results show that the choice of probability distribution, among the ones recommended for uncertainty modelling, has a small effect on the result. Thus, the standard uncertainty of point dose computations is the main parameter affecting dose expected volume histograms.

USE OF THE BOOTSTRAP FOR ERROR PROPAGATION IN ESTIMATING ZOOPLANKTON PRODUCTION

Error propagation is the calculation of statistical error in a quantity that comprises multiple components each with associated error. Such quantities constitute the end point of many ecological studies, but composite errors are rarely incorporated so that uncertainty in the final estimate is either unknown or underestimated. In this study we present the use of both parametric (observations are resampled from standard probability distributions) and nonparametric (raw observations are resampled) bootstrap techniques to propagate errors through the many steps involved in the egg-ratio estimation of seasonal production for the freshwater zooplankter Bythotrephes longimanus. We first compute parametric and nonparametric bootstrap estimates of the standard deviation in seasonal production of B. longimanus, showing that it ranges from 21% to 27% of observed seasonal production. We demonstrate that our bootstrapping procedures are robust by developing a theoretical model and showing that the true variance of a parameter is included in 96% of the simulated 95% confidence intervals. We also show that the choice of probability distribution in the parametric bootstrap can change the standard deviation of seasonal production by up to 90%. We argue that ecologists should use such error propagation techniques more routinely than is currently the case.

Regions of convergence of the Rayleigh quotient iteration method

AbstractThe Rayleigh quotient iteration method finds an eigenvector and the corresponding eigenvalue of a symmetric matrix. This is a fundamental problem in science and engineering. Parlett and Kahan have shown, in 1968, that for almost any initial vector in the unit sphere, the Rayleigh quotient iteration method converges to some eigenvector. In this paper, the regions of the unit sphere which include all possible initial vectors converging to a specific eigenvector are studied. The generalized eigenvalue problem Ax = λBx is considered. It is shown that the regions do not change when the matrix is shifted or multiplied by a scalar. These regions are completely characterized in the three‐dimensional case. It is shown that, in this case, the area of the region of convergence corresponding to the interior eigenvalue is larger than the area of those corresponding to any extreme one. This can be interpreted, with the appropriate choice of probability distribution, as: the probability of converging to an eigenvector corresponding to the interior eigenvalue is larger than the probability of converging to an eigenvector corresponding to any extreme eigenvalue. It is conjectured that the same is true for matrices of any order. Experiments in higher dimensions are presented which conform with the conjecture.