Data-Driven Stabilized Forgetting Design Using the Geometric Mean of Normal Probability Densities

Background Scales for evaluating insomnia differ in number of items, response format, and result in different scores distributions and score ranges and may not facilitate meaningful comparisons. Objectives Transform ordinal item-scores of three scales of insomnia to continuous, equidistant, monotonic, normally distributed scores, avoiding limitations of summative scoring of Likert scales. Methods Equidistant item-scores by weighted sum using data-driven weights to different levels of different items, considering cell frequencies of Item-Levels matrix, followed by normalization and conversion to [1, 10]. Equivalent test-scores (as sum of transformed item- scores) for a pair of scales were found by Normal Probability curves. Empirical illustration given. Results Transformed test-scores are continuous, monotonic and followed Normal distribution with no outliers and tied scores. Such test-scores facilitate ranking, better classification and meaningful comparison of scales of different lengths and formats and finding equivalent score combinations of two scales. For a given value of transformed test-score of a scale, easy alternate method avoiding integration proposed to find equivalent scores of another scales. Equivalent scores of scales help to relate various cut-off scores of different scales and uniformity in interpretations. Integration of various scales of insomnia is achieved by finding one-to-one correspondence among the equivalent score of various scales with correlation over 0.99 Conclusion Resultant test-scores facilitated undertaking analysis in parametric set up. Considering the theoretical advantages including meaningfulness of operations, better comparison, use of such method of transforming scores of Likert items/test is recommended test and items, Future studies were suggested.

Download Full-text

On the Jensen–Shannon Symmetrization of Distances Relying on Abstract Means

Entropy ◽

10.3390/e21050485 ◽

2019 ◽

Vol 21 (5) ◽

pp. 485 ◽

Cited By ~ 7

Author(s):

Frank Nielsen

Keyword(s):

Closed Form ◽

Exponential Family ◽

Geometric Mean ◽

Mixture Distribution ◽

Exponential Families ◽

Leibler Divergence ◽

Probability Densities ◽

The Mean ◽

Jensen Shannon Divergence ◽

Closed Form Formula

The Jensen–Shannon divergence is a renowned bounded symmetrization of the unbounded Kullback–Leibler divergence which measures the total Kullback–Leibler divergence to the average mixture distribution. However, the Jensen–Shannon divergence between Gaussian distributions is not available in closed form. To bypass this problem, we present a generalization of the Jensen–Shannon (JS) divergence using abstract means which yields closed-form expressions when the mean is chosen according to the parametric family of distributions. More generally, we define the JS-symmetrizations of any distance using parameter mixtures derived from abstract means. In particular, we first show that the geometric mean is well-suited for exponential families, and report two closed-form formula for (i) the geometric Jensen–Shannon divergence between probability densities of the same exponential family; and (ii) the geometric JS-symmetrization of the reverse Kullback–Leibler divergence between probability densities of the same exponential family. As a second illustrating example, we show that the harmonic mean is well-suited for the scale Cauchy distributions, and report a closed-form formula for the harmonic Jensen–Shannon divergence between scale Cauchy distributions. Applications to clustering with respect to these novel Jensen–Shannon divergences are touched upon.

Download Full-text

Random response of a simple system with stochastic uncertainty and noise in parameters

Applied and Computational Mechanics ◽

10.24132/acm.2021.575 ◽

2021 ◽

Vol 15 (2) ◽

Author(s):

Jiří Náprstek ◽

Cyril Fischer

Keyword(s):

White Noise ◽

Probability Density ◽

Random Perturbation ◽

Random Response ◽

Additive White Noise ◽

Normal Probability ◽

Probability Densities ◽

Parametric Noise ◽

Truncated Normal ◽

General Explanation

The paper is concerned with the analysis of the simultaneous effect of a random perturbation and white noise in the coefficient of the system on its response. The excitation of the system of the 1st order is described by the sum of a deterministic signal and additive white noise, which is partly correlated with a parametric noise. The random perturbation in the parameter is considered statistics in a set of realizations. It reveals that the probability density of these perturbations must be limited in the phase space, otherwise the system would lose the stochastic stability in probability, either immediately or after a certain time. The width of the permissible zone depends on the intensity of the parametric noise, the extent of correlation with the additive excitation noise, and the type of probability density. The general explanation is demonstrated on cases of normal, uniform, and truncated normal probability densities.

Download Full-text

Statistical measures of the central tendency for H+ activity and pH

Soil Science Annual ◽

10.1515/ssa-2017-0022 ◽

2017 ◽

Vol 68 (4) ◽

pp. 174-181 ◽

Cited By ~ 1

Author(s):

Izabela Kuna-Broniowska ◽

Halina Smal

Keyword(s):

Probability Distributions ◽

Geometric Mean ◽

Arithmetic Mean ◽

Central Tendency ◽

Statistical Measures ◽

Normal Probability ◽

Explicit Opinion ◽

Normal Probability Distribution ◽

Theoretical Considerations

Abstract Despite the numerous papers on the statistical analyses of pH, there is no explicit opinion on the use of arithmetic mean as a measure of the central tendency for pH and H+ activity. The problem arises because the transformation of the arithmetic mean for one does not give the arithmetic mean for the other. The paper presents 1) the theoretical considerations on the distribution of pH and H+ activity and relation between them, properties of these distributions, the choice of distributions which should be consistent with the distribution of pH and the distribution of H+ activity and measures of central tendency for features of such distributions and 2) examples of calculations of measures of central tendency for pH and H+ activity based on the literature data on soil and lake water pH. These data analyses included distributions of pH and H+ activities, properties of distribution, descriptive statistics for pH and for the H+ activity and comparison of arithmetic mean with the geometric mean. From the results, it could be concluded that a uniform approach to the choice of measure for the central tendency of pH and H+ activity requires the determination of the type of measure (mean) for one of them and then consistent transformation of this measure. The choice of measure of the central tendency for the variable should be preceded by determination of its distribution. Normal probability distribution of pH and thus lognormal distribution of H+ activity indicate that the arithmetic mean, and its corresponding geometric mean should be used as proper measures of the central tendency for pH and for H+ activity. Besides, the position statistic that is a median can be used for each of those variables, irrespective of their probability distributions.

Download Full-text

Using Test Positivity and Reported Case Rates to Estimate State-Level COVID-19 Prevalence and Seroprevalence in the United States

10.1101/2020.10.07.20208504 ◽

2020 ◽

Author(s):

Weihsueh A. Chiu ◽

Martial L. Ndeffo-Mbah

Keyword(s):

Public Health ◽

United States ◽

Geometric Mean ◽

State Level ◽

Credible Interval ◽

Assistant Professor ◽

The United States ◽

Data Driven ◽

Infection Prevalence ◽

Public Health Decision

AbstractAccurate estimates of infection prevalence and seroprevalence are essential for evaluating and informing public health responses needed to address the ongoing spread of COVID-19 in the United States. A data-driven Bayesian single parameter semi-empirical model was developed and used to evaluate state-level prevalence and seroprevalence of COVID-19 using daily reported cases and test positivity ratios. COVID-19 prevalence is well-approximated by the geometric mean of the positivity rate and the reported case rate. As of December 8, 2020, we estimate nation-wide a prevalence of 1.4% [Credible Interval (CrI): 0.8%-1.9%] and a seroprevalence of 11.1% [CrI: 10.1%-12.2%], with state-level prevalence ranging from 0.3% [CrI: 0.2%-0.4%] in Maine to 3.0% [CrI: 1.1%-5.7%] in Pennsylvania, and seroprevalence from 1.4% [CrI: 1.0%-2.0%] in Maine to 22% [CrI: 18%-27%] in New York. The use of this simple and easy-to-communicate model will improve the ability to make public health decisions that effectively respond to the ongoing pandemic.Biographical Sketch of AuthorsDr. Weihsueh A. Chiu, is a professor of environmental health sciences at Texas A&M University. He is an expert in data-driven Bayesian modeling of public health related dynamical systems. Dr. Martial L. Ndeffo-Mbah, is an Assistant Professor of Epidemiology at Texas A&M University. He is an expert in mathematical and computational modeling of infectious diseases.Summary LineRelying on reported cases and test positivity rates individually can result in incorrect inferences as to the spread of COVID-19, and public health decision-making can be improved by instead using their geometric mean as a measure of COVID-19 prevalence and transmission.

Download Full-text