Platykurtic Distributions Research Articles

Three Parameter Generalized Gaussian Type Distribution

Three Parameter Generalized Gaussian Type Distribution

LITHOLOGY CONTROLS AMMONOID SIZE DISTRIBUTIONS

ABSTRACT Body-size distributions of organisms across environments in space and time are a powerful source of information on ecological and evolutionary processes. However, most studies only focus on selected parameters of size distributions (e.g., central tendency or extremes) and rarely take into account entire distributions and how they are affected by the collection style and facies. Here we analyze the impact of facies, region, taxonomy, and collection style over size distributions using diameter as a proxy of Late Devonian ammonoids in their entirety using non-metric multidimensional scaling and PERMANOVA based on Kolmogorov distance. The effects are then compared with effects on mean sizes. In all analyses, lithology was the dominant effect, with sizes greater by 59% in marls and by 33% in limestones, as compared to black shales. The effect of complete sampling style was a decrease in size by 11%. Kurtosis was an important parameter differentiating size distributions, with platykurtic distributions in marls and leptokurtic distributions in limestones, suggesting that this parameter may reflect different degrees of time averaging. Most size distributions were positively skewed, but most strongly in marls. Complete sampling led to skewness values close to zero (symmetrical distributions) and high kurtosis. Samples from higher paleolatitudes were on average smaller, but contained outliers with the largest sizes, highlighting the need to analyze entire distributions. Lithology and collection differences need to be accounted for when evaluating size differences across space (polar gigantism) and time (Lilliput effect). Similarly, differences in facies may affect species determination.

Working together: Implications of leadership style for the music ensemble

Music leaders who conduct ensembles in education and community settings, create an ecology of participation depending on their style of leadership. The paradigm of master/apprentice that has long been pilloried in music education academia is still a popular modus operandi for many conductors. Understanding the effect that leadership has on group identity and ensemble cohesion can help participant musicians to continue lifelong engagement with ensemble music making. This exploratory research consists of three phases. First, the relevant literature regarding group work, leadership theories, and musical ensembles was explored. Second, utilizing a purpose-designed questionnaire instrument issued to three musical ensembles. The data was collected and analyzed to determine which leadership factors were considered most important in developing a positive ecology for ensemble members. Data were collected from three musical ensembles, resulting in a convenience sample of n = 73. Some unexpected results emerged from several analytical techniques that explored the cross-tabulation of questionnaire data using Pearson’s kurtosis and skewness coefficients and platykurtic distributions. Finally, this research recommends a more egalitarian ecology in order to maintain a strong interest in ensemble music making for participant musicians.

Root mean square error or mean absolute error? Use their ratio as well

The key statistical properties of the Root Mean Square Error (RMSE) and the Mean Absolute Error (MAE) estimators were derived in this study for zero mean symmetric error distributions. A density function, named the Approximate Root Normal Distribution (ARND), was developed to approximate the distribution of a square root of a normal random variable. This enabled approximating the distribution of the RMSE estimator. The theoretical derivations and the demonstrations on common distributions, a benchmark time series, and a real world data set (with prediction errors generated from ANN and ARIMA models) lead to the following practically useful findings. When comparing errors having the same distribution type, RMSE was shown to be preferred for platykurtic distributions, MAE for leptokurtic distributions, and either RMSE or MAE for mesokurtic distributions. For different distribution types, however, using the two estimators alone was shown to lead to erroneous conclusions. The revelation that the estimated RMSE/MSE ratio could identify whether the errors came from platykurtic/mesokurtic/leptokurtic distributions was a useful complementary result. Comparison of errors based on, error distributions, sample size and the standard errors of the estimators, was discussed. The proposed procedure for deriving the statistical properties of the two estimators has scope for extension for other distribution types.

Transfer from soil to grass and statistical analysis of naturally occurring radionuclides in soil from phosphate mining and processing sites in Maritime Region of Togo

Natural radionuclides activity concentrations of 226Ra, 228Ra, and 40K in soil and grass samples, statistical analysis of these radionuclides, and transfer factors (TFs) from soil to the grass were carried out from phosphate mining and processing sites in southern areas of Togo using gamma-ray spectrometry-based high-purity germanium (HPGe) detector. The average activity concentrations of 226Ra, 228Ra, and 40K were found within the range usually measured worldwide. Statistical analysis was done by investigating the skewness, the kurtosis, and the density distribution of radionuclides in the study areas. Both skewness and kurtosis displayed that the activity concentrations of soil and grass samples in the discharge waste site (Kpémé) and mining site (Hahotoé) exhibited slightly positively skewed, negatively skewed, leptokurtic, and platykurtic distributions. The TFs for 226Ra, 228Ra, and 40K from discharge waste soil to grass were evaluated. The average transfer factors of 226Ra, 228Ra, and 40K in the phosphate processing area were found to be 0.16, 0.32, and 8.42, respectively. 226Ra and 228Ra TFs values were found to be within the limit of worldwide measure values, while that of 40K was found to be larger than the values measured in other countries. The average TF value of 8.42 for 40K was eight times higher than the average value of unity, showing its large intake by vegetables. The high TFs for 40K could be explained by the extensive application of phosphate fertilizers to soils when growing vegetables.

Quantification of the Life Time Flight Capabilities of the South American Palm Weevil, Rhynchophorus palmarum (L.) (Coleoptera: Curculionidae).

Simple SummaryThe South American palm weevil, Rhynchophorus palmarum, is an invasive pest that has killed thousands of ornamental palms in San Diego County in California, USA. Emerging management plans for this pest need to consider the flight capabilities of this insect, which are not well understood. To address this shortcoming, flight mills, a type of computerized insect “merry-go round” that measures how far weevils can fly in the laboratory, were used to quantify the flight distances of 101 weevils that were flown repeatedly over the course of their lifetimes. The results indicate that weevils are strong flyers capable of flying numerous times before dying of natural causes. Over their lifetimes, weevils, on average, covered distances that cumulatively totaled >220 km. One female weevil flew an impressive cumulative distance of approximately 806 km over the course of nine consecutive flights before dying.The life time flight capabilities of an invasive palm pest, Rhynchophorus palmarum, were assessed using flight mill assays under controlled conditions in the laboratory. A total of 101 weevils were used for experiments and subjected to repeat flight assays. A total of 17 flight trials were run, of which the first 14 provided useful data prior to weevil death. Male and female weevils exhibited a strong capacity for repeat long distance flights. Flight metrics of interest were not affected by weevil sex or mating status. Cumulative lifetime flight distances for male and female R. palmarum averaged ~268 km and ~220 km, respectively. A maximum lifetime cumulative flight distance of ~758 km and ~806 km was recorded for one male of unknown mating status and one unmated female weevil, respectively. Dispersal data for individual flights (i.e., trials 1 through 9, 10–14 combined) and all flight trial data (i.e., flights 1–14 combined) exhibited platykurtic distributions. The results presented here may have important implications for modeling the spread of this invasive pest and for the development of monitoring and management plans.

Statistical Analysis of AC Dielectric Strength for Palm Oil under the Influence of Moisture

This paper presents a statistical study on the AC Breakdown Voltage (BDV) of Refined, Bleached and Deodorized Palm Oil (RBDPO) olein under the influence of moisture. Different moisture contents of the RBPDPO were prepared by drying the RBDPO in the oven at different time intervals. The AC BDV test of RBDPO was performed whereby the distance between 2 electrodes was set to 1 mm. The statistical analyses of AC BDV data for RBDPO were carried out based on normal, lognormal and Weibull distributions. It was shown that as the moisture content increased, the AC BDV of RDBPO decreased exponentially. Statistical analyses revealed that the AC BDV data with different moisture contents had platykurtic distributions. Moisture could influence the skewness of the distribution whereby the tail shifted from right to left as the content increased. At different moisture contents of RBDPO, most of the AC BDV data of RBDPO could be represented by Weibull distribution.

Omnibus test for normality based on the Edgeworth expansion.

Statistical inference in the form of hypothesis tests and confidence intervals often assumes that the underlying distribution is normal. Similarly, many signal processing techniques rely on the assumption that a stationary time series is normal. As a result, a number of tests have been proposed in the literature for detecting departures from normality. In this article we develop a novel approach to the problem of testing normality by constructing a statistical test based on the Edgeworth expansion, which approximates a probability distribution in terms of its cumulants. By modifying one term of the expansion, we define a test statistic which includes information on the first four moments. We perform a comparison of the proposed test with existing tests for normality by analyzing different platykurtic and leptokurtic distributions including generalized Gaussian, mixed Gaussian, α-stable and Student’s t distributions. We show for some considered sample sizes that the proposed test is superior in terms of power for the platykurtic distributions whereas for the leptokurtic ones it is close to the best tests like those of D’Agostino-Pearson, Jarque-Bera and Shapiro-Wilk. Finally, we study two real data examples which illustrate the efficacy of the proposed test.

Particle Size Characteristics of Fluvial Suspended Sediment in Proglacial Streams, King George Island, South Shetland Island

In this study, the characterization of particle size distribution of suspended sediment that is transported by streams (Ornithologist Creek, Ecology Glacier Creeks, Petrified Forest Creek, Czech Creek, Vanishing Creek, Italian Creek) in the area of the Arctowski Polish Antarctic Station is presented. During the first period of the summer season, the aforementioned streams are supplied by the melting snow fields, while later on, by thawing permafrost. The water samples were collected from the streams at monthly intervals during the Antarctic summer season (January - March) of 2016. The particle size distribution was measured in the laboratory with a LISST-25X laser diffraction particle size analyser. According to Sequoia Scientific Inc., LISST-25X can measure particle sizes (Sauter Mean Diameter) between 2.50 and 500 μm. The results of particle size measurements were analysed in relation to flow velocity (0.18–0.89 m/s), the cross-sectional parameters of the streams, suspended sediment concentration (0.06–167.22 mg/dm3) and the content of particulate organic matter (9.8–84.85%). Overall, the mean particle size ranged from 28.8 to 136 μm. The grain size of well-sorted sediments ranged from 0.076 to 0.57, with the skewness and kurtosis values varying from -0.1 to 0.4, and from 0.67 to 1.3, respectively. Based on the particle size characteristics of suspended sediment, the streams were divided into two groups. For most of the streams, the sediment was very well sorted, while fine sand and very fine sand were dominant fractions displaying symmetric and platykurtic distributions, respectively. Only in two streams, the suspended sediment consisted of silt-size grains, well or moderately well sorted, with coarse-skewness and mostly mesokurtic distribution. The C-M chart suggested that the transportation processes of suspended sediment included the suspended mode only. The grain-size distribution of suspended sediment was mainly influenced by the stream runoff, surface sediment type and biological processes.

MP estimation applied to platykurtic sets of geodetic observations

AbstractMPestimation is a method which concerns estimating of the location parameters when the probabilistic models of observations differ from the normal distributions in the kurtosis or asymmetry. The system of Pearson’s distributions is the probabilistic basis for the method. So far, such a method was applied and analyzed mostly for leptokurtic or mesokurtic distributions (Pearson’s distributions of types IV or VII), which predominate practical cases. The analyses of geodetic or astronomical observations show that we may also deal with sets which have moderate asymmetry or small negative excess kurtosis. Asymmetry might result from the influence of many small systematic errors, which were not eliminated during preprocessing of data. The excess kurtosis can be related with bigger or smaller (in relations to the Hagen hypothesis) frequency of occurrence of the elementary errors which are close to zero. Considering that fact, this paper focuses on the estimation with application of the Pearson platykurtic distributions of types I or II. The paper presents the solution of the corresponding optimization problem and its basic properties.Although platykurtic distributions are rare in practice, it was an interesting issue to find out what results can be provided by MPestimation in the case of such observation distributions. The numerical tests which are presented in the paper are rather limited; however, they allow us to draw some general conclusions.

Heart Rate and Systolic Blood Pressure Variability in the Time Domain in Patients with Recent and Long-Standing Diabetes Mellitus.

Diabetes Mellitus (DM) affects the cardiovascular response of patients. To study this effect, interbeat intervals (IBI) and beat-to-beat systolic blood pressure (SBP) variability of patients during supine, standing and controlled breathing tests were analyzed in the time domain. Simultaneous noninvasive measurements of IBI and SBP for 30 recently diagnosed and 15 long-standing DM patients were compared with the results for 30 rigorously screened healthy subjects (control). A statistically significant distinction between control and diabetic subjects was provided by the standard deviation and the higher moments of the distributions (skewness, and kurtosis) with respect to the median. To compare IBI and SBP for different populations, we define a parameter, α, that combines the variability of the heart rate and the blood pressure, as the ratio of the radius of the moments for IBI and the same radius for SBP. As diabetes evolves, α decreases, standard deviation of the IBI detrended signal diminishes (heart rate signal becomes more “rigid”), skewness with respect to the median approaches zero (signal fluctuations gain symmetry), and kurtosis increases (fluctuations concentrate around the median). Diabetes produces not only a rigid heart rate, but also increases symmetry and has leptokurtic distributions. SBP time series exhibit the most variable behavior for recently diagnosed DM with platykurtic distributions. Under controlled breathing, SBP has symmetric distributions for DM patients, while control subjects have non-zero skewness. This may be due to a progressive decrease of parasympathetic and sympathetic activity to the heart and blood vessels as diabetes evolves.

Improving absolute gravity estimates by the Lp-norm approximation of the ballistic trajectory

Iteratively re-weighted least squares (IRLS) were used to simulate the Lp-norm approximation of the ballistic trajectory in absolute gravimeters. Two iterations of the IRLS delivered sufficient accuracy of the approximation without a significant bias. The simulations were performed on different samplings and perturbations of the trajectory. For the platykurtic distributions of the perturbations, the Lp-approximation with 3 < p < 4 was found to yield several times more precise gravity estimates compared to the standard least-squares. The simulation results were confirmed by processing real gravity observations performed at the excessive noise conditions.

Statistical Investigation and Mapping of Monthly Modified Refractivity Gradient over Pakistan at the 700 Hectopascal Level

This study is an effort to investigate the spatial-temporal variability of the modified refractivity gradient at the 700 hPa pressure level over Pakistan and its neighbouring regions of Afghanistan, India, Iran and the Arabian Sea using the remote sensing data of the AQUA (AIRX3STM) satellite from 2008 to 2012. Trapping conditions only found in December were spread over Khyber Pakhtunkhwa (Gilgit-Baltistan, Pakistan) with an average value of -182.042 M/Km and showing Leptokurtic distributions. The lowest monthly average value super-refractive conditions existed in the autumn season with a strong monthly correlation (>0.91 M/Km). A very high monthly correlation (0.9 M/Km) was found for the super-refractive conditions over the whole time period. The largest spatial and temporal normal conditions appeared in January with the average value for normal conditions being 132.72 M/Km (found over Zabul, Afghanistan) with Leptokurtic distributions. During May normal conditions were the smallest in spatial extent over Pakistan, India and Afghanistan, showing Platykurtic distributions. Sub-refractive conditions mostly prevailed at all times. The probability for extreme sub-refractive conditions was very high in 2008-2012. The highest average sub-refractive conditions appeared in the winter and autumn seasons (spread around Quetta and Kalam, Pakistan). The highest monthly average sub-refractive conditions with a value of 1,265,188 M/Km were found in January and spread around the Sarbaz River Iran. Correlations for the existence of sub-refractive conditions varied from 0.8 M/Km (moderate strong) to 0.4 M/Km during the autumn to winter season. Permanent super-refractive conditions existed over Baluchistan from February to September.

Reference statistics for the structure of measurement series of groundwater levels (Wielkopolska Lowland, western Poland)

This study analyses the statistical and dynamic properties of time series of mean annual groundwater levels for the years 1961–2000, with the purpose of preparing reference statistics for the study period. Data came from 62 measurement stations of the groundwater observation network of the Institute of Meteorology and Water Management—the National Research Institute IMGW-PIB—located across the Wielkopolska Lowland (western Poland). Station-grouping criteria were as follows: the availability of a 40-year measurement series, spatial location, as well as separation of the aquifer from the surface. For estimating the time series structure of groundwater levels, the following stationarity tests were conducted: the augmented Dickey-Fuller (ADF) test and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test, with positive results in 44% of the stations, the majority comprising a group that represents confined aquifers. Their non-stationarity was attributed to the presence of a long-term stochastic trend. In the majority of cases the groundwater-level frequency distributions were within the group of platykurtic distributions, right asymmetrical. In the period studied, 62% of the series showed a downward tendency of groundwater levels, including 45% for which the trend was statistically significant; a rising trend was only statistically significant in half of the series. The masking effect of anthropogenic factors was indicated as a fundamental element interfering with the statistical properties of the groundwater observation series in the period 1961–2000 in the Wielkopolska Lowland. This information can be of use for the management and protection of water resources and in prognostic studies on modifying aquiferous systems with recorded water deficits.Editor D. Koutsoyiannis; Associate editor T. Ouarda

Distributional Properties of Relative Phase in Bimanual Coordination

Studies of bimanual coordination have typically estimated the stability of coordination patterns through the use of the circular standard deviation of relative phase. The interpretation of this statistic depends upon the assumption of a von Mises distribution. The present study tested this assumption by examining the distributional properties of relative phase in three bimanual coordination patterns. There were significant deviations from the von Mises distribution due to differences in the kurtosis of distributions. The kurtosis depended upon the relative phase pattern performed, with leptokurtic distributions occurring in the in-phase and antiphase patterns and platykurtic distributions occurring in the 30° pattern. Thus, the distributional assumptions needed to validly and reliably use the standard deviation are not necessarily present in relative phase data though they are qualitatively consistent with the landscape properties of the intrinsic dynamics.

Effect of platykurtic and leptokurtic distributions in the random-field Ising model: Mean-field approach

The influence of the tail features of the local magnetic field probability density function (PDF) on the ferromagnetic Ising model is studied in the limit of infinite range interactions. Specifically, we assign a quenched random field whose value is in accordance with a generic distribution that bears platykurtic and leptokurtic distributions depending on a single parameter tau<3 to each site. For tau<5/3, such distributions, which are basically Student-t and r distribution extended for all plausible real degrees of freedom, present a finite standard deviation, if not the distribution has got the same asymptotic power-law behavior as a alpha-stable Lévy distribution with alpha=(3-tau)/(tau-1). For every value of tau, at specific temperature and width of the distribution, the system undergoes a continuous phase transition. Strikingly, we impart the emergence of an inflexion point in the temperature-PDF width phase diagrams for distributions broader than the Cauchy-Lorentz (tau=2) which is accompanied with a divergent free energy per spin (at zero temperature).

Contributions to composite sampling

The initial use of composite sampling involved the analysis of many negative samples with relatively high laboratory cost (Dorfman sampling). We propose a method of double compositing and compare its efficiency with Dorfman sampling. The variability of composite measurement samples has environmental interest (hot spots). The precision of these estimates depends on the kurtosis of the distribution; leptokurtic distributions (γ2 > 0) have increased precision as the number of field samples is increased. The opposite effect is obtained for platykurtic distributions. In the lognormal case, coverage probabilities are reasonable for σ < 0.5. The Poisson distribution can be associated with temporal compositing, of particular interest where radioactive measurements are taken. Sample size considerations indicate that the total sampling effort is directly proportional to the length of time sampled. If there is background radiation then increasing levels of this radiation require larger sample sizes to detect the same difference in radiation.

An Investigation of the Raudenbush (1988) Test for Studying Variance Heterogeneity

Abstract The meta-analytic method proposed by Raudenbush (1988) for studying variance heterogeneity was investigated. This method uses the difference in two weighted log-standard deviations (e.g., treatment vs. control group) as an effect size and provides a test of the homogeneity of effect sizes. The relationship between study characteristics and effect sizes can also be investigated. Although appealing, the performance of this procedure for realistic data sets characterized by nonnormal data and small study sample sizes is unknown. The results of a Monte Carlo study indicated that the Type I error rate of the test was sensitive to even modestly platykurtic score distributions and to the ratio of study sample size to the number of studies, even if the data were normally distributed. On the whole, this test cannot be recommended to meta-analysts.

Minimum Sample Size Ensuring Validity of Classical Confidence Intervals for Means of Skewed and Platykurtic Distributions

COCHRAN (1953) and BARTCH (1957) gave formulae for the magnitude of the sample size (n) ensuring the validity of the limiting normal distribution of the sample mean x(n) obtained from a non-normal distribution with marked asymmetry and kurtosis. These formulae have been checked empirically in this paper using (a) simulated data with given asymmetry and kurtosis and (b) real data gathered from a coronary heart disease study. We find that our results are in general agreement with Bartch's formula. However, in a number of cases, the asymptotic normal distribution is attained for smaller sample size than that required by Bartch's formula.

Scaled test statistics and robust standard errors for non‐normal data in covariance structure analysis: A Monte Carlo study

Research studying robustness of maximum likelihood (ML) statistics in covariance structure analysis has concluded that test statistics and standard errors are biased under severe non-normality. An estimation procedure known as asymptotic distribution free (ADF), making no distributional assumption, has been suggested to avoid these biases. Corrections to the normal theory statistics to yield more adequate performance have also been proposed. This study compares the performance of a scaled test statistic and robust standard errors for two models under several non-normal conditions and also compares these with the results from ML and ADF methods. Both ML and ADF test statistics performed rather well in one model and considerably worse in the other. In general, the scaled test statistic seemed to behave better than the ML test statistic and the ADF statistic performed the worst. The robust and ADF standard errors yielded more appropriate estimates of sampling variability than the ML standard errors, which were usually downward biased, in both models under most of the non-normal conditions. ML test statistics and standard errors were found to be quite robust to the violation of the normality assumption when data had either symmetric and platykurtic distributions, or non-symmetric and zero kurtotic distributions.