Gaussian Population Research Articles

Testing for normality: a user's (cautionary) guide.

The normality assumption postulates that empirical data derives from a normal (Gaussian) population. It is a pillar of inferential statistics that enables the theorization of probability functions and the computation of p-values thereof. The breach of this assumption may not impose a formal mathematical constraint on the computation of inferential outputs (e.g., p-values) but may make them inoperable and possibly lead to unethical waste of laboratory animals. Various methods, including statistical tests and qualitative visual examination, can reveal incompatibility with normality and the choice of a procedure should not be trivialized. The following minireview will provide a brief overview of diagrammatical methods and statistical tests commonly employed to evaluate congruence with normality. Special attention will be given to the potential pitfalls associated with their application. Normality is an unachievable ideal that practically never accurately describes natural variables, and detrimental consequences of non-normality may be safeguarded by using large samples. Therefore, the very concept of preliminary normality testing is also, arguably provocatively, questioned.

Short-term wind power output prediction using hybrid-enhanced seagull optimization algorithm and support vector machine: A high-precision method

ABSTRACT Affected by the uncertainty of external environmental factors, wind power generation has significant characteristics of randomness and non-stationarity. Accurately predicting wind power is a necessary technical means to improve the efficiency of wind energy utilization and reduce wind abandonment. Integrating large-scale wind power into the grid has put higher requirements for wind power prediction accuracy. Hence, this study proposes a combined prediction model based on wavelet denoising (WD), hybrid enhanced seagull optimization algorithm (HESOA) and support vector machine (SVM) to predict short-term wind power. The WD technology obtains effective signals from original wind power data, providing a reasonable basis for prediction research. The HESOA is proposed by introducing the Tent mapping, individual self-learning factor and Gaussian population strategy into the original seagull optimization algorithm to optimize SVM’s prediction ability. The case study results reflect that the prediction method is feasible and universal in suppressing wind power generation volatility in spring and winter. The RMSE of short-term wind power prediction is controlled below 65 kW, and the value of R2 is above 98%. This study improves the accuracy of wind power prediction and helps to strengthen the renewable energy consumption level and reliability of wind power grid-connected generation systems.

A Universal Test on Spikes in a High-Dimensional Generalized Spiked Model and Its Applications

This paper aims to test the number of spikes in a generalized spiked covariance matrix, the spiked eigenvalues of which may be extremely larger or smaller than the non-spiked ones. For a high-dimensional problem, we first propose a general test statistic and derive its central limit theorem by random matrix theory without a Gaussian population constraint. We then apply the result to estimate the noise variance and test the equality of the smallest roots in generalized spiked models. Simulation studies showed that the proposed test method was correctly sized, and the power outcomes showed the robustness of our statistic to deviations from a Gaussian population. Moreover, our estimator of the noise variance resulted in much smaller mean absolute errors and mean squared errors than existing methods. In contrast to previously developed methods, we eliminated the strict conditions of diagonal or block-wise diagonal form of the population covariance matrix and extend the work to a wider range without the assumption of normality. Thus, the proposed method is more suitable for real problems.

Correlations between Anthropometric Measurements and Sports Discipline Aptitude

Background: Sports specialization is required for the advancement of elite-level skills of a competitor. Therefore, this study attempted to assess the applicability of anthropometric measurements for a tailored selection of sports disciplines. Methods: The sports disciplines studied in this report were wrestling, triple jumping, badminton, and tennis. The data used in this study were obtained from a PubMed search. Literature-derived data were used as a template to build a random Gaussian population of N = 500 subjects used for ratio calculation using the error propagation approach. The obtained ratios encompassed height/sitting height, height/length, height/arm length, height/waist circumference, height/chest circumference, sitting height/leg length, sitting height/arm length, sitting height/waist circumference, sitting height/chest circumference, arm/leg length, and arm length/forearm length. Results: There is a clear relationship between a sports discipline and the distribution of the anthropometric ratio. The anthropometric measurements of wrestlers are the most outstanding among the disciplines studied. The use of machine learning algorithms, that is, the decision tree classifier, allows for building a model able to distinguish between the disciplines of sports studied. Conclusions: The presented approach allows for selection of a specific sports discipline for a young person. Moreover, an extension of the model built by other sports disciplines and anthropometric measurement may be a practical tool for selecting sports subjects.

High-dimensional Edgeworth expansion of LR statistic for testing block circular symmetry covariance structure and its errors

The paper considers the asymptotical expansion on the likelihood ratio (LR) statistic for testing the block circular symmetric (BCS) covariance structure of a multivariate Gaussian population. When the number of blocks u and the dimension of each block p satisfy and as the sample size the Edgeworth expansion of the null distribution of the LR test statistic and its uniform Berry-Esseen type bound are established. Some numerical simulations indicate that the proposed approximation is more accurate than the traditional Chi-square approximate method on dealing with the high-dimensional test.

Demonstration of Stochastic Resonance, Population Coding, and Population Voting Using Artificial MoS2 Based Synapses.

Fast detection of weak signals at low energy expenditure is a challenging but inescapable task for the evolutionary success of animals that survive in resource constrained environments. This task is accomplished by the sensory nervous system by exploiting the synergy between three astounding neural phenomena, namely, stochastic resonance (SR), population coding (PC), and population voting (PV). In SR, the constructive role of synaptic noise is exploited for the detection of otherwise invisible signals. In PC, the redundancy in neural population is exploited to reduce the detection latency. Finally, PV ensures unambiguous signal detection even in the presence of excessive noise. Here we adopt a similar strategies and experimentally demonstrate how a population of stochastic artificial neurons based on monolayer MoS2 field effect transistors (FETs) can use an optimum amount of white Gaussian noise and population voting to detect invisible signals at a frugal energy expenditure (∼10s of nano-Joules). Our findings can aid remote sensing in the emerging era of the Internet of things (IoT) that thrive on energy efficiency.

Mapping the Trap‐State Landscape in 2D Metal‐Halide Perovskites Using Transient Photoluminescence Microscopy (Advanced Optical Materials 18/2021)

This cover image illustrates a transient microscopy measurement of the excited state energy transport in a perovskite lattice. An initial Gaussian population of excitations flows outwards, here illustrated by the flow of water. The energy transport is obstructed by defects along the way and transient microscopy measurements can therefore be used to accurately map out the trap-state landscape in the perovskite lattice. For further information, see article number 2001875 by Ferry Prins and co-workers.

Estimation of canonical correlation directions: From Gaussian to sub-Gaussian population

Cai and Zhang (2018) established separate perturbation upper bound estimators for canonical correlation directions under centered Gaussian population and some conditions on the minimum singular value σr(S) of a correlation matrix S . They posed an open problem for the optimality of their estimators. In this paper, the optimality of Cai and Zhang’s estimation is firstly proved up to some multiplicated constants. Then motivated by Ma and Li’s work (Ma and Li, 2020), we give an upper bound estimation for centered sub-Gaussian population, and a better estimate for bounded sub-Gaussian population. Finally, all estimates are extended from centered population to non-centered one.

AdaReg: data adaptive robust estimation in linear regression with application in GTEx gene expressions.

The Genotype-Tissue Expression (GTEx) project provides a valuable resource of large-scale gene expressions across multiple tissue types. Under various technical noise and unknown or unmeasured factors, how to robustly estimate the major tissue effect becomes challenging. Moreover, different genes exhibit heterogeneous expressions across different tissue types. Therefore, we need a robust method which adapts to the heterogeneities of gene expressions to improve the estimation for the tissue effect. We followed the approach of the robust estimation based on γ-density-power-weight in the works of Fujisawa, H. and Eguchi, S. (2008). Robust parameter estimation with a small bias against heavy contamination. J. Multivariate Anal. 99: 2053-2081 and Windham, M.P. (1995). Robustifying model fitting. J. Roy. Stat. Soc. B: 599-609, where γ is the exponent of density weight which controls the balance between bias and variance. As far as we know, our work is the first to propose a procedure to tune the parameter γ to balance the bias-variance trade-off under the mixture models. We constructed a robust likelihood criterion based on weighted densities in the mixture model of Gaussian population distribution mixed with unknown outlier distribution, and developed a data-adaptive γ-selection procedure embedded into the robust estimation. We provided a heuristic analysis on the selection criterion and found that our practical selection trend under various γ's in average performance has similar capability to capture minimizer γ as the inestimable mean squared error (MSE) trend from our simulation studies under a series of settings. Our data-adaptive robustifying procedure in the linear regression problem (AdaReg) showed a significant advantage in both simulation studies and real data application in estimating tissue effect of heart samples from the GTEx project, compared to the fixed γ procedure and other robust methods. At the end, the paper discussed some limitations on this method and future work.

Inferior Occipital Gyrus Is Organized along Common Gradients of Spatial and Face-Part Selectivity.

The ventral visual stream of the human brain is subdivided into patches with categorical stimulus preferences, like faces or scenes. However, the functional organization within these areas is less clear. Here, we used functional magnetic resonance imaging and vertex-wise tuning models to independently probe spatial and face-part preferences in the inferior occipital gyrus (IOG) of healthy adult males and females. The majority of responses were well explained by Gaussian population tuning curves for both retinotopic location and the preferred relative position within a face. Parameter maps revealed a common gradient of spatial and face-part selectivity, with the width of tuning curves drastically increasing from posterior to anterior IOG. Tuning peaks clustered more idiosyncratically but were also correlated across maps of visual and face space. Preferences for the upper visual field went along with significantly increased coverage of the upper half of the face, matching recently discovered biases in human perception. Our findings reveal a broad range of neural face-part selectivity in IOG, ranging from narrow to “holistic.” IOG is functionally organized along this gradient, which in turn is correlated with retinotopy.SIGNIFICANCE STATEMENT Brain imaging has revealed a lot about the large-scale organization of the human brain and visual system. For example, occipital cortex contains map-like representations of the visual field, while neurons in ventral areas cluster into patches with categorical preferences, like faces or scenes. Much less is known about the functional organization within these areas. Here, we focused on a well established face-preferring area—the inferior occipital gyrus (IOG). A novel neuroimaging paradigm allowed us to map the retinotopic and face-part tuning of many recording sites in IOG independently. We found a steep posterior–anterior gradient of decreasing face-part selectivity, which correlated with retinotopy. This suggests the functional role of ventral areas is not uniform and may follow retinotopic “protomaps.”

High-dimensional Edgeworth expansion of the determinant of sample correlation matrix and its error bound

The paper considers the asymptotic distribution on the determinant of the high-dimensional sample correlation matrix of the Gaussian population with independent components. In particular, when the dimension p and the sample size N satisfy , and , the asymptotic expansion and a uniform error bound of the distribution function of the logarithmic determinant of the sample correlation matrix are obtained by the Edgeworth expansion method. An application of the result to high-dimensional independence test is also proposed, some numerical simulations reveal that the proposed method outperforms the traditional chi-square approximation method and performs as efficient as the method introduced by Jiang and Yang [Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions, Ann. Statist. 41(4) (2013), pp. 2029–2074].

Asymptotic normality and moderate deviation principle for high-dimensional likelihood ratio statistic on block compound symmetry covariance structure

ABSTRACTThe paper considers a high-dimensional likelihood ratio (LR) test on the block compound symmetric (BCS) covariance structure of a multivariate Gaussian population. When the dimension of each block p, the number of blocks u and the sample size n satisfy that and pu<n−1 as , the asymptotic normality and the moderate deviation principle of the logarithmic LR statistic are obtained. Some numerical simulations demonstrate that the proposed method in high-dimensional BCS test outperforms the traditional Chi-square approximation method, and it is as efficient as the Edgeworth expansion method by Mitsui et al. [Likelihood ratio test statistic for block compound symmetry covariance structure and its asymptoic expansion. Technical Report No.15-03, Statistical Research Group, Hiroshima University, Japan; 2015]. In addition, the proposed method is more applicable because the asymptotic distribution of the test statistic is more concise and the restriction on parameters p, u is milder.

General methods for the characterization of the nonuniformity of a radiometric calibration source.

The spatial nonuniformity of a radiometric calibration source is key to its performance, uncertainty budget, and utility. Multiple disparate approaches are currently used to assess nonuniformity; thus, it is critical to understand how nonuniformity is calculated when comparing radiometric calibration sources. Two factors are key in assessing nonuniformity: (1) the methodology chosen to calculate the nonuniformity and (2) how the intensity of the light is sampled. Here, we present and compare different methodologies of calculating spatial nonuniformity. In general, we find that these methods can be categorized into two groups: those in which the equations assume a Gaussian population distribution and those in which the equations can be applied to any shape of distribution. Additionally, we compare different methods of sampling the light intensity within the region of interest. We show that peak deviation nonuniformity analysis, in which the pixels of the image of the illumination source are resampled into 69 equally sized bins encompassing the region of interest, can be used to effectively assess the nonuniformity of any radiance distribution. Furthermore, the nonuniformity determined by this method allows for an accurate comparison to industry standard nonuniformity metrics.

A test for block circular symmetric covariance structure with divergent dimension

The paper considers the likelihood ratio (LR) test on the block circular symmetric covariance structure of a multivariate Gaussian population with divergent dimension. When the sample sizen, the dimension of each blockpand the number of blocksusatisfypu&lt;n− 1 andp=p(n) →∞asn→∞, the asymptotic distribution and the moderate deviation principle of the logarithmic LR test statistic under the null hypothesis are established. Some numerical simulations indicate that the proposed LR test method performs well in the divergent-dimensional block circular symmetric covariance structure test.

Detailed somatotopy in primary motor and somatosensory cortex revealed by Gaussian population receptive fields

The relevance of human primary motor cortex (M1) for motor actions has long been established. However, it is still unknown how motor actions are represented, and whether M1 contains an ordered somatotopy at the mesoscopic level. In the current study we show that a detailed within-limb somatotopy can be obtained in M1 during finger movements using Gaussian population Receptive Field (pRF) models. Similar organizations were also obtained for primary somatosensory cortex (S1), showing that individual finger representations are interconnected throughout sensorimotor cortex. The current study additionally estimates receptive field sizes of neuronal populations, showing differences between finger digit representations, between M1 and S1, and additionally between finger digit flexion and extension. Using the Gaussian pRF approach, the detailed somatotopic organization of M1 can be obtained including underlying characteristics, allowing for the in-depth investigation of cortical motor representation and sensorimotor integration.

Classification of Exoplanets According to Density

Considering probability distribution as a function of the average density $\bar{\rho}$ computed for 424 extrasolar planets we identify three log-normal Gaussian population components. The two most populous components at $\bar{\rho}\simeq0.7$ g/cc and $\bar{\rho}\simeq7$ g/cc are the ice/gas giants and iron/rock super-Earths, respectively. A third component at $\bar{\rho}\simeq30$ g/cc is consistent with brown dwarfs, i.e., electron degeneracy supported objects. We note presence of several extreme density planetary objects.

Commentary: Fisher’s infinitesimal model: A story for the ages

Mendel (1866) suggested that if many heritable “factors” contribute to a trait, near-continuous variation could result. Fisher (1918) clarified the connection between Mendelian inheritance and continuous trait variation by assuming many loci, each with small effect, and by informally invoking the central limit theorem. Barton et al. (2017) rigorously analyze the approach to a multivariate Gaussian distribution of the genetic effects for descendants of parents who may be related. This commentary distinguishes three nested approximations, referred to as “infinitesimal genetics,” “Gaussian descendants” and “Gaussian population,” each plausibly called “the infinitesimal model.” The first and most basic is Fisher’s “infinitesimal” approximation of the underlying genetics – namely, many loci, each making a small contribution to the total variance. As Barton et al. (2017) show, in the limit as the number of loci increases (with enough additivity), the distribution of genotypic values for descendants approaches a multivariate Gaussian, whose variance–covariance structure depends only on the relatedness, not the phenotypes, of the parents (or whether their population experiences selection or other processes such as mutation and migration). Barton et al. (2017) call this rigorously defensible “Gaussian descendants” approximation “the infinitesimal model.” However, it is widely assumed that Fisher’s genetic assumptions yield another Gaussian approximation, in which the distribution of breeding values in a population follows a Gaussian — even if the population is subject to non-Gaussian selection. This third “Gaussian population” approximation, is also described as the “infinitesimal model.” Unlike the “Gaussian descendants” approximation, this third approximation cannot be rigorously justified, except in a weak-selection limit, even for a purely additive model. Nevertheless, it underlies the two most widely used descriptions of selection-induced changes in trait means and genetic variances, the “breeder’s equation” and the “Bulmer effect.” Future generations may understand why the “infinitesimal model” provides such useful approximations in the face of epistasis, linkage, linkage disequilibrium and strong selection.

Sensitivity and specificity of normality tests and consequences on reference interval accuracy at small sample size: a computer-simulation study.

According to international guidelines, parametric methods must be chosen for RI construction when the sample size is small and the distribution is Gaussian. However, normality tests may not be accurate at small sample size. The purpose of the study was to evaluate normality test performance to properly identify samples extracted from a Gaussian population at small sample sizes, and assess the consequences on RI accuracy of applying parametric methods to samples that falsely identified the parent population as Gaussian. Samples of n = 60 and n = 30 values were randomly selected 100 times from simulated Gaussian, lognormal, and asymmetric populations of 10,000 values. The sensitivity and specificity of 4 normality tests were compared. Reference intervals were calculated using 6 different statistical methods from samples that falsely identified the parent population as Gaussian, and their accuracy was compared. Shapiro-Wilk and D'Agostino-Pearson tests were the best performing normality tests. However, their specificity was poor at sample size n = 30 (specificity for P < .05: .51 and .50, respectively). The best significance levels identified when n = 30 were 0.19 for Shapiro-Wilk test and 0.18 for D'Agostino-Pearson test. Using parametric methods on samples extracted from a lognormal population but falsely identified as Gaussian led to clinically relevant inaccuracies. At small sample size, normality tests may lead to erroneous use of parametric methods to build RI. Using nonparametric methods (or alternatively Box-Cox transformation) on all samples regardless of their distribution or adjusting, the significance level of normality tests depending on sample size would limit the risk of constructing inaccurate RI.

Whether Gaussian Nucleus Entropy Helps? Case in Point is Prediction of Number of Cesarean Births

In this article, entropy in the collected data about the Gaussian population mean is traced from its embryonic stage as new data are periodically collected. The traditional Shannon's entropy has shortcomings from the data analytics point of view and it creates a necessity to refine the Shannon's entropy. Its refined version is named Gaussian Nucleus Entropy in this article. Advantages of the refined version are pointed out. The Prior, likelihood, Posterior and predictive nucleus entropies are derived, interconnected and interpreted. The results are illustrated using data on cesarean births in thirteen countries in the period [1987, 2007]. The medical communities and families are alarmed, as the cesarean births are increasing not due to emergency or necessity basis but rather for monetary or convenience basis. Nucleus entropy based data analysis answers whether their alarm is baseless.

Measurement of the roughness of nano-scale surfaces, both unannealed and with limited anneal, by atomic force microscopy

The roughness and bias measured for surfaces scanned by a parabolic atomic-force microscope tip are analyzed computationally for surfaces exhibiting a random, Gaussian population of heights, uncorrelated from pixel to pixel. The fraction of the surface with which the probe is in contact is then analyzed in detail to show the area fraction of the surface that is, in practice, contacted within a defined tolerance. This shows that it is very difficult to measure much of these surfaces for any significant roughness even though the roughness value may sometimes be approximately correct. The surface is then allowed a limited anneal, and the improvement in the validity of the measurement is determined. Annealing removes the very short-range structure and permits more meaningful roughnesses to be determined with a greater fraction of the surface contacted by the probe. This limited anneal may simulate the behaviour of hot atoms that are incident or impacted by energetic ions and that rapidly cool at the surface. Asymmetric, non-Gaussian height distributions are also analyzed, and these generate a significantly different measurement bias that can be more sensitive to the tip radius. The height distribution shapes alter with the tip radius, so accurate estimation of the originating height distribution is not possible. Simple equations are provided to describe the effects.