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Concept: Probability density function


Distributed robustness is thought to influence the buffering of random phenotypic variation through the scale-free topology of gene regulatory, metabolic, and protein-protein interaction networks. If this hypothesis is true, then the phenotypic response to the perturbation of particular nodes in such a network should be proportional to the number of links those nodes make with neighboring nodes. This suggests a probability distribution approximating an inverse power-law of random phenotypic variation. Zero phenotypic variation, however, is impossible, because random molecular and cellular processes are essential to normal development. Consequently, a more realistic distribution should have a y-intercept close to zero in the lower tail, a mode greater than zero, and a long (fat) upper tail. The double Pareto-lognormal (DPLN) distribution is an ideal candidate distribution. It consists of a mixture of a lognormal body and upper and lower power-law tails.

Concepts: Evolution, Fungus, Yeast, Saccharomyces cerevisiae, Saccharomyces pastorianus, Normal distribution, Probability distribution, Probability density function


Laser speckle contrast analysis (LASCA) is limited to being a qualitative method for the measurement of blood flow and tissue perfusion as it is sensitive to the measurement configuration. The signal intensity is one of the parameters that can affect the contrast values due to the quantization of the signals by the camera and analog-to-digital converter (ADC). In this paper we deduce the theoretical relationship between signal intensity and contrast values based on the probability density function (PDF) of the speckle pattern and simplify it to a rational function. A simple method to correct this contrast error is suggested. The experimental results demonstrate that this relationship can effectively compensate the bias in contrast values induced by the quantized signal intensity and correct for bias induced by signal intensity variations across the field of view.

Concepts: Scientific method, Diffraction, Laser, Probability theory, Normal distribution, Probability density function, Digital signal processing, Speckle pattern


Hypersensitivity to DNaseI digestion is a hallmark of open chromatin and DNaseI-seq allows the genome-wide identification of regions of open chromatin. Interpreting these data is challenging, largely because of inherent variation in signal-to-noise ratio (SNR) between datasets. We have developed PeaKDEck, a peak calling program that distinguishes signal from noise by randomly sampling read densities and employing kernel density estimation to generate a dataset-specific probability distribution of random background signal. PeaKDEck uses this probability distribution to select an appropriate read density threshold for peak calling in each dataset. We benchmark PeaKDEck using published ENCODE DNaseI-seq data, and other peak calling programs, and demonstrate superior performance in low SNR datasets.Availability and implementation: PeaKDEck is written in standard Perl and runs on any platform with Perl installed. PeaKDEck is also available as a standalone application written in Perl/Tk, which does not require Perl to be installed. Files, including a user guide can be downloaded at: CONTACT: SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Concepts: Statistics, Data, Non-parametric statistics, Probability density function, Random variable, Histogram, Signal-to-noise ratio, Kernel density estimation


Recent progress in Affective Computing (AC) has enabled integration of physiological cues and spontaneous expressions to reveal a subject’s emotional state. Due to the lack of an effective technique for evaluating multimodal correlations, experience and intuition play a main role in present AC studies when fusing affective cues or modalities, resulting in unexpected outcomes. This study seeks to demonstrate a dynamic correlation between two such affective cues, physiological changes and spontaneous expressions, which were obtained by a combination of stereo vision based tracking and imaging photoplethysmography (iPPG), with a designed protocol involving 20 healthy subjects. The two cues obtained were sampled into a Statistical Association Space (SAS) to evaluate their dynamic correlation. It is found that the probability densities in the SAS increase as the peaks in two cues are approached. Also the complex form of the high probability density region in the SAS suggests a nonlinear correlation between two cues. Finally the cumulative distribution on the zero time-difference surface is found to be small (<0.047) demonstrating a lack of simultaneity. These results show that the two cues have a close interrelation, that is both asynchronous and nonlinear, in which a peak of one cue heralds a peak in the other.

Concepts: Fundamental physics concepts, Correlation and dependence, Probability theory, Emotion, Probability density function, Feeling, Cumulative distribution function


Lyme disease occurs in specific geographic regions of the United States. We present a method for defining high-risk counties based on observed versus expected number of reported human Lyme disease cases. Applying this method to successive periods shows substantial geographic expansion of counties at high risk for Lyme disease.

Concepts: United States, U.S. state, Probability density function, Random variable, Economic inequality


We explore a spatially implicit patch-occupancy model of a population on a landscape with continuous-valued heterogeneous habitat quality, primarily considering the case where the habitat quality of a site affects the mortality rate but not the fecundity of individuals at that site. Two analytical approaches to the model are constructed, by summing over the sites in the landscape and by integrating over the range of habitat quality. We obtain results relating the equilibrium population density and all moments of the probability distribution of the habitat quality of occupied sites, and relating the probability distributions of total habitat quality and occupied habitat quality. Special cases are considered for landscapes where habitat quality has either a uniform or a linear probability density function. For these cases, we demonstrate habitat association, where the quality of occupied sites is higher than the overall mean quality of all sites; the discrepancy between the two is reduced at larger population densities. The variance of the quality of occupied sites may be greater or less than the overall variance of habitat quality, depending on the distribution of habitat quality across the landscape. Increasing the variance of habitat quality is also shown to increase the ability of a population to persist on a landscape.

Concepts: Population, Population density, Probability theory, Normal distribution, Probability distribution, Probability density function, Random variable, Cumulative distribution function


The optimum sequence parameters of diffusion spectrum MRI (DSI) on clinical scanners were investigated previously. However, the scan time of approximately 30min is still too long for patient studies. Additionally, relatively large sampling interval in the diffusion-encoding space may cause aliasing artifact in the probability density function when Fourier transform is undertaken, leading to estimation error in fiber orientations. Therefore, this study proposed a non-Cartesian sampling scheme, body-centered-cubic (BCC), to avoid the aliasing artifact as compared to the conventional Cartesian grid sampling scheme (GRID). Furthermore, the accuracy of DSI with the use of half-sphere sampling schemes, i.e. GRID102 and BCC91, was investigated by comparing to their full-sphere sampling schemes, GRID203 and BCC181, respectively. In results, smaller deviation angle and lower angular dispersion were obtained by using the BCC sampling scheme. The half-sphere sampling schemes yielded angular precision and accuracy comparable to the full-sphere sampling schemes. The optimum b(max) was approximately 4750s/mm(2) for GRID and 4500s/mm(2) for BCC. In conclusion, the BCC sampling scheme could be implemented as a useful alternative to the GRID sampling scheme. Combination of BCC and half-sphere sampling schemes, that is BCC91, may potentially reduce the scan time of DSI from 30min to approximately 14min while maintaining its precision and accuracy.

Concepts: Variance, Fundamental physics concepts, Probability theory, Normal distribution, Accuracy and precision, Probability density function, Random variable, Dirac delta function


We propose an evidence synthesis approach through a degradation model to estimate causal influences of physiological factors on myocardial infarction (MI) and coronary heart disease (CHD). For instance several studies give incidences of MI and CHD for different age strata, other studies give relative or absolute risks for strata of main risk factors of MI or CHD. Evidence synthesis of several studies allows incorporating these disparate pieces of information into a single model. For doing this we need to develop a sufficiently general dynamical model; we also need to estimate the distribution of explanatory factors in the population. We develop a degradation model for both MI and CHD using a Brownian motion with drift, and the drift is modeled as a function of indicators of obesity, lipid profile, inflammation and blood pressure. Conditionally on these factors the times to MI or CHD have inverse Gaussian ([Formula: see text]) distributions. The results we want to fit are generally not conditional on all the factors and thus we need marginal distributions of the time of occurrence of MI and CHD; this leads us to manipulate the inverse Gaussian normal distribution ([Formula: see text]) (an [Formula: see text] whose drift parameter has a normal distribution). Another possible model arises if a factor modifies the threshold. This led us to define an extension of [Formula: see text] obtained when both drift and threshold parameters have normal distributions. We applied the model to results published in five important studies of MI and CHD and their risk factors. The fit of the model using the evidence synthesis approach was satisfactory and the effects of the four risk factors were highly significant.

Concepts: Estimator, Myocardial infarction, Heart, Normal distribution, Probability density function, Brownian motion, Carl Friedrich Gauss, Gaussian function


Gompertz-related distributions have dominated mortality studies for 187 years. However, nonrelated distributions also fit well to mortality data. These compete with the Gompertz and Gompertz-Makeham data when applied to data with varying extents of truncation, with no consensus as to preference. In contrast, Gaussian-related distributions are rarely applied, despite the fact that Lexis in 1879 suggested that the normal distribution itself fits well to the right of the mode. Study aims were therefore to compare skew-t fits to Human Mortality Database data, with Gompertz-nested distributions, by implementing maximum likelihood estimation functions (mle2, R package bbmle; coding given). Results showed skew-t fits obtained lower Bayesian information criterion values than Gompertz-nested distributions, applied to low-mortality country data, including 1711 and 1810 cohorts. As Gaussian-related distributions have now been found to have almost universal application to error theory, one conclusion could be that a Gaussian-related distribution might replace Gompertz-related distributions as the basis for mortality studies.

Concepts: Estimation theory, Maximum likelihood, Ronald Fisher, Normal distribution, Probability density function, Likelihood function, Uniform distribution


Accurate and precise displacement estimation has been a hallmark of clinical ultrasound. Displacement estimation accuracy has largely been considered to be limited by the Cramer¿Rao lower bound (CRLB). However, the CRLB only describes the minimum variance obtainable from unbiased estimators. Unbiased estimators are generally implemented using Bayes¿ theorem, which requires a likelihood function. The classic likelihood function for the displacement estimation problem is not discriminative and is difficult to implement for clinically relevant ultrasound with diffuse scattering. Because the classic likelihood function is not effective, a perturbation is proposed. The proposed likelihood function was evaluated and compared against the classic likelihood function by converting both to posterior probability density functions (PDFs) using a noninformative prior. Example results are reported for bulk motion simulations using a 6¿ tracking kernel and 30 dB SNR for 1000 data realizations. The canonical likelihood function assigned the true displacement a mean probability of only 0.070 ± 0.020, whereas the new likelihood function assigned the true displacement a much higher probability of 0.22 ± 0.16. The new likelihood function shows improvements at least for bulk motion, acoustic radiation force induced motion, and compressive motion, and at least for SNRs greater than 10 dB and kernel lengths between 1.5 and 12λ.

Concepts: Estimation theory, Probability theory, Accuracy and precision, Conditional probability, Probability density function, Bayes' theorem, Likelihood function, Bayesian statistics