Lognormal Probability Distribution Function Research Articles

Physics-based distributions for randomly scattered signals and their Bayesian conjugate priors

Sound waves propagating through the atmosphere and ocean are randomly scattered by turbulence, internal waves, surface roughness, and other processes. In some limiting cases, probability density functions (pdfs) for the scattered signal variations can be derived, e.g., the log-normal pdf for weak scattering (in the Rytov approximation) and the exponential pdf for strong scattering. A variety of more general, usually empirically based, distributions are available which reduce to these limiting cases, such as the Rice, gamma, and generalized gamma. For situations involving multiple receivers, multivariate log-normal, Wishart, and matrix gamma pdfs may be employed. Parametric uncertainties and spatial/temporal variability in the scattering process can be addressed with a compound pdf formulation, which involves an additional distribution for the uncertain or variable parameters. From a Bayesian perspective, the scattering pdf corresponds to the likelihood function, the pdf for the uncertain parameters to the prior/posterior, and the compound pdf to the marginal likelihood. Many common scattering pdfs possess Bayesian conjugate priors, which lend themselves to simple updating equations and analytical solutions for the posteriors and marginal likelihoods. This presentation summarizes important pdfs for randomly scattered signals and their conjugate priors when available.

Assessment of Wind Power Potential Based on Raleigh Distribution Model: An Experimental Investigation for Coastal Zone

When compared with other renewable energy resources (RER), the wind energy share in the global energy production is increasing rapidly. Currently, the Government of Pakistan (GoP) is moving towards RER, specifically wind and solar energy. In this paper, the wind energy potential of Tando Ghulam Ali, Sindh, Pakistan is explored. For this purpose, one-year wind speed data is considered at various heights through various probability distribution functions (PDFs). Statistical comparison of Rayleigh, gamma, generalized extreme value (GEV) and lognormal PDFs have been done with two methods, namely root mean square error and (R^2) in order to select the best PDF. Results showed that the Rayleigh distribution function is the best at the above mentioned area for finding various factors like site selection and wind power cost per kWh.

Hydrodynamic study of an emulsion liquid membrane containing carbon nanotube in a mixer–settler: Mean size and size distribution of emulsion globules

Mean size and size distribution of emulsion globules are indicative of interfacial area available for mass transfer in an emulsion liquid membrane (ELM) system. In this study, emulsion globules size of an ELM containing different amounts of multi-walled carbon nanotube (MWCNT) in a horizontal mixer–settler has been measured and analyzed. For this purpose, MWCNTs were dispersed in the membrane phase composed of kerosene as diluent and Span80 as surfactant. The influences of the presence and concentration of MWCNT and also key hydrodynamic parameters such as turbulent energy dissipation per unit mass and emulsion phase volume fraction on globule size have been investigated. According to the results, adding MWCNT in ELM caused 11.42–14.85% decrease in the Sauter mean diameter (D32) of emulsion globules. It was also observed that presence of MWCNT led to increase in the density of small globules and narrower globule size distribution. Also in agreement with these observations, stability results showed that introducing MWCNT to the system led to 62.26% reduction in breakage ratio. Furthermore, a new empirical correlation for prediction of D32 was proposed with an average absolute relative error (AARE) of 3.36%. Finally, the normal and log-normal probability distribution functions were used to predict the globule size distribution and their parameters (α and β) were well predicted by the dimensionless correlations wherein the MWCNT concentration introduced as a new independent parameter.

Probability distribution functions of gas surface density in M 33

Aims. We examine the interstellar medium (ISM) of M 33 to unveil fingerprints of self-gravitating gas clouds throughout the star-forming disk. Methods. The probability distribution functions (PDFs) for atomic, molecular, and total gas surface densities are determined at a resolution of about 50 pc over regions that share coherent morphological properties and considering cloud samples at different evolutionary stages in the star formation cycle. Results. Most of the total gas PDFs are well fit by log-normal functions whose width decreases radially outward. Because the HI velocity dispersion is approximately constant throughout the disk, the decrease in PDF width is consistent with a lower Mach number for the turbulent ISM at large galactocentric radii where a higher fraction of HI is in the warm phase. The atomic gas is found mostly at face-on column densities below NHlim = 2.5 × 1021 cm−2, with small radial variations of NHlim. The molecular gas PDFs do not show strong deviations from log-normal functions in the central region where molecular fractions are high. Here the high pressure and rate of star formation shapes the PDF as a log-normal function, dispersing self-gravitating complexes with intense feedback at all column densities that are spatially resolved. Power-law PDFs for the molecules are found near and above NHlim, in the southern spiral arm and in a continuous dense filament extending at larger galactocentric radii. In the filament nearly half of the molecular gas departs from a log-normal PDF, and power laws are also observed in pre-star-forming molecular complexes. The slope of the power law is between −1 and −2. This slope, combined with maps showing where the different parts of the power law PDFs come from, suggests a power-law stratification of the density within molecular cloud complexes, in agreement with the dominance of self-gravity.

Irradiance fluctuations of an indoor FSO channel based on hot air chambers

In this paper the fourth order statistical momentum behavior of irradiance fluctuations through hot air chambers which is crucial for evaluating the proper performance of optical turbulence generator (OTG) as a laser communication channel is studied. Basic principles of designing a developed free space optical wave propagation channel are presented. In order to validate the OTG performance with respect to the real atmosphere, beside Kolmogrov statistics presumptions like homogeneity, isotropy, stationarity and ergodicity, random fluctuations of the signal due to atmospheric turbulence is investigated and compared with log-normal and gamma-gamma PDF model. We have shown the above properties for our OTG and 2/3 power law behavior for the temperature structure function according to the Kolmogrov theory. Experimental data of irradiance fluctuations are well corresponds to the log-normal and gamma-gamma distributions under proper wind velocities and chamber active area. Therefore the chamber irradiance fluctuations resemble more accurately to the real atmosphere. Atmospheric parameters of the turbulence in different wind speeds and channel widths are estimated by new method of fitting the theoretical PDF to the experimental intensity irradiance distribution. The irradiance fluctuations again temperature difference and turbulence strength is measured and compared with the theory.

Residential inter-zonal ventilation rates for exposure modeling

ABSTRACTResidential inter-zonal (e.g., between rooms) ventilation is comprised of fresh air infiltration in and exfiltration out of the whole house plus the “fresh” air that is entering (and exiting) the room of interest from other rooms or areas within the house. Clearly, the inter-zone ventilation rate in any room of interest will be greater than the infiltration/exfiltration ventilation rate of outdoor air for the whole house. The purpose of this study is to determine how much greater the inter-zonal ventilation rate is in typical U.S. residences compared to the whole house ventilation rate from outdoor air. The data for this statistical analysis came from HouseDB, a 1995 EPA database of residential ventilation rates. Analytical results indicate that a lognormal distribution provides the best fit to the data. Lognormal probability distribution functions (PDFs) are provided for various inter-zonal ventilation rates for comparison to the PDF for the whole house ventilation rates. All ventilation rates are expressed as air change rates per hour (ACH). These PDFs can be used as inputs to exposure models. This analysis suggests that if one were performing a deterministic analysis for unknown housing stocks in the U.S., a default mean and median ACH values of 0.4/hr and 0.3/hr, respectively, for whole house ventilation would be appropriate; and 0.7/hr and 0.6/hr, respectively, for inter-zonal ventilation.

Parametric Theory of Cutoff Grade Estimation in Mining

Estimation of cutoff grade (fractional assay), x(C), in mining of mineral resources is crucial for profit maximization and sustainable growth [7, 8, 9]. It is very complex involving several geological and economic random variables which can be solved assuming the parameters are time-invariant (Static Models) or are time-varying (Dynamic Models). Parametric estimation of cutoff grade, x(C), involves modeling of the assay distribution of the ore (globally and locally) and of the economic random variables such as sale value of marketable ores (s/ton) and cost of production of mineral resource (c/ton). Fractional assay, x where 0< x < 1, of minerals/oxides/elements in rocks and ores is known to possess globally/locally log-normal pdf or some log function of assay is Normal/Gaussian [1,3,4,6] under proper geological/statistical sampling, with the two parameters: mean(μ) and variance(σ2). However, estimation of cutoff grade, x(C), where(x(C) = inverse cdf, F-1 (x(C))), is a very complex problem for both static and dynamic modeling. A high cutoff induces lower profits as less ore material can be extracted for sale whereas a low cutoff also induces lower profits as a much larger quantity of lower grade ore and waste materials having little sale value have to be handled which increase the cost of mining and processing. Since life of mine is about 15 years or more, all profits must be reduced to net present value (NPV) for economic comparisons and decisions [5]. Static model assumes time-invariant geological, spatial and economic random variables (rv.s) or random vectors needing simpler statistical analysis [4] to estimate the required parameters, whereas dynamic modeling though more realistic and desirable) requires rather complex time series analysis and forecasting procedure [3]. Forecasted values of the predictors from the concerned dynamic model parameters [3] are then used as inputs to the (linearized) regression equation in dynamic model situations. Optimal cutoff grade, x(CO), is therefore a time-step specific and/or block specific random variable which does not have a global value to be computed. Therefore, general solution to optimal grade, x(CO) , under dynamic model is not feasible and hence, not pursued further.

Simulation of the minimum annual river flows based on the RCP climatic scenario, time horizon up to 2060-2080 and the Kaczawa River

In this paper a new simulations of minimum daily flow for Kaczawa River a left side tributary of the Odra River in south-west Poland are presented. Generated data were made based on very long series of 35 years of observed data and 24 sites of meteorological stations for south-west Poland gathered from the the Institute of Meteorology and Water Management National Research Institute (IMGW). For the data generation the spatial weather generator SWGEN producing the multisite daily time series was applied. Data were generated for the present (the year 2000 are used as a background) as well for future climate condition for 2060 and 2080 according Representative Concentration Pathways (RCPs) scenarios. The flow simulation in the river catchment is made using MIKE SHE hydrological model. Simulations are done for 2060 and 2080. The large number of new simulated series determined by the lead time, two climate change scenarios (RCP4.5 and RCP6.0), and number of generated years (1000 for each case) is equal to 5000 for a single station. Finally, Lognormal Pdf function for the minimum flow is presented as well probability of exceedance of minimum values.

Scattered signal distributions, parametric uncertainties, and Bayesian sequential updating

A variety of probability density functions (pdfs) have been proposed for scattered signals, which have varying analytical advantages and ranges of physical applicability. We discuss here situations modeled by a compound pdf, in which a basic pdf, attributable to the underlying scattering process, has uncertain parameters or is modulated by variability in the environment. The parameters of the modulating pdf are termed hyperparameters. Some previous examples of compound formulations include the K-distribution, for which strong scattering (exponential pdf) is modulated by a gamma pdf for the mean signal power, and scattering by intermittent turbulence, for which strong scattering is modulated by a log-normal pdf for the structure-function parameter. We describe some alternative formulations, including strong scattering modulated by a gamma pdf for the inverse mean power, and Rytov (log-normal) scattering modulated by a normal pdf for the log-mean of the signal. These lead to relatively simple marginalized signal power distributions (Lomax and log-normal, respectively). Furthermore, the conditional scattered signal pdf may be viewed as a likelihood function in which the modulating pdf is the Bayesian conjugate prior. Hence the hyperparameters of the modulating process can be refined by simple sequential Bayesian updating as additional transmission data become available.

Lensing and the Warm-hot Intergalactic Medium

Abstract The correlation of weak lensing and Cosmic Microwave Anisotropy (CMB) data traces the pressure distribution of the hot, ionized gas and the underlying matter density field. The measured correlation is dominated by baryons residing in halos. Detecting the contribution from unbound gas by measuring the residual cross-correlation after masking all known halos requires a theoretical understanding of this correlation and its dependence with model parameters. Our model assumes that the gas in filaments is well described by a log-normal probability distribution function, with temperatures 105–7 K and overdensities ξ ≤ 100. The lensing-Comptonization cross-correlation is dominated by gas with overdensities in the range ξ ≈ [3–33]; the signal is generated at redshifts z ≤ 1. If only 10% of the measured cross-correlation is due to unbound gas, then the most recent measurements set an upper limit of on the mean temperature of the intergalactic Medium. The amplitude is proportional to the baryon fraction stored in filaments. The lensing-Comptonization power spectrum peaks at a different scale than the gas in halos, making it possible to distinguish both contributions. To trace the distribution of the low-density and low-temperature plasma on cosmological scales, the effect of halos will have to be subtracted from the data, requiring observations with larger signal-to-noise ratios than are currently available.

Stochastic assessment of the energy performance of buildings

A building’s energy performance is a complex multi-dimensional metric consisting of a variety of parameters. Presented herein are the results of a stochastic analysis of the factors affecting a building’s energy performance. The analysis is based on the Dwelling Energy Assessment Procedure (DEAP) (amended for cooling loads) and the general guidelines prescribed by the European Energy Performance of Buildings (EPBD) Directive 2010/31/EU. Modifications to the DEAP model are made for investigating the effect of variable external weather conditions on a building’s energy performance, and to incorporate the additional energy requirement for cooling. Subsequently, a stochastic analysis for three dwelling types is performed to assess the impact of 68 factors on the energy performance of buildings, for 12 different regions in Europe. It is concluded that (1) the factors with the greatest impact on energy use are in descending order, the floor area, external weather conditions, dwelling’s envelope u value (roof, window, walls, and floors), the space heating system, ventilation, windows area and walls area; (2) the energy performance of a building follows a lognormal probability distribution function; (3) buildings in colder EU regions exhibit higher energy profiles and higher variability in their energy profiles than those in warmer regions.

Optimal power flow solutions incorporating stochastic wind and solar power

Generations from several sources in an electrical network are to be optimally scheduled for economical and efficient operation of the network. Optimal power flow problem is formulated with all relevant system parameters including generator outputs and solved subsequently to obtain the optimal settings. The network may consist of conventional fossil fuel generators as well as renewable sources like wind power generators and solar photovoltaic. Classical optimal power flow itself is a highly non-linear complex problem with non-linear constraints. Incorporating intermittent nature of solar and wind energy escalates the complexity of the problem. This paper proposes an approach to solve optimal power flow combining stochastic wind and solar power with conventional thermal power generators in the system. Weibull and lognormal probability distribution functions are used for forecasting wind and solar photovoltaic power output respectively. The objective function considers reserve cost for overestimation and penalty cost for underestimation of intermittent renewable sources. Besides, emission factor is also included in objectives of selected case studies. Success history based adaptation technique of differential evolution algorithm is adopted for the optimization problem. To handle various constraints in the problem, superiority of feasible solutions constraint handling technique is integrated with success history based adaptive differential evolution algorithm. The algorithm thus combined and constructed gives optimum results satisfying all network constraints.

Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI.

The discovery of new exoplanets makes us wonder where each new exoplanet stands along its way to develop life as we know it on Earth. Our Evo-SETI Theory is a mathematical way to face this problem. We describe cladistics and evolution by virtue of a few statistical equations based on lognormal probability density functions (pdf) in the time. We call b-lognormal a lognormal pdf starting at instant b (birth). Then, the lifetime of any living being becomes a suitable b-lognormal in the time. Next, our “Peak-Locus Theorem” translates cladistics: each species created by evolution is a b-lognormal whose peak lies on the exponentially growing number of living species. This exponential is the mean value of a stochastic process called “Geometric Brownian Motion” (GBM). Past mass extinctions were all-lows of this GBM. In addition, the Shannon Entropy (with a reversed sign) of each b-lognormal is the measure of how evolved that species is, and we call it EvoEntropy. The “molecular clock” is re-interpreted as the EvoEntropy straight line in the time whenever the mean value is exactly the GBM exponential. We were also able to extend the Peak-Locus Theorem to any mean value other than the exponential. For example, we derive in this paper for the first time the EvoEntropy corresponding to the Markov-Korotayev (2007) “cubic” evolution: a curve of logarithmic increase.

Methodology to Backcalculate Individual Speed Data Originally Aggregated by Road Detectors

Highway and traffic engineers collect vehicular speed data with detectors based on a variety of fixed and mobile device technologies, to support analysis and design activities. Most acquisition units aggregate speed data into speed classes for ease of management and storage. An unfortunate result of this practice is a significant loss of content associated with individual speed data. Moreover, the use of individual speeds is often necessary to support road safety analysis and speed management decisions. For bridging of this gap, this paper introduces an algorithm that disaggregates speed data collected with automatic road detectors that can measure speed frequency only in intervals. The objective is to obtain backcalculated individual speeds that operate with continuous distribution functions rather than discrete ones. This information allows the derivation of more robust, basic descriptive measures (average, variance, and percentiles) according to normal, lognormal, and gamma probability distribution functions. Therefore, the information produced is more useful than that calculated from standard aggregated speed reports. In this investigation, individual speed data collected from video cameras were used to derive reference distributions and descriptive measures on the same road sections where inductive double-loop detectors were installed. Comparisons of the backcalculated individual speeds and those collected from video cameras support the validity of the proposed algorithm.

Uncertainty quantification of environmental performance metrics in heterogeneous aquifers with long-range correlations

We investigate how the uncertainty stemming from disordered porous media that display long-range correlation in the hydraulic conductivity (K) field propagates to predictions of environmental performance metrics (EPMs). In this study, the EPMs are quantities that are of relevance to risk analysis and remediation, such as peak flux-averaged concentration, early and late arrival times among others. By using stochastic simulations, we quantify the uncertainty associated with the EPMs for a given disordered spatial structure of the K-field and identify the probability distribution function (PDF) model that best captures the statistics of the EPMs of interest. Results indicate that the probabilistic distribution of the EPMs considered in this study follows lognormal PDF. Finally, through the use of information theory, we reveal how the persistent/anti-persistent correlation structure of the K-field influences the EPMs and corresponding uncertainties.

Testing the lognormality of the galaxy and weak lensing convergence distributions from Dark Energy Survey maps

It is well known that the probability distribution function (PDF) of galaxy density contrast is approximately lognormal; whether the PDF of mass fluctuations derived from weak lensing convergence (kappa_WL) is lognormal is less well established. We derive PDFs of the galaxy and projected matter density distributions via the Counts in Cells (CiC) method. We use maps of galaxies and weak lensing convergence produced from the Dark Energy Survey (DES) Science Verification data over 139 deg^2. We test whether the underlying density contrast is well described by a lognormal distribution for the galaxies, the convergence and their joint PDF. We confirm that the galaxy density contrast distribution is well modeled by a lognormal PDF convolved with Poisson noise at angular scales from 10-40 arcmin (corresponding to physical scales of 3-10 Mpc). We note that as kappa_WL is a weighted sum of the mass fluctuations along the line of sight, its PDF is expected to be only approximately lognormal. We find that the kappa_WL distribution is well modeled by a lognormal PDF convolved with Gaussian shape noise at scales between 10 and 20 arcmin, with a best-fit chi^2/DOF of 1.11 compared to 1.84 for a Gaussian model, corresponding to p-values 0.35 and 0.07 respectively, at a scale of 10 arcmin. Above 20 arcmin a simple Gaussian model is sufficient. The joint PDF is also reasonably fitted by a bivariate lognormal. As a consistency check we compare the variances derived from the lognormal modelling with those directly measured via CiC. Our methods are validated against maps from the MICE Grand Challenge N-body simulation.

Bit error rate analysis of MISO FSO systems

Multiple-input single-output (MISO) systems are employed in free space optical (FSO) links to mitigate the degrading effects of atmospheric turbulence. In this paper, we consider a MISO FSO system with practical transmitter and receiver configuration that consists of radial laser array with Gaussian beams and a Gaussian receiver aperture function. We have employed our previously derived formulation of the power scintillation in which Huygens–Fresnel principle was employed. Therefore, we choose system parameters within the range of validity of the wave structure functions. Using the on-off keying modulation and the log-normal probability distribution function, we quantify the average bit error rate (〈BER〉) of laser array beams in weak turbulence. It is observed that the radial array beams at the transmitter are more advantageous than the single Gaussian beam. However, increasing the number of array beamlets to more than three seems to have negligible effects on 〈BER〉 . It is further observed that 〈BER〉 decreases when the source size, the ring radius and the receiver aperture radius increase.

Regionalization of Rainfall Intensity-Duration-Frequency (IDF) Curves in Botswana

A regional analysis of design storms, defined as the expected rainfall intensity for given storm duration and return period, is conducted to determine storm Rainfall Intensity-Duration-Frequency (IDF) relationships. The ultimate purpose was to determine IDF curves for homogeneous regions identified in Botswana. Three homogeneous regions were identified based on topographic and rainfall characteristics which were constructed with the K-Means Clustering algorithm. Using the mean annual rainfall and the 24 hr annual maximum rainfall as an indicator of rainfall intensity for each homogeneous region, IDF curves and maps of rainfall intensities of 1 to 24 hr and above durations were produced. The Gamma and Lognormal probability distribution functions were able to provide estimates of rainfall depths for low and medium return periods (up to 100 years) in any location in each homogeneous region of Botswana.

A Study of Optical Properties of Soot Aggregates Composed of Poly-Disperse Monomers Using the Superposition T-Matrix Method

The diameters of soot monomers may not be constant in the single fractal aggregated soot particle. The optical properties of light absorbing soot particles aggregated with poly-disperse monomers were studied using the superposition T-matrix method. Soot aggregates were generated with different log-normal probability distribution functions (PDF) of soot monomer diameter, according to the same soot volumes and monomer numbers. The single scattering properties of soot particles were calculated at a wavelength of 550 nm, assuming a soot refractive index of 1.95 + 0.79i and a mass density of 1.8 g/cm3. The random-orientation averaging results indicated that the optical properties of soot aggregates were fairly varied for the different distributions of the monomer diameters. In these simulations, the extinction and absorption of soot aggregates were slightly (<10%) affected by the monomer poly-dispersity. The simulated mass absorption cross-sections (MAC) of fresh dry soot particles aggregated with poly-disperse monomers reached up to 6.62 ± 0.07 m2/g, which was closer to the measurement (7.5 ± 1.2 m2/g) than the assumption of volume-equivalent mono-disperse monomer (6.36 ± 0.06 m2/g). Moreover, the optical properties of soot coated with an organic shell were calculated, and the optical results showed that the absorption cross-sections of the internally mixed soot particles were also slightly (<8%) influenced by the monomer poly-dispersity. We found that the effect of the monomer poly-dispersity on the light scattering and the single scattering albedo may be considerably large (up to ˜50% in extreme cases) for fresh dry soot aggregates. This effect on light scattering should be taken into account for those aggregates composed of monomers with widely distributed diameters.Copyright 2015 American Association for Aerosol Research

Understanding star formation in molecular clouds

We analyse column density and temperature maps derived from Herschel dust continuum observations of a sample of massive infrared dark clouds (G11.11-0.12, G18.82-0.28, G28.37+0.07, G28.53-0.25). We disentangle the velocity structure of the clouds using 13CO 1-0 and 12CO 3-2 data, showing that these IRDCs are the densest regions in massive giant molecular clouds and not isolated features. The probability distribution function (PDF) of column densities for all clouds have a power-law distribution over all (high) column densities, regardless of the evolutionary stage of the cloud: G11.11-0.12, G18.82-0.28, and G28.37+0.07 contain (proto)-stars, while G28.53-0.25 shows no signs of star formation. This is in contrast to the purely log-normal PDFs reported for near/mid-IR extinction maps. We only find a log-normal distribution for lower column densities, if we perform PDFs of the column density maps of the whole GMC in which the IRDCs are embedded. By comparing the PDF slope and the radial column density profile, we attribute the power law to the effect of large-scale gravitational collapse and to local free-fall collapse of pre- and protostellar cores. Independent from the PDF analysis, we find infall signatures in the spectral profiles of 12CO for G28.37+0.07 and G11.11-0.12, supporting the scenario of gravitational collapse. IRDCs are the densest regions within GMCs, which may be the progenitors of massive stars or clusters. At least some of the IRDCs are probably the same features as ridges (high column density regions with N>1e23 cm-2 over small areas), which were defined for nearby IR-bright GMCs. Because IRDCs are only confined to the densest (gravity dominated) cloud regions, the PDF constructed from this kind of a clipped image does not represent the (turbulence dominated) low column density regime of the cloud.