Lognormal Character Research Articles

Human lipoproteins comprise at least 12 different classes that are lognormally distributed.

This study presents the results of HPLC, a gentler and rapid separation method in comparison with the conventional ultracentrifugation, for 55 human serum samples. The elution patterns were analysed parametrically, and the attribute of each class was confirmed biochemically. Human samples contained 12 classes of lipoproteins, each of which may consist primarily of proteins. There are three classes of VLDLs. The level of each class was distributed lognormally, and the standard amount and the 95% range were estimated. Some lipoprotein classes with a narrow range could become ideal indicators of specific diseases. This lognormal character suggests that the levels are controlled by the synergy of multiple factors; multiple undesirable lifestyle habits may drastically increase the levels of specific lipoprotein classes. Lipoproteins in medical samples have been measured by enzymatic methods that coincide with conventional ultracentrifugation; however, the high gravity and time required for ultracentrifugation can cause sample degradation. Actually, the enzymatic methods measured the levels of several mixed classes. The targets of enzymatic methods have to be revised.

Swift Observations of Mrk 421 in Selected Epochs. III. Extreme X-Ray Timing/Spectral Properties and Multiwavelength Lognormality during 2015 December–2018 April

Abstract We present the results from the timing and spectral study of Mrk 421 based mainly on the Swift data in the X-ray energy range obtained during the time interval 2015 December–2018 April. The most extreme X-ray flaring activity on long-term, daily, and intraday timescales was observed during the 2 month period that started in 2017 December, when the 0.3–10 keV flux exceeded a level of 5 × 10−9 erg cm−2 s−1, recorded only twice previously. While the TeV-band and X-ray variabilities were mostly correlated, the source often varied in a complex manner in the MeV–GeV and radio–UV energy ranges, indicating that the multifrequency emission of Mrk 421 could not always be generated in a single zone. The longer-term flares at X-rays and γ-rays showed a lognormal character, possibly indicating a variability imprint of the accretion disk onto the jet. A vast majority of the 0.3–10 keV spectra were consistent with the log-parabolic model, showing relatively low spectral curvature and correlations between the different spectral parameters, predicted in the case of the first- and second-order Fermi processes. The position of the synchrotron spectral energy distribution peak showed an extreme variability on diverse timescales between the energies E p < 0.1 and >15 keV, with 15% of the spectra peaking at the hard X-ray, and was related to the peak height as with α ∼ 0.6, which is expected for the transition from Kraichnan-type turbulence into the “hard sphere” one. The 0.3–300 GeV spectra showed features of the hadronic contribution, jet–star interaction, and upscatter in the Klein–Nishina regime in different time intervals.

Solar Flare Distributions: Lognormal Instead of Power Law?

Abstract In many statistical solar flare studies, power laws are claimed and exponents derived by fitting a line to a log–log histogram. It is well known that this approach is statistically unstable, and very large statistics are needed to produce reliable exponent estimates. This may explain part of the observed divergence in power-law exponents in various studies. Moreover, the question is seldom addressed to what extent the data really do support power-law behavior. In this paper, we perform a comprehensive study of 6924 flares detected in SDO/AIA 9.4 nm images by the Solar Demon flare detection software between 2010 May 13 and 2018 March 16 and 9601 flares detected during the same period in GOES/XRS data by the LYRAFF flare detection software. We apply robust statistics to the SDO/AIA 9.4 nm peak intensity and the GOES/XRS raw peak flux, background-subtracted peak flux, and background-subtracted fluence and find clear indications that most background-corrected data are not well described by a power law and that all are better described by a lognormal distribution, while the raw GOES/XRS peak flux is best described by a power law. This may explain the success of power-law fits in flare studies using uncorrected data. The behavior of flare distributions has important implications for large-scale science questions such as coronal heating and the nature of solar flares. The apparent lognormal character of flare distributions in our data sets suggests that the assumed power-law nature of flares and its consequences need to be reexamined with great care.

Relation of plasma flow structures to passive particle tracer orbits

The properties of plasma flow topological structures are compared with those of passive tracer particles within a framework of the continuous random walk (CTRW) approach. Vortices may cause some of the trapping of particles, while large scale flows may carry them from vortex to vortex. The results indicate that most of the trappings that are completed during the calculation correspond to tracers trapped on broken filaments, including possible multiple trappings. The probability distribution function of the trapping times is then a function of the filament length, and has a lognormal character, like the distribution of filament lengths.

Estimating Common Parameters of Lognormally Distributed Environmental and Biomonitoring Data: Harmonizing Disparate Statistics from Publications

The progression of science is driven by the accumulation of knowledge and builds upon published work of others. Another important feature is to place current results into the context of previous observations. The published literature, however, often does not provide sufficient direct information for the reader to interpret the results beyond the scope of that particular article. Authors tend to provide only summary statistics in various forms, such as means and standard deviations, median and range, quartiles, 95% confidence intervals, and so on, rather than providing measurement data. Second, essentially all environmental and biomonitoring measurements have an underlying lognormal distribution, so certain published statistical characterizations may be inappropriate for comparisons. The aim of this study was to review and develop direct conversions of different descriptions of data into a standard format comprised of the geometric mean (GM) and the geometric standard deviation (GSD) and then demonstrate how, under the assumption of lognormal distribution, these parameters are used to answer questions of confidence intervals, exceedance levels, and statistical differences among distributions. A wide variety of real-world measurement data sets was reviewed, and it was demonstrated that these data sets are indeed of lognormal character, thus making them amenable to these methods. Potential errors incurred from making retrospective estimates from disparate summary statistics are described. In addition to providing tools to interpret “other people’s data,” this review should also be seen as a cautionary tale for publishing one’s own data to make it as useful as possible for other researchers.

Lognormal Distribution of Apples in Relation to Maturity Restriction at Harvest

A model linking maturity indicators and the restrictions they impose on the apple yield assigned to a long-term storage (AppleLS) was studied. Using a three-step approach “maturity restrictions – parameterization - empirical distribution functions”, parameters of theoretical distributions for fitting AppleLS were estimated. Packinghouse data for three varieties over five years was used to building separate models for three periods of harvest. Normal and lognormal fits to empirical distribution functions of AppleLS were tested. A good agreement between the empirical distribution functions and their lognormal fits (high p-values for test) was observed in all studied cases. The established lognormal character of the distribution of fruit quantity in relation to maturity restrictions at harvest can be applied to modeling the quality of fruit handled in packinghouses.

Statistical Properties of the Proportional Voting Process

We propose a model of elections based on the Sznajd model of social interactions in the stochastic Ravasz{Barabasi hierarchical network. The results of numerical calculations for the population described by this model were compared with statistical analysis of elections' results to the lower house of Parliament held in Poland in the years of 2001, 2005 and 2007. It is shown that the distribution of votes among candidates for the whole country as well as for the district of Warsaw has the log-normal character. The results from Poland are compared with other countries: Brazil and India. For all the cases log-normal distribution of votes for the candidates from each political party is shown. The probability of gaining a number of votes as a function of the number of the candidate on the list is presented.

Superorbital variability of X-ray and radio emission of Cyg X-1 - II. Dependence of the orbital modulation and spectral hardness on the superorbital phase

We discover a pronounced dependence of the strength of the X-ray orbital modulation and the hardness in Cyg X-1 in the hard state on its superorbital phase. Our results can be well modelled as a combination of two effects: the precession of the accretion disc (which causes the superorbital flux modulation) and the orbital-phase dependent X-ray absorption in an accretion bulge, located at the accretion disc edge close to the supergiant companion but displaced from the line connecting the stars by about 25^o. Our findings are supported by the distribution of the X-ray dips showing concentration towards zero superorbital phase, which corresponds to the bulge passing through the line of sight. We Fourier analyse our model, and find it explains the previous finding of asymmetric beat (between the orbital and superorbital modulations) frequencies in the observed power spectrum, provided the disc precession is prograde. We find no statistically significant changes of the orbital modulation with the superorbital phase in the 15-GHz radio data. This absence is consistent with the radio being emitted by a jet in the system, in which case the orbital modulation is caused by wind absorption far away from the disc. We also find that both the X-ray and radio fluxes of Cyg X-1 in the hard state on time scales >10^4-s have lognormal distributions, which complements a previous finding of a lognormal flux distribution in the hard state on 1-s time scales. We point out that the lognormal character of the flux distribution requires that flux logarithms rather than fluxes themselves should be used for averaging and error analysis. We also provide a correct formula for the uncertainty of rms of a light curve for the case when the uncertainty is higher than the measurement.

Spatiotemporal characterization of Ensemble Prediction Systems – the Mean-Variance of Logarithms (MVL) diagram

Abstract. We present a novel approach to characterize and graphically represent the spatiotemporal evolution of ensembles using a simple diagram. To this aim we analyze the fluctuations obtained as differences between each member of the ensemble and the control. The lognormal character of these fluctuations suggests a characterization in terms of the first two moments of the logarithmic transformed values. On one hand, the mean is associated with the exponential growth in time. On the other hand, the variance accounts for the spatial correlation and localization of fluctuations. In this paper we introduce the MVL (Mean-Variance of Logarithms) diagram to intuitively represent the interplay and evolution of these two quantities. We show that this diagram uncovers useful information about the spatiotemporal dynamics of the ensemble. Some universal features of the diagram are also described, associated either with the nonlinear system or with the ensemble method and illustrated using both toy models and numerical weather prediction systems.

Distribution of the Magnetic Flux in Elements of the Magnetic Field in Active Regions

The unsigned magnetic flux content in the flux concentrations of two active regions is calculated by using a set of 248 high-resolution Solar and Heliospheric Observatory Michelson Doppler Imager magnetograms for each active region. Data for flaring active region NOAA 9077 (2000 July 14) and nonflaring active region NOAA 0061 (2002 August 9) were analyzed. We present an algorithm to automatically select and quantify magnetic flux concentrations above a threshold p. Each active region is analyzed using four different values of the threshold p (p = 25, 50, 75, and 100 G). Probability distribution functions and cumulative distribution functions of the magnetic flux were calculated and approximated by the lognormal, exponential, and power-law functions in the range of flux Φ > 1019 Mx. The Kolmogorov-Smirnov test, applied to each of the approximations, showed that the observed distributions are consistent with the lognormal approximation only. Neither exponential nor power-law functions can satisfactorily approximate the observed distributions. The parameters of the lognormal distribution do not depend on the threshold value; however, they are different for the two active regions. For flaring active region 9077, the expectation value of the magnetic flux content is μ = 28.1 × 1018 Mx, and the standard deviation of the lognormal distribution is σ = 79.0 × 1018 Mx. For nonflaring active region NOAA 0061, these values are μ = 23.8 × 1018 and σ = 29.6 × 1018 Mx. The lognormal character of the observed distribution functions suggests that the process of fragmentation dominates over the process of concentration in the formation of the magnetic structure in an active region.

Suspended minerogenic particle distributions in high‐energy coastal environments: Optical implications

This paper examines suspended minerogenic particle distributions in the near‐coastal ocean, Oceanside, California. The environment is dominated by resuspension of particles from a well‐sorted sandy sediment. We obtain information on the suspended mineral matter from the relocatable numerical sedimentation model, TRANS98. Inputs for the model include bottom particle size distribution, wind speed, wave period and height, and near‐bottom current speed. Model output is a vertical profile of mineral particle concentrations and size distributions at selected depths. The mineral particle size distribution allows the calculation of the total scattering coefficient of suspended minerogenic matter. The particle size distribution for the bed at Oceanside is approximately Gaussian lognormal in character. The predicted size distribution near bottom is a skewed Gaussian lognormal; then the distribution shifts toward the increasing importance of smaller size classes as we approach the water surface. Deeper stations maintain a lognormal character up into the surface layers while the shallower stations switch to a particle size distribution of the smallest size class being the modal class in the surface layers. Hyperbolic size distributions have been proposed for the Oceanside site from optical observations. These distributions would have a slope of 2.1 for coastal water which would decrease to as low as 1.3 for shallow nearshore stations and near the bottom. None of the proposed hyperbolic slopes provides an adequate approximation of the particle distributions predicted by TRANS98. The larger particle size classes are significantly overestimated by hyperbolic models, which causes the total scattering coefficient of the minerogenic matter to be overestimated. These results indicate that hyperbolic slope models cannot be used to retrieve particle size distributions by inversions involving the total scattering coefficient nor are they accurate estimators of the minerogenic scattering coefficient.

Concentration of 226Ra in rocks of the southern part of Lower Silesia (SW Poland)

The aim of the article is to present a preliminary description of rocks in the southern part of Lower Silesia and the Sudety Mountains in particular, with regard to 226Ra content. The research demonstrates that the average content of this isotope was 40.4 Bq/kg in the rocks of the southern part of Lower Silesia, and 41.7 Bq/kg in the rocks of the Sudetes. These values are slightly higher than the mean 226Ra content in the upper part of the Earth’s crust, while the measured maximum content of this isotope (244 Bq/kg) is more than twice as high as the upper range of the values most frequently recorded in the upper part of the Earth’s crust. The minimum values were lower than the detection limit, which was about 1 Bq/kg. These results reflect the mosaic-like geological structure of Lower Silesia, and particularly the Sudety Mountains, the occurrence of SiO 2-rich igneous rocks and the products of their metamorphism, as well as numerous manifestations of uranium mineralisation or even deposital concentrations of this element. The rocks with the highest 226Ra contents include (in decreasing order): aplites, granites, gneisses and leucogranites, granite-gneisses, granodiorites and rhyolites, and, finally, mudstones. The lowest values of 226Ra content, on the other hand, were measured in sandstones, marls and conglomerates, and extremely low—in marbles and quartzites. The results show that background values of 226Ra content in the rocks of the southern part of Lower Silesia fall within a range from several to about 100 Bq/kg, which is the same as the range most frequently recorded in the upper part of the Earth’s crust. Distribution of these values has log-normal character. The research demonstrates that the southern part of Lower Silesia, and the Sudetes in particular, may be marked by an increased radon potential. Particularly liable areas are: the Karkonosze granite massif, especially in its border zones, the L a ̨ dek-Śnieżnik and the Izera massifs, especially in their gneiss parts, as well as the Orlica massif. There is an urgent necessity to undertake more extensive and detailed research into this subject in connection with radiological hazard of radon. These statements are compliant with observed high radon concentrations in some groundwaters and tourist underground objects on these areas.

Work hardening behaviour and subgrain size distribution of hot worked aluminium

The subgrain structure of hot worked aluminium was found to remain geometrically self similar for various deformation conditions, undergoing only alteration in size as the flow stress changes. The stress and temperature related alterations of the subgrain size distributions have been described mathematically by the two parameters of the lognormal probability function which fits the experimental data. It has been found that the rate of scaling of the structure changes in stage IV work hardening. Experimental observations and statistical considerations suggest that the lognormal character of the subgrain size distribution arises as a result of a multiplicative interaction of recovery processes.

Work hardening behaviour and subgrain size distribution of hot worked aluminium

The subgrain structure of hot worked aluminium was found to remain geometrically self similar for various deformation conditions, undergoing only alteration in size as the flow stress changes. The stress and temperature related alterations of the subgrain size distributions have been described mathematically by the two parameters of the lognormal probability function which fits the experimental data. It has been found that the rate of scaling of the structure changes in stage IV work hardening. Experimental observations and statistical considerations suggest that the lognormal character of the subgrain size distribution arises as a result of a multiplicative interaction of recovery processes.

Dealing with Uncertainty in Risk Assessment

This paper is a commentary on Hattis’ three laws of risk assessment. The first law, that “application of standard statistical techniques to a single data set will nearly always reveal only a trivial proportion of the overall uncertainty in the parameter value” is illustrated both by examining the relevance of animal models to man and by a retrospective view of exposure conditions whose importance has only recently been recognized to be important. The second law, that “any estimate of the uncertainty of a parameter value will always itself be more uncertain than the estimate of the parameter value,” is examined in terms of a model addressing multiple levels of uncertainty, e.g., the “uncertainty in the uncertainty”. A argument is made that the number of terms needed for convergence of this uncertainty hierarchy depends on how far from the central tendency of the risk distribution one goes. The further out the “tail” of the distribution, the more terms in the uncertainty hierarchy are needed for convergence. The third law, that “nearly all parameter distribu tions look lognormal, as long as you don't look too closely,” is illustrated with a number of examples. Several reasons are put forward as to why risk variables appear so frequently to be lognormal. Recognition of the lognormal character of variable distributions can provide insight into the proper form for the associated uncertainty distributions.

Distribution of sunspot umbral areas - 1917-1982

Over 24,000 measurements of individual sunspot umbral areas taken from the Mount Wilson white-light plate collection covering the period 1917-1982 are used to determine the relative size distribution of sunspot umbras. In the range 1.5-141 millionths of a solar hemisphere, the sunspot umbral areas are found to be distributed lognormally. Moreover, the same distribution is obtained for all phases of the solar cycle (maximum, minimum, ascending, descending), as well as for various individual cycles, between 1917 and 1982. Both the mean and the geometric logarithmic standard deviation of this distribution appear to be intrinsically constant over the entire data set; only the number of spots exhibits the familiar solar cycle variations. If the observed lognormal umbral size distribution is not a particular attribute of the sunspot umbras but is instead of a more fundamental property of emerging magnetic flux, then the data would predict a maximum in the size spectrum of photospheric magnetic structures for flux tubes with radii in the range 500-800 km. The absence of solar cycle variations in the relative distribution of umbral areas and especially the lognormal character of this distribution may both argue for the fragmentation of magnetic elements in the solar envelope.

On frequency distributions of air pollutant concentrations

Observed frequency distributions of air pollutant concentration levels are critically analyzed with respect to their statistical description. It is demonstrated that several common distributions can be used to fit observed data, one of which is the popular log-normal distribution. The observation that concentration distributions for all averaging times are approximately log-normal can be explained if the short averaging time data are themselves assumed to be log-normally distributed. The near log-normality of pollutant concentration frequency distributions can be explained on the basis of the near log-normality of wind speed distributions, although this explanation does not establish that wind speed distributions are solely responsible for observed concentration distributions. It is concluded that pollutant concentration frequency distributions are the result of complex phenomena and cannot be predicted exactly, but that the approximate log-normal character of the distributions is useful from a practical point of view and can be understood qualitatively on the basis of the relation between wind speed and concentration.

Quantitative micrography of carbon black microstructure

The d-spacing distribution in a number of carbon blacks was accurately measured by means of an internal calibration standard. The distribution appears to be random, following a log-normal pattern for all blacks, except for thermal blacks, which are concentrically arranged around the particle nucleus. In all furnace blacks and in channel blacks the d-spacing are identical with an identical distribution pattern, regardless of particle size and degree of particle coalescence. Except for thermal blacks, all other carbon blacks investigated appear to be built up from layer planes oriented around a plurality of nuclei randomly distributed in the particles. Layer planes follow the particle contour, generally only at the periphery of the particle. Graphitization leads to growth as well as motion of layer planes, resulting in a marked narrowing of the d-spacing distribution without changing its log-normal character. It is shown that X-ray data are intrinsically restricted to a limited number of carbon planes, while electron micrography has no such restrictions in revealing the carbon black microstructure.

Tail of the distribution of sums of log-normal variates

It is shown that the asymptotic behavior of the tail of the sum distribution of a finite number of individually log-normal distributed variates displays the log-normal character of those variates with maximum logarithmic variance. A quantitative definition of the sum-distribution "tail" is established in terms of upper bounds on the relative error or deviation from the asymptotic log-normal property.

Structural aspects of atmospheric radio noise in the tropics

The amplitude probability and pulse rate distributions of noise envelope have been made the basis of a study of some structural aspects of atmospheric radio noise in a tropical area. The lognormal character of radio noise in the tropics has also been investigated. Direct comparisons are made with results obtained in temperate latitudes. The differences in the structural characteristics of noise have been related to intense local thunderstorm activity on tropical land masses.