Lognormal Probability Density Function Research Articles

Flood peak regionalization using mixed-mode estimation of the parameters of the log-normal distribution

Most hydrologic design problems occur at sites where stream-gage data are not available. At these ungaged locations, therefore, it is, necessary to use either an uncalibrated estimation method or an empirical equation that is based on a regionalization of peak-discharge estimates from gaged data. Traditionally, empirical equations have been calibrated by regressing estimated peak discharges, which have been obtained from a log-Pearson Type-III analysis of gaged data, on watershed and precipitation characteristics. This method is sensitive to both extreme events and small sample sizes. In an effort to minimize the problems with outliers and data variance, we derived a particular form of the three-parameter log-normal distribution by utilizing a variate transform of the annual peak-flow series for fitting to a histogram. This mixed-mode fitting uses nonlinear least squares to optimize on the class errors between the observed and computed histograms of the data. The sample variability is controlled by using a fixed probability range rather than the sample range. The mixed-mode estimator technique was used to derive parameters for 132 gaged sites in Pennsylvania. Using the mixed-mode parameters for a log-normal probability density function, peak discharges of five different exceedance probabilities were computed for each station and then regionalized for use at ungaged sites. Regression techniques were used to relate the peak discharges to watershed and precipitation characteristics; drainage area was the only predictor variable that was necessary for accurate estimates. The significant predictor variable in a log-form was drainage area. For comparative purposes, the log-Pearson Type-III peak-discharge estimates were also regionalized. The average error for the mixed-mode log-normal method did not differ significantly from the log-Pearson Type-III regionalized equation error. The mixed-mode method provided correlation coefficients ranging from 0.91 to 0.94, whereas the traditional method provided correlations ranging from 0.88 to 0.96 for different return periods.

A Statistical Analysis of Bell System Building Fires, 1971-1977

About 200 fires in Bell System buildings and adjacent grounds (excluding Western Electric) are reported to ATT the actual number of fires that occur may be somewhat higher. The dollar damage of reported fires (excluding only the $60 million fire in New York City on February 27, 1975) is reasonably modeled above the median by a log normal probability density function. This paper introduces a detailed taxonomy of fires, showing substantial differences in their frequency and costliness. The paper concludes with various special topics: (i) an analysis of employee injuries and service interruptions caused by fire; (ii) the correlation of business hours with fire frequency and building occupancy with fire severity; (iii) the methods used to fight fires; (iv) an analysis of multiple fires in buildings and of a cluster of fires in the Greater New York area in March 1975.

Prototype model for volume backscatter strength in the upper ocean caused by light-sensitive biological scatterers

It is known that biological scatterers in near-surface sound-scattering layers often tend to follow isolumes. By assuming a distribution of preferred isolumes tracked by the scatterers, exponential attenuation of downwelling light intensity with depth, and a uniform depth-invariant backscatter cross section for each scatterer, a theoretical model for volume backscatter strength S(z) can be derived. Thus, if truncated normal, log-normal probability density functions are assumed to describe the distribution of preferred isolumes, then S(z) is a quadratic function of exp(αz); z, where α is the optical extinction coefficient. Despite the simplicity of the physical assumptions, these two particular models together provide a good phenomenological description of a number of experimental midnight and mid-day 3.85–15.5-kHz profiles obtained at the Ocean Acre site near Bermuda. [Work supported by Naval Material Command.]

On the psychophysical function

Certain mathematical principles for the phenomenological description of data are introduced and applied to establish a new psychophysical function for the sensation of brightness. This function, which is the integral of the lognormal probability density function, has the psychophysical functions of Fechner and Stevens as limiting cases. The predictions of the new function are compared with experimental evidence on the discrimination of light intensity by the human eye, and an estimate of the total number of distinct steps of brightness discrimination is determined.

Detection Probabilities for Log-Normally Distributed Signals

The amplitude and power of a large family of radio signals are observed to have log-normal probability density functions. Among these are signals propagated through random inhomogeneous media, a notable example being low frequency atmospheric radio noise. Of greater importance are certain radar targets that have been observed to have essentially log-normal density functions. Both ships and space vehicles may fall into this category. Curves of probability of detection vs. signal-to-noise ratio for the case of log-normal signals in Gaussian noise have been computed and are presented in this paper. The curves apply for square-law detection with varying degrees of postdetection linear integration. Both fully correlated and completely uncorrelated fluctuating signals are considered. It is shown that for log-normal signal distributions having large variances, the probability of detection differs significantly from that obtained using curves based on an assumed Rayleigh signal distribution.

Characters of Log-normal Distribution and Probability Density Function whose Spectrum has Many Peaks

According to the time-series theory, the probability function can be expressed as followsp (x) =1/√2π {εex2/2ε2+√1-ε2xe-x2/2∫-∞√1-ε2/εxe-u2/2du} (Theory of maxima, Cartwright and Longuet-Higgins) xe-x2+a2/2Io (xa) (Theory of envelope, Rice) Both extremity of ε=1 and a= ∞, correspond to Gauss-Laplace's distribution, namelyP (x)=1/√2πe-x2/2;1/√2πe- (x-a) 2/2Now. if we change independent variable from x to Z, whereZ= log x, so probability function becomes : P (Z) =2/√2πexp {Z-e2z/2} _??_2/√2πee-2Z2/e;1/√2πexp {Z- (eZ-a) 2/2} _??_a/√2πe-a2/2 (Z-loga) 2, which are summarized macroscopically as follows : P (Z) =1/√2πσexp- (Z-Z) 2/2σ2, where σ is dispersion and Z is total mean of Z.This is log-normal distribution, and we find that this annexes both theory of maxima and envelope.Spectrum has sometimes many peaks. Consequently in the frequency distribution of oscillation appear also many peaks. If we express irregular motion asξ=∑Nn=1Cncos (ωnt+ψn) and letting R be the amplitude of envelope, then probability density isP (R) = R∫o∞r ΠNn=1Jo (Cnr) Jo (Rr) drwhose number of peaks is about equal to N.