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Frits Agterberg - One of the best experts on this subject based on the ideXlab platform.
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New Method of Fitting Pareto–Lognormal Size-Frequency Distributions to Worldwide Cu and Zn Deposit Size Data
Natural Resources Research, 2018Co-Authors: Frits AgterbergAbstract:Earlier methods of fitting Pareto–Lognormal distributions to large samples of worldwide metal deposit size data are improved by using a sliding window method for estimating upper-tail Pareto coefficients and constructing best-fitting Lognormal Q – Q plots with their corresponding probability-density curves. Lower-tail Pareto distributions are fitted to some extent as well. Copper and Zn deposits of the world are taken as example in this paper. Three principal statistical laws resulting in the basic Lognormal with two Pareto tails are thought to underlie the generation of Pareto–Lognormals for amounts of metal in primarily hydrothermal ore deposits. Historical trends in mining and exploration are thought to create an excess of smaller deposits with respect to the basic Lognormal that decreases steadily with increasing deposit size until it changes into a deficit slightly before median size is reached. This deficit decreases for the largest metal deposit sizes for which the upper-tail Pareto and extrapolated basic Lognormal show similar size frequencies again. The Pareto–Lognormal model can also be used to describe metal size-frequency distributions for smaller geographically coherent regions on the continents. A new version of the original model of de Wijs is considered to help explain why regional Pareto–Lognormal distributions with lesser logarithmic variances and Pareto coefficients can be combined to form worldwide size-frequency distributions of the same type.
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Pareto–Lognormal Modeling of Known and Unknown Metal Resources
Natural Resources Research, 2017Co-Authors: Frits AgterbergAbstract:Recently, large worldwide databases with statistics on amounts of metal in mineral deposits have become available. Frequently, most metal is contained in the largest deposits for a metal. A major problem in meaningful modeling of the size–frequency distributions of the largest deposits is that they are very rare. Until now it was rather difficult to establish the exact form of their size–frequency distribution. However, because of the new very large databases it can now be concluded that two commonly used approaches (Lognormal and Pareto) thought to be mutually incompatible in the past, are both correct with a high probability. One approach does not necessarily exclude validity of the other. Patiño-Douce (Nat Resour Res 25(1):97–124, 2016b ) has shown that metal tonnage frequency distributions for worldwide metal deposits are approximately Lognormal with similar standard deviations ( σ ) of log-transformed data. In this paper, it is assumed that worldwide metals satisfy both Lognormal and Pareto models simultaneously. Copper and Au are taken for example for comparison with results previously obtained for these two metals in the Abitibi area of the Canadian Shield. Worldwide there are 2541 Cu deposits approximately satisfying a Lognormal distribution. Total amount of Cu in these deposits is 2.319 × 10^9 tons of Cu. However, the 45 largest deposits, which together contain 1.281 × 10^9 tons of Cu, satisfy a Pareto distribution. If their Lognormal model would apply in the upper tail as well, these 45 largest deposits should have contained only about 0.076 × 10^9 tons of Cu. It is shown in detail for Cu that the best statistical model for Cu deposits is a worldwide Pareto–Lognormal model in which the basic Lognormal size–frequency distribution is flanked by two juxtaposed Pareto distributions for the largest and smallest Cu deposits, respectively. Both Pareto distributions smoothly change into the central Lognormal by means of bridge functions that can be determined separately. The worldwide Pareto–Lognormal model also was found to be applicable to several other metals, especially Ag, Ni, Pb, and U. For Au, the model does not work as well for the upper tail Pareto distribution as it does for the other metals taken for example.
Norman C. Beaulieu - One of the best experts on this subject based on the ideXlab platform.
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An Extended Limit Theorem for Correlated Lognormal Sums
IEEE Transactions on Communications, 2012Co-Authors: Norman C. BeaulieuAbstract:Recent work has reported a Lognormal limit theorem for sums of identically distributed equicorrelated Lognormal random variables. An extended Lognormal limit theorem for sums of nonidentically distributed correlated Lognormal random variables having a particular correlation structure is derived.
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GLOBECOM - New approximations to the Lognormal characteristic function
2012 IEEE Global Communications Conference (GLOBECOM), 2012Co-Authors: Seyed Ali Saberali, Norman C. BeaulieuAbstract:Computation of the Lognormal characteristic function (CF) is required in analysis of many wireless communication systems. No closed-form expression exists for the Lognormal CF. In this paper, two approximations to the Lognormal CF are derived. The approximations are accurate over specified ranges of parameters. The first approximation is in terms of elementary mathematical functions and is useful for parameters typical in optical communication applications. The second approximation is a finite series in terms of the generalized hypergeometric functions and can be used to evaluate the Lognormal CF with parameters motivated by both optical and wireless communication applications. Theoretical results are corroborated by computer simulations.
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A power series expansion for the truncated Lognormal characteristic function
2010 25th Biennial Symposium on Communications, 2010Co-Authors: Norman C. BeaulieuAbstract:An infinite series expansion for the characteristic function of the Lognormal distribution does not exist, and no infinite series representation of the characteristic function of any modified form of the Lognormal distribution is found in the literature. A power series expansion is derived for the characteristic function of the truncated Lognormal distribution. The series is proved to converge absolutely for any level of truncation. Equivalently, the series converges absolutely for any nonzero value of probability in the missing tail, and the truncated Lognormal can be made arbitrarily close, but not equal to, the Lognormal while retaining convergence. The behaviours of the moments of the truncated Lognormal and Lognormal distributions are examined in detail.
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An optimal Lognormal approximation to Lognormal sum distributions
IEEE Transactions on Vehicular Technology, 2004Co-Authors: Norman C. Beaulieu, Qiong XieAbstract:Sums of Lognormal random variables occur in many problems in wireless communications because signal shadowing is well modeled by the Lognormal distribution. The Lognormal sum distribution is not known in the closed form and is difficult to compute numerically. Several approximations to the distribution have been proposed and employed in applications. Some widely used approximations are based on the assumption that a Lognormal sum is well approximated by a Lognormal random variable. Here, a new paradigm for approximating Lognormal sum distributions is presented. A linearizing transform is used with a linear minimax approximation to determine an optimal Lognormal approximation to a Lognormal sum distribution. The accuracies of the new method are quantitatively compared to the accuracies of some well-known approximations. In some practical cases, the optimal Lognormal approximation is several orders of magnitude more accurate than previous approximations. Efficient numerical computation of the Lognormal characteristic function is also considered.
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Minimax approximation to Lognormal sum distributions
The 57th IEEE Semiannual Vehicular Technology Conference 2003. VTC 2003-Spring., 1Co-Authors: Norman C. Beaulieu, Qiong XieAbstract:Sums of Lognormal random variables occur in many problems in wireless communications because signal shadowing is well modelled by the Lognormal distribution. The Lognormal sum distribution is not known in closed-form and is difficult to compute numerically. Several approximations to the distribution have been proposed and employed in applications. Some widely used approximations are based on the assumption that a Lognormal sum is well approximated by a Lognormal random variable. Here, a new paradigm for approximating Lognormal sum distributions is presented. A linearizing transform is used with a linear minimax approximation to determine an optimal Lognormal approximation to a Lognormal sum distribution. The accuracies of the new method are quantitatively compared to the accuracies of some well-known approximations. In some practical cases, the normal Lognormal approximation is several orders of magnitude more accurate than previous approximations. Efficient numerical computation of the Lognormal characteristic function is considered.
Evangelos Gerasopoulos - One of the best experts on this subject based on the ideXlab platform.
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Multi-modal analysis of aerosol robotic network size distributions for remote sensing applications: dominant aerosol type cases
Atmospheric Measurement Techniques, 2014Co-Authors: Michael Taylor, Stelios Kazadzis, Evangelos GerasopoulosAbstract:Abstract. To date, size distributions obtained from the aerosol robotic network (AERONET) have been fit with bi-Lognormals defined by six secondary microphysical parameters: the volume concentration, effective radius, and the variance of fine and coarse particle modes. However, since the total integrated volume concentration is easily calculated and can be used as an accurate constraint, the problem of fitting the size distribution can be reduced to that of deducing a single free parameter – the mode separation point. We present a method for determining the mode separation point for equivalent-volume bi-Lognormal distributions based on optimization of the root mean squared error and the coefficient of determination. The extracted secondary parameters are compared with those provided by AERONET's Level 2.0 Version 2 inversion algorithm for a set of benchmark dominant aerosol types, including desert dust, biomass burning aerosol, urban sulphate and sea salt. The total volume concentration constraint is then also lifted by performing multi-modal fits to the size distribution using nested Gaussian mixture models, and a method is presented for automating the selection of the optimal number of modes using a stopping condition based on Fisher statistics and via the application of statistical hypothesis testing. It is found that the method for optimizing the location of the mode separation point is independent of the shape of the aerosol volume size distribution (AVSD), does not require the existence of a local minimum in the size interval 0.439 μm ≤ r ≤ 0.992 μm, and shows some potential for optimizing the bi-Lognormal fitting procedure used by AERONET particularly in the case of desert dust aerosol. The AVSD of impure marine aerosol is found to require three modes. In this particular case, bi-Lognormals fail to recover key features of the AVSD. Fitting the AVSD more generally with multi-modal models allows automatic detection of a statistically significant number of aerosol modes, is applicable to a very diverse range of aerosol types, and gives access to the secondary microphysical parameters of additional modes currently not available from bi-Lognormal fitting methods.
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Multi-modal analysis of aerosol robotic network size distributions for remote sensing applications: dominant aerosol type cases
2013Co-Authors: Michael Taylor, Stelios Kazadzis, Evangelos GerasopoulosAbstract:Abstract. To date, size distributions obtained from the aerosol robotic network have been fit with bi-Lognormals defined by six secondary microphysical parameters: the volume concentration, effective radius, and the variance of fine and coarse particle modes. However, since the total integrated volume concentration is easily calculated and can be used as an accurate constraint, the problem of fitting the size distribution can be reduced to that of deducing a single free parameter – the mode separation point. We present a method for determining the mode separation point for equivalent-volume bi-Lognormal distributions based on optimisation of the root mean squared error and the coefficient of determination. The extracted secondary parameters are compared with those provided by AERONET's Level 2.0 Version 2 inversion algorithm for a set of benchmark dominant aerosol types including: desert dust, biomass burning aerosol, urban sulphate and sea salt. The total volume concentration constraint is then also lifted by performing multi-modal fits to the size distribution using nested Gaussian mixture models and a method is presented for automating the selection of the optimal number of modes using a stopping condition based on Fisher statistics and via the application of statistical hypothesis testing. It is found that the method for optimizing the location of the mode separation point is independent of the shape of the AVSD, does not require the existence of a local minimum in the size interval 0.439 μm ≤ r ≤ 0.992 μm, and shows some potential for optimizing the bi-Lognormal fitting procedure used by AERONET particularly in the case of desert dust aerosol. The AVSD of impure marine aerosol is found to require 3 modes. In this particular case, bi-Lognormals fail to recover key features of the AVSD. Fitting the AVSD more generally with multi-modal models allows automatic detection of a statistically-significant number of aerosol modes, is applicable to a very diverse range of aerosol types, and gives access to the secondary microphysical parameters of additional modes currently not available from bi-Lognormal fitting methods.
Jin Zhang - One of the best experts on this subject based on the ideXlab platform.
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approximating the sum of correlated Lognormal or Lognormal rice random variables
International Conference on Communications, 2006Co-Authors: Neelesh B. Mehta, Andreas F. Molisch, Jin ZhangAbstract:A simple and novel method is presented to approximate by the Lognormal distribution the probability density function of the sum of correlated Lognormal random variables. The method is also shown to work well for approximating the distribution of the sum of Lognormal-Rice or Suzuki random variables by the Lognormal distribution. The method is based on matching a low-order Gauss-Hermite approximation of the moment-generating function of the sum of random variables with that of a Lognormal distribution at a small number of points. Compared with methods available in the literature such as the Fenton-Wilkinson method, Schwartz-Yeh method, and their extensions, the proposed method provides the parametric flexibility to address the inevitable trade-off that needs to be made in approximating different regions of the probability distribution function.
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ICC - Approximating the Sum of Correlated Lognormal or, Lognormal-Rice Random Variables
2006 IEEE International Conference on Communications, 2006Co-Authors: Neelesh B. Mehta, Andreas F. Molisch, Jin ZhangAbstract:A simple and novel method is presented to approximate by the Lognormal distribution the probability density function of the sum of correlated Lognormal random variables. The method is also shown to work well for approximating the distribution of the sum of Lognormal-Rice or Suzuki random variables by the Lognormal distribution. The method is based on matching a low-order Gauss-Hermite approximation of the moment-generating function of the sum of random variables with that of a Lognormal distribution at a small number of points. Compared with methods available in the literature such as the Fenton-Wilkinson method, Schwartz-Yeh method, and their extensions, the proposed method provides the parametric flexibility to address the inevitable trade-off that needs to be made in approximating different regions of the probability distribution function.
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GLOBECOM - Flexible Lognormal sum approximation method
GLOBECOM '05. IEEE Global Telecommunications Conference 2005., 2005Co-Authors: Neelesh B. Mehta, Jin ZhangAbstract:A simple and novel method is presented to approximate the distribution of the sum of independent, but not necessarily identical, Lognormal random variables, by the Lognormal distribution. It is shown that matching a short Gauss-Hermite approximation of the moment generating function of the Lognormal sum with that of the Lognormal distribution leads to an accurate Lognormal sum approximation. The advantage of the proposed method over the ones in the literature, such as the Fenton-Wilkinson method, Schwartz-Yeh method, and the recently proposed Beaulieu-Xie method, is that it provides the parametric flexibility to handle the inevitable trade-off that needs to be made in approximating different regions of the probability distribution function. The accuracy is verified using extensive simulations based on a cellular layout
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flexible Lognormal sum approximation method
Global Communications Conference, 2005Co-Authors: Neelesh B. Mehta, Jin ZhangAbstract:A simple and novel method is presented to approximate the distribution of the sum of independent, but not necessarily identical, Lognormal random variables, by the Lognormal distribution. It is shown that matching a short Gauss-Hermite approximation of the moment generating function of the Lognormal sum with that of the Lognormal distribution leads to an accurate Lognormal sum approximation. The advantage of the proposed method over the ones in the literature, such as the Fenton-Wilkinson method, Schwartz-Yeh method, and the recently proposed Beaulieu-Xie method, is that it provides the parametric flexibility to handle the inevitable trade-off that needs to be made in approximating different regions of the probability distribution function. The accuracy is verified using extensive simulations based on a cellular layout
Atsushi Taruya - One of the best experts on this subject based on the ideXlab platform.
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Lognormal Property of Weak Lensing Fields
The Astrophysical Journal, 2002Co-Authors: Atsushi Taruya, Masahiro Takada, Takashi Hamana, Issha Kayo, Toshifumi FutamaseAbstract:The statistical property of the weak lensing fields is studied quantitatively using the ray-tracing simulations. Motivated by the empirical Lognormal model that characterizes the probability distribution function(PDF) of the three-dimensional mass distribution excellently, we critically investigate the validity of Lognormal model in the weak lensing statistics. Assuming that the convergence field, $\kappa$, is approximately described by the Lognormal distribution, we present analytic formulae of convergence for the one-point PDF and the Minkowski functionals. Comparing those predictions with ray-tracing simulations in various cold dark matter models, we find that the one-point Lognormal PDF can describe the non-Gaussian tails of convergence fields accurately up to $\nu\sim10$, where $\nu$ is the level threshold given by $\nu\equiv\kappa/\var^{1/2}$, although the systematic deviation from Lognormal prediction becomes manifest at higher source redshift and larger smoothing scales. The Lognormal formulae for Minkowski functionals also fit to the simulation results when the source redshift is low. Accuracy of the Lognormal-fit remains good even at the small angular scales, where the perturbation formulae by Edgeworth expansion break down. On the other hand, Lognormal models does not provide an accurate prediction for the statistics sensitive to the rare events such as the skewness and the kurtosis of convergence. We therefore conclude that the empirical Lognormal model of the convergence field is safely applicable as a useful cosmological tool, as long as we are concerned with the non-Gaussianity of $\nu\simlt5$ for low source redshift samples.
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probability distribution function of cosmological density fluctuations from a gaussian initial condition comparison of one point and two point Lognormal model predictions with n body simulations
The Astrophysical Journal, 2001Co-Authors: Issha Kayo, Atsushi Taruya, Yasushi SutoAbstract:We quantitatively study the probability distribution function (PDF) of cosmological nonlinear density fluctuations from N-body simulations with a Gaussian initial condition. In particular, we examine the validity and limitations of one-point and two-point Lognormal PDF models against those directly estimated from the simulations. We find that the one-point Lognormal PDF very accurately describes the cosmological density distribution even in the nonlinear regime (rms variance σnl 4, overdensity δ 100). Furthermore, the two-point Lognormal PDFs are also in good agreement with the simulation data from linear to fairly nonlinear regimes, while they deviate slightly from the simulation data for δ -0.5. Thus, the Lognormal PDF can be used as a useful empirical model for the cosmological density fluctuations. While this conclusion is fairly insensitive to the shape of the underlying power spectrum of density fluctuations P(k), models with substantial power on large scales, i.e., n ≡ d ln P(k)/d ln k -1, are better described by the Lognormal PDF. On the other hand, we note that the one-to-one mapping of the initial and evolved density fields, consistent with the Lognormal model, does not approximate the broad distribution of their mutual correlation even on average. Thus, the origin of the phenomenological Lognormal PDF approximation still remains to be understood.
