Theoretical Probability Distribution Functions Research Articles

A Probabilistic Model for Maximum Rainfall Frequency Analysis

As determining the probability of the exceedance of maximum precipitation over a specified duration is critical to hydrotechnical design, particularly in the context of climate change, a model was developed to perform a frequency analysis of maximum precipitation of a specified duration. The PMAXΤP model (Precipitation MAXimum Time (duration) Probability) harbors a pair of computational modules fulfilling different roles: (i) statistical analysis of precipitation series, and (ii) estimation of maximum precipitation for a specified duration and its probability of exceedance. The input data consist of homogeneous 30-element series of precipitation values for 16 different durations: 5, 10, 15, 30, 45, 60, 90, 120, 180, 360, 720, 1080, 1440, 2160, 2880, and 4320 min, obtained through Annual Maximum Precipitation (AMP) and Peaks-Over-Threshold (POT) approaches. The statistical analysis of the precipitation series includes: (i) detecting outliers using the Grubbs-Beck test; (ii) checking for the random variable’s independence using the Wald-Wolfowitz test and the Anderson serial correlation coefficient test; (iii) checking the random variable’s stationarity using nonparametric tests, e.g., the Kruskal-Wallis test and Spearman rank correlation coefficient test for trends of mean and variance; (iv) identifying the trend of the random variables using correlation and regression analysis, including an evaluation of the form of the trend function; and (v) checking for the internal correlation of the random variables using the Anderson autocorrelation coefficient test. To estimate maximum precipitations of a specified duration and with a specified probability of exceedance, three-parameter theoretical probability distributions were used: a shifted gamma distribution (Pearson type III), a log-normal distribution, a Weibull distribution (Fisher-Tippett type III), a log-gamma distribution, as well as a two-parameter Gumbel distribution. The best distribution was selected by: (i) maximum likelihood estimation of parameters; (ii) tests of the hypothesis of goodness of fit of the theoretical probability distribution function with the empirical distribution using Pearson’s χ2 test; (iii) selection of the best-fitting function within each type according to the criterion of minimum Kolmogorov distance; (iv) selection of the most credible probability distribution function from the set of various types of best-fitting functions according to the Akaike information criterion; and (v) verification of the most credible function using single-dimensional tests of goodness of fit: the Kolmogorov-Smirnov test, the Anderson-Darling test, the Liao-Shimokawa test, and Kuiper’s test. The PMAXTP model was tested on data from two meteorological stations in northern Poland (Chojnice and Bialystok) drawn from a digital database of high-resolution precipitation records for the period of 1986 to 2015, available for 100 stations in Poland (i.e., the Polish Atlas of Rainfall Intensities (PANDa)). Values of maximum precipitation with a specified probability of exceedance obtained from the PMAXTP model were compared with values obtained from the probabilistic Bogdanowicz-Stachý model. The comparative analysis was based on the standard error of fit, graphs of the density function for the probability of exceedance, and estimated quantile errors. The errors of fit were lower for the PMAXTP compared to the Bogdanowicz-Stachý model. For both stations, the smallest errors were obtained for the quantiles determined on the basis of maximum precipitation POT using PMAXTP.

Wavelet Packet-Genetic Programming: A New Model for Meteorological Drought Hindcasting

This study presents developing procedures and verification of a new hybrid model, namely wavelet packet-genetic programming (WPGP) for short-term meteorological drought forecast. To this end, the multi-temporal standardized precipitation evapotranspiration index (SPEI) has been used as the drought quantifying parameter at two meteorological stations at Ankara province, Turkey. The new WPGP model comprises two main steps. In the first step, the wavelet packet, which is a generalization of the well-known wavelet transform, is used to decompose the SPEI series into deterministic and stochastic sub-signals. Then, classic genetic programming (GP) is applied to formulate the deterministic sub-signal considering its effective lags. To characterize the stochastic component, different theoretical probability distribution functions were assessed, and the best one was selected to integrate with the GP-evolved function. The efficiency of the new model was cross-validated with the first order autoregressive (AR1), GP, and random forest (RF) models developed as the benchmarks in the present study. The results showed that the WPGP is a robust model, superior to AR1 and RF, and significantly increases the predictive accuracy of the standalone GP model.

Nuw CDM cosmology from the weak-lensing convergence PDF

ABSTRACT Pinning down the total neutrino mass and the dark energy equation of state is a key aim for upcoming galaxy surveys. Weak lensing is a unique probe of the total matter distribution whose non-Gaussian statistics can be quantified by the one-point probability distribution function (PDF) of the lensing convergence. We calculate the convergence PDF on mildly nonlinear scales from first principles using large-deviation statistics, accounting for dark energy and the total neutrino mass. For the first time, we comprehensively validate the cosmology dependence of the convergence PDF model against large suites of simulated lensing maps, demonstrating its per cent level precision and accuracy. We show that fast simulation codes can provide highly accurate covariance matrices, which can be combined with the theoretical PDF model to perform forecasts and eliminate the need for relying on expensive N-body simulations. Our theoretical model allows us to perform the first forecast for the convergence PDF that varies the full set of ΛCDM parameters. Our Fisher forecasts establish that the constraining power of the convergence PDF compares favourably to the two-point correlation function for a Euclid-like survey area at a single source redshift. When combined with a cosmic microwave background prior from Planck, the PDF constrains both the neutrino mass Mν and the dark energy equation of state w0 more strongly than the two-point correlation function.

Probability distributions of daily rainfall extremes in Lazio and Sicily, Italy, and design rainfall inferences

Study regionWe investigate samples from two Italian regions, i.e. Lazio and Sicily, located in central and south Italy, respectively, and characterized by two diverse climates. Study focusIn engineering practice, the study of maxima daily rainfall values is commonly dealt with light-tailed probability distribution functions, such as the Gumbel. The choice of a distribution rather than another may cause estimation errors of rainfall values associated to specific return periods. Recently, several studies demonstrate that heavy-tailed distributions are preferable for extreme events modelling. Here, we opt for six theoretical probability distribution functions and evaluate their performance in fitting extreme precipitation samples. We select the samples with two common methods, i.e. the Peak-Over-Threshold and the Annual Maxima. We assess the best fitting distribution to the empirical samples of extreme values through the Ratio Mean Square Error Method and the Kolmogorov-Smirnov test. New hydrological insights for the regionThe assessment of the best fitting distribution to daily rainfall of the two different areas investigated here leads to interesting remarks. Despite the diversity of their climate, results suggest that heavy-tailed distributions describe more accurately empirical data rather than light-tailed ones. Therefore, extreme events may have been largely underestimated in the past in both areas. The proposed investigation can prompt the choice of the best fitting probability distribution to evaluate the design hydrological quantities supporting common engineering practice.

A new risk probability calculation method for urban ecological risk assessment

The ecological risk associated with urbanization is of great concern where multiple stressors and risk receptors co-exist. Probabilistic risk characterization methods were rarely applied in past urban ecological risk assessments because of the difficulties in the derivation of theoretical probability distribution functions and the definite integral calculation. Therefore, we proposed a new method which is based on computer simulation and able to facilitate the calculation of risk probabilities. This method quantifies multiple ecological risk-related indicators using ecological models, implements Monte Carlo simulation to calculate the risk probability of single indicators, and applies the copula model to calculate the joint risk probability of multiple indicators. We conducted an assessment of urban ecological risk related to urban surface water environment in Beijing as a case study to validate this method. The results show that the means of surface runoff risk probability, total nitrogen pollutant load risk probability, and comprehensive (joint) risk probability were 0.33, 0.44, and 0.23, respectively, in the areas within Beijing Sixth Ring Road. All three types of risk were at moderate levels in the study areas, but exhibited high spatial heterogeneity and urban–suburban gradient. The average contributions of the three risk types were 25% (surface runoff risk), 32% (total nitrogen pollutant load risk), and 43% (comprehensive risk), indicating that the joint risk was overall the major risk type. In conclusion, our method considering multiple indicators and their probabilistic attributes can handle the uncertainties in ecological models and thus has potential to evaluate different types of urban ecological risks.

Annual Daily Maximum Rainfall-Based IDF Curve Derivation Methodology

Intensity–duration–frequency (IDF) curves are among the most demandable information in meteorology, hydrology and engineering water resources design, planning, operation, and management works. This paper suggests their generation based on annual daily maximum rainfall (ADMR) records from Turkey, Ceylanpinar region meteorology stations next to Syrian border. First, available storm rainfall records are used for dimensionless–intensity–duration (DID) curve development, then, the suitable theoretical probability distribution function (PDF) is obtained for the ADMR record series, which provides return periods’ identification for a given set of risk values, and finally, the ADMR data combination with the DID curve results in IDF curve set. The application of the methodology is presented for four stations in the Southeastern Province of Turkey, Ceylanpinar drainage basin, next to the Syrian border. The comparison of the newly developed IDF curves with the available measurement-based IDF curves indicated the validity of the proposed approaches with less than 10% error.

A hybrid probabilistic bi-sector fuzzy regression based methodology for normal distributed hydrological variable

An advantage of the probabilistic approach is the exploitation of the observed probability values in order to test the goodness-of-fit for the examined theoretical probability distribution function (pdf). Since, in fact, the interest of the engineers is to achieve a relation between the hydrological variable and the corresponding probability which corresponds to a selected return period, a fuzzy linear relation between the standardized normal variable Z and the examined hydrologic random variable is achieved in condition that the hydrological variable is normally distributed. In this article, primary, the implementation of the fuzzy linear regression of Tanaka is proposed regarding the annual cumulative precipitation. Thus, all the historical data will be included in the produced fuzzy band. However, since many times the question is about the inverse process, that is, the determination of the return period for a given hydrological value, then, for this purpose, a fuzzy bi-sector regression is developed. The proposed bi-sector fuzzy regression incorporates the inclusion property regarding the produced fuzzy band as the fuzzy regression of Tanaka does. The proposed innovative methodology provides the opportunity to achieve simultaneously a fuzzy assessment of the mean value and the standard deviation based on the solution of the fuzzy linear regression. To test the suitability of the produced fuzzy band, several measures are proposed which incorporate the magnitude of the produced fuzzy band and the comparison between the estimated fuzzy mean value and standard deviation with the unbiased crisp estimation of the same variables and the median of the sample.

Statistical analysis of annual maximum daily rainfall for Nelspruit and its environs.

Nelspruit and its environs frequently experience extreme high annual maximum daily rainfall (AMDR) events resulting in flood hazards. These flood hazards have caused flood disasters that have resulted in loss of property and lives. The main objective of this study was to carry out statistical analysis of extreme high AMDR events that have caused flood hazards, which in turn have caused flood disasters in Nelspruit and its environs. Empirical continuous probability distribution functions (ECPDF) and theoretical continuous probability distribution functions (TCPDF) were applied to carry out the statistical analysis of the extreme high AMDR events. Annual maximum daily rainfall event of magnitude 100 mm was identified as a threshold. Events > 100 mm were considered as extreme high events resulting in flood disasters. The results of empirical frequency analysis showed that the return period of flood disasters was 10 years. The occurrence probability of flood disaster event at least once in 1, 2, 3, 4 and 5 years was 0.10, 0.19, 0.27, 0.34 and 0.41, respectively. Generalised logistic PDF was identified as the best-fit theoretical PDF for statistical analysis of the extreme high AMDR events in Nelspruit and its environs. The results of this study contributed to the understanding of frequency and magnitude of extreme high AMDR events that could lead to flood disasters. The results could be applied in developing flood disaster management strategies in Nelspruit and its environs.

A semi-analytical computation of the theoretical uncertainties of the solar neutrino flux

We present a comparison between Monte Carlo simulations and a semi-analytical approach that reproduces the theoretical probability distribution functions of the solar neutrino fluxes, stemming from the $pp$, $pep$, $hep$, $^7\mathrm{Be}$, $^8\mathrm{B}$, $^{13}\mathrm{N}$, $^{15}\mathrm{O}$, and $^{17}\mathrm{F}$ source reactions. We obtain good agreement between the two approaches. Thus, the semi-analytical method yields confidence intervals that closely match those found, based on Monte Carlo simulations, and points towards the same general symmetries of the investigated probability distribution functions. Furthermore, the negligible computational cost of this method is a clear advantage over Monte Carlo simulations, making it trivial to take new observational constraints on the input parameters into account.

Probability distribution functions of turbulence in seepage-affected alluvial channel

The present experimental study is carried out on the probability distribution functions (PDFs) of turbulent flow characteristics within near-bed-surface and away-from-bed surfaces for both no seepage and seepage flow. Laboratory experiments were conducted in the plane sand bed for no seepage (NS), 10% seepage (10%S) and 15% seepage (15%) cases. The experimental calculation of the PDFs of turbulent parameters such as Reynolds shear stress, velocity fluctuations, and bursting events is compared with theoretical expression obtained by Gram–Charlier (GC)-based exponential distribution. Experimental observations follow the computed PDF distributions for both no seepage and seepage cases. Jensen-Shannon divergence (JSD) method is used to measure the similarity between theoretical and experimental PDFs. The value of JSD for PDFs of velocity fluctuation lies between 0.0005 to 0.003 while the JSD value for PDFs of Reynolds shear stress varies between 0.001 to 0.006. Even with the application of seepage, the PDF distribution of bursting events, sweeps and ejections are well characterized by the exponential distribution of the GC series, except that a slight deflection of inward and outward interactions is observed which may be due to weaker events. The value of JSD for outward and inward interactions ranges from 0.0013 to 0.032, while the JSD value for sweep and ejection events varies between 0.0001 to 0.0025. The theoretical expression for the PDF of turbulent intensity is developed in the present study, which agrees well with the experimental observations and JSD lies between 0.007 and 0.015. The work presented is potentially applicable to the probability distribution of mobile-bed sediments in seepage-affected alluvial channels typically characterized by the various turbulent parameters. The purpose of PDF estimation from experimental data is that it provides a complete numerical description in the areas of turbulent flow either at a single or finite number of points.

A study of simulated normal probability functions using Microsoft Excel

Simulations are used in many calculations and forecasting procedures, from market analysis to estimation of measurement uncertainty. Microsoft Excel offers an innate procedure to simulate a Gaussian distribution and a pseudorandom number generator (PRNG). Using the latter, five additional simulations models with a potential to simulate the normal probability distribution function (PDF) were explored and compared with the theoretical Gaussian PDF based on a defined average (location) and standard deviation (width). The simulated distributions appeared “displaced” from the Gaussian function. Simulations of the distributions by a large number of observations confirmed different properties of the six simulation procedures. The range of the distributions differed, i.e. the difference between the maximum and minimum simulated values. The innate procedure yielded an unsymmetrical distribution that was shifted to the right (higher values). A miniscule divergence from expected values was demonstrated. A systematic difference between the simulated distributions and the theoretical Gauss distribution may not be of any practical importance in applied metrology, whereas it may be devastating in other applications, e.g. encoding and market simulations. There were differences between the models which may influence their use. It is suggested that a root cause may be found in the PRNG.

Probability density functions for the variable solar wind near the solar cycle minimum

AbstractUnconditional and conditional statistics are used for studying the histograms of magnetic field multiscale fluctuations in the solar wind near the solar cycle minimum in 2008. The unconditional statistics involves the magnetic data during the whole year in 2008. The conditional statistics involves the magnetic field time series split into concatenated subsets of data according to a threshold in dynamic pressure. The threshold separates fast‐stream leading edge compressional and trailing edge uncompressional fluctuations. The histograms obtained from these data sets are associated with both multiscale (B) and small‐scale (δB) magnetic fluctuations, the latter corresponding to time‐delayed differences. It is shown here that, by keeping flexibility but avoiding the unnecessary redundancy in modeling, the histograms can be effectively described by a limited set of theoretical probability distribution functions (PDFs), such as the normal, lognormal, kappa, and log‐kappa functions. In a statistical sense the model PDFs correspond to additive and multiplicative processes exhibiting correlations. It is demonstrated here that the skewed small‐scale histograms inherent in turbulent cascades are better described by the skewed log‐kappa than by the symmetric kappa model. Nevertheless, the observed skewness is rather small, resulting in potential difficulties of estimation of the third‐order moments. This paper also investigates the dependence of the statistical convergence of PDF model parameters, goodness of fit, and skewness on the data sample size. It is shown that the minimum lengths of data intervals required for the robust estimation of parameters is scale, process, and model dependent.

Distribution choice for the assessment of design rainfall for the city of London (Ontario, Canada) under climate change

With the effect of global climate change the rainfall intensity is changing, and in many places it is drastically increasing. The use of intensity–duration–frequency (IDF) curves based on historic rainfall data might, therefore, underestimate the risk associated with the design and assessment of drainage systems. The theoretical probability distribution function used in the establishment of IDF curves based on historical observations might need to be different for the future conditions. The Gumbel (EV1) distribution is the currently recommended distribution for use in Canada and the EV1 and the Log-Pearson type 3 (LP3) are routinely used in the US. This study investigates potential utility of the generalized extreme value (GEV) distribution for use in climate change impact studies by the City of London that is located in the Upper Thames River Basin. All results point that GEV seems to be the best choice for the use with the Upper Thames River Basin data. We would like to use results of this study and open the discussion on the choice of most appropriate distribution for the development of IDF curves under changing climate conditions in Canada.

Geoeffectiveness of solar wind interplanetary magnetic structures

We address the geoeffectiveness of three interplanetary structures in the interplanetary space: magnetic clouds (MCs), interplanetary shocks (IPSs), and corotating interaction regions (CIRs). The geoeffectiveness is evaluated using the geomagnetic indices Kp, AE, and Dst. We find that MCs are more geoeffective than IPSs, or CIRs. The average values of magnetic indices are significantly enhanced during disturbed periods associated with MCs, IPSs and CIRs, compared to the whole interval. The highest effect is noted for MC disturbed periods. Results obtained for the three data sets are used to derive a theoretical (continuous) probability distribution function (PDF) by fitting the histograms representing the percentage of events against the intervals of magnetic index. PDFs allow estimation of the probability of a given level of geomagnetic activity to be reached after the detection, by in situ solar wind observations, of a given interplanetary structure approaching the Earth.

A tool for determining urban emission characteristics to be used in exposure assessment

The exposure of citizens to elevated air pollution concentrations is one of the major factors leading to the deterioration of the quality of life and possibly to health problems in urban areas. The concentration of air pollutants depends largely on pollutant emission levels. If the statistical probability distribution function of the concentration of an air pollutant is known, it is possible to estimate how many times this concentration exceeds the air quality standards, or estimate changes in the emission levels in an area. It can be also used to estimate the long term exposure of population to certain pollutants. In this paper fifteen theoretical probability distribution functions, were used to fit the actual concentration frequency distributions of CO, NO 2, O 3, SO 2, and Black Smoke (BS) in Athens, Greece for a 23-year period. The results showed that the theoretical distribution type best describing the distribution of the pollutants is Inverse Gaussian followed by the Extreme value distribution. The number of exceedances of air quality limits was used to validate the performance of the theoretical distributions that were best fitted to the observed ones. The temporal evolution of emission strength was estimated through the temporal evolution of the parameters of the probability distribution functions. Missing periods were accounted for by estimating the respective distribution functions through interpolation or extrapolation from the existing ones. The derived variation of emission levels consistently represents the emission reduction strategies enforced over the years, as well as the escalating growth of the passenger car fleet volume, and could be a useful tool for the design and assessment of emission control strategies.

Floods in Serbia in the 1999-2009 period: Hydrological analysis and flood protection measures

The review on greatest floods recorded in Vojvodina and central Serbia within the period from 1999 to 2009 is given in this paper. For 13 hydrological stations, that recorded the greatest floods for the present period, probability of occurrence of these floods has been accomplished. Based on analysis of time series of discharge and water level maximum, performed by applying probability theory and mathematical statistics, and calculated theoretical probability distribution function of floods, probability of occurrence of flood has been obtained. Most often the best agreement with the empirical distribution function had a Log-Pearson III, Pearson III distribution. These results can be used for dimensioning of hydro-technical objects for flood protection. The most significant causes for floods recorded in this period were melting of snow and intensive rainfall. In this paper the current situation of flood protection and future development of flood protection measures were also presented. .

Computationally efficient algorithms for the two-dimensional Kolmogorov–Smirnov test

Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2d-1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of Ω(n2lgn), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test.

Evolution of the multiscale statistical properties of corotating streams from 1 to 95 AU

This paper discusses the multiscale structure of the large‐scale speed fluctuations between 1 and 95 AU during the declining phase of the solar cycle, when corotating streams are dominant structures close to the Sun. A deterministic, multifluid, one‐dimensional MHD model with the WIND data at 1 AU during 1995 as input was used to compute the time series of daily averages of speed, V(ti), with a length on the order of 1 year at R = 5, 10, … 95 AU, from which we calculated various statistical functions describing the solar wind speed. The probability distribution functions (PDFs) of the running speed differences dVn(ti) were calculated at scales τ from 1 to 64 days. The theoretical PDFs at 50 AU have the same qualitative forms as those observed by Voyager 2 (V2) between 46 and 49 AU. The theoretical PDF at a scale of 1 day at 15 AU agrees with the corresponding PDF observed by V2 between 13 and 16 AU. With increasing distance from the Sun, the width of the predicted PDF with τ = 1 decreases and the tail becomes more prominent. The standard deviation of dVn(ti) at various R computed from the model, SD(τ, R), decreases nearly exponentially with increasing R, and it is consistent with the observations of SD(τ, R) made by V2 near 15 and 50 AU. The skewness at a scale of 1 day increases almost linearly with R beyond ≈35 AU, but it shows structure between 5 and 35 AU; it is consistent with the observations of V2 near 15 and 50 AU. The theoretical power spectral density of the speed fluctuations, PSD(f, R), agrees with the observations of V2 near 15 and 50 AU. For frequencies f > 8.5 × 10−7 Hz, PSD(f, R) ∝ f−s(R). The observed and predicted s are ≤−2 beyond 40 AU, consistent with the jump‐ramp structure of V(t) in the distant heliosphere. We predict that the PDF for τ = 1 which will be observed by V2 at 70 AU in 2003 will have a core whose width is only ≈5 km/s and a tail extending up to 60 km/s that represents the major jumps in the speed profile. The predicted PDF at 95 AU is very similar to that at 70 AU. The model predicts that most speed jumps at 15 and 70 AU will have the signature of forward shocks and reverse shocks. The amplitudes of the large and moderate speed jumps diminish as they move from 45 to 95 AU. Some speed jumps move through this region without interacting with other speed jumps. In a few cases, neighboring speed jumps merge between 45 and 95 AU. The structure of the solar wind changes slowly as it moves from 45 to 95 AU.

Classical stochastic approach to cosmology revisited

The classical stochastic model of cosmology recently developed by us is reconsidered. In that approach the parameterw defined by the equation of statep = wp was taken to be fluctuating with mean zero and we compared the theoretical probability distribution function (PDF) for the Hubble parameter with observational data corresponding to a universe with matter and vacuum energy. Even though qualitative agreement between the two was obtained, an attempt is herein made to introduce a more realistic assumption for the mean ofw and use it for the calculations. In the present theory the mean values of bothp andw are taken to be nonzero. The theoretical and observational PDFs are compared for different epochs and values of the Hubble parameter. The corresponding values of the diffusion constantD obtained are approximately constant. We use the scatter in the observed redshift-magnitude data of Type Ia supernova to place limits on the stochastic variation in expansion rate and consequently, on the stochastic variation of the equation of state.

Decision support for the single‐period inventory problem

The development of a decision support tool for the single‐period inventory problem is presented. The support tool allows a consideration of the following factors: empirical frequency distributions, theoretical probability distribution functions and managerial probability estimates of total demand over the period; piece‐wise linear (possibly discontinuous) cost functions. Such functions allow for the possibility of “fixed cost” elements and/or “economies/diseconomies” of scale and account for most, if not all, of the purchase cost, holding cost, salvage revenue and shortage cost functions that arise in practice; choice of performance measures; and “what‐if” analysis on the problem parameters. The support tool, which uses the Monte Carlo simulation option of Visual IFPS/Plus, is transparent and constructively simple and thus readily facilitates understanding, acceptance and implementation by management. The support tool runs under the Windows operating system and has easy access to appropriate spreadsheet and word‐processing packages for further processing and report presentation.