# CumFreq:   Cumulative frequency analysis with probability distribution fitting Free download of software

 Summary: The CumFreq model program calculates the cumulative (no exceedance, non-exceedance) frequency and it does probability distribution fitting of data series, e.g. crop production, soil properties, salinity, depth to watertable (water-table), rainfall, river and drain discharge, groundwater (ground-water) and river level, hydraulic conductivity and soil permeability for water.       The computer program fits various linear, logarithmic, exponential and double exponential cumulative frequency distributions, including the normal, log-normal (lognormal), (log)logistic, Cauchy, Pareto, Weibull, Frechet (Fisher-Tippett type II), Kumaraswamy, Gumbel, GEV, Erlang (Gamma), Burr, Dagum, Laplace, Student, Gompertz, exponential, Poisson and Rayleigh probability distributions, and selects the most appropriate distribution using the method of best fit. Some of the distributions are useful for extreme value analysis.       Preference for a specific distribution can also be expressed. The Cumfreq calculator model not only uses logarithmic but also exponential transformations of the data. The exponent (greater than 0) is optimized automatically. These so called generalized distributions have great flexibility, but they are mostly not used in other distribution fitting calculators, except the Burr distribution which is a generalization of Pareto.       Further, the Cumfreq application (app) uses mirrored (inverese, complementary) distributions that reverse the skewness, so that the number of distributions that are possibly applicable is enlarged. One of the few existing examples is the inverse (mirrored, complementary) characteristic of the Burr and Dagum distributions. The Cumfreq program calculator allows negative data. In case of distributions that do not support negative data, the distribution is shifted in positive direction so that they are made applicable.       The download and use of the CumFreq model program is totally free. It is freeware. Details: When the mathematical model program detects a discontnuous probability function, it gives the user the choice to introduce a breakpoint (break-point) or threshold value, yielding a segmented or composite frequency and probability distribution.       The resulting probability distribution is accompanied by 90% confidence limits and belts (bands) and return periods (recurrence intervals). The program also divides the range of data into a number of equal intervals and calculates an interval frequency distribution or histogram. The number of intervals is to be given by the user.       On 22 March 2018 the option to see a list of distributions ranked according to their goodness of fit was introduced. On 6 April 2018, next to the cumulative distribution function (CDF), the probability density function (PDF) was inlcluded in the histogram graph.       Examples are given. Start: The program starts clicking on CumFreq.Exe. More details are given in the program itself.

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 Documentation: A lecture note ("Data Analysis") on statistical analysis with examples of CumFreq applications is found on the articles page. Further, this page gives an article binomial confidence limits for cumulative probability distributions.           The principles of frequency analysis can be found in a lecture note on the FAQ's page. Further, this page gives a summary paper on binomial confidence limits for cumulative frequency distributions.           An article using CumFreqA has appeared in the International Journal of Mathematical Methods, see this site. An article specialised in application of the generalized logistic distribution can be found as PDF document or on ResearchGate           A paper entitled: "The generalized standard and mirrored Gumbel probability distributions, composite or not, are applicable to many datasets, either symmetrical, skew to the left, or skew to the right" can be read at composite paper or at generalized           A a publication entitled: "How to derive a probability distribution from a data set using the simple method of plotting positions and the free CumFreq model" can be consulted at simple method or at this place           Another publication entitled: "Comparing the results of transmuted Gumbel probability distributions found in literature with those obtained by the free CumFreq program" is to be seen at transmuted Gumbel or at this site           Stil another publication entitled: "Comparing the results of Extended Gumbel Type-2 probability distributions found in literature with those obtained by the free CumFreq program" can be found at extended Gumbel or over here           In case you wish to read: "Comparing the results of Truncated Generalized Frechet Weibull distribution found in literature with those obtained by the free CumFreq program" use the link Frechet Weibull           A list of publications in which CumFreq is used can be seen in this paper           There is an article comparing results of extended probbility distributions found in literature with those obtained using the CumFreqA program that finds best fitting bimodal solutions in: this bimodal article. Experiences: For improvement, I am interested to learn about your experiences with CumFreq. For this there is a contact form. Acknowlegments: Thanks to a suggestion by Madhu Kandampadath the generalized extreme value (GEV) distribution was included in the list of options on 17 February 2018.           Two days later the composite Laplace distribution was added. On 27 May 2018, thanks to observations of John Haddy, the Burr and Dagum (Burr mirrored) distributions were included. Three days later the Gompertz distribution was incorporated.

 Hints: CumFreq can create histograms and probability density functions as illustrated on the density page. An amplified version of CumFreq named CumFreqA can be seen on: composite distributions. This version contains extra facilities regarding composite probability distributions. These are not often found with an exception of the Laplace distribution.           The versatility of generalization makes it possible, for example, to fit approximately normally distributed data sets to a large number of different probability distributions, see multple fit, while negatively skewed distributions can be fitted to square normal and mirrored Gumbel distributions, as in left skew, and finally the various possibilities of curved regression are show in curvatures.

 Figure 1. Introduction screen to CumFreq program (model). The CumFreq calculator models the cumulative (non-exceedance) frequency distribution and fits it to a probability distribution. It uses, amongst other, the normal, logistic, exponential and Gumbel distributions. For distribution fitting, the CumFreq calculator app linearizes the probability distribution.For a list of linearizations, based on logarithmic and other transformations, see: transformations

 Figure 2. Input tabsheet The input tabsheet shows the probability distributions that are being used in the CumFreq calculator model to fit the data to. In this application program one can select "best of all" or indicate a preference. The number of intervals needed to prepare the histogram can be entered, as well as a cut-off (threshold) velue.

 Figure 3. Cumulative distribution fitting Probability distribution fitting is based on plotting positions (the observed data). A 90% confidence interval of the fitted probability distribution is shown. It is a specialty of the CumFreq calculator model calculator to apply "generalized" distributions, which, in this application program, makes them fit better than the standard ones. They are based on an exponential transformation of the data to obtain a closer fit. The curve shown is also called cumulative distribution function (PDF).

 Figure 4. Histogram of fitted distribution + density function The histogram provides the observed and calculated frequencies by interval. The histogram gives an impression of the symmetry of the probability distribution and whether it is skew to the left or to the right. The probability density function (PDF) is shown as the product of density and lenhth of the intervals.

 Figure 5. Probability distributions ranked by goodness of fit CumFreq provides the option to produce a list of probability distributions ranked by goodness of fit.

 Figure 5. Calculator CumFreq provides the option to calculate probabilities of data values and vice versa. The same holds for return periods. In addition confidence intervals are given.

 Figure 6. Composite probabilty distributions. The amplified CumFreq calculator (CumFreqA) provides the possibility to select a composite probability distribution (like the Laplace distribution) or to find the best fitting of all. See: composite distribution.

 Figure 7. Bimodal probabilty distribution. The amplified CumFreq calculator (CumFreqA) in some cases comes up with a bimodal probability disribution. See also: bimodal distribution.