Originally the method was developed for the analysis of hydrological measurements of spatially varying magnitudes (e.g. hydraulic conductivity of the soil) and of magnitudes varying in time (e.g. rainfall, river discharge) to find their return periods. However, it can be used for many other types of phenomena, including those that contain negative values.
Software features
Screenprint of input tabsheet
CumFreq uses the plotting position approach to estimate the cumulative frequency of each of the observed magnitudes in a data series of the variable.[2]
Another characteristic of CumFreq is that it provides the option to use two different probability distributions, one for the lower data range, and one for the higher. The ranges are separated by a break-point. The use of such composite (discontinuous) probability distributions can be useful when the data of the phenomenon studied were obtained under different conditions.[3]
Composite (discontinuous) distribution with confidence belt[4]
During the input phase, the user can select the number of intervals needed to determine the histogram. He may also define a threshold to obtain a truncated distribution.
The program can produce generalizations of the normal, logistic, and other distributions by transforming the data using an exponent that is optimized to obtain the best fit.
This feature is not common in other distribution-fitting software which normally include only a logarithmic transformation of data obtaining distributions like the lognormal and loglogistic.
Generalization of symmetrical distributions (like the normal and the logistic) makes them applicable to data obeying a distribution that is skewed to the right (using an exponent <1) as well as to data obeying a distribution that is skewed to the left (using an exponent >1). This enhances the versatility of symmetrical distributions.
(A) Gumbel probability distribution skew to right and (B) Gumbel mirrored skew to left
Inverting distributions
Skew distributions can be mirrored by distribution inversion (see survival function, or complementary distribution function) to change the skewness from positive to negative and vice versa. This amplifies the number of applicable distributions and increases the chance of finding a better fit. CumFreq makes use of that opportunity.
Shifting distributions
When negative data are present that are not supported by a probability distribution, the model performs a distribution shift to the positive side while, after fitting, the distribution is shifted back.
Nine return period curves of 50-year samples from a theoretical 1000-year record (base line)
The prediction of the return period, which is of interest in time series, is also accompanied by a confidence belt. The construction of confidence belts is not found in most other software.
The figure to the right shows the variation that may occur when obtaining samples of a variate that follows a certain probability distribution. The data were provided by Benson.[6]
List of probability distributions ranked by goodness of fit, example
The confidence belt around an experimental cumulative frequency or return period curve gives an impression of the region in which the true distribution may be found.
Also, it clarifies that the experimentally found best fitting probability distribution may deviate from the true distribution.
Histogram and probability density of a data set fitting the GEV distribution[7]
Goodness of fit
Cumfreq produces a list of distributions ranked by goodness of fit.
Probability distribution calculator as used in the CumFreq software
The software offers the option to use a probability distribution calculator. The cumulative frequency and the return period are give as a function of data value as input. In addition, the confidence intervals are shown. Reversely, the value is presented upon giving the cumulative frequency or the return period.
^ abFrequency and Regression Analysis. Chapter 6 in: H.P.Ritzema (ed., 1994), Drainage Principles and Applications, Publ. 16, pp. 175–224, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. ISBN90-70754-33-9 . Free download as PDF from : ILRI website or from : [1]
^Software for Generalized and Composite Probability Distributions. International Journal of Mathematical and Computational Methods, 4, 1-9. On line: [2]
^Drainage research in farmers' fields: analysis of data, 2002. Contribution to the project "Liquid Gold" of the International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. [3]
^Benson, M.A. 1960. Characteristics of frequency curves based on a theoretical 1000 year record. In: T.Dalrymple (ed.), Flood frequency analysis. U.S. Geological Survey Water Supply paper 1543−A, pp. 51–71
