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Probability distribution fitting or simply distribution fitting is the fitting of a probability distribution to a series of data concerning the repeated measurement of a variable phenomenon.
Contents
- Selection of distribution
- Techniques of fitting
- Generalization of distributions
- Inversion of skewness
- Shifting of distributions
- Uncertainty of prediction
- References
The aim of distribution fitting is to predict the probability or to forecast the frequency of occurrence of the magnitude of the phenomenon in a certain interval.
There are many probability distributions (see list of probability distributions) of which some can be fitted more closely to the observed frequency of the data than others, depending on the characteristics of the phenomenon and of the distribution. The distribution giving a close fit is supposed to lead to good predictions.
In distribution fitting, therefore, one needs to select a distribution that suits the data well.
Selection of distribution
The selection of the appropriate distribution depends on the presence or absence of symmetry of the data set with respect to the mean value.
Symmetrical distributions
When the data are symmetrically distributed around the mean while the frequency of occurrence of data farther away from the mean diminishes, one may for example select the normal distribution, the logistic distribution, or the Student's t-distribution. The first two are very similar, while the last, with one degree of freedom, has "heavier tails" meaning that the values farther away from the mean occur relatively more often (i.e. the kurtosis is higher). The Cauchy distribution is also symmetric.
Skew distributions to the right
When the larger values tend to be farther away from the mean than the smaller values, one has a skew distribution to the right (i.e. there is positive skewness), one may for example select the log-normal distribution (i.e. the log values of the data are normally distributed), the log-logistic distribution (i.e. the log values of the data follow a logistic distribution), the Gumbel distribution, the exponential distribution, the Pareto distribution, the Weibull distribution, or the Fréchet distribution. The last three distributions are bounded to the left.
Skew distributions to the left
When the smaller values tend to be farther away from the mean than the larger values, one has a skew distribution to the left (i.e. there is negative skewness), one may for example select the square-normal distribution (i.e. the normal distribution applied to the square of the data values), the inverted (mirrored) Gumbel distribution, or the Gompertz distribution, which is bounded to the left.
Techniques of fitting
The following techniques of distribution fitting exist:
Generalization of distributions
It is customary to transform data logarithmically to fit symmetrical distributions (like the normal and logistic) to data obeying a distribution that is positively skewed (i.e. skew to the right, with mean > mode, and with a right hand tail that is longer than the left hand tail), see lognormal distribution and the loglogistic distribution. A similar effect can be achieved by taking the square root of the data.
To fit a symmetrical distribution to data obeying a negatively skewed distribution (i.e. skewed to the left, with mean < mode, and with a right hand tail this is shorter than the left hand tail) one could use the squared values of the data to accomplish the fit.
More generally one can raise the data to a power p in order to fit symmetrical distributions to data obeying a distribution of any skewness, whereby p < 1 when the skewness is positive and p > 1 when the skewness is negative. The optimal value of p is to be found by a numerical method. The numerical method may consist of assuming a range of p values, then applying the distribution fitting procedure repeatedly for all the assumed p values, and finally selecting the value of p for which the sum of squares of deviations of calculated probabilities from measured frequencies (chi square) is minimum, as is done in CumFreq.
The generalization enhances the flexibility of probability distributions and increases their applicability in distribution fitting.
Inversion of skewness
Skewed distributions can be inverted (or mirrored) by replacing in the mathematical expression of the cumulative distribution function (F) by its complement: F'=1-F, obtaining the complementary distribution function (also called survival function) that gives a mirror image. In this manner, a distribution that is skewed to the right is transformed into a distribution that is skewed to the left and vice versa.
The technique of skewness inversion increases the number of probability distributions available for distribution fitting and enlarges the distribution fitting opportunities.
Shifting of distributions
Some probability distributions, like the exponential, do not support data values (X) equal to or less than zero. Yet, when negative data are present, such distributions can still be used replacing X by Y=X-Xm, where Xm is the minimum value of X. This replacement represents a shift of the probability distribution in positive direction, i.e. to the right, because Xm is negative. After completing the distribution fitting of Y, the corresponding X-values are found from X=Y+Xm, which represents a back-shift of the distribution in negative direction, i.e. to the left.
The technique of distribution shifting augments the chance to find a properly fitting probability distribution.
Uncertainty of prediction
Predictions of occurrence based on fitted probability distributions are subject to uncertainty, which arises from the following conditions:
An estimate of the uncertainty in the first and second case can be obtained with the binomial probability distribution using for example the probability of exceedance Pe (i.e. the chance that the event X is larger than a reference value Xr of X) and the probability of non-exceedance Pn (i.e. the chance that the event X is smaller than or equal to the reference value Xr, this is also called cumulative probability). In this case there are only two possibilities: either there is exceedance or there is non-exceedance. This duality is the reason that the binomial distribution is applicable.
With the binomial distribution one can obtain a confidence interval of the prediction. Such an interval also estimates the risk of failure, i.e. the chance that the predicted event still remains outside the confidence interval. The confidence or risk analysis may include the return period T=1/Pe as is done in hydrology.