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. 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 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. 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, one may for example select the log-normal
distribution, the log-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, one may for
example select the square-normal distribution, the inverted Gumbel
distribution, or the Gompertz distribution, which is bounded to the
left. Techniques of fitting
The following techniques of distribution fitting exist:
Parametric methods, by which the parameters of the distribution are
calculated from the data series. The parametric methods are:
method of moments method of L-moments
Maximum likelihood method Regression method, using a
transformation of the cumulative distribution function so that a linear
relation is found between the cumulative probability and the values of the data,
which may also need to be transformed, depending on the selected probability
distribution. In this method the cumulative probability needs to be
estimated by the plotting position. Generalization of distributions
It is customary to transform data logarithmically to fit symmetrical
distributions to data obeying a distribution that is positively skewed,
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 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 by
replacing in the mathematical expression of the cumulative distribution function
by its complement: F'=1-F, obtaining the complementary distribution 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
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: The true probability distribution of
events may deviate from the fitted distribution, as the observed data
series may not be totally representative of the real probability of occurrence of
the phenomenon due to random error The occurrence of events in another
situation or in the future may deviate from the fitted distribution as this
occurrence can also be subject to random error
A change of environmental conditions may cause a change in the probability of
occurrence of the phenomenon 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 and the
probability of non-exceedance Pn. 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. See also
Density estimation References