Keelin (MetaLog) distribution

New to Analytica 5.0

Keelin( values , percentiles, I, lb, ub, nTerms, flags, over)

Also analytic distribution functions (see Analytic distribution functions below):

CumKeelinInv( p, values, percentiles, I, lb, ub, nTerms, flags)

CumKeelin( x, values, percentiles, I, lb, ub, nTerms, flags)

DensKeelin( x,p, values, percentiles, I, lb, ub, nTerms, flags)

The Keelin distribution, also known as the Keelin MetaLog distribution. This is a smooth, continuous distribution that can be specified in one of three ways:

• From a set of representative points (a data sample).
• From a list of (value,percentile) pairs, where the «values» are sample values for the quantity and the «percentiles» are the percentile (aka quantile or fractile) levels, all between 0 and 1. Another way of saying this is that «values», «percentiles» are points on the cumulative probability curve.
• From a coefficient vector. These coefficients are typically obtained from data «values», or from (value, percentile) pairs using the function KeelinCoefficients. The coefficient vector may be a must shorter description of the distribution than the original data. When passing coefficients, you must specify the «flags»=1 bit, and then «I» index indexes the basis terms rather than the data.

The Keelin distribution is introduced in the paper:

• Thomas W. Keelin (Nov. 2016), "The Metalog Distribution", Decision Analytics, 13(4):243-277,

The Keelin distribution is a highly flexible distribution that is capable of taking on the shape of almost all common distributions. It is among the most versatile of all distributions, with an ability to produce unbounded, semi-bounded and bounded distributions with nearly any theoretically possible combination of skewness and kurtosis. In this respect, it is even more flexible that the family of Pearson distributions.

If you have a data sample that is representative of your quantity, and you wonder which distribution you should fit to your data, the Keelin is a good option. Instead of worrying about finding which parametric form you need, the Keelin distribution usually adapts to the data quite nicely.

If you need a Keelin distribution based on 3 symmetric fractiles, such as based on 10-50-90 percentile estimates, use the UncertainLMH() function. UncertainLMH() is a more-convenient special case of Keelin() for that purpose.

Parameters

• «values»: This can be either: (1) A representative sample of data points, with «percentiles» omitted, (2) a collection of fractile estimates (corresponding to the quantile levels in «percentiles»), or (3) a Keelin coefficient vector with the «flags»=1 bit set. In all cases, «values» must be indexed by «I».
• «percentiles»: (Optional): The percentile levels (also called quantile or fractile levels) for the values in «values», also indexed by «I». Each number must be between 0 and 1. For example, when a value in «percentiles» is 0.05, the corresponding value in «values» is the 5th percentile.
• «I»: (Optional): The index of «values» and «percentile». This can be omitted when either «values» or «percentile» is itself an index.
• «lb», «ub»: (Optional) Upper and lower bound. Set one or both of these to a single number if you know in advance that your quantity is bounded. When neither is specified, the distribution is unbounded (i.e., with tails going to -INF and INF). When one is set the distribution is semi-bounded, and when both are set it is fully bounded.
• «nTerms»: (Optional) The number of basis terms used for the fit. This should be 2 or greater. See #Number of terms below.
• «flags»: (Optional) A bit-field, where any of the following flags can be added together.

  1 = «values» contains coefficients (as obtained from the KeelinCoefficients function). When not set, «values» contains sample values.

  8 = Do not test for or issue a warning when infeasible. (See Infeasibility below. No validation of feasibility is performed when coefficients are passed in (i.e., when «flags»=1 is set).

  16 = Return a sample even when infeasible. When this bit is not set, Null is returned if infeasible. When this is set, a mid-value or sample is returned anyway.

• «over»: (Optional) A list of indexes to sample independently across.

Examples

Fit to data

Suppose you've collected data on the weights of fish caught last year in the Columbia river, and now you want to fit a distribution to these measurements. Since you know that a fish's weight cannot be negative, you'll use a semi-bounded distribution. Suppose the data is in a variable named Fish_weight which in indexed by Fish_ID. Use

Keelin( Fish_weight, I:Fish_ID, lb: 0)

Using fractiles

You find a published table stating the 500-year, 100-year, 10-year and median rain fall levels for a town of interest (where the 500-year level is a level so big that it is experienced only once every 500 years).

Index Fractile := [ 1/2, 1/10, 1/100, 1/500 ]

Variable Rainfall_level := Table(Fractile)(5, 12, 25, 60)

Chance Rainfall := Keelin( Rainfall_level, Fractile, lb:0 )

The resulting CDF is plotted here on a log-X scale:

Infeasibility

Some combinations of «values», «percentiles» and «nTerms» are considered to be infeasible. These are situations in which the Keelin algorithm cannot find a sensible distribution function that fits the data. See Keelin (2016) for mathematical details.

Infeasible data sets often have sharp changes or gaps. In some cases, varying «nTerms» can alter fealibility. If your data set turns out to be infeasible with 10 terms, you might find that with 9 or 11 terms (or some other number) it becomes feasible.

By default, when it fits a Keelin distribution to the data, it tests whether the fit is feasible. This test is not performed when coefficients are passed in, but the test for feasibility will have been performed by KeelinCoefficients when that function fit the data. The test for feasibility can be skipped by setting the «flags»=8 bit. If the fit is found to be infeasible, a warning is issued (unless you have warnings suppressed) and the function returns Null. If either of the «flags»=8 or «flags»=16 bits is set, then it returns a sample even when infeasible. In theory, a sample from an infeasible fit is not guaranteed to match the data, but in practice it still tends to be pretty good. However, the results from the analytic functions such as CumKeelinInv, CumKeelin and DensKeelin don't make sense (specifically, CumKeelinInv is not monotonically increasing, and DensKeelin has negative values).

Number of terms

The optional «nTerms» parameter varies the number of basis terms used for the fit. A larger number of terms results in a more detailed fit, but may also overfit when the data has randomness. Varying «nTerms» may also impact whether the data is or is not infeasible.

With 2 terms, the smallest that should be considered, the distribution is limited to a Logistic distribution (or Log-Logistic or Logit-Logistic when «lb» or «ub» are set), which gives it enough flexibility to match mean and standard deviation, but not skewness or kurtosis. With 3 terms skewness can be adjusted, but not kurtosis. With 4 terms, the median, variance, skewness and kurtosis can all be adjusted. In most cases, increasing «nTerms» enables it to fit your target distribution more closely.

You may find it useful to create a panel of fit distributions by varying «nTerms», making it possible to see what detail is revealed by the addition of terms, and also where the addition of more terms doesn't add useful detail. In many cases I've observed that there is an improved "fit" up to a point, followed by a plateau with very little change as «nTerms» increases, eventually followed by obvious over-fitting where it starts capturing the random spacing of samples. Often the plateau lasts for a long time. In these cases, it would make sense to set «nTerms» to a value on the plateau.

To create a "panel", first create an index: Index NumTerms := 2..50

Then use your data (say x indexed by I) to explore these: Variable fit_x := Keelin( x, ,I, nTerms:NumTerms)

In the following experiment, a data set with 100 measurements (not from any know distribution) was fit. A histogram of the data itself is shown here:

Here are four "fit" Keelin distributions as «nTerms» was varies from 2 to 20:

At 10 terms, a bi-modal effect starts to appear, which may actually be there in the data, and which is not visible below 10 terms, However, at 20 terms there appears to be more variation than is probably warranted, which we might interpret as the onset of overfitting.

Analytic distribution functions

The Keelin function is a distribution function, meaning that it returns the median when evaluated in Mid-mode, a sample for Monte Carlo analysis when evaluated in Sample-mode, or a random variate when it is used in Random(). There are also analytic distribution functions for Keelin described in this section. The analytic distribution functions compute the exact value for a given Keelin without sampling error. These use the same parameters as Keelin( ), but also include a point of percentile where the analytic function is to be evaluated at, and this parameter comes first. It is common to pass an array of values in for the point or percentile parameter.

CumKeelinInv( p, values, percentiles, I, lb, ub, nTerms, flags)

The inverse cumulative distribution function, also called the quantile function. Given a percentile (quantile) level «p», returns the value x such that there is a «p» probability that the actual value (or a value sampled at random from the distribution) is less than or equal to x.

Given a data set xi indexed by I, the 10-25-50-75-90 percentiles are obtained using:

Index percent=[10%, 25%, 50%, 75%, 90%] Do CumKeelinInv( percent, xi, , I)

Passing a Uniform(0,1) for «p» is equivalent to the Keelin distribution function:

CumKeelinInv( Uniform(0,1), xi )

CumKeelin( x, values, percentiles, I, lb, ub, nTerms, flags)

The cumulative distribution function. For a given value «x», returns the probability that the true value (or a value sampled at random from the distribution) is less than or equal to «x».

Index quartile := [10%, 25%, 50%, 75%, 90%]

Index estimate := Table(quartile)( 20, 40, 70, 100, 130 )

This returns the probability that the quantity is less than 50:

CumKeelin( 50, estimate, quartile, quartile) → 0.33

DensKeelin( x,p, values, percentiles, I, lb, ub, nTerms, flags)

The probability density function. Either «x» or «p», but not both, must be specified. When «x» is specified, returns the probability density at «x». When «p» is specified, returns the probability density at the «p»th quartile.

Using quartile and estimate shown in the example for CumKeelin above,

DensKeelin( 50, values:estimate, percentiles:quartile, I:quartile ) → 8.16m

DensKeelin( p:33%, values:estimate, percentiles:quartile, I:quartile ) → 8.16m

A good way to graph the probability density function is to create an X-Y plot of DensKeelin versus CumKeelinInv. This is illustrated here, again using the quartile and estimate example from above.

Index p := Sequence(0.1%, 99.9%, 0.1%);
Index K := 1..5;
Var a := KeelinCoefficients( estimate, quartile, quartile, K );
Var x := CumKeelinInv( p, a, I:K, flags:1 );
Var density := DensKeelin( p:p, values:a, I:K, flags:1 );
Index AxisLabel := ['X', 'Probability Density'];
Array( AxisLabel, [x,density] )

You need to plot this as an X-Y chart by pressing the [XY] button in the result window and enabling Use a comparison index. The comparison index here will be .AxisLabel.

In the graph, an efficiency is obtained by calling KeelinCoefficients so that the data fit is performed only once, so that it doesn't have to be repeated in both successive calls to CumKeelinInv and DensKeelin. It evaluates both at 999 points (the number of points in p. Zero and 1 are not included in p since these would be -Inf and Inf in a distribution with tails like this. A nice thing about this approach is that it is easy to vary over the full range of p without having to know the actual range of the quantity itself.

See Also

• UncertainLMH -- Simpler version when specifying say 10-50-90 or 25-50-75 estimates.
• KeelinCoefficients
• DensKeelin, CumKeelin, CumKeelinInv -- analytic distribution functions for Keelin
• CumDist, Smooth_Fractile
• ProbDist