RELATIVE DISPERSION INDEX

RELATIVE DISPERSION INDEX (LET)

Let Subcommand

Given a vector of counts, compute the difference from uniformity based on a transformation of the variational distance.

In spatial analysis, it is sometimes desired to determine if the points in the given space are consistent with a uniform distribution. One such measure is based on the variational distance which is defined as
\( d = \frac{1}{2} \sum_{k=0}^{\infty}{|P(Unif = k) - P(data=k)|} \)

Given that the points have been converted to a set of N counts, X_k, this formula becomes

\( d = \frac{1}{2} \frac{\sum_{k=1}^{N}{|\frac{1}{N} - X_{k}|}} {\sum_{k=1}^{N}{X_{k}}} \)

The value of the variational distance is between zero and one with values closer to zero indicating greater consistency with a uniform distribution.

The relative dispersion index is a scaled version of the variational distance

\( RDI = 100 (1 - d) \)

where d is the variational distance statistic. This transforms the zero to one scale to a zero to 100 scale. Values close to 100 indicate consistency with a uniform distribution.

LET <a> = RELATIVE DISPERSION INDEX <y>
                  <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <a> is a parameter where the computed statistic is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = RELATIVE DISPERSION INDEX Y1
LET A = RELATIVE DISPERSION INDEX Y1 SUBSET TAG > 2

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

None

VARIATIONAL DISTANCE	= Compute the variational distance of a variable containing a set of counts.
UNIFORM CHISQUARE STAT	= Compute the chi-square statistic for uniformity for a variable containing a set of counts.
GOODNESS OF FIT	= Perform a goodness of fit test.

Kashiwagi, Fagan, Douglas, Yamamoto, Heckert, Leigh, Obrzut, Du, Lin-Gibson, Mu, Winey, Haggennueller (2007), "Relationship between dispersion metric and properties of PMMA/SWNT nanocomposites", Polymer Journal, Vol. 48, pp. 4855 - 4866.

Spatial Statistics

2014/3

 
LET Y  = UNIFORM RANDOM NUMBERS FOR I = 1 1 1000
LET Y1 X1 = BINNED Y
LET Y  = NORMAL RANDOM NUMBERS FOR I = 1 1 1000
LET Y2 X2 = BINNED Y
LET Y  = EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 1000
LET Y3 X3 = BINNED Y
LET A1 = RELATIVE DISPERSION INDEX Y1
LET A2 = RELATIVE DISPERSION INDEX Y2
LET A3 = RELATIVE DISPERSION INDEX Y3
SET WRITE DECIMALS 4
PRINT A1 A2 A3
    

The following output is generated.

 PARAMETERS AND CONSTANTS--

    A1      --        92.5333
    A2      --        60.7571
    A3      --        48.1910