VARIATIONAL DISTANCE

VARIATIONAL DISTANCE (LET)

Let Subcommand

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

LET <a> = VARIATIONAL DISTANCE <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 = VARIATIONAL DISTANCE Y1
LET A = VARIATIONAL DISTANCE Y1 SUBSET TAG > 2

The relative dispersion index is a scaled version of the variational distance. Enter HELP RELATIVE DISPERSION INDEX for details.

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

None

None

RELATIVE DISPERSION INDEX	= Compute the relative dispersion index 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 = VARIATIONAL DISTANCE Y1
LET A2 = VARIATIONAL DISTANCE Y2
LET A3 = VARIATIONAL DISTANCE Y3
SET WRITE DECIMALS 4
PRINT A1 A2 A3
    

The following output is generated.

 PARAMETERS AND CONSTANTS--

    A1      --         0.0747
    A2      --         0.3924
    A3      --         0.5181