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The VARIOGRAM Procedure |

The basic starting point in
computing the semivariogram is the enumeration
of pairs of points for the spatial data.
Figure 70.11 shows a spatial domain in which a set
of measurements are made at the indicated locations.
Two points *P _{1}* and

For three classes, .When the example directed line segment *P _{1}*

Note that if the designated points *P _{1}* and

If you specify an angle tolerance less than the
default, for example, *ATOL*=15^{o},
some point pairs might be excluded. For example, the selected
point pair *P _{1}*

On the other hand, you can specify an angle tolerance *greater*
than the default. This can result in a point pair being counted
in more than one angle class. This has a smoothing effect on the
variogram and is useful when there is a small amount of data
available.

An alternative way to specify angle classes
and angle tolerances
is with the DIRECTIONS statement.
The DIRECTIONS statement is useful when angle classes are not
equally spaced. When you specify the DIRECTIONS statement,
you should also specify the angle tolerance.
The default value
of the angle tolerance is 45^{o} when a DIRECTIONS
statement is used instead of the NDIRECTIONS= option in
the COMPUTE statement.
This may not be appropriate for a particular set of angle
classes. See the "DIRECTIONS Statement" section
for more details on the
DIRECTIONS statement.

When the directed line segment *P _{1}*

Because pairwise distances are positive, lag class zero
is smaller than lag classes 1, ... , *MAXLAG*-1.
For example, if you specify LAGD=1.0 and MAXLAG=10,
and you do not specify a LAGTOL= value in the COMPUTE statement
in PROC VARIOGRAM,
the ten lag classes generated
by the preceding equation are

- [0,.5), [.5,1.5), [1.5,2.5), ... , [8.5,9.5)

This is because the default lag tolerance is one-half the LAGD= value, resulting in no gaps between the distance class intervals. This is shown in Figure 70.14.

On the other hand, if you do specify a distance tolerance with the DTOL= option in the COMPUTE statement, a further check is performed to see if the point pair falls within this tolerance of the nearest lag. In the preceding example, if you specify LAGD=1.0 and MAXLAG=10 (as before) and also specify LAGTOL=0.25, the intervals become

- [0,0.25), [0.75,1.25), [1.75,2.25), ... , [8.75,9.25)

Note that this specification results in gaps in the lag classes;
a point pair *P _{1}*

For example, suppose two points *P _{3}*,

The endpoint of vector *P _{3}*

Finally, a pair *P*_{i}*P*_{j} that falls in a lag class larger
than the value of the MAXLAG= option is excluded
from the semivariogram calculation.

From this description, it is clear that the number of pairs within each angle/distance class is strongly affected by the angle and lag tolerances. Since it is desirable to have the maximum number of point pairs within each class, the angle tolerance and the distance tolerance should usually be the default values.

Denote all pairs *P*_{i}*P*_{j} belonging to angle class
and distance class *L*=*L*(*P*_{i}*P*_{j}) by . For example,
in the preceding illustration, *P _{1}*

Let
denote the *number* of such pairs. Let *V*_{i}, *V*_{j} be the
measured values at points *P*_{i}, *P*_{j}. The component of
the standard (or method of moments) semivariogram corresponding
to angle/distance class is given by

The robust version of the semivariogram, as suggested by Cressie (1993), is given by

This robust version of the semivariogram is computed when you specify the ROBUST option in the COMPUTE statement in PROC VARIOGRAM.

PROC VARIOGRAM computes and writes to the OUTVAR= data set the quantities , and .

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