Variance Clouds

The variance cloud is a graph of the variances for all individual pairs of points in an autocorrelation analysis. Like the h-Scattergram, it is particularly useful for discovering outliers that may inappropriately skew the average value for a lag class. By placing the mouse on top of individual variogram points you can determine which pairs of points in the data set are suspect, as for the upper right-hand point in the window below.

Variance cloud analysis is closely related to h-Scattergram analysis. Both analyses present pair-by-pair information on variances within lag classes arranged by separation distances.

The formula to calculate variance for any given pair of points at locations i and j reduces to the mean squared difference between those points:

$images\formula_variance_cloud_wmf.gif$

Any given pair(i,j) are separated by a specific distance; this distance is plotted along the x axis of the variance cloud graph. All pairs on a specific graph are in the same separation distance (lag) class.

Note that a variance cloud is specific to both direction (isotropic or a specific anisotropic direction) and to a particular lag class. In the variogram below, the cursor is on the point representing lag class 7 of the isotropic variogram, which one might suspect contains an outlier because it is so different from the other points:

$images\variogram_with_outlier_present.gif$

Clicking on this variogram point brings up the variance cloud for lag class 7 below, and it becomes apparent that a number of pairs are very different from the others –placing the cursor over each point reveals that all of the points high on the graph contain record 4 as a member of the pair (the cursor below is over the point represented by records 4 and 96 with a separation distance of 52.77):

$images\variance_cloud_with_outlier_present.gif$

Clicking on this variance cloud point brings up the Sample Details window, which gives us the option to temporarily mask one of the data records that make up this pair:

$images\sample_details_variance_cloud_window.gif$

Since record 4 is a member of all of these outlier pairs, we choose to mask record 4, which gives us a much more reasonable variogram:

$images\variogram_with_outlier_removed.gif$

Re-examination of the variance cloud for lag class 7 reveals that the highest y-axis values are substantially lower than before (0.35 vs. 6.12) and, more importantly, all of the major outlier pairs are gone. This was accomplished by removing a single data record from the analysis:

$images\variance_cloud_with_outlier_removed.gif$

Lag Class

The variogram lag class for which the variance cloud is created. The variance for every pair of points in the lag class is plotted against the distance interval separating that pair.

Other Actions

The right-click menu and other commands work here as they do for the Variograms Window.