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### Global Spatial Statistics

Global Spatial Statistics

Global spatial statistics look for an overall pattern between proximity and the similarity of pixel values. These statistics provide a single value that describes the spatial autocorrelation of the dataset as a whole. ENVI offers three global spatial statistics, consisting of the spatial autocorrelation statistics known as Moral’s I, Geary’s C, and semivariance.

• The Moran’s I index compares the differences between neighboring pixels and the mean to provide a measure of local homogeneity. The value range is between +1 and -1, where +1 = strong positive spatial autocorrelation, 0 = spatially uncorrelated data, and -1 = strong negative spatial autocorrelation.
• The Geary’s C index compares the differences between neighboring pixels to the standard deviation to provide a measure of dissimilarity within a dataset. The value range is between 0 and 2, where 0 = strong positive spatial autocorrelation, 1 = spatially uncorrelated data, and 2 = strong negative spatial autocorrelation.
• The semivariance statistic uses the squared difference between neighboring pixel values to provide another measure of dissimilarity within a dataset. It differs from the unitless Moran’s I and Geary’s C indices in that it is in the same units of the input dataset, and the value range is only constrained to be greater than or equal to 0.

While autocorrelation statistics provide an indication of the local homogeneity of a dataset, it is sometimes interesting to understand how that autocorrelation decreases as distance increases. You can examine this using the correlogram or semivariogram. These plots consist of an autocorrelation statistics calculated at various lag distances displayed against the lag. When using the Moran’s I or Geary’s C statistics, this plot is called a correlogram; when using the semivariance statistics, this plot is called a semivariogram.

1. From the Toolbox, select Statistics > Compute Global Spatial Statistics. The Global Spatial Statistics Input File dialog appears.
2. Select an input file and perform optional spatial and spectral subsetting, and/or masking, then click OK. The Global Spatial Statistics Parameters dialog appears.
3. In the Global Spatial Statistics Parameters dialog, set the following choices:
4. From the Neighborhood Rule drop-down list, select which adjacency rule to use in the calculation. This rule defines which adjacent pixels to compare to the central pixel. The choices are:
• Rook’s Case (default): Selects the pixels on the top, bottom, left, and right.
• Bishop’s Case: Selects four diagonal neighboring pixels.
• Queen’s Case: Selects all eight neighboring pixels.
• Horizontal: Selects two neighboring pixels in the same row.
• Vertical: Selects two neighboring pixels in the same column.
• Positive Slope: Selects two neighboring pixels in opposite corners in a positive diagonal.
• Negative Slope: Selects two neighboring pixels in opposite corners in a negative diagonal.
5. Select the Output Semivariogram and Correlogram check box to calculate a semivariogram and correlogram. The correlogram plots the autocorrelation value at multiple lag distances. The semivariogram plots the semivariance at increasing distances. Selecting this check box causes the Select Maximum Lag (Pixels) option to display.
6. Use Select Maximum Lag (Pixels) with the Output Semivariogram and Correlogram option to specify the maximum lag distance, in pixels, to use in the correlogram and semivariogram calculation. The autocorrelation statistics are calculated at each lag distance, up to the specified maximum lag. For example, a value of 5 means that autocorrelation will be calculated for lags of 5, 4, 3, 2, and for each pixel’s nearest neighbors. This value must be greater than 1, but less than the lesser value of the number of rows and columns in the source dataset. The default is 2 pixels.
7. Select the Output to the Screen check box to send the output to the Global Spatial Statistics Results dialog. The dialog displays automatically after the statistics calculation is complete. Using either this option, the Output to a Text Report File option, or both is required. This is the default selection.
8. Select the Output to a Text Report File check box to save the output to a text report file. The format of the text report file is identical to the format of the text report in the Global Spatial Statistics Results dialog. If selected, enter a filename in the Enter Output Report Filename field. Using this option, the Output to the Screen option, or both is required.
9. Click OK. If you selected Output to Screen, the Global Spatial Statistics Results plot window displays. If you selected Output to a Text Report, ENVI saves the report to a text file.

## View Global Spatial Statistics Data

In the Global Spatial Statistics Parameters plot window:

• If you selected Output Semivariogram and Correlogram in the Global Spatial Statistics Parameters dialog, the Global Spatial Statistics Results plot window shows graphical output in a plot, correlogram data in tabular format, and the statistics report in text format (see figure below).
• If you selected Output Semivariogram and Correlogram in the Global Spatial Statistics Parameters dialog, the Global Spatial Statistics Results plot window shows the statistics report in text format and band data in tabular format.

When you have the plot view, select from the Moran’s I, Geary’s C, or Semivariogram radio buttons in the upper portion of the Global Spatial Statistics Results dialog to change the view. The Moran’s I and Geary’s C plots show the autocorrelation for each lag; the semivariogram view plots the semivariance for each lag. To view the plot key, right-click in the plot and select Plot Key.

In the lower portion of the Global Spatial Statistics Results plot window, the View Correlogram tab is the default display when the input file has multiple bands. When the input file has a single band, the Text Report tab is the default display. You can switch between the two views by selecting the appropriate tab. When you select the Moran’s I, Geary’s C, or Semivariogram radio buttons, the correlogram data updates to reflect the current plot view.

If you did not generate plot output, the View By Band and Text Report tabs display. When the input file has a single band, the Text Report tab is the default display.

When viewing information in the Text Report tab, you can view statistics for a specific band in the report. To do this, click Select Stat, then select the band to view.

When the input file has multiple bands, the View By Band tab is the default display. The View By Band tab shows the Moran’s I and Geary’s C indices, and semivariance statistics by band.

For right-click menu options, see ENVI Classic Plot Functions.

References

Daniel A. Griffith, 1987. Spatial Autocorrelation – A Primer. Association of American Geographers, Washington D.C.

Curran, P.J., 1988. The Semivariogram in Remote Sensing: An Introduction. Remote Sensing of Environment, 24:493-507.

Woodcock, C.E. and A.H. Strahler, 1987. The Factor of Scale in Remote Sensing. Remote Sensing of Environment, 21:311-332.

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