Overview of grid-based statistics

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Overview of grid-based statistics

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In the earlier discussion of local, focal, zonal and global analysis of grids (Section 4.6.1, Operations on single and multiple grids) we described how basic statistical measures may be obtained from grid datasets. These measures are similar to those applied to univariate attribute data, but may operate at various levels of spatial grouping of a single grid or may be applied across grids. As an example, many of Idrisi’s statistics facilities, which are a rather mixed collection provided under a single menu item, are listed in Table 5‑4.

Table 5‑4 Sample statistical tools for grid data — Idrisi

Facility

Description

Histo

Single grid — create histogram on n classes from cell values. Classes may cover a spatial subset of the grid (e.g. by masking), have excluded values (e.g. 0, ‑999), have user defined boundaries etc.

Pattern

Single grid — generate selected Landscape metrics (e.g. richness, dominance, diversity, fragmentation) — as discussed in Section 5.3.4, Landscape Metrics

Center

Computes the weighted mean center of an input image (grid) where cells are regarded as point counts or weights (see also, Section 4.2.5, Centroids and centers). This function also generates the standard distance radius (see Section 4.5.3, Directional analysis of point datasets) for the point set, and the Coefficient of Relative Dispersion (CRD) which is the ratio of the standard distance measure to the radius of a circle having the same area as the study region, expressed as a percentage

Crosstab

The standard form of Crosstab compares two classified images, typically matching in spatial extent (e.g. time slices of the same region, images of adjacent zones of the same size and resolution), each with up to 127 classes. The principal output is a crosstabulation table, showing each class in image 1 as the rows, each class of image 2 as the columns, and counts of values in the table. Hence if image 1 contains 234 cells classified as A and image 2 contains 370 cells classified as A, then at most 234 cells will appear in row A column A of the output table. For common classification schemes (with square crosstabs) the diagonal of this crosstab matrix shows matching (e.g. unchanged) grid values. Potentially up to MxN combinations of classes are possible (taking AB as different from BA). Class combinations may be mapped providing a spatial representation of the crosstabulation that is a form of spatial overlay (AND) operation. Two measures of association (correlation) between the images are provided: Cramer’s V and the Kappa index (described in more detail in Section 5.3.2, Crosstabulated grid data, the Kappa Index and Cramer’s V statistic). The latter requires that the classification scheme for both images match

Quadrat

Quadrat (grid based) point pattern analysis. In this case each grid cell is regarded as a separate quadrat with cell values corresponding to point counts. Masking may be applied, and may well be desirable. Standard statistical measures are provided, with the variance/mean (V/M) ratio being used as an indicator of spatial clustering. Values of V/M=1 may indicate a random pattern, assuming an underlying independent, homogeneous random process, whilst V/M<1 suggests more uniform patterns under this model. These interpretations are not valid if the process is not stationary. Note that the results are significantly affected by quadrat (grid cell) size and overall sampling area. If D=study area/number of points, ideal quadrat sizes are suggested as being in the range [D,2D]

Regress

Multireg

Logisticreg

Provides regression modeling between multiple image files or attribute files, with or without image masking. Interpretation of image (grid) file regression may be difficult, since spatial autocorrelation within images is almost always strong (and may even have been generated by resampling or other interpolation methods), resulting in misleading values for degrees of freedom and associated statistical measures

Within many GIS packages statistical tools that apply to grid datasets are grouped together with other, non-statistical facilities, or embedded within application-specific functionality, such as image classification. For example, in ArcGIS the majority of these tools are located in the Spatial Analyst (Neighborhood analysis and Multivariate analysis subsections); in ENVI statistics facilities are located under the Basic Tools|Statistics menu and the Basic Tools|Spatial Statistics menu, the latter providing global and local spatial autocorrelation measures of the types described in Section 5.5.3, Local indicators of spatial association (LISA). In the following subsections we examine a number of these tools in greater detail.