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Tool Reference › Home Range › Calculate Statistical Matrices

Calculate Statistical Matrices

Home Range · geoprocessing tool · by Jeff Jenness
Works at every ArcGIS Pro license level
Learn more About Mahalanobis distances covers the theory — including what the mean vector and covariance matrix do inside the Mahalanobis formula.

Summary

Computes the mean vector, covariance matrix, inverse covariance matrix, and/or correlation matrix for a set of landscape variables and writes them as tables. This is the producer tool of the Mahalanobis suite: compute the statistics once from your best reference sample, then supply the mean and covariance tables to Mahalanobis Distance Raster or at Points through their Existing tables source — the statistics aren't recomputed for every run, and every run is guaranteed to use exactly the same reference.

What it computes

The Mahalanobis formula D² = (x − μ)′ Σ⁻¹ (x − μ) needs two ingredients: the mean vector μ and the covariance matrix Σ. This tool derives both, either from sample point features (the landscape variables sampled at the points, optionally only the selected ones) or from the cells of a categorical raster matching a chosen category. Categories can be picked by any field of the raster's attribute table — choose Landfire's broad EVT_LF = “Tree” and every tree-type cell value joins the sample automatically — or by the literal cell value when there is no attribute table (the fuller discussion is on the Raster page). The landscape variables are specified exactly as in the other Mahalanobis tools — rasters (multiband rasters expand to one variable per band), plus optional polygon variables rasterized from a numeric field.

Two further tables are optional extras for inspection rather than reuse: the inverse covariance (computed by your chosen inversion method — the consuming tools invert the covariance themselves, so this table is never required) and the correlation matrix, the natural place to screen for collinear variables before they cause trouble. A pair of strongly correlated variables contributes little independent information and pushes the covariance toward singularity — the tool reports the covariance condition number, and very large values are the warning sign.

Order matters The variables become the columns of the tables in the order listed (rasters first, then polygons), and each output table embeds that ordered variable list in its metadata. When the tables are reused, the consuming tool validates that the same variables arrive in the same order — because a different order silently produces incorrect distances.

Environments

The tool honors the Output Coordinate System, Cell Size, Processing Extent, Snap Raster, and Mask environments when aligning the landscape variables to a common analysis grid. Other geoprocessing environments do not affect it.

A tour of the dialog

Here the statistics are computed from a sample-point layer over three landscape variables — slope, curvature and elevation, in that order — with all four output tables requested:

The Home Range Tools gallery open on the ribbon, with the Calculate Statistical Matrices button, in the Mahalanobis Tools row, outlined in blue
Where to find it: Calculate Statistical Matrices is in the Mahalanobis Tools row of the Home Range Tools gallery, in the Home Range group of the Wildlife and Forestry tab.
The Calculate Statistical Matrices pane: slope, curvature and elevation as raster landscape variables, statistics calculated directly from the Sample_Points_for_Mahal layer with bilinear sampling, and all four output tables named
The dialog: variables and sample points in, four statistics tables out.
The four output tables open in ArcGIS Pro: the mean vector with one row per variable, and the covariance, inverse covariance and correlation matrices, every row labeled by variable name in the analysis order slope, curvature, elevation
The four tables. Notice every row is labeled by variable in the order the variables were listed — slope, curvature, elevation — and that same order is embedded in the tables' metadata for the consuming tools to validate. The correlation matrix is the collinearity screen: here the strongest correlations are curvature's, at about 0.105 with slope and 0.104 with elevation — far too weak to cause trouble, so all three variables are pulling their own weight.

ModelBuilder

All four tables are model outputs. The natural pattern: compute the mean and covariance once at the head of a model, then fan those two tables out to several Mahalanobis Raster or at Points runs — different landscapes, years, or study areas, all scored against the identical reference.

A ModelBuilder diagram: the slope, curvature and elevation rasters and the sample points feeding Calculate Statistical Matrices, which outputs the means, covariance, inverse covariance and correlation tables
The tool in a model: variables and sample points in, the four statistics tables out — the mean and covariance pair ready to feed downstream Mahalanobis runs through their Existing-tables source.

Parameters

LabelExplanationData type
Raster Landscape VariablesRequired · raster_vars The rasters defining the multivariate space; at least one is required and sets the analysis grid. Raster Layer (multiple)
Polygon Landscape VariablesOptional · poly_vars Polygon feature class + numeric field rows, rasterized to the analysis grid. Value Table
Bands to excludeOptional · exclude_bands Multiband raster + 1-based band number rows to drop. Value Table
Source of mean vector and covariance matrixRequired · stats_source Sample point features or categorical raster; choose the source first and the relevant inputs below enable accordingly. (The Existing-tables option is intentionally absent here — this tool produces those tables.) String
Sample pointsOptional · points The reference points; points outside the landscape data or on NoData cells are excluded and reported. Feature Layer
Use only the selected pointsOptional · use_selected Restrict the statistics to the layer's active selection. Boolean
Categorical rasterOptional · cat_raster The raster whose matching cells form the reference sample. Raster Layer
Category attribute fieldOptional · cat_field A field from the raster's attribute table naming the categories the way you want to choose them; leave blank to enter a literal cell value instead. Field
Category valueOptional · cat_value The category whose cells form the reference sample — a dropdown of the chosen field's values, with every matching cell value gathered automatically. With no field, literal cell value(s), semicolon-separated for several (“3;4;5”). String
Cell value samplingOptional · sample_method Exact (nearest cell) or Interpolated (bilinear); bilinear suits continuous surfaces, exact suits categorical rasters. String
Covariance inversion methodRequired · inversion Standard (inverse) or Pseudo-inverse (SVD); used only when the inverse covariance table is requested. String
Output mean vector tableOptional · out_mean The mean vector (one row per variable), auto-named Mahalanobis_Means; clear to skip. Table
Output covariance matrix tableOptional · out_cov The covariance matrix, auto-named Mahalanobis_Covariance; clear to skip. Together with the mean vector, this is what the consuming tools read. Table
Output inverse covariance matrix tableOptional · out_invcov For inspection; never required for reuse. Table
Output correlation matrix tableOptional · out_corr For screening collinearity among the variables. Table

Python

Mean and covariance from three rasters sampled at a point layer, with the two optional inspection tables:

import arcpy
arcpy.ImportToolbox(r"C:\path\to\JennessEnterprisesTools.pyt")  # your install path
out_gdb = r"C:\Project\Mahalanobis.gdb"
arcpy.jenness.MahalanobisCovariance(
    raster_vars="elev_clip;slope_clip;curvature",
    sample_method="Interpolated (bilinear)",
    stats_source="Calculate directly from sample point features",
    points="Sample_Points_for_Mahal",
    inversion="Standard (inverse)",
    out_mean=out_gdb + r"\Mahalanobis_Means",
    out_cov=out_gdb + r"\Mahalanobis_Covariance",
    out_invcov=out_gdb + r"\Mahalanobis_Inv_Covariance",
    out_corr=out_gdb + r"\Mahalanobis_Correlation")

For categorical-raster statistics, pass the attribute field and category value exactly as in the dialog (stats_source="Categorical raster", cat_raster="Landfire", cat_field="EVT_LF", cat_value="Tree") — the tool gathers the matching cell values itself. Or skip the field and give literal cell value(s), semicolon-separated for several (cat_field=None, cat_value="7016;7019;7023"); the full example is on the Raster page.

Recommended citation

Jenness, J. 2026. Calculate Statistical Matrices. Wildlife and Forestry Tools add-in for ArcGIS Pro, v. 1.99 (September 2026). Jenness Enterprises. Available at: https://github.com/JeffJenness/Wildlife_Tools.

Credits and references

By Jeff Jenness, Jenness Enterprises (www.jennessent.com), ported from his ArcView Mahalanobis Distances extension and the ArcMap Land Facet Corridor Tools.

Licensing information

Works at every ArcGIS Pro license level (Basic, Standard, Advanced). No extension licenses are required.