1Geography, in part through cartography, and now through visualization in geographic information systems (GISs), maintains a historical connection to art (Rees, 1980; Cartwright et al., 2009). Spatial autocorrelation—attribute values of neighboring geographic locations are far more similar than those of more distant locations—a fundamental concept of geography, can be found everywhere, including in art. This connection may be extended to spatial statistics; a relatively new methodology known as eigenvector spatial filtering (ESF; Griffith 2003) surprisingly links to paintings by artist Susie Rosmarin (http://www.artnet.com/artists/susie-rosmarin/biography).
2ESF exploits the notion of spatial autocorrelation as map pattern by using a set of synthetic proxy variables, which are extracted as eigenvectors from a binary 0-1 matrix, called a spatial weights or connectivity matrix, say C, that represents how geographic objects are tied together in space. This is the adjacency matrix of a graph representing the dual of a surface partitioned into a set of mutually exclusive and collectively exhaustive polygons. Such a surface commonly constitutes the shape file in a GIS.
3An ESF is a linear combination of eigenvectors that visualizes the spatial autocorrelation latent in a georeferenced variable (i.e., a variable that is tagged to the polygon locations constituting a shape file), replicating the map pattern resulting from this spatial autocorrelation. In a regression specification, an ESF identifies and isolate the stochastic spatial dependencies among a set of georeferenced observations, filtering this spatial autocorrelation out of regression residuals and transferring it to the corresponding regression intercept term (making it a varying rather than constant intercept).
4Various indices of spatial autocorrelation exist. The Moran Coefficient (MC) is the most popular one, in part because it tends to have better statistical properties. This index can be written in matrix form as
MC = (n/1T C1)YT(I – 11T/n)C(I – 11T/n)Y/ YT(I – 11T/n)Y ,
where n is the number of polygons in a shape file, Y is an n-by-1 vector of attribute values, I is an n-by-n identity matrix, 1 is an n-by-1 vector of ones, and superscript T denotes matrix transpose. The eigenfunctions (i.e., the paired eigenvalues and eigenvectors) are extracted from the following matrix appearing in the numerator:
(I – 11T/n)C(I – 11T/n) . (1)
5When multiplied by (n/1T C1), an eigenvalue is converted to the MC measuring the spatial autocorrelation in its associated eigenvector (Tiefelsdorf and Boots, 1995; Griffith, 1996). The sign of an eigenvalue indicates the nature of spatial autocorrelation represented by its corresponding eigenvector; the magnitude indicates the degree of correlation.
6Extracting the eigenfunctions from expression (1) constitutes a spectral decomposition of that matrix. Accordingly, the extracted eigenvectors representing other than negligible spatial autocorrelation can be viewed as portraying global, regional, or local components of spatial autocorrelation. These components take on particular patterns for a regular square tessellation forming a complete square region. Here the eigenvectors of a P-by-P, P > 1, set of square pixels are known analytically to be either exact (if either h or k is even) or asymptotically converge on (Griffith, 2000)
where i = 1, 2, …, P and j = 1, 2, …, P represent the (i, j) pixel locations on a map, and h = 1, 2, …, P and k = 1, 2, …, P represent the (h, k) eigenvectors. The corresponding MCs are known analytically to be either exact or asymptotically converge on
7The basic modification here is that the eigenfunction for h = k = 1 is replace by an eigenvalue of 0 and an eigenvector proportional to the vector 1.
8The spectral decomposition results in two eigenfunctions portraying maximum spatial autocorrelation, one with a global east-west trend (Figure 1a), and the other with a global north-south trend (Figure 1b); for P = 30, their MC is 1.02124. This is a global pattern because it is a trend surface across the image, but one that is distorted into elliptical patterns because of edge effects. Regional patterns contain large clusters of relatively similar values (Figure 1c); for P = 30, 900-by-1 eigenvector E4,4 has a MC of 0.95065. Local patterns contain small clusters of relatively similar values (Figure 1d); for P = 30, 900-by-1 eigenvector E12,12 has a MC of 0.35181. The map patterns are less conspicuous for cases where h and k are not equal.
9The visualization of Ek,k is interesting in that the clusters reflect subdivisions of the regular square tessellation into h-by-k rectangles forming global, regional, or local clusters. Figures 1a and 1b respectively are eigenvectors E1,2 and E2,1. Both exhibit a division of the 30-by-30 square tessellation into 2 regions along one axis, and 1 region along the other axis. Figure 1c depicts a division of the landscape into 4 regions along each axis; each is roughly a 7-by-7 square. Figure 1d reveals a division of the landscape into 12 regions along each axis; many 2-by-2 squares of similar values are conspicuous.
Figure 1. Spatial autocorrelation map patterns. Right: (a) global east-west trend (maximum positive). Middle right: (b) global north-south trend (maximum positive). Middle left: (c) regional clusters. Left: (d) local clusters.
10Figure 2 demonstrates how spatial interpolation (e.g., kriging) smooths the type of choropleth map patterns appearing in Figure 1.
Figure 2. A 100-by-100 square tessellation. From upper left to lower right. (a) E2,2. (b) E3,3. (c) E4,4. (d) E5,5. (e) E10,10. (f) E20,20.
11Figures 1 and 2 furnish ideal map patterns for surface partitioning such as those for remotely sensed satellite images. Many surface partitionings are not for regular square tessellations forming complete square or rectangular regions. Figure 3 furnishes examples for a range of polygon numbers: Puerto Rico counties, n = 73; Texas counties, n = 254; and, China counties, n = 2,379. A global, a regional, and a local component is conspicuous for each surface partitioning. For all three surface partitioning, the global map pattern has a MC slightly greater than 1, the regional map pattern has a MC of roughly 0.8, and the local map pattern has a MC of roughly 0.3.
12In November of 2010, the San Antonio Museum of Art publicized the exhibition entitled Psychedelic: Optical and Visionary Art since the 1960s1 that it organized for a national tour. This exhibition included paintings by the artist Susie Rosmarin. Her canvases are complete square regions. Table 1 portrays examples of her paintings. A comparison of these selected paintings with the graphics in Figure 1 reveals a striking resemblance between her paintings and the eigenvectors of a regular square tessellation forming a complete square region.
Figure 3. Global (left), regional (middle), and local (right) specimen spatial autocorrelation components for real world surface partitioning. Top: Puerto Rico. Middle: Texas. Bottom: China.
13The red, green, and blue spectral bands upon which .jpg images are built can be used to uncover the eigenvectors related to Rosmarin’s paintings. Figure 4 illustrates this procedure for the painting in the second column of Table 1, which comprises a 240-by-240 regular square tessellation. Its spectral band numbers range from 0 to 255 for red and green, and 68 to 255 for blue. An ESF for this painting contains two prominent eigenvectors, E2,2, which captures the coarse resolution pattern of this painting, and E209,209, which captures the fine resolution pattern of this painting. The resulting ESF closely mirrors the blue spectral band, which dominates the painting. This ESF accounts for 9% of the variation in the red spectral values across the canvas, whereas it accounts for 18% in the green and 12% in the blue spectral values.
Figure 4. From left to right: respectively, the (a) red, (b) green, and (c) blue spectral bands underlying the .jpg image of the painting; (d) E2,2, (e) E209,209, and (f) ESF.
14Table 1 tabulates results for a range of Rosmarin paintings that qualitatively differ together with only their most prominent eigenvectors. In each of these cases, the few eigenvectors account for considerable variability across the red and green spectral bands, and across the blue band for five of the paintings.
Table 1. Percentage of variance accounted for by selected eigenvectors
15This paper summarizes surprising, and until now unknown, connections between spatial statistics, through GIS, and art, adding to the literature about geography and art. The history of cartography and art already documents selected linkages between these two disciplines. The latent eigenvectors in Susie Rosmarin’s paintings are a surprise, especially given the newness of ESF methodology, emphasizing that spatial autocorrelation can be found everywhere. GIS visualization of selected eigenvectors helps document this contention.