Figure 13.5 (continued) Spatial autocorrelation analysis of artificial spatial structures shown on the left: (d) gradient; (e) step. (h) All-directional correlogram of random values. (f) Sampling grid used on each of the artificial spatial structures to obtain 225 "observed values" for spatial autocorrelation analysis. (g) Histogram showing the number of pairs in each distance class. Distances, from 1 to 19.8 in units of the sampling grid, were grouped into 16 distance classes. Spatial autocorrelation statistics (I or c) are not shown for distance classes 15 and 16; see text.

correlograms were computed. According to Oden's test, the correlograms are globally significant at the a = 5% level since several individual values are significant at the Bonferroni-corrected level a' = 0.05/16 = 0.00312. In each correlogram, the progressive Bonferroni correction method was applied to identify significant spatial autocorrelation coefficients: the coefficient for distance class 1 was tested at the a = 0.05 level; the coefficient for distance class 2 was tested at the a' = 0.05/2 level; and, more generally, the coefficient for distance class k was tested at the a' = 0.05/k level. Spatial autocorrelation coefficients are not reported for distance classes 15 and 16 (60 and 10 pairs, respectively) because they only include the pairs of points bordering the surface, to the exclusion of all other pairs.

There is a correspondence between individual significant spatial autocorrelation coefficients and the main elements of the spatial structure. The correspondence can clearly be seen in this example, where the data generating process is known. This is not the case when analysing field data, in which case the existence and nature of the spatial structures must be confirmed by mapping the data. The presence of several equispaced patches produces an alternation of significant positive and negative values along the correlograms. The first spatial autocorrelation coefficient, which is above 0 in Moran's correlogram and below 1 in Geary's, indicates positive spatial autocorrelation in the first distance class; the first class contains the 420 pairs of points that are at distance 1 of each other on the grid (i.e. the first neighbours in the N-S or E-W directions of the map). Positive and significant spatial autocorrelation in the first distance class confirms that the distance between first neighbours is smaller than the patch size; if the distance between first neighbours in this example was larger than the patch size, first neighbours would be dissimilar in values and autocorrelation would be negative for the first distance class. The next peaking positive autocorrelation value (which is smaller than 1 in Geary's correlogram) occurs at distance class 5, which includes distances from 4.95 to 6.19 in grid units; this corresponds to positive autocorrelation between points located at similar positions on neighbouring bumps, or neighbouring troughs; distances between successive peaks are 5 grid units in the E-W or N-S directions. The next peaking positive autocorrelation value occurs at distance class 9 (distances from 9.90 to 11.14 in grid units); it includes value 10, which is the distance between second-neighbour bumps in the N-S and E-W directions. Peaking negative autocorrelation values (which are larger than 1 in Geary's correlogram) are interpreted in a similar way. The first such value occurs at distance class 3 (distances from 2.48 to 3.71 in grid units); it includes value 2.5, which is the distance between peaks and troughs in the N-S and EW directions on the map. If the bumps were unevenly spaced, the correlograms would be similar for the small distance classes, but there would be no other significant values afterwards.

The main problem with all-directional correlograms is that the diagonal comparisons are included in the same calculations as the N-S and E-W comparisons. As distances become larger, diagonal comparisons between, say, points located near the top of the nine bumps tend to fall in different distance classes than comparable N-S or E-W comparisons. This blurs the signal and makes the spatial autocorrelation coefficients for larger distance classes less significant and interpretable.

• Wave (Fig. 13.5b) — Each crest was generated as a normal curve. Crests were separated by five grid units; the surface was constructed in this way to make it comparable to Fig. 13.5a. The correlograms are nearly indistinguishable from those of the nine bumps. All-directional correlograms alone cannot tell apart regular bumps from regular waves; directional correlograms or maps are required.

• Single bump (Fig. 13.5c) — One of the normal curves of Fig. 13.5a was plotted alone at the centre of the study area. Significant negative autocorrelation, which reaches distance classes 6 or 7, delimits the extent of the "range of influence" of this single bump, which covers half the study area. It is not limited here by the rise of adjacent bumps, as this was the case in (a).

• Linear gradient (Fig. 13.5d) — The correlogram is monotonic decreasing. Nearly all autocorrelation values in the correlograms are significant.

True, false There are actually two kinds of gradients (Legendre, 1993). "True gradients", on the one gradient hand, are deterministic structures. They correspond to generating model 2 of Subsection 1.1.1

(eq. 1.2) and can be modelled using trend-surface analysis (Subsection 13.2.1). The observed values have independent error terms, i.e. error terms which are not autocorrelated. "False gradients", on the other hand, are structures that may look like gradients, but actually correspond to autocorrelation generated by some spatial process (model 1 of Subsection 1.1.1; eq. 1.1). When the sampling area is small relative to the range of influence of the generating process, the data generated by such a process may look like a gradient.

In the case of "true gradients", spatial autocorrelation coefficients should not be tested for significance because the condition of second-order stationarity is not satisfied (definition in previous Subsection); the expected value of the mean is not the same over the whole study area. In the case of "false gradients", however, tests of significance are warranted. For descriptive purposes, correlograms may still be computed for "true gradients" (without tests of significance) because the intrinsic assumption is satisfied. One may also choose to extract a "true gradient" using trend-surface analysis, compute residuals, and look for spatial autocorrelation among the residuals. This is equivalent to trend extraction prior to time series analysis (Section 12.2).

How does one know whether a gradient is "true" or "false"? This is a moot point. When the process generating the observed structure is known, one may decide whether it is likely to have generated spatial autocorrelation in the observed data, or not. Otherwise, one may empirically look at the target population of the study. In the case of a spatial study, this is the population of potential sites in the larger area into which the study area is embedded, the study area representing the statistical population about which inference can be made. Even from sparse or indirect data, a researcher may form an opinion as to whether the observed gradient is deterministic ("true gradient") or is part of a landscape displaying autocorrelation at broader spatial scale ("false gradient").

• Step (Fig. 13.5e) — A step between two flat surfaces is enough to produce a correlogram which is indistinguishable, for all practical purposes, from that of a gradient. Correlograms alone cannot tell apart regular gradients from steps; maps are required. As in the case of gradients, there are "true steps" (deterministic) and "false steps" (resulting from an autocorrelated process), although the latter is rare. The presence of a sharp discontinuity in a surface generally indicates that the two parts should be subjected to separate analyses. The methods of boundary detection and constrained clustering (Section 13.3) may help detect such discontinuities and delimit homogeneous areas prior to spatial autocorrelation analysis.

• Random values (Fig. 13.5h) — Random numbers, drawn from a standard normal distribution, were generated for each point of the grid and used as the variable to be analysed. Random data are said to represent a "pure nugget effect" in geostatistics. The autocorrelation coefficients were small and non-significant at the 5% level. Only the Geary correlogram is presented.

Sokal (1979) and Cliff & Ord (1981) describe, in general terms, where to expect significant values in correlograms, for some spatial structures such as gradients and large or small patches. Their summary tables are in agreement with the test examples above. The absence of significant coefficients in a correlogram must be interpreted with caution, however:

• It may indicate that the surface under study is free of spatial autocorrelation at the study scale. Beware: this conclusion is subject to type II (or P) error. Type II error depends on the power of the test which is a function of (1) the a significance level,

(2) the size of effect (i.e. the minimum amount of autocorrelation) one wants to detect,

(3) the number of observations (n), and (4) the variance of the sample (Cohen, 1988):

Is the test powerful enough to warrant such a conclusion? Are there enough observations to reach significance? The easiest way to increase the power of a test, for a given variable and fixed a, is to increase n.

• It may indicate that the sampling design is inadequate to detect the spatial autocorrelation that may exist in the system. Are the grain size, extent and sampling interval (Section 13.0) adequate to detect the type of autocorrelation one can hypothesize from knowledge about the biological or ecological process under study?

Ecologists can often formulate hypotheses about the mechanism or process that may have generated a spatial phenomenon and deduct the shape that the resulting surface should have. When the model specifies a value for each geographic position (e.g. a spatial gradient), data and model can be compared by correlation analysis. In other instances, the biological or ecological model only specifies process generating the spatial autocorrelation, not the exact geographic position of each resulting value. Correlograms may be used to support or reject the biological or ecological hypothesis. As in the examples of Fig. 13.5, one can construct an artificial model-surface corresponding to the hypothesis, compute a correlogram of that surface, and compare the correlograms of the real and model data. For instance, Sokal et al. (1997a) generated data corresponding to several gene dispersion mechanisms in populations and showed the kind of spatial correlogram that may be expected from each model. Another application concerning phylogenetic patterns of human evolution in Eurasia and Africa (space-time model) is found in Sokal et al. (1997b).

Bj0rnstad & Falck (1997) and Bj0rnstad et al. (1998) proposed a spline correlogram which provides a continuous and model-free function for the spatial covariance. The spline correlogram may be seen as a modification of the nonparametric covariance function of Hall and co-workers (Hall & Patil, 1994; Hall et al., 1994). A bootstrap algorithm estimates the confidence envelope of the entire correlogram or derived statistics. This method allows the statistical testing of the similarity between correlograms of real and simulated (i.e. model) data.

During a study of the factors potentially responsible for the choice of settling sites of Balanus crenatus larvae (Cirripedia) in the St. Lawrence Estuary (Hudon et al., 1983), plates of artificial substrate (plastic laminate) were subjected to colonization in the infralittoral zone. Plates were positioned vertically, parallel to one another. A picture was taken of one of the plates after a 3-month immersion at a depth of 5 m below low tide, during the summer 1978. The picture was divided into a (10 x 15) grid, for a total of 150 pixels of 1.7 x 1.7 cm. Barnacles were counted by C. Hudon and P. Legendre for the present Ecological application (Fig. 13.6a; unpublished in op. cit.). The hypothesis to be tested is that barnacles have a patchy distribution. Barnacles are gregarious animals; their larvae are chemically attracted to settling sites by arthropodine secreted by settled adults (Gabbott & Larman, 1971).

A gradient in larval concentration was expected in the top-to-bottom direction of the plate because of the known negative phototropism of barnacle larvae at the time of settlement (Visscher, 1928). Some kind of border effect was also expected because access to the centre of the plates located in the middle of the pack was more limited than to the fringe. These large-scale effects create violations to the condition of second-order stationarity. A trend-surface equation (Subsection 13.2.1) was computed to account for it, using only the Y coordinate (top-

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