*Ecological Archives* E093-137-A6

Anton J. Flügge, Sofia C. Olhede, and David J. Murrell. 2012. The memory of spatial patterns: changes in local abundance and aggregation in a tropical forest. *Ecology* 93:1540–1549. http://dx.doi.org/10.1890/11-1004.1

Appendix F. Description of the estimation procedure for the weights used in the weighted regression models.

A plot of the residuals of the linear regression model using current abundance and change of abundance between 2000 and 2005 versus current abundance *n*_{2005,i} (see Fig. F1) indicates that the variance of the residuals depends on *n*_{2005,i}. Empirically we model the variance as:

var{ε_{2005,i}} = *a*^{2}(log*n*_{2005,i})^{2b}

This implies that:

log|ε_{2005,i}| = log*a* + *b* log |log*n*_{2005,i}| + *v*_{2005,i}

Where *v*_{2005,i} has constant mean and variance (under some circumstances the mean is zero, if it is not, there is no effects of this, as we shall subsequently use weighted least squares). We fit *a* and *b* using least squares, and then use weighted regression with this fitted model, with a weighting function given by:

*w*_{i} = *â*^{-2}(log*n*_{2005,i})^{-2b}

One outlier observation is ignored in this fit.

Fig. F1. The relationship between local abundance and the residuals of the linear regression model for all 143 species from BCI that meet our selection criteria (top-canopy trees Δ; middle canopy trees O; understory trees ∇; shrubs *). The black line shows the estimated standard deviation *w*_{i}^{0.5} of the error that was used for the weighted regression in the our analysis.