TABLE 1.

Symbols used for indices, data, fixed effects, random effects, and derived quantities

Symbol	Description	Dimensions
Index
i	Sample	–
s	Spatial location (“knot”)	–
t	Time interval (year)	–
Data
b	Data for observed forest bird abundance (i.e., biomass; g)	ni
a	Area sampled	1
tit	Covariate data for observed titmouse abundance (i.e., biomass; g)	$n_i\times n_t$
PC	Covariate data for environmental principal component (PC1 or PC2)	$n_i\times n_t$
control	Covariate data for observed control group species abundance (i.e., biomass; g)	$n_i\times n_t$
x	Number of locations (“knots”) in the spatial mesh used in spatial interpolation	n
Fixed effects
β	Intercept for expected forest bird density	ni
${\sigma }_m^2$	Variance in expected forest bird abundance	ni
γ 1	Estimated effect of titmouse abundance covariate	1
γ 2	Estimated effect of [titmouse abundance]2 covariate	1
γ 3	Estimated effect of environmental principal component (PC1 or PC2) covariate	1
γ 4	Estimated effect of a control group (from a random draw) abundance covariate	1
ρε	Temporal autoregressive correlation in spatio‐temporal variation of forest bird density	1
Random effects
ω	Spatial variation in expected forest bird density	ns
ε	Spatio‐temporal variation in expected forest bird density	$n_s\times n_t$
${\sigma }_{\omega }^2$	Variance parameter for spatial variation of expected forest bird density	1
${\sigma }_{\epsilon }^2$	Variance parameter for spatio‐temporal variation of expected forest bird density	1
Derived quantities
d	Expected forest bird density (g per km2)	$n_s\times n_t$
R	Matrix of spatial correlations in expected forest bird density	$n_s\times n_s$