Table I.

Stationary correlation structures that are continuous functions of distance.

Structure	(j, k)th element*, j ≠ k	Params	Data types†
LEAR	ρ dmin+δ[(djk−dmin)/(dmax−dmin)]	2	L/T,S,O
AR(1)	ρ djk	1	L/T,S,O
DE	${\rho }^{d_{\mathit{jk}}^v}$	2	L/T,S,O
GAR(1)	${\rho }^{d_{\mathit{jk}}}\frac{\Gamma (d_{\mathit{jk}}+\delta )F(\delta ,d_{\mathit{jk}}+\delta ;d_{\mathit{jk}}+1;{\rho }^2)}{\Gamma \left(\delta \right)\Gamma (d_{\mathit{jk}}+1)F(\delta ,\delta ;1;{\rho }^2)}$	2	L/T
Exponential	exp(−djk/ϕ)	1	S
Gaussian	$\exp (-d_{\mathit{jk}}^2∕{\varphi }^2)$	1	S
Linear	(1 − ϕdjk)I(ϕdjk≤1)	1	S
Matern	[1/Γ(ν)](djk/2ϕ)ν2Kv(djk/ϕ)	2	S
Spherical	$[1-(3d_{\mathit{jk}}∕2\varphi )+(d_{\mathit{jk}}^3∕2{\varphi }^3\left)\right]∕(d_{\mathit{jk}}\le \varphi )$	1	S

^*d_jk, distance between jth and kth measurement of ith subject; Γ(·), gamma function; F(θ₁, θ₂; θ₃; θ), hypergeometric function; and K_v(·), modified Bessel function of the second kind of (real) order v > 0.

^†L/T, longitudinal/time series; S, spatial; and O, other.