Table 1.

Structured functional models. For nested models, i = 1, 2,..., I; j = 1, 2,..., J_i; k = 1, 2,..., K_ij; i₁ = 1, 2,..., I₁, i₂ = 1, 2,..., I_2i1,..., i_r = 1, 2,..., I_{ri1i2...ir–1}. For crossed designs, i = 1, 2,..., I; j = 1, 2,..., J; k = 1, 2,..., n_ij; (C2s) “Two-way sub” stands for “Two-way crossed design with subsampling”; (CM) contains combinations of any s (s = 1, 2,..., r) subset of the r latent processes, as 'well as repeated measurements within each cell. ${\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_d\in \{i_{k_1}{i}_{k_2}\cdots i_{k_s}u:k_1,k_2,\dots ,k_s\in (1,2,\dots ,r),u\in (\varnothing ,1,2,\dots ,I_{i_1{i}_2\dots i_r}),s\le r\}$, u is the index for repeated observation in cell (i_k1,i_k2,...i_kr). ε_.t, is the white noise distributed as (0, σ²).

Nested	(N1) One-way	Yi(t) = μ(t) + Xi(t) + εit
	(N2) Two-way	Yij(t) = μ(t) + Xi(t) + Uij(t) + εijt
	(N3) Three-way	Yijk(t) = μ(t) + Xi(t) + Uij(t) + Wijk (t) + εijkt
	(NM) Multi-way	$Y_{i_1{i}_2\cdot i_r}\left(t\right)=\mu \left(t\right)+R_{i_1}^{\left(1\right)}\left(t\right)+\cdots +R_{i_1\cdots i_r}^{\left(r\right)}\left(t\right)+{∊}_{i_1{i}_2\cdots i_{r}t}$
Crossed	(C2) Two-way	Yij(t) = μ(t) + Xi(t) + Zj(t) + Wij(t) + εijt
	(C2s) Two-way sub	Yijk(t) = μ(t) + Xi(t) + Zj(t) + Wij(t) + Uijk(t) + εijkt
	(CM) Multi-way	$Y_{i_1{i}_2\cdots i_{r}u}\left(t\right)=\mu \left(t\right)+R_{{\mathcal{S}}_1}\left(t\right)+\cdots +R_{{\mathcal{S}}_d}\left(t\right)+{∊}_{i_1{i}_2\cdots i_{r}ut}$