with Gaussian factors, because there are no monotone Gaussian processes. Therefore, a more indirect approach is needed. We will assume the w$_{ij}$ s belong to the family of interpolating Hermite cubic splines W τ 0 for a specified knot vector τ$_{0}$ . We have seen in Section 2 that a w$_{ij}$ ∈ W τ 0 is parameterized by a vector τ$_{ij}$ that can be treated as a random effect. However, due to the restriction a < τ$_{ij}$ 1 < · · · < τ$_{ijr}$ < b we cannot assume τ$_{ij}$ is Normal. So we follow the approach of Brumback and Lindstrom (2004) and use the Jupp (1978) transform θ$_{ij}$ = J ( τ$_{ij}$ ) , defined as θ$_{ijk}$ = log { ( τ$_{ij,k}$ +1 − τ$_{ijk}$ ) / ( τ$_{ijk}$ − τ$_{ij,k}$ − 1 ) } for k = 1 , . . . , r , which is an invertible transformation that maps vectors τ$_{ij}$ with increasing coordinates into unconstrained vectors θ$_{ij}$ . For the unconstrained vector θ$_{ij}$ we can assume a multivariate Normal distribution and an additive ANOVA model: with η$_{i}$ ∼ N ( 0 , Σ ) and ξ$_{ij}$ ∼ N ( 0 , Ω ) independent of each other and among themselves. We will also assume the θ$_{ij}$ s are independent of the amplitude factors α$_{i}$ ( t ) and β$_{ij}$ ( t ) , although a model with correlations between amplitude and warping factors can be set up (see below). We take θ$_{0}$ = J ( τ$_{0}$ ) ; the covariance matrices Σ and Ω will be estimated from the data. In analogy with (9) we define which is the proportion of the warping variability explained by the main factor. Putting together the models for z$_{ij}$ ( t ) , w$_{ij}$ ( t ) and the observational model (5), we can derive the likelihood function for the observed data vectors y$_{ij}$ = ( y$_{ij}$ 1 , . . . , y$_{ijν}$$_{ij}$ ) . Given a realization of the random effect θ$_{ij}$ , which is determined by realizations of η$_{i}$ and ξ$_{ij}$ , the corresponding warped time grids are t ∗ $_{ijk}$( θ$_{ij}$ ) = w − 1 ij ( t$_{ijk}$ ) , k = 1 , . . . , ν$_{ij}$ , and the corresponding warped B-spline matrices B ∗ $_{ij}$( θ$_{ij}$ ) ∈ R ν$_{ij}$ × s are given by [ B ∗ $_{ij}$( θ$_{ij}$ )]$_{kl}$ = 9 h$_{w}$ = tr( Σ ) tr( Σ + Ω ) , (11) θ$_{ij}$ = θ$_{0}$ + η$_{i}$ + ξ$_{ij}$ , (10)