The mvRWTS function fits a multivariate Random Walk (RW; i.e., the time series counterpart of the Brownian motion process).
The "method" argument allows the user to try different algorithms for computing the log-likelihood. The "rpf" and "sparse" methods use fast GLS algorithms based on factorization for avoiding the computation of the inverse of the variance-covariance matrix and its determinant involved in the log-likelihood estimation. The "inverse" approach uses the "stable" standard explicit computation of the inverse and determinant of the matrix and is therefore slower. The "pseudoinverse" method uses a generalized inverse that is safer for matrix near singularity but highly time consuming. See ?mvLL for more details on these computational methods.
Arguments in the "param" list are:
"constraint" - The "constraint" argument in the "param" list allows the user to compute the joint likelihood for each trait by assuming they evolved independently ( constraint="diagonal", or constraint="equaldiagonal"). If constraint="equal", the sigma values are constrained to be the same for each trait using the constrained Cholesky decomposition proposed by Adams (2013) or a separation strategy based on spherical parameterization when p>2 (Clavel et al. 2015).
User-defined constraints can be specified through a numeric matrix (square and symmetric) with integer values taken as indices of the parameters.
For instance, for three traits:
constraint=matrix(c(1, 3, 3, 3, 2, 3, 3, 3, 2), 3).
Covariances constrained to be zero are introduced by NA values, e.g.,
constraint=matrix(c(1, 4, 4, 4, 2, NA, 4, NA, 3), 3).
Difference between two nested fitted models can be assessed using the "LRT" function. See example below and ?LRT.
"decomp" - For the general case (unconstrained models), the sigma matrix is parameterized by various methods to ensure its positive definiteness (Pinheiro and Bates, 1996). These methods are the "cholesky", "eigen+", and "spherical" parameterizations.
"trend" - Default set to FALSE. If TRUE, the ancestral state is allowed to drift leading to a directional random walk. Note that it is possible to provide a vector of integer indices to constraint the estimated trends when p>1 (see the vignettes).
"sigma" - Starting values for the likelihood estimation. By default the trait covariances are used as starting values for the likelihood optimization. The user can specify starting values as square symmetric matrices or a simple vector of values for the upper factor of the sigma matrix. The parameterization is done using the factorization determined through the "decomp" argument (Pinheiro and Bates, 1996). Thus, you should provide p*(p+1)/2 values, with p the number of traits (e.g., random numbers or the values from the cholesky factor of a symmetric positive definite sigma matrix; see example below). If a constrained model is used, the number of starting values is (p*(p-1)/2)+1.