MARSS_vectorized: Vectorized Multivariate AR-1 State-space Model

In MARSS: Multivariate Autoregressive State-Space Modeling

Vectorized Multivariate AR-1 State-space Model

Description

The EM algorithm (MARSSkem) in the MARSS package works by converting the more familiar MARSS model in matrix form into the vectorized form which allows general linear constraints (Holmes 2012). The vectorized form is:

\mathbf{x}(t) = (\mathbf{x}(t-1)^\top \otimes \mathbf{I}_m)(\mathbf{f}_b(t)+\mathbf{D}_b(t)\beta) + (\mathbf{f}_u(t)+\mathbf{D}_u(t)\upsilon) + \mathbf{w}(t), \textrm{ where } \mathbf{W}(t) \sim \textrm{MVN}(0,\textbf{Q}(t))

\mathbf{y}(t) = (\mathbf{x}(t)^\top \otimes \mathbf{I}_n)(\mathbf{f}_z(t)+\mathbf{D}_z(t)\zeta) + (\mathbf{f}_a(t)+\mathbf{D}_a(t)\alpha) + \mathbf{v}(t), \textrm{ where } \mathbf{V}(t) \sim \textrm{MVN}(0,\textbf{R}(t))

\mathbf{x}(1) \sim \textrm{MVN}(x0, V0) \textrm{ or } \mathbf{x}(0) \sim \textrm{MVN}(x0, V0)

where \beta, \upsilon, \zeta, and \alpha are column vectors of estimated values, the \mathbf{f} are column vectors of inputs (fixed values), and the \mathbf{D} are perturbation matrices that align the estimated values into the right rows. The \mathbf{f} and \mathbf{D} are potentially time-varying. \otimes means kronecker product and \mathbf{I}_p is a p x p identity matrix.

Normally the user will specify their model in "marxss" form, perhaps with text short-cuts. The "marxss" form is then converted to "marss" form using the conversion function marxss_to_marss(). In "marss" form, the D, d, C, and c information is put in A and U respectively. If there are inputs (d and c), then this will make A and U time-varying. This is unfortunate, because this slows down the EM algorithm considerably due to the unfortunate decision (early on) to store time-varying parameters as 3-dimensional. The functions for the "marss" form (in the file MARSS_marss.R) convert the "marss" form model into vectorized form and prepares the f (fixed) and D (free) matrices that are at the heart of the model specification.

Note, "marss" is a model form. A model form is defined by a collection of form functions discussed in marssMODEL. These functions are not exported to the user, but are called by MARSS() using the argument form. These internal functions convert the users model list into the vectorized form of a MARSS model and do extensive error-checking. "marxss" is also a model form and these models are also stored in vectorized form (See examples below).

Details

See Holmes (2012) for a discussion of MARSS models in vectorized form.

Author(s)

Eli Holmes, NOAA, Seattle, USA.

References

Holmes, E. E. (2012). Derivation of the EM algorithm for constrained and unconstrained multivariate autoregressive state-space (MARSS) models. Technical Report. arXiv:1302.3919 [stat.ME]

See Also

marssMODEL, MARSS.marss(), MARSS.marxss()

Examples

dat <- t(harborSealWA)
dat <- dat[2:4, ] 
MLEobj <- MARSS(dat)

# free (D) and fixed (f) matrices
names(MLEobj$model$free)
names(MLEobj$model$fixed)
# In marss form, the D, C, d, and c matrices are found in A and U
# If there are inputs, this makes U time-varying
names(MLEobj$marss$free)
names(MLEobj$marss$fixed)

# par is in marss form so does not have values for D, C, d, or c
names(MLEobj$par)
# if you need the par in marxss form, you can use print
tmp <- print(MLEobj, what="par", form="marxss", silent=TRUE)
names(tmp)

MARSS documentation built on May 31, 2023, 9:28 p.m.