inits is a list specifying the starting
values for parameters in the model. By default, all of these values are
specified according to the literature, but RSTr allows the capability of
specifying your own initial values. If you wish to provide initial
values, note that you don’t have to specify inits for
all parameters if you only want to specify some of them - any
undefined inits will be defined by the default values. For
example, you can specify only the initial values for lambda
and all other values will be generated on their own. However, if one
value is specified for a certain parameter in inits, all
values must be specified for that parameter in inits: you
cannot, for example, define initial values for just one year of
lambda. Finally, any values included in your
inits list that aren’t aligned with the above names will be
ignored.
The models in RSTr share many inits, but a couple of
models have inits that are unique to them. All potential
initial values are presented here.
Here are the possible initial value parameters for the MSTCAR model:
lambda: The estimated spatially smoothed rate for
each region-group-time. lambda is an array of
real numbers with dimensions n_region x n_group x n_time.
Has support (0, 1) for method = "binomial" and
support (0, Inf) for `method = “poisson”;
beta: The mean rate for each island-group-year on a
logit- or log-transformed scale. Islands are sets of regions that
exclusively share adjacency information. For example, in
miadj, there are two islands that represent the counties of
the Upper Peninsula and the Lower Peninsula. These islands don’t touch
each other, and thus don’t share adjacency information. Each island is
assigned its own beta. beta is an
array of real numbers with dimensions
n_island x n_group x n_time;
Z: The spatiotemporal random effects. These are the
parameters that induce smoothing on the counties, with the intensity of
the smoothing dictated by the spatial covariance matrices
G. Z is an array of real numbers
with dimensions n_region x n_group x n_time;
G: The spatial covariance matrices. This parameter
determines the intensity of the spatial smoothing performed by
Z and represents the strength of the relationship between
each group in a given time period. G is an
array of temporally-evolving positive-definite symmetric
matrices with dimensions
n_group x n_group x n_time;
rho: The temporal correlation. This parameter
decides the strength of the relationship between values in time period
t to values in time period t-1. It is a
matrix of size n_group x 1 of real numbers
with support [0,1];
tau2: The non-spatial variance. This parameter picks
up any variance in values of lambda for each group. It is a
matrix of size n_group x 1 of positive real
numbers; and
Ag: The general spatial covariance matrix. This
parameter describes the overall relationship between groups across the
entire model and is used in the prior distribution for the matrices in
G. Ag is a positive-definite symmetric matrix
with dimensions n_group x n_group.
The MCAR model utilizes a majority of the initial values of the
MSTCAR model. However, MCAR does not include inits for
rho or Ag. Note that specification for the
MCAR model is slightly different than that of the MSTCAR model. If an
MCAR model is run with data containing several time periods,
tau2 will require values for every time period along with
every group.
The CAR models have the smallest set of initial values, using only
lambda, beta, Z, and
tau2 from the MCAR model. Similar to the MCAR, if a CAR
model is run with multiple groups and time periods, tau2
requires values for every group and time period present. The only new
initial value for the CAR models is sig2, which takes the
place of G in the MCAR and MSTCAR models:
sig2 represents the spatial variance of a CAR/RCAR
model. This parameter picks up any variance in values of Z
for each group. It is a matrix of size
n_group x n_time of positive real numbers.