Create a covariance parameter initial object that specifies
initial and/or known values to use while estimating specific covariance parameters
with ssn_lm() or ssn_glm(). See spmodel::randcov_initial() for documentation regarding
random effect covariance parameter initial objects.
tailup_initial(tailup_type, de, range, known)taildown_initial(taildown_type, de, range, known)
euclid_initial(euclid_type, de, range, rotate, scale, known)
nugget_initial(nugget_type, nugget, known)
A list with two elements: initial and is_known.
initial is a named numeric vector indicating the spatial covariance parameters
with specified initial and/or known values. is_known is a named
numeric vector indicating whether the spatial covariance parameters in
initial are known or not. The class of the list
matches the the relevant spatial covariance type.
The tailup covariance function type. Available options
include "linear", "spherical", "exponential",
"mariah", "epa", and "none".
The spatially dependent (correlated) random error variance. Commonly referred to as a partial sill.
The correlation parameter.
A character vector indicating which covariance parameters are to be
assumed known. The value "given" is shorthand for assuming all
covariance parameters given to *_initial() are assumed known.
The taildown covariance function type. Available options
include "linear", "spherical", "exponential",
"mariah", "epa", and "none".
The euclidean covariance function type. Available options
include "spherical", "exponential", "gaussian",
"cosine", "cubic", "pentaspherical", "wave",
"jbessel", "gravity", "rquad", "magnetic", and
"none".
Anisotropy rotation parameter (from 0 to \(\pi\) radians) for the euclidean portion of the covariance. A value of 0 (the default) implies no rotation.
Anisotropy scale parameter (from 0 to 1) for the euclidean portion of the covariance. A value of 1 (the default) implies no scaling.
The nugget covariance function type. Available options
include "nugget" or "none".
The spatially independent (not correlated) random error variance. Commonly referred to as a nugget.
Create an initial object for use with ssn_lm() or ssn_glm().
NA values can be given for ie, rotate, and scale, which lets
these functions find initial values for parameters that are sometimes
otherwise assumed known (e.g., rotate and scale with ssn_lm() and ssn_glm().
Parametric forms for each spatial covariance type are presented below.
tailup_type Details: Let \(D\) be a matrix of hydrologic distances,
\(W\) be a diagonal matrix of weights from additive, \(r = D / range\),
and \(I\) be
an identity matrix. Then parametric forms for flow-connected
elements of \(R(zu)\) are given below:
linear: \((1 - r) * (r <= 1) * W\)
spherical: \((1 - 1.5r + 0.5r^3) * (r <= 1) * W\)
exponential: \(exp(-r) * W\)
mariah: \(log(90r + 1) / 90r * (D > 0) + 1 * (D = 0) * W\)
epa: \((D - range)^2 * F * (r <= 1) * W / 16range^5\)
none: \(I\) * W
Details describing the F matrix in the epa covariance are given in Garreta et al. (2010).
Flow-unconnected elements of \(R(zu)\) are assumed uncorrelated.
Observations on different networks are also assumed uncorrelated.
taildown_type Details: Let \(D\) be a matrix of hydrologic distances,
\(r = D / range\),
and \(I\) be an identity matrix. Then parametric forms for flow-connected
elements of \(R(zd)\) are given below:
linear: \((1 - r) * (r <= 1)\)
spherical: \((1 - 1.5r + 0.5r^3) * (r <= 1)\)
exponential: \(exp(-r)\)
mariah: \(log(90r + 1) / 90r * (D > 0) + 1 * (D = 0)\)
epa: \((D - range)^2 * F1 * (r <= 1) / 16range^5\)
none: \(I\)
Now let \(A\) be a matrix that contains the shorter of the two distances between two sites and the common downstream junction, \(r1 = A / range\), \(B\) be a matrix that contains the longer of the two distances between two sites and the common downstream junction, \(r2 = B / range\), and \(I\) be an identity matrix. Then parametric forms for flow-unconnected elements of \(R(zd)\) are given below:
linear: \((1 - r2) * (r2 <= 1)\)
spherical: \((1 - 1.5r1 + 0.5r2) * (1 - r2)^2 * (r2 <= 1)\)
exponential: \(exp(-(r1 + r2))\)
mariah: \((log(90r1 + 1) - log(90r2 + 1)) / (90r1 - 90r2) * (A =/ B) + (1 / (90r1 + 1)) * (A = B)\)
epa: \((B - range)^2 * F2 * (r2 <= 1) / 16range^5\)
none: \(I\)
Details describing the F1 and F2 matrices in the epa
covariance are given in Garreta et al. (2010).
Observations on different networks are assumed uncorrelated.
euclid_type Details: Let \(D\) be a matrix of Euclidean distances,
\(r = D / range\), and \(I\) be an identity matrix. Then parametric
forms for elements of \(R(ze)\) are given below:
exponential: \(exp(- r )\)
spherical: \((1 - 1.5r + 0.5r^3) * (r <= 1)\)
gaussian: \(exp(- r^2 )\)
cubic: \((1 - 7r^2 + 8.75r^3 - 3.5r^5 + 0.75r^7) * (r <= 1)\)
pentaspherical: \((1 - 1.875r + 1.25r^3 - 0.375r^5) * (r <= 1)\)
cosine: \(cos(r)\)
wave: \(sin(r) * (h > 0) / r + (h = 0)\)
jbessel: \(Bj(h * range)\), Bj is Bessel-J function
gravity: \((1 + r^2)^{-0.5}\)
rquad: \((1 + r^2)^{-1}\)
magnetic: \((1 + r^2)^{-1.5}\)
none: \(I\)
nugget_type Details: Let \(I\) be an identity matrix and \(0\)
be the zero matrix. Then parametric
forms for elements the nugget variance are given below:
nugget: \(I\)
none: \(0\)
In short, the nugget effect is modeled when nugget_type is "nugget"
and omitted when nugget_type is "none".
Dispersion and random effect initial objects are specified via
spmodel::dispersion_initial() and spmodel::randcov_initial(), respectively.
Peterson, E.E. and Ver Hoef, J.M. (2010) A mixed-model moving-average approach to geostatistical modeling in stream networks. Ecology 91(3), 644--651.
Ver Hoef, J.M. and Peterson, E.E. (2010) A moving average approach for spatial statistical models of stream networks (with discussion). Journal of the American Statistical Association 105, 6--18. DOI: 10.1198/jasa.2009.ap08248. Rejoinder pgs. 22--24.
tailup_initial("exponential", de = 1, range = 20, known = "range")
tailup_initial("exponential", de = 1, range = 20, known = "given")
euclid_initial("spherical", de = 2, range = 4, scale = 0.8, known = c("range", "scale"))
dispersion_initial("nbinomial", dispersion = 5)
Run the code above in your browser using DataLab