ssn_glm {SSN2} | R Documentation |
Fitting Generalized Linear Models for Spatial Stream Networks
Description
This function works on spatial stream network objects to fit generalized linear models with spatially autocorrelated errors using likelihood methods, allowing for non-spatial random effects, anisotropy, partition factors, big data methods, and more. The spatial formulation is described in Ver Hoef and Peterson (2010) and Peterson and Ver Hoef (2010).
Usage
ssn_glm(
formula,
ssn.object,
family,
tailup_type = "none",
taildown_type = "none",
euclid_type = "none",
nugget_type = "nugget",
tailup_initial,
taildown_initial,
euclid_initial,
nugget_initial,
dispersion_initial,
additive,
estmethod = "reml",
anisotropy = FALSE,
random,
randcov_initial,
partition_factor,
...
)
Arguments
formula |
A two-sided linear formula describing the fixed effect structure
of the model, with the response to the left of the |
ssn.object |
A spatial stream network object with class |
family |
The generalized linear model family for use with |
tailup_type |
The tailup covariance function type. Available options
include |
taildown_type |
The taildown covariance function type. Available options
include |
euclid_type |
The euclidean covariance function type. Available options
include |
nugget_type |
The nugget covariance function type. Available options
include |
tailup_initial |
An object from |
taildown_initial |
An object from |
euclid_initial |
An object from |
nugget_initial |
An object from |
dispersion_initial |
An object from |
additive |
The name of the variable in |
estmethod |
The estimation method. Available options include
|
anisotropy |
A logical indicating whether (geometric) anisotropy should
be modeled. Not required if |
random |
A one-sided linear formula describing the random effect structure
of the model. Terms are specified to the right of the |
randcov_initial |
An optional object specifying initial and/or
known values for the random effect variances. See |
partition_factor |
A one-sided linear formula with a single term specifying the partition factor. The partition factor assumes observations from different levels of the partition factor are uncorrelated. |
... |
Other arguments to |
Details
The generalized linear model for spatial stream networks can be written as
g(\mu) = \eta = X \beta + zu + zd + ze + n
, where \mu
is the expectation
of the response given the random errors, y
, g()
is a function that links the mean
and \eta
(and is called a link function), X
is the fixed effects design
matrix, \beta
are the fixed effects, zu
is tailup random error,
zd
is taildown random error, and ze
is Euclidean random error,
and n
is nugget random error.
There are six generalized linear model
families available: poisson
assumes y
is a Poisson random variable
nbinomial
assumes y
is a negative binomial random
variable, binomial
assumes y
is a binomial random variable,
beta
assumes y
is a beta random variable,
Gamma
assumes y
is a gamma random
variable, and inverse.gaussian
assumes y
is an inverse Gaussian
random variable.
The supports for y
for each family are given below:
family: support of
y
Gaussian:
-\infty < y < \infty
poisson:
0 \le y
;y
an integernbinomial:
0 \le y
;y
an integerbinomial:
0 \le y
;y
an integerbeta:
0 < y < 1
Gamma:
0 < y
inverse.gaussian:
0 < y
The generalized linear model families and the parameterizations of their link functions are given below:
family: link function
Gaussian:
g(\mu) = \eta
(identity link)poisson:
g(\mu) = log(\eta)
(log link)nbinomial:
g(\mu) = log(\eta)
(log link)binomial:
g(\mu) = log(\eta / (1 - \eta))
(logit link)beta:
g(\mu) = log(\eta / (1 - \eta))
(logit link)Gamma:
g(\mu) = log(\eta)
(log link)inverse.gaussian:
g(\mu) = log(\eta)
(log link)
The variance function of an individual y
(given \mu
)
for each generalized linear model family is given below:
family:
Var(y)
Gaussian:
\sigma^2
poisson:
\mu \phi
nbinomial:
\mu + \mu^2 / \phi
binomial:
n \mu (1 - \mu) \phi
beta:
\mu (1 - \mu) / (1 + \phi)
Gamma:
\mu^2 / \phi
inverse.gaussian:
\mu^2 / \phi
The parameter \phi
is a dispersion parameter that influences Var(y)
.
For the poisson
and binomial
families, \phi
is always
one. Note that this inverse Gaussian parameterization is different than a
standard inverse Gaussian parameterization, which has variance \mu^3 / \lambda
.
Setting \phi = \lambda / \mu
yields our parameterization, which is
preferred for computational stability. Also note that the dispersion parameter
is often defined in the literature as V(\mu) \phi
, where V(\mu)
is the variance
function of the mean. We do not use this parameterization, which is important
to recognize while interpreting dispersion estimates.
For more on generalized linear model constructions, see McCullagh and
Nelder (1989).
In the generalized linear model context, the tailup, taildown, Euclidean, and
nugget covariance affect the modeled mean of an observation (conditional on
these effects). On the link scale, the tailup random errors capture spatial
covariance moving downstream (and depend on downstream distance), the taildown
random errors capture spatial covariance moving upstream (and depend on upstream)
distance, the Euclidean random errors capture spatial covariance that depends on
Euclidean distance, and the nugget random errors captures variability
independent of spatial locations. \eta
is modeled using a
spatial covariance function expressed as
de(zu) * R(zu) + de(zd) * R(zd) + de(ze) * R(ze) + nugget * I
.
de(zu)
, de(zu)
, and de(zd)
represent the tailup, taildown, and Euclidean
variances, respectively. R(zu)
, R(zd)
, and R(ze)
represent the tailup,
taildown, and Euclidean correlation matrices, respectively. Each correlation
matrix depends on a range parameter that controls the distance-decay behavior
of the correlation. nugget
represents the nugget variance and
I
represents an identity matrix.
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 functiongravity:
(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"
.
estmethod
Details: The various estimation methods are
-
reml
: Maximize the restricted log-likelihood. -
ml
: Maximize the log-likelihood.
anisotropy
Details: By default, all Euclidean covariance parameters except rotate
and scale
are assumed unknown, requiring estimation. If either rotate
or scale
are given initial values other than 0 and 1 (respectively) or are assumed unknown
in euclid_initial()
, anisotropy
is implicitly set to TRUE
.
(Geometric) Anisotropy is modeled by transforming a Euclidean covariance function that
decays differently in different directions to one that decays equally in all
directions via rotation and scaling of the original Euclidean coordinates. The rotation is
controlled by the rotate
parameter in [0, \pi]
radians. The scaling
is controlled by the scale
parameter in [0, 1]
. The anisotropy
correction involves first a rotation of the coordinates clockwise by rotate
and then a
scaling of the coordinates' minor axis by the reciprocal of scale
. The Euclidean
covariance is then computed using these transformed coordinates.
random
Details: If random effects are used (the estimation method must be "reml"
or
"ml"
), the model
can be written as g(\mu) = \eta = X \beta + W1\gamma 1 + ... Wj\gamma j + zu + zd + ze + n
,
where each Z is a random effects design matrix and each u is a random effect.
partition_factor
Details: The partition factor can be represented in matrix form as P
, where
elements of P
equal one for observations in the same level of the partition
factor and zero otherwise. The covariance matrix involving only the
spatial and random effects components is then multiplied element-wise
(Hadmard product) by P
, yielding the final covariance matrix.
Other Details: Observations with NA
response values are removed for model
fitting, but their values can be predicted afterwards by running
predict(object)
.
Value
A list with many elements that store information about
the fitted model object and has class ssn_glm
. Many generic functions that
summarize model fit are available for ssn_glm
objects, including
AIC
, AICc
, anova
, augment
, coef
,
cooks.distance
, covmatrix
, deviance
, fitted
, formula
,
glance
, glances
, hatvalues
, influence
,
labels
, logLik
, loocv
, model.frame
, model.matrix
,
plot
, predict
, print
, pseudoR2
, summary
,
terms
, tidy
, update
, varcomp
, and vcov
.
This fitted model list contains the following elements:
-
additive
: The name of the additive function value column. -
anisotropy
: Whether euclidean anisotropy was modeled. -
call
: The function call. -
coefficients
: Model coefficients. -
contrasts
: Any user-supplied contrasts. -
cooks_distance
: Cook's distance values. -
crs
: The geographic coordinate reference system. -
deviance
: The model deviance. -
diagtol
: A tolerance value that may be added to the diagonal of ovariance matrices to encourage decomposition stability. -
estmethod
: The estimation method. -
euclid_max
: The maximum euclidean distance. -
family
: The generalized linear model family -
fitted
: Fitted values. -
formula
: The model formula. -
hatvalues
: The hat (leverage) values. -
is_known
: An object that identifies which parameters are known. -
local_index
: An index identifier used internally for sorting. -
missing_index
: Which rows in the "obs" object had missing responses. -
n
: The sample size. -
npar
: The number of estimated covariance parameters. -
observed_index
: Which rows in the "obs" object had observed responses. -
optim
: The optimization output. -
p
: The number of fixed effects. -
partition_factor
: The partition factor formula. -
pseudoR2
: The pseudo R-squared. -
random
: The random effect formula. -
residuals
: The residuals. -
sf_column_name
: The name of the geometry columnsssn.object
-
size
: The size of the binomial trials if relevant. -
ssn.object
: An updatedssn.object
. -
tail_max
: The maximum stream distance. -
terms
: The model terms. -
vcov
: Variance-covariance matrices -
xlevels
: The levels of factors in the model matrix. -
y
: The response.
These list elements are meant to be used with various generic functions
(e.g., residuals()
that operate on the model object.
While possible to access elements of the fitted model list directly, we strongly
advise against doing so when there is a generic available to return the element
of interest. For example, we strongly recommend using residuals()
to
obtain model residuals instead of accessing the fitted model list directly via
object$residuals
.
Note
This function does not perform any internal scaling. If optimization is not stable due to large extremely large variances, scale relevant variables so they have variance 1 before optimization.
References
Garreta, V., Monestiez, P. and Ver Hoef, J.M. (2010) Spatial modelling and prediction on river networks: up model, down model, or hybrid? Environmetrics 21(5), 439–456.
McCullagh P. and Nelder, J. A. (1989) Generalized Linear Models. London: Chapman and Hall.
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.
Examples
# Copy the mf04p .ssn data to a local directory and read it into R
# When modeling with your .ssn object, you will load it using the relevant
# path to the .ssn data on your machine
copy_lsn_to_temp()
temp_path <- paste0(tempdir(), "/MiddleFork04.ssn")
mf04p <- ssn_import(temp_path, overwrite = TRUE)
ssn_gmod <- ssn_glm(
formula = Summer_mn ~ ELEV_DEM,
ssn.object = mf04p,
family = "Gamma",
tailup_type = "exponential",
additive = "afvArea"
)
summary(ssn_gmod)