Fit spatial linear models for point-referenced data (i.e., geostatistical models) using a variety of estimation methods, allowing for random effects, anisotropy, partition factors, and big data methods.
Usage
splm(
formula,
data,
spcov_type,
xcoord,
ycoord,
spcov_initial,
estmethod = "reml",
weights = "cressie",
anisotropy = FALSE,
random,
randcov_initial,
partition_factor,
local,
...
)
Arguments
- formula
A two-sided linear formula describing the fixed effect structure of the model, with the response to the left of the
~
operator and the terms on the right, separated by+
operators.- data
A data frame or
sf
object object that contains the variables infixed
,random
, andpartition_factor
as well as geographical information. If ansf
object is provided withPOINT
geometries, the x-coordinates and y-coordinates are used directly. If ansf
object is provided withPOLYGON
geometries, the x-coordinates and y-coordinates are taken as the centroids of each polygon.- spcov_type
The spatial covariance type. Available options include
"exponential"
,"spherical"
,"gaussian"
,"triangular"
,"circular"
,"cubic"
,"pentaspherical"
,"cosine"
,"wave"
,"jbessel"
,"gravity"
,"rquad"
,"magnetic"
,"matern"
,"cauchy"
,"pexponential"
, and"none"
. Parameterizations of each spatial covariance type are available in Details. Multiple spatial covariance types can be provided as a character vector, and thensplm()
is called iteratively for each element and a list is returned for each model fit. The default forspcov_type
is"exponential"
. Whenspcov_type
is specified, all unknown spatial covariance parameters are estimated.spcov_type
is ignored ifspcov_initial
is provided.- xcoord
The name of the column in
data
representing the x-coordinate. Can be quoted or unquoted. Not required ifdata
is ansf
object.- ycoord
The name of the column in
data
representing the y-coordinate. Can be quoted or unquoted. Not required ifdata
is ansf
object.- spcov_initial
An object from
spcov_initial()
specifying initial and/or known values for the spatial covariance parameters. Multiplespcov_initial()
objects can be provided in a list. Thensplm()
is called iteratively for each element and a list is returned for each model fit.- estmethod
The estimation method. Available options include
"reml"
for restricted maximum likelihood,"ml"
for maximum likelihood,"sv-wls"
for semivariogram weighted least squares, and"sv-cl"
for semivariogram composite likelihood. The default is"reml"
.- weights
Weights to use when
estmethod
is"sv-wls"
. Available options include"cressie"
,"cressie-dr"
,"cressie-nopairs"
,"cressie-dr-nopairs"
,"pairs"
,"pairs-invd"
,"pairs-invrd"
, and"ols"
. Parameterizations for each weight are available in Details. The default is"cressie"
.- anisotropy
A logical indicating whether (geometric) anisotropy should be modeled. Not required if
spcov_initial
is provided with 1)rotate
assumed unknown or assumed known and non-zero or 2)scale
assumed unknown or assumed known and less than one. Whenanisotropy
isTRUE
, computational times can significantly increase. The default isFALSE
.- random
A one-sided linear formula describing the random effect structure of the model. Terms are specified to the right of the
~ operator
. Each term has the structurex1 + ... + xn | g1/.../gm
, wherex1 + ... + xn
specifies the model for the random effects andg1/.../gm
is the grouping structure. Separate terms are separated by+
and must generally be wrapped in parentheses. Random intercepts are added to each model implicitly when at least one other variable is defined. If a random intercept is not desired, this must be explicitly defined (e.g.,x1 + ... + xn - 1 | g1/.../gm
). If only a random intercept is desired for a grouping structure, the random intercept must be specified as1 | g1/.../gm
. Note thatg1/.../gm
is shorthand for(1 | g1/.../gm)
. If only random intercepts are desired and the shorthand notation is used, parentheses can be omitted.- randcov_initial
An optional object specifying initial and/or known values for the random effect variances.
- 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.
- local
An optional logical or list controlling the big data approximation. If omitted,
local
is set toTRUE
orFALSE
based on the sample size (the number of non-missing observations indata
) -- if the sample size exceeds 5,000,local
is set toTRUE
. Otherwise it is set toFALSE
.local
is also set toFALSE
whenspcov_type
is"none"
and there are no random effects specified viarandom
. IfFALSE
, no big data approximation is implemented. If a list is provided, the following arguments detail the big data approximation:index:
The group indexes. Observations in different levels ofindex
are assumed to be uncorrelated for the purposes of estimation. Ifindex
is not provided, it is determined by specifyingmethod
and eithersize
orgroups
.method
: The big data approximation method used to determineindex
. Ignored ifindex
is provided. Ifmethod = "random"
, observations are randomly assigned toindex
based onsize
. Ifmethod = "kmeans"
, observations assigned toindex
based on k-means clustering on the coordinates withgroups
clusters. The default is"kmeans"
. Note that both methods have a random component, which means that you may get different results from separate model fitting calls. To ensure consistent results, specifyindex
or set a seed viabase::set.seed()
.size
: The number of observations in eachindex
group whenmethod
is"random"
. If the number of observations is not divisible bysize
, some levels getsize - 1
observations. The default is 100.groups:
The number ofindex
groups. Ifmethod
is"random"
,size
is \(ceiling(n / groups)\), where \(n\) is the sample size. Automatically determined ifsize
is specified. Ifmethod
is"kmeans"
,groups
is the number of clusters.var_adjust:
The approach for adjusting the variance-covariance matrix of the fixed effects."none"
for no adjustment,"theoretical"
for the theoretically-correct adjustment,"pooled"
for the pooled adjustment, and"empirical"
for the empirical adjustment. The default is"theoretical"
.parallel
: IfTRUE
, parallel processing via the parallel package is automatically used. The default isFALSE
.ncores
: Ifparallel = TRUE
, the number of cores to parallelize over. The default is the number of available cores on your machine.
When
local
is a list, at least one list element must be provided to initialize default arguments for the other list elements. Iflocal
isTRUE
, defaults forlocal
are chosen such thatlocal
is transformed intolist(size = 100, method = "kmeans", var_adjust = "theoretical", parallel = FALSE)
.- ...
Other arguments to
esv()
orstats::optim()
.
Value
A list with many elements that store information about
the fitted model object. If spcov_type
or spcov_initial
are
length one, the list has class splm
. Many generic functions that
summarize model fit are available for splm
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
. If
spcov_type
or spcov_initial
are length greater than one, the
list has class splm_list
and each element in the list has class
splm
. glances
can be used to summarize splm_list
objects, and the aforementioned splm
generics can be used on each
individual list element (model fit).
Details
The spatial linear model for point-referenced data (i.e., geostatistical model) can be written as \(y = X \beta + \tau + \epsilon\), where \(X\) is the fixed effects design matrix, \(\beta\) are the fixed effects, \(\tau\) is random error that is spatially dependent, and \(\epsilon\) is random error that is spatially independent. Together, \(\tau\) and \(\epsilon\) are modeled using a spatial covariance function, expressed as \(de * R + ie * I\), where \(de\) is the dependent error variance, \(R\) is a correlation matrix that controls the spatial dependence structure among observations, \(ie\) is the independent error variance, and \(I\) is an identity matrix.
spcov_type
Details: Parametric forms for \(R\) are given below, where \(\eta = h / range\)
for \(h\) distance between observations:
exponential: \(exp(- \eta )\)
spherical: \((1 - 1.5\eta + 0.5\eta^3) * I(h <= range)\)
gaussian: \(exp(- \eta^2 )\)
triangular: \((1 - \eta) * I(h <= range)\)
circular: \((1 - (2 / \pi) * (m * sqrt(1 - m^2) + sin^{-1}(m))) * I(h <= range), m = min(\eta, 1)\)
cubic: \((1 - 7\eta^2 + 8.75\eta^3 - 3.5\eta^5 + 0.75\eta^7) * I(h <= range)\)
pentaspherical: \((1 - 1.875\eta + 1.25\eta^3 - 0.375\eta^5) * I(h <= range)\)
cosine: \(cos(\eta)\)
wave: \(sin(\eta) / \eta * I(h > 0) + I(h = 0)\)
jbessel: \(Bj(h * range)\), Bj is Bessel-J function
gravity: \((1 + \eta^2)^{-0.5}\)
rquad: \((1 + \eta^2)^{-1}\)
magnetic: \((1 + \eta^2)^{-1.5}\)
matern: \(2^{1 - extra}/ \Gamma(extra) * \alpha^{extra} * Bk(\alpha, extra)\), \(\alpha = (2extra * \eta)^{0.5}\), Bk is Bessel-K function with order \(1/5 \le extra \le 5\)
cauchy: \((1 + \eta^2)^{-extra}\), \(extra > 0\)
pexponential: \(exp(h^{extra}/range)\), \(0 < extra \le 2\)
none: \(0\)
All spatial covariance functions are valid in one spatial dimension. All
spatial covariance functions except triangular
and cosine
are
valid in two dimensions.
estmethod
Details: The various estimation methods are
reml
: Maximize the restricted log-likelihood.ml
: Maximize the log-likelihood.sv-wls
: Minimize the semivariogram weighted least squares loss.sv-cl
: Minimize the semivariogram composite likelihood loss.
anisotropy
Details: By default, all spatial covariance parameters except rotate
and scale
as well as all random effect variance parameters
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 spcov_initial()
, anisotropy
is implicitly set to TRUE
.
(Geometric) Anisotropy is modeled by transforming a covariance function that
decays differently in different directions to one that decays equally in all
directions via rotation and scaling of the original 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 spatial
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 \(y = X \beta + Z1u1 + ... Zjuj + \tau + \epsilon\),
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.
local
Details: The big data approximation works by sorting observations into different levels
of an index variable. Observations in different levels of the index variable
are assumed to be uncorrelated for the purposes of model fitting. Sparse matrix methods are then implemented
for significant computational gains. Parallelization generally further speeds up
computations when data sizes are larger than a few thousand. Both the "random"
and "kmeans"
values of method
in local
have random components. That means you may get slightly different
results when using the big data approximation and rerunning splm()
with the same code. For consistent results,
either set a seed via base::set.seed()
or specify index
to local
.
Observations with NA
response values are removed for model
fitting, but their values can be predicted afterwards by running
predict(object)
.
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.
Examples
spmod <- splm(z ~ water + tarp,
data = caribou,
spcov_type = "exponential", xcoord = x, ycoord = y
)
summary(spmod)
#>
#> Call:
#> splm(formula = z ~ water + tarp, data = caribou, spcov_type = "exponential",
#> xcoord = x, ycoord = y)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.41281 -0.20763 -0.11205 0.02956 0.45429
#>
#> Coefficients (fixed):
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 2.04981 0.31093 6.592 4.33e-11 ***
#> waterY -0.08310 0.06449 -1.289 0.197563
#> tarpnone 0.08005 0.07759 1.032 0.302166
#> tarpshade 0.28654 0.07667 3.737 0.000186 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Pseudo R-squared: 0.3963
#>
#> Coefficients (exponential spatial covariance):
#> de ie range
#> 0.1109 0.0226 19.1168