Description Usage Arguments Details Value Author(s) References See Also Examples
The function computes the (tapered) covariance matrix for a spatial (temporal or spatio-temporal) covariance model and a set of spatial (temporal or spatio-temporal) points.
1 2 3 4 |
coordx |
A numeric (d x 2)-matrix (where
|
coordy |
A numeric vector giving 1-dimension of
spatial coordinates; |
coordt |
A numeric vector giving 1-dimension of
temporal coordinates. At the moment implemented only for the
Gaussian case. Optional argument, the default is |
corrmodel |
String; the name of a correlation model, for the description see the Section Details. |
distance |
String; the name of the spatial distance. The default
is |
grid |
Logical; if |
iskrig |
Logical: the default value is |
maxdist |
Numeric; an optional positive value indicating the
marginal spatial compact support. See |
maxtime |
Numeric; an optional positive value indicating the
marginal temporal compact support. See |
param |
A list of parameter values required for the correlation
model. See |
taper |
String; the name of the taper correlation
function if type is |
tapsep |
Numeric; an optional value indicating the separabe parameter in the space-time quasi taper (see Details). |
type |
String; the type of covariance matrix
|
The parameter param
is a list including all the parameters of a
covariance function model.
In particular, the covariance models share the following paramaters: the
sill
that
represents the common variance
of the random field,
the nugget
that represents the local variation (white
noise) at the origin.
For each correlation model you can check the list of the specific parameters using CorrelationParam
.
Here there is the list of all the implemented space and space-time
correlation models. The list of space-time correlation functions
includes separable
and non-separable models.
Purerly spatial correlation models:
cauchy
R(h) = (1+h^2)^(-β)
The parameter β is positive. It is a special case of the gencauchy
model.
exponential
R(h) = exp(-h)
This model is a special case of the whittle
and the stable
model.
gauss
R(h)=exp(-h^2)
This model is a special case of the stable
model.
gencauchy
(generalised cauchy
)
R(h) = ( 1+h^α )^{-\frac{β}{α}}
The parameter α is in (0,2], and β
is positive.
spherical
R(h) = (1- 1.5 h+0.5 h^3) 1_{[0,1]}(h)
This isotropic covariance function is valid only for dimensions less than or equal to 3.
stable
R(h)=exp(-h ^α)
The parameter α is in (0,2].
wave
R(h)=sin(h)/h if h>0 and C(0)=1
This isotropic covariance function is valid only for dimensions less than or equal to 3.
matern
R(h) = 2^{1-ν} Γ(ν)^{-1} x^ν K_ν(h)
The parameter ν is positive.
This is the model of choice if the smoothness of a random field is to
be parametrised: if ν > m then the
graph is m times differentiable.
Spatio-temporal correlation models:
Non-separable models:
gneiting
(non-separabel space time model)
R(h, u) = \frac{e^{ \frac{-h^ν} { (1+u^λ)^{0.5 γ ν }}}} {1+u^λ}
The parameters ν and λ take values in [0,2]; the parameter γ take values in [0,1]. For γ=0 it is a separable model.
gneiting_GC
(non-separabel space time model with great circle distances)
R(h, u) = \frac{ e^{ \frac{-u^λ }{1+h^ν)^{0.5 γ λ}} }} { 1+h^ν}
iacocesare
(non-separabel space time model)
R(h, u) = (1+h^ν+u^λ)^{-δ}
The parameters ν and λ take values in [1,2]; the parameters δ must be greater than or equal to half the space-time dimension.
porcu
(non-separabel space time model)
R(h, u) = (0.5 (1+h^ν)^γ +0.5 (1+u^λ)^γ)^{-γ^{-1}}
The parameters ν and λ take values in [0,2]; the paramete γ take values in [0,1]. The limit of the correlation model as γ tends to zero leads to a separable model.
porcu2
(non-separabel space time model)
R(h, u) =\frac{ e^{ -h^ν ( 1+u^λ)^{0.5 γ ν}}} { (1+u^λ)^{1.5}}
The parameters ν and λ take values in [0,2]; the parameter γ take values in [0,1]. For γ=0 it is a separable model.
Separable models.
Space-time separable correlation models are easly obtained as the product of a spatial and a temporal correlation model, that is
R(h,u)=R(h) R(u)
Several combinations are possible:
exp_exp
: spatial exponential model and temporal exponential model
exp_cauchy
: spatial exponential model and temporal cauchy model
matern_cauchy
: spatial matern model and temporal cauchy model
stable_stable
: spatial stabel model and temporal stable model
Note that some models are nested. (The exp_exp
with the stable_stable
for instance.)
Spatial taper function models.
For spatial covariance tapering the tapered correlation functions are:
Wendland1
R(h) = (1-h)^2 (1+0.5 h) 1_{[0,1]}(h)
Wendland2
R(h) = (1-h)^4 (1+4 h) 1_{[0,1]}(h)
Wendland3
R(h) = (1-h)^6 (1+6 h + 35 h^2 /3) 1_{[0,1]}(h)
Spatio-tempora tapered correlation models.
For space-time covariance tapering likelihood the taper functions are obtained as the product of a spatial and a temporal taper (Separable taper). Several combinations are possible:
Wendlandi_Wendlandj
: spatial Wendlandi
taper and temporal Wendlandj
taper with i,j=1,2,3.
Space-time non separable adaptive-taper with dynamically space-time compact support is:
qt_time
and qt_space
. In The case of qt_time
the space-time quasi taper is:
T(h,u) = (arg)^{-6} (1+7 x) (1-x)^7 1_{[0,\frac{maxtime}{arg}]}(u)
arg=(1+\frac{h}{maxdist} )^β, x=u \frac{ arg}{maxtime}
where 0<=β<=1 is a fixed parameter of separability (tapsep
), maxtime
the fixed temporal compact support and maxdist
the
fixed spatial scale parameter.
The adaptive-taper qt_space
is the same taper but changing the time with the space.
Remarks:
Let R(h)
be a spatial correlation model given in standard notation.
Then the covariance model
applied with arbitrary variance and scale equals to:
C(h)=sill +nugget , \quad if \quad h=0
C(h)=sill * R( \frac{h}{scale},...) , \quad if \quad h>0
Similarly if R(h,u)
is a spatio-temporal correlation model given in standard notation,
then the covariance model is:
C(h,u)=sill +nugget , \quad if \quad h=0,u=0
C(h,u)=sill * R( \frac{h}{scale_s},\frac{u}{scale_t},...) , \quad if \quad h>0 \quad or \quad u>0
Here ‘...’ stands for additional parameters.
Let R(h)
be a spatial taper given in standard notation.
Then the taper function applied with an arbitrary compact support (maxdist) equals to:
T(h)= R( \frac{h}{maxdist})
Similarly if R(h,u)
is a spatio-temporal taper given in standard notation, then the taper
function applied with arbitrary compact supports (maxdist, maxtime) equals to:
T(h,u)= R( \frac{h}{maxdist},\frac{u}{maxtime})
Then the tapered covariance matrix is obtained as:
C_{tap}(h,u)= T(h,u)C(h,u)
Returns an object of class CovMat
.
An object of class CovMat
is a list containing
at most the following components:
coordx |
A d-dimensional vector of spatial coordinates; |
coordy |
A d-dimensional vector of spatial coordinates; |
coordt |
A t-dimensional vector of temporal coordinates; |
covmatrix |
The covariance matrix if |
corrmodel |
String: the correlation model; |
distance |
String: the type of spatial distance; |
grid |
Logical: |
nozero |
In the case of tapered matrix the percentage of non zero values in the covariance matrix. Otherwise is NULL. |
maxdist |
Numeric: the marginal spatial compact support if |
maxtime |
Numeric: the marginal temporal compact support if |
namescorr |
String: The names of the correlation parameters; |
numcoord |
Numeric: the number of spatial coordinates; |
numtime |
Numeric: the number the temporal coordinates; |
param |
Numeric: The covariance parameters; |
tapmod |
String: the taper model if |
spacetime |
|
In the space-time case covmatrix
is the covariance matrix of the random vector
Z(s_1,t_1),Z(s_1,t_2),..Z(s_n,t_1),..,Z(s_n,t_m)
for n
spatial locatione sites and m
temporal instants.
Simone Padoan, simone.padoan@unibocconi.it, http://faculty.unibocconi.it/simonepadoan; Moreno Bevilacqua, moreno.bevilacqua@uv.cl, https://sites.google.com/a/uv.cl/moreno-bevilacqua/home.
Bevilacqua, M., Mateu, J., Porcu, E., Zhang, H. and Zini, A. (2010). Weighted composite likelihood-based tests for space-time separability of covariance functions. Statistics and Computing, 20(3), 283-293.
Gaetan, C. and Guyon, X. (2010) Spatial Statistics and Modelling. Spring Verlang, New York.
Gneiting, T. (2002). Nonseparable, stationary covariance functions for space-time data. Journal of the American Statistical Association, 97, 590–600.
Gneiting, T., Genton, M. G. and Guttorp, P. (2007). Geostatistical space-time models, stationarity, separability and full symmetry. In Finkenstadt, B., Held, L. and Isham, V. (eds.), Statistical Methods for Spatio-Temporal Systems, Chapman & Hall/CRC, Boca Raton, pp. 151-175
Schlather, M. (1999) An introduction to positive definite functions and to unconditional simulation of random fields. Technical report ST 99–10, Dept. of Maths and Statistics, Lancaster University
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | library(CompRandFld)
library(spam)
################################################################
###
### Example 1. Covariance matrix associated to
### a Matern correlation model
###
###############################################################
# Define the spatial-coordinates of the points:
x <- runif(500, 0, 2)
y <- runif(500, 0, 2)
matrix1 <- Covmatrix(x, y, corrmodel="matern", param=list(smooth=0.5,
sill=1,scale=0.2,mean=0))
dim(matrix1$covmatrix)
################################################################
###
### Example 3. Covariance matrix associated to
### a space-time double exponential correlation model
###
###############################################################
# Define the temporal-coordinates:
times <- c(1,2,3)
# Define correlation model
corrmodel="exp_exp"
# Define covariance parameters
param=list(scale_s=0.3,scale_t=0.5,sill=1,mean=0)
# Simulation of a spatial Gaussian random field:
matrix3 <- Covmatrix(x, y, times, corrmodel=corrmodel,
param=param)
dim(matrix3$covmatrix)
################################################################
###
### Example 2. Tapered Covariance matrix associated to
### a Matern correlation model
###
###############################################################
# Define the spatial-coordinates of the points:
#x <- runif(500, 0, 2)
#y <- runif(500, 0, 2)
#matrix2 <- Covmatrix(x, y, corrmodel="matern", param=list(smooth=0.5,
# sill=1,scale=0.2,mean=0),maxdist=0.3,taper="Wendland1",
# type="Tapering")
# Tapered covariance matrix
#as.matrix(matrix2$covmatrix)[1:15,1:15]
# Percentage of no zero values in the tapered matrix
#matrix2$nozero
################################################################
###
### Example 4. Tapered Covariance matrix associated to
### a space-time double exponential correlation model
###
###############################################################
#param <- list(scale_s=2,scale_t=1,sill=1,mean=0)
#matrix4 <- Covmatrix(x, y, times, corrmodel="exp_exp", param=param, maxdist=0.3,
# maxtime=2,taper="Wendland2_Wendland2",type="Tapering")
# Tapered space time covariance matrix
#as.matrix(matrix4$covmatrix)[1:10,1:10]
# Percentage of no zero values in the tapered matrix
#matrix4$nozero
|
Loading required package: dotCall64
Loading required package: grid
Spam version 2.1-1 (2017-07-02) is loaded.
Type 'help( Spam)' or 'demo( spam)' for a short introduction
and overview of this package.
Help for individual functions is also obtained by adding the
suffix '.spam' to the function name, e.g. 'help( chol.spam)'.
Attaching package: 'spam'
The following objects are masked from 'package:base':
backsolve, forwardsolve
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