fitmaxstab: Fits a max-stable process to data
In SpatialExtremes: Modelling Spatial Extremes
Description
Usage
Arguments
Details
Value
Warning
Author(s)
References
Examples
Description
This function fits max-stable processes to data using
pairwise likelihood. Two max-stable characterisations are available:
the Smith and Schlather representations.
Usage
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fitmaxstab(data, coord, cov.mod, loc.form, scale.form,
shape.form, marg.cov = NULL, temp.cov = NULL, temp.form.loc = NULL,
temp.form.scale = NULL, temp.form.shape = NULL, iso = FALSE, ...,
fit.marge = FALSE, warn = TRUE, method = "Nelder", start, control =
list(), weights = NULL, corr = FALSE, check.grad
= FALSE)
Arguments
data
A matrix representing the data. Each column corresponds to
one location.
coord
A matrix that gives the coordinates of each
location. Each row corresponds to one location.
cov.mod
A character string corresponding to the covariance
model in the max-stable representation. Must be one of "gauss" for
the Smith's model; "whitmat", "cauchy", "powexp", "bessel" or
"caugen" for the Whittle-Matern, the Cauchy, the Powered Exponential, the
Bessel and the Generalized Cauchy correlation families with the
Schlather's model; "brown" for Brown-Resnick processes. The
geometric Gaussian and Extremal-t models with a Whittle-Matern
correlation function can be fitted by passing respectively
"gwhitmat" or "twhitmat". Other correlation function families are
considered in a similar way.
loc.form, scale.form, shape.form
R formulas defining the
spatial linear model for the GEV parameters. May be missing. See
section Details.
marg.cov
Matrix with named columns giving additional covariates
for the GEV parameters. If NULL, no extra covariates are
used.
temp.cov
Matrix with names columns giving additional *temporal*
covariates for the GEV parameters. If NULL, no temporal trend
are assume for the GEV parameters — see section Details.
temp.form.loc, temp.form.scale, temp.form.shape
R formulas
defining the temporal trends for the GEV parameters. May be
missing. See section Details.
iso
Logical. If TRUE an isotropic model is fitted to
data. Otherwise (default), anisotropy is allowed. Currently, this is
only implemented for the Smith's model.
...
Several arguments to be passed to the
optim, nlm or nlminb
functions. See section details.
fit.marge
Logical. If TRUE, the GEV parameters
are estimated pointwise or using the formulas given by
loc.form, scale.form and shape.form. If
FALSE, observations are supposed to be unit Frechet
distributed. Note that when formulas are given, fit.marge is
automatically set to TRUE.
warn
Logical. If TRUE (default), users are warned if
the log-likelihood is infinite at starting values and/or problems
arised while computing the standard errors.
method
The method used for the numerical optimisation
procedure. Must be one of BFGS, Nelder-Mead,
CG, L-BFGS-B, SANN, nlm or
nlminb. See optim for details. Please note that
passing nlm or nlminb will use the nlm
or nlminb functions instead of optim.
start
A named list giving the initial values for the
parameters over which the pairwise likelihood is to be minimized. If
start is omitted the routine attempts to find good starting
values - but might fail.
control
A list giving the control parameters to be passed to
the optim function.
weights
A numeric vector specifying the weights in the pairwise
likelihood - and so has length the number of pairs. If NULL
(default), no weighting scheme is used.
corr
Logical. If TRUE (non default), the asymptotic
correlation matrix is computed.
check.grad
Logical. If TRUE (non default), the analytic
gradient of the pairwise likelihood will be compared to the
numerical one. Such a checking might be usefull for ill-conditionned
situation diagnosis.
Details
As spatial data often deal with a large number of locations, it is
impossible to write analytically the joint distribution. Consequently,
the fitting procedure substitutes the "full likelihood" for the
pairwise likelihood.
Let define L_{i,j}(x_{i,j}, θ) the
likelihood for site i and j, where i = 1, …, N-1, j=i+1, …,
N, N is the number of site within the region and
x_{i,j} are the joint observations for site i
and j. Then the pairwise likelihood
PL(θ) is defined by:
llik_P = log PL(θ) = ∑_{i = 1}^{N-1}
∑_{j=i+1}^{N} log L_{i,j} (x_{i,j}, θ)
As pairwise likelihood is an approximation of the “full likelihood”,
standard errors cannot be computed directly by the inverse of the
Fisher information matrix. Instead, a sandwich estimate must be used
to account for model mispecification e.g.
hat(θ) ~
N(θ, H^{-1} J H^{-1})
where H is the Fisher information matrix (computed from the
misspecified model) and J is the variance of the score
function.
There are two different kind of covariates : "spatial" and
"temporal".
A "spatial" covariate may have different values accross station but
does not depend on time. For example the coordinates of the stations
are obviously "spatial". These "spatial" covariates should be used
with the marg.cov and loc.form, scale.form, shape.form.
A "temporal" covariates may have different values accross time but
does not depend on space. For example the years where the annual
maxima were recorded is "temporal". These "temporal" covariates should
be used with the temp.cov and temp.form.loc,
temp.form.scale, temp.form.shape.
As a consequence note that marg.cov must have K rows (K being
the number of sites) while temp.cov must have n rows (n being
the number of observations).
Value
This function returns a object of class maxstab. Such objects
are list with components:
fitted.values
A vector containing the estimated parameters.
std.err
A vector containing the standard errors.
fixed
A vector containing the parameters of the model that
have been held fixed.
param
A vector containing all parameters (optimised and fixed).
deviance
The (pairwise) deviance at the maximum pairwise
likelihood estimates.
corr
The correlation matrix.
convergence, counts, message
Components taken from the
list returned by optim - for the mle method.
data
The data analysed.
model
The max-stable characterisation used.
fit.marge
A logical that specifies if the GEV margins were
estimated.
cov.fun
The estimated covariance function - for the Schlather
model only.
extCoeff
The estimated extremal coefficient function.
cov.mod
The covariance model for the spatial structure.
Warning
When using reponse surfaces to model spatially the GEV parameters, the
likelihood is pretty rough so that the general purpose optimization
routines may fail. It is your responsability to check if the
numerical optimization succeeded or not. I tried, as best as I can, to
provide warning messages if the optimizers failed but in some cases,
no warning will appear!
Author(s)
Mathieu Ribatet
References
Cox, D. R. and Reid, N. (2004) A note on pseudo-likelihood constructed
from marginal densities. Biometrika 91, 729–737.
Demarta, S. and McNeil, A. (2005) The t copula and Related Copulas
International Statistical Review 73, 111-129.
Gholam–Rezaee, M. (2009) Spatial extreme value: A composite
likelihood. PhD Thesis. Ecole Polytechnique Federale de Lausanne.
Kabluchko, Z., Schlather, M. and de Haan, L. (2009) Stationary
max-stable fields associated to negative definite functions
Annals of Probability 37:5, 2042–2065.
Padoan, S. A. (2008) Computational Methods for Complex Problems in
Extreme Value Theory. PhD Thesis. University of Padova.
Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010) Likelihood-based
inference for max-stable processes. Journal of the American
Statistical Association (Theory and Methods) 105:489,
263-277.
Schlather, M. (2002) Models for Stationary Max-Stable Random
Fields. Extremes 5:1, 33–44.
Smith, R. L. (1990) Max-stable processes and spatial
extremes. Unpublished.
Examples
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## Not run:
##Define the coordinate of each location
n.site <- 30
locations <- matrix(runif(2*n.site, 0, 10), ncol = 2)
colnames(locations) <- c("lon", "lat")
##Simulate a max-stable process - with unit Frechet margins
data <- rmaxstab(40, locations, cov.mod = "whitmat", nugget = 0, range = 3,
smooth = 0.5)
##Now define the spatial model for the GEV parameters
param.loc <- -10 + 2 * locations[,2]
param.scale <- 5 + 2 * locations[,1] + locations[,2]^2
param.shape <- rep(0.2, n.site)
##Transform the unit Frechet margins to GEV
for (i in 1:n.site)
data[,i] <- frech2gev(data[,i], param.loc[i], param.scale[i],
param.shape[i])
##Define a model for the GEV margins to be fitted
##shape ~ 1 stands for the GEV shape parameter is constant
##over the region
loc.form <- loc ~ lat
scale.form <- scale ~ lon + I(lat^2)
shape.form <- shape ~ 1
##Fit a max-stable process using the Schlather's model
fitmaxstab(data, locations, "whitmat", loc.form, scale.form,
shape.form)
## Model without any spatial structure for the GEV parameters
## Be careful this could be *REALLY* time consuming
fitmaxstab(data, locations, "whitmat")
## Fixing the smooth parameter of the Whittle-Matern family
## to 0.5 - e.g. considering exponential family. We suppose the data
## are unit Frechet here.
fitmaxstab(data, locations, "whitmat", smooth = 0.5, fit.marge = FALSE)
## Fitting a penalized smoothing splines for the margins with the
## Smith's model
data <- rmaxstab(40, locations, cov.mod = "gauss", cov11 = 100, cov12 =
25, cov22 = 220)
## And transform it to ordinary GEV margins with a non-linear
## function
fun <- function(x)
2 * sin(pi * x / 4) + 10
fun2 <- function(x)
(fun(x) - 7 ) / 15
param.loc <- fun(locations[,2])
param.scale <- fun(locations[,2])
param.shape <- fun2(locations[,1])
##Transformation from unit Frechet to common GEV margins
for (i in 1:n.site)
data[,i] <- frech2gev(data[,i], param.loc[i], param.scale[i],
param.shape[i])
##Defining the knots, penalty, degree for the splines
n.knots <- 5
knots <- quantile(locations[,2], prob = 1:n.knots/(n.knots+1))
knots2 <- quantile(locations[,1], prob = 1:n.knots/(n.knots+1))
##Be careful the choice of the penalty (i.e. the smoothing parameter)
##may strongly affect the result Here we use p-splines for each GEV
##parameter - so it's really CPU demanding but one can use 1 p-spline
##and 2 linear models.
##A simple linear model will be clearly faster...
loc.form <- y ~ rb(lat, knots = knots, degree = 3, penalty = .5)
scale.form <- y ~ rb(lat, knots = knots, degree = 3, penalty = .5)
shape.form <- y ~ rb(lon, knots = knots2, degree = 3, penalty = .5)
fitted <- fitmaxstab(data, locations, "gauss", loc.form, scale.form, shape.form,
control = list(ndeps = rep(1e-6, 24), trace = 10),
method = "BFGS")
fitted
op <- par(mfrow=c(1,3))
plot(locations[,2], param.loc, col = 2, ylim = c(7, 14),
ylab = "location parameter", xlab = "latitude")
plot(fun, from = 0, to = 10, add = TRUE, col = 2)
points(locations[,2], predict(fitted)[,"loc"], col = "blue", pch = 5)
new.data <- cbind(lon = seq(0, 10, length = 100), lat = seq(0, 10, length = 100))
lines(new.data[,1], predict(fitted, new.data)[,"loc"], col = "blue")
legend("topleft", c("true values", "predict. values", "true curve", "predict. curve"),
col = c("red", "blue", "red", "blue"), pch = c(1, 5, NA, NA), inset = 0.05,
lty = c(0, 0, 1, 1), ncol = 2)
plot(locations[,2], param.scale, col = 2, ylim = c(7, 14),
ylab = "scale parameter", xlab = "latitude")
plot(fun, from = 0, to = 10, add = TRUE, col = 2)
points(locations[,2], predict(fitted)[,"scale"], col = "blue", pch = 5)
lines(new.data[,1], predict(fitted, new.data)[,"scale"], col = "blue")
legend("topleft", c("true values", "predict. values", "true curve", "predict. curve"),
col = c("red", "blue", "red", "blue"), pch = c(1, 5, NA, NA), inset = 0.05,
lty = c(0, 0, 1, 1), ncol = 2)
plot(locations[,1], param.shape, col = 2,
ylab = "shape parameter", xlab = "longitude")
plot(fun2, from = 0, to = 10, add = TRUE, col = 2)
points(locations[,1], predict(fitted)[,"shape"], col = "blue", pch = 5)
lines(new.data[,1], predict(fitted, new.data)[,"shape"], col = "blue")
legend("topleft", c("true values", "predict. values", "true curve", "predict. curve"),
col = c("red", "blue", "red", "blue"), pch = c(1, 5, NA, NA), inset = 0.05,
lty = c(0, 0, 1, 1), ncol = 2)
par(op)
## End(Not run)
Example output
Computing appropriate starting values
Starting values are defined
Starting values are:
nugget range smooth locCoeff1 locCoeff2
3.201048e-06 5.921723e+00 3.788132e-01 -9.772706e+00 1.518939e+00
scaleCoeff1 scaleCoeff2 scaleCoeff3 shapeCoeff1
6.793378e+00 1.398357e+00 1.023208e+00 1.723752e-01
Estimator: MPLE
Model: Schlather
Weighted: FALSE
Pair. Deviance: 360388.4
TIC: 362212.8
Covariance Family: Whittle-Matern
Estimates
Marginal Parameters:
Location Parameters:
locCoeff1 locCoeff2
-9.855 1.403
Scale Parameters:
scaleCoeff1 scaleCoeff2 scaleCoeff3
6.236 1.505 1.010
Shape Parameters:
shapeCoeff1
0.1688
Dependence Parameters:
nugget range smooth
5.870e-06 5.817e+00 3.788e-01
Standard Errors
nugget range smooth locCoeff1 locCoeff2 scaleCoeff1
0.11827 3.81939 0.20012 1.37896 0.93183 1.17829
scaleCoeff2 scaleCoeff3 shapeCoeff1
0.21399 0.08817 0.06780
Asymptotic Variance Covariance
nugget range smooth locCoeff1 locCoeff2 scaleCoeff1
nugget 0.013989 -0.331757 0.022205 -0.005838 -0.002202 -0.019901
range -0.331757 14.587768 -0.678794 0.280155 0.394891 1.679716
smooth 0.022205 -0.678794 0.040049 -0.009161 -0.010904 -0.049476
locCoeff1 -0.005838 0.280155 -0.009161 1.901540 -0.193194 0.756507
locCoeff2 -0.002202 0.394891 -0.010904 -0.193194 0.868303 0.268001
scaleCoeff1 -0.019901 1.679716 -0.049476 0.756507 0.268001 1.388376
scaleCoeff2 -0.005278 0.159521 -0.007570 -0.029934 0.079761 -0.063395
scaleCoeff3 -0.001956 0.163800 -0.004720 0.014052 0.037416 0.059547
shapeCoeff1 -0.003732 0.140531 -0.006708 0.003818 -0.007618 0.017012
scaleCoeff2 scaleCoeff3 shapeCoeff1
nugget -0.005278 -0.001956 -0.003732
range 0.159521 0.163800 0.140531
smooth -0.007570 -0.004720 -0.006708
locCoeff1 -0.029934 0.014052 0.003818
locCoeff2 0.079761 0.037416 -0.007618
scaleCoeff1 -0.063395 0.059547 0.017012
scaleCoeff2 0.045793 0.004016 0.001383
scaleCoeff3 0.004016 0.007774 0.001956
shapeCoeff1 0.001383 0.001956 0.004597
Optimization Information
Convergence: successful
Function Evaluations: 717
Estimator: MPLE
Model: Schlather
Weighted: FALSE
Pair. Deviance: 419537.9
TIC: 419686.5
Covariance Family: Whittle-Matern
Estimates
Marginal Parameters:
Assuming unit Frechet.
Dependence Parameters:
nugget range smooth
0.02268 2.81872 0.40923
Standard Errors
nugget range smooth
0.2083 1.4979 0.2931
Asymptotic Variance Covariance
nugget range smooth
nugget 0.04337 -0.28536 0.05976
range -0.28536 2.24363 -0.42030
smooth 0.05976 -0.42030 0.08590
Optimization Information
Convergence: successful
Function Evaluations: 144
Estimator: MPLE
Model: Schlather
Weighted: FALSE
Pair. Deviance: 419538
TIC: 419674.7
Covariance Family: Whittle-Matern
Estimates
Marginal Parameters:
Assuming unit Frechet.
Dependence Parameters:
nugget range
0.092 2.595
Standard Errors
nugget range
0.05206 0.58715
Asymptotic Variance Covariance
nugget range
nugget 0.00271 0.01916
range 0.01916 0.34474
Optimization Information
Convergence: successful
Function Evaluations: 55
Computing appropriate starting values
Starting values are defined
Starting values are:
cov11 cov12 cov22 locCoeff1 locCoeff2
122.035044305 21.417488646 292.111808323 8.968679178 -0.614563792
locCoeff3 locCoeff4 locCoeff5 locCoeff6 locCoeff7
0.023502186 0.031913850 0.010914613 -0.199661352 0.101344148
scaleCoeff1 scaleCoeff2 scaleCoeff3 scaleCoeff4 scaleCoeff5
19.082278205 -2.271890250 0.062481422 0.086448566 -0.223780602
scaleCoeff6 scaleCoeff7 shapeCoeff1 shapeCoeff2 shapeCoeff3
0.061938906 -0.013621861 0.396210581 -0.052537202 0.004254419
shapeCoeff4 shapeCoeff5 shapeCoeff6 shapeCoeff7
-0.003011063 0.025965291 -0.050155178 0.018978756
initial value 115622.796665
iter 10 value 114871.435594
iter 20 value 114609.795040
iter 30 value 114387.699316
iter 40 value 114387.050636
iter 40 value 114387.050333
iter 40 value 114387.050333
final value 114387.050333
converged
Estimator: MPLE
Model: Smith
Weighted: FALSE
Pair. Deviance: 228774.1
TIC: 232208.3
Covariance Family: Gaussian
Estimates
Marginal Parameters:
Location Parameters:
locCoeff1 locCoeff2 locCoeff3 locCoeff4 locCoeff5 locCoeff6 locCoeff7
9.459766 -0.618229 0.023920 -0.008505 0.097868 -0.266904 0.122033
Scale Parameters:
scaleCoeff1 scaleCoeff2 scaleCoeff3 scaleCoeff4 scaleCoeff5 scaleCoeff6
16.31512 -1.67953 0.04482 0.03944 -0.04788 -0.13452
scaleCoeff7
0.06571
Shape Parameters:
shapeCoeff1 shapeCoeff2 shapeCoeff3 shapeCoeff4 shapeCoeff5 shapeCoeff6
0.289157 -0.048372 0.006142 -0.007464 0.022317 -0.037363
shapeCoeff7
0.013554
Dependence Parameters:
cov11 cov12 cov22
151.654 8.533 304.588
Standard Errors
cov11 cov12 cov22 locCoeff1 locCoeff2 locCoeff3
5.591e+01 3.499e+01 1.107e+02 2.292e+00 3.241e-01 8.252e-03
locCoeff4 locCoeff5 locCoeff6 locCoeff7 scaleCoeff1 scaleCoeff2
2.500e-02 5.793e-02 6.960e-02 2.954e-02 2.474e+00 2.844e-01
scaleCoeff3 scaleCoeff4 scaleCoeff5 scaleCoeff6 scaleCoeff7 shapeCoeff1
6.969e-03 1.857e-02 3.342e-02 3.808e-02 1.659e-02 8.571e-02
shapeCoeff2 shapeCoeff3 shapeCoeff4 shapeCoeff5 shapeCoeff6 shapeCoeff7
1.140e-02 1.080e-03 1.671e-03 4.898e-03 7.192e-03 2.507e-03
Asymptotic Variance Covariance
cov11 cov12 cov22 locCoeff1 locCoeff2
cov11 3.126e+03 1.737e+02 4.032e+03 4.055e+01 -5.172e+00
cov12 1.737e+02 1.224e+03 7.479e+02 1.076e+01 -1.618e+00
cov22 4.032e+03 7.479e+02 1.226e+04 6.163e+01 -8.861e+00
locCoeff1 4.055e+01 1.076e+01 6.163e+01 5.252e+00 -6.996e-01
locCoeff2 -5.172e+00 -1.618e+00 -8.861e+00 -6.996e-01 1.051e-01
locCoeff3 7.071e-02 2.065e-02 2.344e-01 1.694e-02 -2.221e-03
locCoeff4 5.159e-01 9.331e-02 1.924e-01 1.525e-02 -3.510e-03
locCoeff5 -9.540e-01 -2.056e-01 -8.443e-01 -5.106e-02 1.152e-02
locCoeff6 6.018e-01 1.210e-01 5.719e-01 2.413e-02 -9.808e-03
locCoeff7 -2.175e-01 -4.064e-02 -2.217e-01 -9.944e-03 4.131e-03
scaleCoeff1 1.022e+02 3.131e-01 1.838e+02 1.572e+00 -1.029e-01
scaleCoeff2 -1.032e+01 7.982e-01 -1.651e+01 -9.757e-02 -1.183e-03
scaleCoeff3 2.839e-01 2.253e-03 4.035e-01 6.112e-04 1.725e-04
scaleCoeff4 1.195e-01 -1.297e-01 5.376e-01 1.297e-02 -8.230e-04
scaleCoeff5 -8.521e-02 3.547e-01 -9.323e-02 -3.252e-03 -1.704e-03
scaleCoeff6 -1.006e+00 -4.752e-01 -2.439e+00 -3.076e-02 5.639e-03
scaleCoeff7 4.619e-01 2.049e-01 1.124e+00 1.577e-02 -2.714e-03
shapeCoeff1 2.423e+00 7.510e-01 5.528e+00 -1.298e-03 -1.549e-03
shapeCoeff2 -4.116e-01 -1.253e-01 -8.000e-01 -6.096e-03 6.461e-04
shapeCoeff3 4.128e-02 6.060e-03 8.652e-02 8.036e-04 -1.008e-04
shapeCoeff4 -4.487e-02 -5.333e-03 -8.555e-02 -9.546e-04 1.460e-04
shapeCoeff5 1.331e-01 2.243e-03 2.163e-01 3.359e-03 -4.804e-04
shapeCoeff6 -2.353e-01 -1.078e-02 -4.117e-01 -5.816e-03 7.352e-04
shapeCoeff7 8.691e-02 5.505e-03 1.578e-01 2.142e-03 -2.503e-04
locCoeff3 locCoeff4 locCoeff5 locCoeff6 locCoeff7
cov11 7.071e-02 5.159e-01 -9.540e-01 6.018e-01 -2.175e-01
cov12 2.065e-02 9.331e-02 -2.056e-01 1.210e-01 -4.064e-02
cov22 2.344e-01 1.924e-01 -8.443e-01 5.719e-01 -2.217e-01
locCoeff1 1.694e-02 1.525e-02 -5.106e-02 2.413e-02 -9.944e-03
locCoeff2 -2.221e-03 -3.510e-03 1.152e-02 -9.808e-03 4.131e-03
locCoeff3 6.810e-05 -2.006e-05 -5.919e-05 -2.408e-05 9.132e-06
locCoeff4 -2.006e-05 6.250e-04 -1.346e-03 1.470e-03 -6.086e-04
locCoeff5 -5.919e-05 -1.346e-03 3.356e-03 -3.833e-03 1.605e-03
locCoeff6 -2.408e-05 1.470e-03 -3.833e-03 4.844e-03 -2.051e-03
locCoeff7 9.132e-06 -6.086e-04 1.605e-03 -2.051e-03 8.728e-04
scaleCoeff1 5.020e-03 -3.876e-03 1.820e-02 -4.657e-02 2.129e-02
scaleCoeff2 -3.090e-04 1.064e-03 -3.970e-03 7.126e-03 -3.211e-03
scaleCoeff3 -3.691e-06 1.328e-05 2.865e-05 -9.177e-05 4.590e-05
scaleCoeff4 7.302e-05 -2.168e-04 4.395e-04 -6.294e-04 2.622e-04
scaleCoeff5 -5.229e-05 4.556e-04 -1.132e-03 1.478e-03 -6.267e-04
scaleCoeff6 -7.269e-05 -4.768e-04 1.263e-03 -1.328e-03 5.540e-04
scaleCoeff7 4.240e-05 1.946e-04 -5.304e-04 5.367e-04 -2.240e-04
shapeCoeff1 -2.095e-05 3.011e-04 -7.140e-04 8.279e-04 -3.437e-04
shapeCoeff2 -1.324e-05 -3.581e-05 4.328e-05 4.494e-05 -2.387e-05
shapeCoeff3 2.701e-06 2.555e-06 -7.151e-06 6.717e-07 2.024e-08
shapeCoeff4 -3.377e-06 -5.226e-06 1.770e-05 -1.552e-05 6.495e-06
shapeCoeff5 1.235e-05 1.409e-05 -5.032e-05 4.035e-05 -1.668e-05
shapeCoeff6 -2.132e-05 -1.326e-05 4.886e-05 -1.324e-05 4.216e-06
shapeCoeff7 7.896e-06 2.146e-06 -9.349e-06 -7.438e-06 3.759e-06
scaleCoeff1 scaleCoeff2 scaleCoeff3 scaleCoeff4 scaleCoeff5
cov11 1.022e+02 -1.032e+01 2.839e-01 1.195e-01 -8.521e-02
cov12 3.131e-01 7.982e-01 2.253e-03 -1.297e-01 3.547e-01
cov22 1.838e+02 -1.651e+01 4.035e-01 5.376e-01 -9.323e-02
locCoeff1 1.572e+00 -9.757e-02 6.112e-04 1.297e-02 -3.252e-03
locCoeff2 -1.029e-01 -1.183e-03 1.725e-04 -8.230e-04 -1.704e-03
locCoeff3 5.020e-03 -3.090e-04 -3.691e-06 7.302e-05 -5.229e-05
locCoeff4 -3.876e-03 1.064e-03 1.328e-05 -2.168e-04 4.556e-04
locCoeff5 1.820e-02 -3.970e-03 2.865e-05 4.395e-04 -1.132e-03
locCoeff6 -4.657e-02 7.126e-03 -9.177e-05 -6.294e-04 1.478e-03
locCoeff7 2.129e-02 -3.211e-03 4.590e-05 2.622e-04 -6.267e-04
scaleCoeff1 6.123e+00 -6.777e-01 1.528e-02 2.347e-02 -3.932e-02
scaleCoeff2 -6.777e-01 8.091e-02 -1.789e-03 -2.643e-03 5.571e-03
scaleCoeff3 1.528e-02 -1.789e-03 4.856e-05 1.618e-05 -5.863e-05
scaleCoeff4 2.347e-02 -2.643e-03 1.618e-05 3.447e-04 -5.437e-04
scaleCoeff5 -3.932e-02 5.571e-03 -5.863e-05 -5.437e-04 1.117e-03
scaleCoeff6 -2.241e-02 2.926e-04 -6.544e-05 2.524e-04 -8.593e-04
scaleCoeff7 1.233e-02 -4.034e-04 3.088e-05 -7.828e-05 3.291e-04
shapeCoeff1 7.460e-02 -5.078e-03 2.275e-04 -1.388e-04 6.411e-04
shapeCoeff2 -1.668e-02 1.463e-03 -5.050e-05 1.326e-05 -6.637e-05
shapeCoeff3 2.007e-03 -1.856e-04 4.875e-06 4.620e-06 -8.170e-07
shapeCoeff4 -2.175e-03 2.010e-04 -5.428e-06 -2.363e-06 -5.140e-06
shapeCoeff5 6.866e-03 -6.830e-04 1.561e-05 2.174e-05 -1.534e-05
shapeCoeff6 -1.239e-02 1.235e-03 -2.785e-05 -4.703e-05 4.632e-05
shapeCoeff7 4.672e-03 -4.669e-04 1.046e-05 1.902e-05 -2.079e-05
scaleCoeff6 scaleCoeff7 shapeCoeff1 shapeCoeff2 shapeCoeff3
cov11 -1.006e+00 4.619e-01 2.423e+00 -4.116e-01 4.128e-02
cov12 -4.752e-01 2.049e-01 7.510e-01 -1.253e-01 6.060e-03
cov22 -2.439e+00 1.124e+00 5.528e+00 -8.000e-01 8.652e-02
locCoeff1 -3.076e-02 1.577e-02 -1.298e-03 -6.096e-03 8.036e-04
locCoeff2 5.639e-03 -2.714e-03 -1.549e-03 6.461e-04 -1.008e-04
locCoeff3 -7.269e-05 4.240e-05 -2.095e-05 -1.324e-05 2.701e-06
locCoeff4 -4.768e-04 1.946e-04 3.011e-04 -3.581e-05 2.555e-06
locCoeff5 1.263e-03 -5.304e-04 -7.140e-04 4.328e-05 -7.151e-06
locCoeff6 -1.328e-03 5.367e-04 8.279e-04 4.494e-05 6.717e-07
locCoeff7 5.540e-04 -2.240e-04 -3.437e-04 -2.387e-05 2.024e-08
scaleCoeff1 -2.241e-02 1.233e-02 7.460e-02 -1.668e-02 2.007e-03
scaleCoeff2 2.926e-04 -4.034e-04 -5.078e-03 1.463e-03 -1.856e-04
scaleCoeff3 -6.544e-05 3.088e-05 2.275e-04 -5.050e-05 4.875e-06
scaleCoeff4 2.524e-04 -7.828e-05 -1.388e-04 1.326e-05 4.620e-06
scaleCoeff5 -8.593e-04 3.291e-04 6.411e-04 -6.637e-05 -8.170e-07
scaleCoeff6 1.450e-03 -6.265e-04 -1.703e-03 2.599e-04 -2.586e-05
scaleCoeff7 -6.265e-04 2.753e-04 7.417e-04 -1.135e-04 1.220e-05
shapeCoeff1 -1.703e-03 7.417e-04 7.347e-03 -7.728e-04 7.079e-05
shapeCoeff2 2.599e-04 -1.135e-04 -7.728e-04 1.300e-04 -9.395e-06
shapeCoeff3 -2.586e-05 1.220e-05 7.079e-05 -9.395e-06 1.165e-06
shapeCoeff4 3.654e-05 -1.730e-05 -9.002e-05 9.920e-06 -1.607e-06
shapeCoeff5 -7.899e-05 3.991e-05 1.995e-04 -1.937e-05 4.357e-06
shapeCoeff6 1.164e-04 -5.957e-05 -3.297e-04 3.672e-05 -6.935e-06
shapeCoeff7 -3.919e-05 2.023e-05 1.201e-04 -1.420e-05 2.470e-06
shapeCoeff4 shapeCoeff5 shapeCoeff6 shapeCoeff7
cov11 -4.487e-02 1.331e-01 -2.353e-01 8.691e-02
cov12 -5.333e-03 2.243e-03 -1.078e-02 5.505e-03
cov22 -8.555e-02 2.163e-01 -4.117e-01 1.578e-01
locCoeff1 -9.546e-04 3.359e-03 -5.816e-03 2.142e-03
locCoeff2 1.460e-04 -4.804e-04 7.352e-04 -2.503e-04
locCoeff3 -3.377e-06 1.235e-05 -2.132e-05 7.896e-06
locCoeff4 -5.226e-06 1.409e-05 -1.326e-05 2.146e-06
locCoeff5 1.770e-05 -5.032e-05 4.886e-05 -9.349e-06
locCoeff6 -1.552e-05 4.035e-05 -1.324e-05 -7.438e-06
locCoeff7 6.495e-06 -1.668e-05 4.216e-06 3.759e-06
scaleCoeff1 -2.175e-03 6.866e-03 -1.239e-02 4.672e-03
scaleCoeff2 2.010e-04 -6.830e-04 1.235e-03 -4.669e-04
scaleCoeff3 -5.428e-06 1.561e-05 -2.785e-05 1.046e-05
scaleCoeff4 -2.363e-06 2.174e-05 -4.703e-05 1.902e-05
scaleCoeff5 -5.140e-06 -1.534e-05 4.632e-05 -2.079e-05
scaleCoeff6 3.654e-05 -7.899e-05 1.164e-04 -3.919e-05
scaleCoeff7 -1.730e-05 3.991e-05 -5.957e-05 2.023e-05
shapeCoeff1 -9.002e-05 1.995e-04 -3.297e-04 1.201e-04
shapeCoeff2 9.920e-06 -1.937e-05 3.672e-05 -1.420e-05
shapeCoeff3 -1.607e-06 4.357e-06 -6.935e-06 2.470e-06
shapeCoeff4 2.792e-06 -7.817e-06 1.114e-05 -3.730e-06
shapeCoeff5 -7.817e-06 2.399e-05 -3.436e-05 1.152e-05
shapeCoeff6 1.114e-05 -3.436e-05 5.173e-05 -1.787e-05
shapeCoeff7 -3.730e-06 1.152e-05 -1.787e-05 6.287e-06
Optimization Information
Convergence: successful
Function Evaluations: 221
Gradient Evaluations: 40
SpatialExtremes documentation built on Sept. 1, 2020, 3:01 a.m.
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Description Usage Arguments Details Value Warning Author(s) References Examples
Description
This function fits max-stable processes to data using pairwise likelihood. Two max-stable characterisations are available: the Smith and Schlather representations.
Usage
1 2 3 4 5 6 | fitmaxstab(data, coord, cov.mod, loc.form, scale.form,
shape.form, marg.cov = NULL, temp.cov = NULL, temp.form.loc = NULL,
temp.form.scale = NULL, temp.form.shape = NULL, iso = FALSE, ...,
fit.marge = FALSE, warn = TRUE, method = "Nelder", start, control =
list(), weights = NULL, corr = FALSE, check.grad
= FALSE)
|
Arguments
data |
A matrix representing the data. Each column corresponds to one location. |
coord |
A matrix that gives the coordinates of each location. Each row corresponds to one location. |
cov.mod |
A character string corresponding to the covariance model in the max-stable representation. Must be one of "gauss" for the Smith's model; "whitmat", "cauchy", "powexp", "bessel" or "caugen" for the Whittle-Matern, the Cauchy, the Powered Exponential, the Bessel and the Generalized Cauchy correlation families with the Schlather's model; "brown" for Brown-Resnick processes. The geometric Gaussian and Extremal-t models with a Whittle-Matern correlation function can be fitted by passing respectively "gwhitmat" or "twhitmat". Other correlation function families are considered in a similar way. |
loc.form, scale.form, shape.form |
R formulas defining the spatial linear model for the GEV parameters. May be missing. See section Details. |
marg.cov |
Matrix with named columns giving additional covariates
for the GEV parameters. If |
temp.cov |
Matrix with names columns giving additional *temporal*
covariates for the GEV parameters. If |
temp.form.loc, temp.form.scale, temp.form.shape |
R formulas defining the temporal trends for the GEV parameters. May be missing. See section Details. |
iso |
Logical. If |
... |
Several arguments to be passed to the
|
fit.marge |
Logical. If |
warn |
Logical. If |
method |
The method used for the numerical optimisation
procedure. Must be one of |
start |
A named list giving the initial values for the
parameters over which the pairwise likelihood is to be minimized. If
|
control |
A list giving the control parameters to be passed to
the |
weights |
A numeric vector specifying the weights in the pairwise
likelihood - and so has length the number of pairs. If |
corr |
Logical. If |
check.grad |
Logical. If |
Details
As spatial data often deal with a large number of locations, it is impossible to write analytically the joint distribution. Consequently, the fitting procedure substitutes the "full likelihood" for the pairwise likelihood.
Let define L_{i,j}(x_{i,j}, θ) the likelihood for site i and j, where i = 1, …, N-1, j=i+1, …, N, N is the number of site within the region and x_{i,j} are the joint observations for site i and j. Then the pairwise likelihood PL(θ) is defined by:
llik_P = log PL(θ) = ∑_{i = 1}^{N-1} ∑_{j=i+1}^{N} log L_{i,j} (x_{i,j}, θ)
As pairwise likelihood is an approximation of the “full likelihood”, standard errors cannot be computed directly by the inverse of the Fisher information matrix. Instead, a sandwich estimate must be used to account for model mispecification e.g.
hat(θ) ~ N(θ, H^{-1} J H^{-1})
where H is the Fisher information matrix (computed from the misspecified model) and J is the variance of the score function.
There are two different kind of covariates : "spatial" and "temporal".
A "spatial" covariate may have different values accross station but
does not depend on time. For example the coordinates of the stations
are obviously "spatial". These "spatial" covariates should be used
with the marg.cov and loc.form, scale.form, shape.form.
A "temporal" covariates may have different values accross time but
does not depend on space. For example the years where the annual
maxima were recorded is "temporal". These "temporal" covariates should
be used with the temp.cov and temp.form.loc,
temp.form.scale, temp.form.shape.
As a consequence note that marg.cov must have K rows (K being
the number of sites) while temp.cov must have n rows (n being
the number of observations).
Value
This function returns a object of class maxstab. Such objects
are list with components:
fitted.values |
A vector containing the estimated parameters. |
std.err |
A vector containing the standard errors. |
fixed |
A vector containing the parameters of the model that have been held fixed. |
param |
A vector containing all parameters (optimised and fixed). |
deviance |
The (pairwise) deviance at the maximum pairwise likelihood estimates. |
corr |
The correlation matrix. |
convergence, counts, message |
Components taken from the
list returned by |
data |
The data analysed. |
model |
The max-stable characterisation used. |
fit.marge |
A logical that specifies if the GEV margins were estimated. |
cov.fun |
The estimated covariance function - for the Schlather model only. |
extCoeff |
The estimated extremal coefficient function. |
cov.mod |
The covariance model for the spatial structure. |
Warning
When using reponse surfaces to model spatially the GEV parameters, the likelihood is pretty rough so that the general purpose optimization routines may fail. It is your responsability to check if the numerical optimization succeeded or not. I tried, as best as I can, to provide warning messages if the optimizers failed but in some cases, no warning will appear!
Author(s)
Mathieu Ribatet
References
Cox, D. R. and Reid, N. (2004) A note on pseudo-likelihood constructed from marginal densities. Biometrika 91, 729–737.
Demarta, S. and McNeil, A. (2005) The t copula and Related Copulas International Statistical Review 73, 111-129.
Gholam–Rezaee, M. (2009) Spatial extreme value: A composite likelihood. PhD Thesis. Ecole Polytechnique Federale de Lausanne.
Kabluchko, Z., Schlather, M. and de Haan, L. (2009) Stationary max-stable fields associated to negative definite functions Annals of Probability 37:5, 2042–2065.
Padoan, S. A. (2008) Computational Methods for Complex Problems in Extreme Value Theory. PhD Thesis. University of Padova.
Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010) Likelihood-based inference for max-stable processes. Journal of the American Statistical Association (Theory and Methods) 105:489, 263-277.
Schlather, M. (2002) Models for Stationary Max-Stable Random Fields. Extremes 5:1, 33–44.
Smith, R. L. (1990) Max-stable processes and spatial extremes. Unpublished.
Examples
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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | ## Not run:
##Define the coordinate of each location
n.site <- 30
locations <- matrix(runif(2*n.site, 0, 10), ncol = 2)
colnames(locations) <- c("lon", "lat")
##Simulate a max-stable process - with unit Frechet margins
data <- rmaxstab(40, locations, cov.mod = "whitmat", nugget = 0, range = 3,
smooth = 0.5)
##Now define the spatial model for the GEV parameters
param.loc <- -10 + 2 * locations[,2]
param.scale <- 5 + 2 * locations[,1] + locations[,2]^2
param.shape <- rep(0.2, n.site)
##Transform the unit Frechet margins to GEV
for (i in 1:n.site)
data[,i] <- frech2gev(data[,i], param.loc[i], param.scale[i],
param.shape[i])
##Define a model for the GEV margins to be fitted
##shape ~ 1 stands for the GEV shape parameter is constant
##over the region
loc.form <- loc ~ lat
scale.form <- scale ~ lon + I(lat^2)
shape.form <- shape ~ 1
##Fit a max-stable process using the Schlather's model
fitmaxstab(data, locations, "whitmat", loc.form, scale.form,
shape.form)
## Model without any spatial structure for the GEV parameters
## Be careful this could be *REALLY* time consuming
fitmaxstab(data, locations, "whitmat")
## Fixing the smooth parameter of the Whittle-Matern family
## to 0.5 - e.g. considering exponential family. We suppose the data
## are unit Frechet here.
fitmaxstab(data, locations, "whitmat", smooth = 0.5, fit.marge = FALSE)
## Fitting a penalized smoothing splines for the margins with the
## Smith's model
data <- rmaxstab(40, locations, cov.mod = "gauss", cov11 = 100, cov12 =
25, cov22 = 220)
## And transform it to ordinary GEV margins with a non-linear
## function
fun <- function(x)
2 * sin(pi * x / 4) + 10
fun2 <- function(x)
(fun(x) - 7 ) / 15
param.loc <- fun(locations[,2])
param.scale <- fun(locations[,2])
param.shape <- fun2(locations[,1])
##Transformation from unit Frechet to common GEV margins
for (i in 1:n.site)
data[,i] <- frech2gev(data[,i], param.loc[i], param.scale[i],
param.shape[i])
##Defining the knots, penalty, degree for the splines
n.knots <- 5
knots <- quantile(locations[,2], prob = 1:n.knots/(n.knots+1))
knots2 <- quantile(locations[,1], prob = 1:n.knots/(n.knots+1))
##Be careful the choice of the penalty (i.e. the smoothing parameter)
##may strongly affect the result Here we use p-splines for each GEV
##parameter - so it's really CPU demanding but one can use 1 p-spline
##and 2 linear models.
##A simple linear model will be clearly faster...
loc.form <- y ~ rb(lat, knots = knots, degree = 3, penalty = .5)
scale.form <- y ~ rb(lat, knots = knots, degree = 3, penalty = .5)
shape.form <- y ~ rb(lon, knots = knots2, degree = 3, penalty = .5)
fitted <- fitmaxstab(data, locations, "gauss", loc.form, scale.form, shape.form,
control = list(ndeps = rep(1e-6, 24), trace = 10),
method = "BFGS")
fitted
op <- par(mfrow=c(1,3))
plot(locations[,2], param.loc, col = 2, ylim = c(7, 14),
ylab = "location parameter", xlab = "latitude")
plot(fun, from = 0, to = 10, add = TRUE, col = 2)
points(locations[,2], predict(fitted)[,"loc"], col = "blue", pch = 5)
new.data <- cbind(lon = seq(0, 10, length = 100), lat = seq(0, 10, length = 100))
lines(new.data[,1], predict(fitted, new.data)[,"loc"], col = "blue")
legend("topleft", c("true values", "predict. values", "true curve", "predict. curve"),
col = c("red", "blue", "red", "blue"), pch = c(1, 5, NA, NA), inset = 0.05,
lty = c(0, 0, 1, 1), ncol = 2)
plot(locations[,2], param.scale, col = 2, ylim = c(7, 14),
ylab = "scale parameter", xlab = "latitude")
plot(fun, from = 0, to = 10, add = TRUE, col = 2)
points(locations[,2], predict(fitted)[,"scale"], col = "blue", pch = 5)
lines(new.data[,1], predict(fitted, new.data)[,"scale"], col = "blue")
legend("topleft", c("true values", "predict. values", "true curve", "predict. curve"),
col = c("red", "blue", "red", "blue"), pch = c(1, 5, NA, NA), inset = 0.05,
lty = c(0, 0, 1, 1), ncol = 2)
plot(locations[,1], param.shape, col = 2,
ylab = "shape parameter", xlab = "longitude")
plot(fun2, from = 0, to = 10, add = TRUE, col = 2)
points(locations[,1], predict(fitted)[,"shape"], col = "blue", pch = 5)
lines(new.data[,1], predict(fitted, new.data)[,"shape"], col = "blue")
legend("topleft", c("true values", "predict. values", "true curve", "predict. curve"),
col = c("red", "blue", "red", "blue"), pch = c(1, 5, NA, NA), inset = 0.05,
lty = c(0, 0, 1, 1), ncol = 2)
par(op)
## End(Not run)
|
Example output
Computing appropriate starting values
Starting values are defined
Starting values are:
nugget range smooth locCoeff1 locCoeff2
3.201048e-06 5.921723e+00 3.788132e-01 -9.772706e+00 1.518939e+00
scaleCoeff1 scaleCoeff2 scaleCoeff3 shapeCoeff1
6.793378e+00 1.398357e+00 1.023208e+00 1.723752e-01
Estimator: MPLE
Model: Schlather
Weighted: FALSE
Pair. Deviance: 360388.4
TIC: 362212.8
Covariance Family: Whittle-Matern
Estimates
Marginal Parameters:
Location Parameters:
locCoeff1 locCoeff2
-9.855 1.403
Scale Parameters:
scaleCoeff1 scaleCoeff2 scaleCoeff3
6.236 1.505 1.010
Shape Parameters:
shapeCoeff1
0.1688
Dependence Parameters:
nugget range smooth
5.870e-06 5.817e+00 3.788e-01
Standard Errors
nugget range smooth locCoeff1 locCoeff2 scaleCoeff1
0.11827 3.81939 0.20012 1.37896 0.93183 1.17829
scaleCoeff2 scaleCoeff3 shapeCoeff1
0.21399 0.08817 0.06780
Asymptotic Variance Covariance
nugget range smooth locCoeff1 locCoeff2 scaleCoeff1
nugget 0.013989 -0.331757 0.022205 -0.005838 -0.002202 -0.019901
range -0.331757 14.587768 -0.678794 0.280155 0.394891 1.679716
smooth 0.022205 -0.678794 0.040049 -0.009161 -0.010904 -0.049476
locCoeff1 -0.005838 0.280155 -0.009161 1.901540 -0.193194 0.756507
locCoeff2 -0.002202 0.394891 -0.010904 -0.193194 0.868303 0.268001
scaleCoeff1 -0.019901 1.679716 -0.049476 0.756507 0.268001 1.388376
scaleCoeff2 -0.005278 0.159521 -0.007570 -0.029934 0.079761 -0.063395
scaleCoeff3 -0.001956 0.163800 -0.004720 0.014052 0.037416 0.059547
shapeCoeff1 -0.003732 0.140531 -0.006708 0.003818 -0.007618 0.017012
scaleCoeff2 scaleCoeff3 shapeCoeff1
nugget -0.005278 -0.001956 -0.003732
range 0.159521 0.163800 0.140531
smooth -0.007570 -0.004720 -0.006708
locCoeff1 -0.029934 0.014052 0.003818
locCoeff2 0.079761 0.037416 -0.007618
scaleCoeff1 -0.063395 0.059547 0.017012
scaleCoeff2 0.045793 0.004016 0.001383
scaleCoeff3 0.004016 0.007774 0.001956
shapeCoeff1 0.001383 0.001956 0.004597
Optimization Information
Convergence: successful
Function Evaluations: 717
Estimator: MPLE
Model: Schlather
Weighted: FALSE
Pair. Deviance: 419537.9
TIC: 419686.5
Covariance Family: Whittle-Matern
Estimates
Marginal Parameters:
Assuming unit Frechet.
Dependence Parameters:
nugget range smooth
0.02268 2.81872 0.40923
Standard Errors
nugget range smooth
0.2083 1.4979 0.2931
Asymptotic Variance Covariance
nugget range smooth
nugget 0.04337 -0.28536 0.05976
range -0.28536 2.24363 -0.42030
smooth 0.05976 -0.42030 0.08590
Optimization Information
Convergence: successful
Function Evaluations: 144
Estimator: MPLE
Model: Schlather
Weighted: FALSE
Pair. Deviance: 419538
TIC: 419674.7
Covariance Family: Whittle-Matern
Estimates
Marginal Parameters:
Assuming unit Frechet.
Dependence Parameters:
nugget range
0.092 2.595
Standard Errors
nugget range
0.05206 0.58715
Asymptotic Variance Covariance
nugget range
nugget 0.00271 0.01916
range 0.01916 0.34474
Optimization Information
Convergence: successful
Function Evaluations: 55
Computing appropriate starting values
Starting values are defined
Starting values are:
cov11 cov12 cov22 locCoeff1 locCoeff2
122.035044305 21.417488646 292.111808323 8.968679178 -0.614563792
locCoeff3 locCoeff4 locCoeff5 locCoeff6 locCoeff7
0.023502186 0.031913850 0.010914613 -0.199661352 0.101344148
scaleCoeff1 scaleCoeff2 scaleCoeff3 scaleCoeff4 scaleCoeff5
19.082278205 -2.271890250 0.062481422 0.086448566 -0.223780602
scaleCoeff6 scaleCoeff7 shapeCoeff1 shapeCoeff2 shapeCoeff3
0.061938906 -0.013621861 0.396210581 -0.052537202 0.004254419
shapeCoeff4 shapeCoeff5 shapeCoeff6 shapeCoeff7
-0.003011063 0.025965291 -0.050155178 0.018978756
initial value 115622.796665
iter 10 value 114871.435594
iter 20 value 114609.795040
iter 30 value 114387.699316
iter 40 value 114387.050636
iter 40 value 114387.050333
iter 40 value 114387.050333
final value 114387.050333
converged
Estimator: MPLE
Model: Smith
Weighted: FALSE
Pair. Deviance: 228774.1
TIC: 232208.3
Covariance Family: Gaussian
Estimates
Marginal Parameters:
Location Parameters:
locCoeff1 locCoeff2 locCoeff3 locCoeff4 locCoeff5 locCoeff6 locCoeff7
9.459766 -0.618229 0.023920 -0.008505 0.097868 -0.266904 0.122033
Scale Parameters:
scaleCoeff1 scaleCoeff2 scaleCoeff3 scaleCoeff4 scaleCoeff5 scaleCoeff6
16.31512 -1.67953 0.04482 0.03944 -0.04788 -0.13452
scaleCoeff7
0.06571
Shape Parameters:
shapeCoeff1 shapeCoeff2 shapeCoeff3 shapeCoeff4 shapeCoeff5 shapeCoeff6
0.289157 -0.048372 0.006142 -0.007464 0.022317 -0.037363
shapeCoeff7
0.013554
Dependence Parameters:
cov11 cov12 cov22
151.654 8.533 304.588
Standard Errors
cov11 cov12 cov22 locCoeff1 locCoeff2 locCoeff3
5.591e+01 3.499e+01 1.107e+02 2.292e+00 3.241e-01 8.252e-03
locCoeff4 locCoeff5 locCoeff6 locCoeff7 scaleCoeff1 scaleCoeff2
2.500e-02 5.793e-02 6.960e-02 2.954e-02 2.474e+00 2.844e-01
scaleCoeff3 scaleCoeff4 scaleCoeff5 scaleCoeff6 scaleCoeff7 shapeCoeff1
6.969e-03 1.857e-02 3.342e-02 3.808e-02 1.659e-02 8.571e-02
shapeCoeff2 shapeCoeff3 shapeCoeff4 shapeCoeff5 shapeCoeff6 shapeCoeff7
1.140e-02 1.080e-03 1.671e-03 4.898e-03 7.192e-03 2.507e-03
Asymptotic Variance Covariance
cov11 cov12 cov22 locCoeff1 locCoeff2
cov11 3.126e+03 1.737e+02 4.032e+03 4.055e+01 -5.172e+00
cov12 1.737e+02 1.224e+03 7.479e+02 1.076e+01 -1.618e+00
cov22 4.032e+03 7.479e+02 1.226e+04 6.163e+01 -8.861e+00
locCoeff1 4.055e+01 1.076e+01 6.163e+01 5.252e+00 -6.996e-01
locCoeff2 -5.172e+00 -1.618e+00 -8.861e+00 -6.996e-01 1.051e-01
locCoeff3 7.071e-02 2.065e-02 2.344e-01 1.694e-02 -2.221e-03
locCoeff4 5.159e-01 9.331e-02 1.924e-01 1.525e-02 -3.510e-03
locCoeff5 -9.540e-01 -2.056e-01 -8.443e-01 -5.106e-02 1.152e-02
locCoeff6 6.018e-01 1.210e-01 5.719e-01 2.413e-02 -9.808e-03
locCoeff7 -2.175e-01 -4.064e-02 -2.217e-01 -9.944e-03 4.131e-03
scaleCoeff1 1.022e+02 3.131e-01 1.838e+02 1.572e+00 -1.029e-01
scaleCoeff2 -1.032e+01 7.982e-01 -1.651e+01 -9.757e-02 -1.183e-03
scaleCoeff3 2.839e-01 2.253e-03 4.035e-01 6.112e-04 1.725e-04
scaleCoeff4 1.195e-01 -1.297e-01 5.376e-01 1.297e-02 -8.230e-04
scaleCoeff5 -8.521e-02 3.547e-01 -9.323e-02 -3.252e-03 -1.704e-03
scaleCoeff6 -1.006e+00 -4.752e-01 -2.439e+00 -3.076e-02 5.639e-03
scaleCoeff7 4.619e-01 2.049e-01 1.124e+00 1.577e-02 -2.714e-03
shapeCoeff1 2.423e+00 7.510e-01 5.528e+00 -1.298e-03 -1.549e-03
shapeCoeff2 -4.116e-01 -1.253e-01 -8.000e-01 -6.096e-03 6.461e-04
shapeCoeff3 4.128e-02 6.060e-03 8.652e-02 8.036e-04 -1.008e-04
shapeCoeff4 -4.487e-02 -5.333e-03 -8.555e-02 -9.546e-04 1.460e-04
shapeCoeff5 1.331e-01 2.243e-03 2.163e-01 3.359e-03 -4.804e-04
shapeCoeff6 -2.353e-01 -1.078e-02 -4.117e-01 -5.816e-03 7.352e-04
shapeCoeff7 8.691e-02 5.505e-03 1.578e-01 2.142e-03 -2.503e-04
locCoeff3 locCoeff4 locCoeff5 locCoeff6 locCoeff7
cov11 7.071e-02 5.159e-01 -9.540e-01 6.018e-01 -2.175e-01
cov12 2.065e-02 9.331e-02 -2.056e-01 1.210e-01 -4.064e-02
cov22 2.344e-01 1.924e-01 -8.443e-01 5.719e-01 -2.217e-01
locCoeff1 1.694e-02 1.525e-02 -5.106e-02 2.413e-02 -9.944e-03
locCoeff2 -2.221e-03 -3.510e-03 1.152e-02 -9.808e-03 4.131e-03
locCoeff3 6.810e-05 -2.006e-05 -5.919e-05 -2.408e-05 9.132e-06
locCoeff4 -2.006e-05 6.250e-04 -1.346e-03 1.470e-03 -6.086e-04
locCoeff5 -5.919e-05 -1.346e-03 3.356e-03 -3.833e-03 1.605e-03
locCoeff6 -2.408e-05 1.470e-03 -3.833e-03 4.844e-03 -2.051e-03
locCoeff7 9.132e-06 -6.086e-04 1.605e-03 -2.051e-03 8.728e-04
scaleCoeff1 5.020e-03 -3.876e-03 1.820e-02 -4.657e-02 2.129e-02
scaleCoeff2 -3.090e-04 1.064e-03 -3.970e-03 7.126e-03 -3.211e-03
scaleCoeff3 -3.691e-06 1.328e-05 2.865e-05 -9.177e-05 4.590e-05
scaleCoeff4 7.302e-05 -2.168e-04 4.395e-04 -6.294e-04 2.622e-04
scaleCoeff5 -5.229e-05 4.556e-04 -1.132e-03 1.478e-03 -6.267e-04
scaleCoeff6 -7.269e-05 -4.768e-04 1.263e-03 -1.328e-03 5.540e-04
scaleCoeff7 4.240e-05 1.946e-04 -5.304e-04 5.367e-04 -2.240e-04
shapeCoeff1 -2.095e-05 3.011e-04 -7.140e-04 8.279e-04 -3.437e-04
shapeCoeff2 -1.324e-05 -3.581e-05 4.328e-05 4.494e-05 -2.387e-05
shapeCoeff3 2.701e-06 2.555e-06 -7.151e-06 6.717e-07 2.024e-08
shapeCoeff4 -3.377e-06 -5.226e-06 1.770e-05 -1.552e-05 6.495e-06
shapeCoeff5 1.235e-05 1.409e-05 -5.032e-05 4.035e-05 -1.668e-05
shapeCoeff6 -2.132e-05 -1.326e-05 4.886e-05 -1.324e-05 4.216e-06
shapeCoeff7 7.896e-06 2.146e-06 -9.349e-06 -7.438e-06 3.759e-06
scaleCoeff1 scaleCoeff2 scaleCoeff3 scaleCoeff4 scaleCoeff5
cov11 1.022e+02 -1.032e+01 2.839e-01 1.195e-01 -8.521e-02
cov12 3.131e-01 7.982e-01 2.253e-03 -1.297e-01 3.547e-01
cov22 1.838e+02 -1.651e+01 4.035e-01 5.376e-01 -9.323e-02
locCoeff1 1.572e+00 -9.757e-02 6.112e-04 1.297e-02 -3.252e-03
locCoeff2 -1.029e-01 -1.183e-03 1.725e-04 -8.230e-04 -1.704e-03
locCoeff3 5.020e-03 -3.090e-04 -3.691e-06 7.302e-05 -5.229e-05
locCoeff4 -3.876e-03 1.064e-03 1.328e-05 -2.168e-04 4.556e-04
locCoeff5 1.820e-02 -3.970e-03 2.865e-05 4.395e-04 -1.132e-03
locCoeff6 -4.657e-02 7.126e-03 -9.177e-05 -6.294e-04 1.478e-03
locCoeff7 2.129e-02 -3.211e-03 4.590e-05 2.622e-04 -6.267e-04
scaleCoeff1 6.123e+00 -6.777e-01 1.528e-02 2.347e-02 -3.932e-02
scaleCoeff2 -6.777e-01 8.091e-02 -1.789e-03 -2.643e-03 5.571e-03
scaleCoeff3 1.528e-02 -1.789e-03 4.856e-05 1.618e-05 -5.863e-05
scaleCoeff4 2.347e-02 -2.643e-03 1.618e-05 3.447e-04 -5.437e-04
scaleCoeff5 -3.932e-02 5.571e-03 -5.863e-05 -5.437e-04 1.117e-03
scaleCoeff6 -2.241e-02 2.926e-04 -6.544e-05 2.524e-04 -8.593e-04
scaleCoeff7 1.233e-02 -4.034e-04 3.088e-05 -7.828e-05 3.291e-04
shapeCoeff1 7.460e-02 -5.078e-03 2.275e-04 -1.388e-04 6.411e-04
shapeCoeff2 -1.668e-02 1.463e-03 -5.050e-05 1.326e-05 -6.637e-05
shapeCoeff3 2.007e-03 -1.856e-04 4.875e-06 4.620e-06 -8.170e-07
shapeCoeff4 -2.175e-03 2.010e-04 -5.428e-06 -2.363e-06 -5.140e-06
shapeCoeff5 6.866e-03 -6.830e-04 1.561e-05 2.174e-05 -1.534e-05
shapeCoeff6 -1.239e-02 1.235e-03 -2.785e-05 -4.703e-05 4.632e-05
shapeCoeff7 4.672e-03 -4.669e-04 1.046e-05 1.902e-05 -2.079e-05
scaleCoeff6 scaleCoeff7 shapeCoeff1 shapeCoeff2 shapeCoeff3
cov11 -1.006e+00 4.619e-01 2.423e+00 -4.116e-01 4.128e-02
cov12 -4.752e-01 2.049e-01 7.510e-01 -1.253e-01 6.060e-03
cov22 -2.439e+00 1.124e+00 5.528e+00 -8.000e-01 8.652e-02
locCoeff1 -3.076e-02 1.577e-02 -1.298e-03 -6.096e-03 8.036e-04
locCoeff2 5.639e-03 -2.714e-03 -1.549e-03 6.461e-04 -1.008e-04
locCoeff3 -7.269e-05 4.240e-05 -2.095e-05 -1.324e-05 2.701e-06
locCoeff4 -4.768e-04 1.946e-04 3.011e-04 -3.581e-05 2.555e-06
locCoeff5 1.263e-03 -5.304e-04 -7.140e-04 4.328e-05 -7.151e-06
locCoeff6 -1.328e-03 5.367e-04 8.279e-04 4.494e-05 6.717e-07
locCoeff7 5.540e-04 -2.240e-04 -3.437e-04 -2.387e-05 2.024e-08
scaleCoeff1 -2.241e-02 1.233e-02 7.460e-02 -1.668e-02 2.007e-03
scaleCoeff2 2.926e-04 -4.034e-04 -5.078e-03 1.463e-03 -1.856e-04
scaleCoeff3 -6.544e-05 3.088e-05 2.275e-04 -5.050e-05 4.875e-06
scaleCoeff4 2.524e-04 -7.828e-05 -1.388e-04 1.326e-05 4.620e-06
scaleCoeff5 -8.593e-04 3.291e-04 6.411e-04 -6.637e-05 -8.170e-07
scaleCoeff6 1.450e-03 -6.265e-04 -1.703e-03 2.599e-04 -2.586e-05
scaleCoeff7 -6.265e-04 2.753e-04 7.417e-04 -1.135e-04 1.220e-05
shapeCoeff1 -1.703e-03 7.417e-04 7.347e-03 -7.728e-04 7.079e-05
shapeCoeff2 2.599e-04 -1.135e-04 -7.728e-04 1.300e-04 -9.395e-06
shapeCoeff3 -2.586e-05 1.220e-05 7.079e-05 -9.395e-06 1.165e-06
shapeCoeff4 3.654e-05 -1.730e-05 -9.002e-05 9.920e-06 -1.607e-06
shapeCoeff5 -7.899e-05 3.991e-05 1.995e-04 -1.937e-05 4.357e-06
shapeCoeff6 1.164e-04 -5.957e-05 -3.297e-04 3.672e-05 -6.935e-06
shapeCoeff7 -3.919e-05 2.023e-05 1.201e-04 -1.420e-05 2.470e-06
shapeCoeff4 shapeCoeff5 shapeCoeff6 shapeCoeff7
cov11 -4.487e-02 1.331e-01 -2.353e-01 8.691e-02
cov12 -5.333e-03 2.243e-03 -1.078e-02 5.505e-03
cov22 -8.555e-02 2.163e-01 -4.117e-01 1.578e-01
locCoeff1 -9.546e-04 3.359e-03 -5.816e-03 2.142e-03
locCoeff2 1.460e-04 -4.804e-04 7.352e-04 -2.503e-04
locCoeff3 -3.377e-06 1.235e-05 -2.132e-05 7.896e-06
locCoeff4 -5.226e-06 1.409e-05 -1.326e-05 2.146e-06
locCoeff5 1.770e-05 -5.032e-05 4.886e-05 -9.349e-06
locCoeff6 -1.552e-05 4.035e-05 -1.324e-05 -7.438e-06
locCoeff7 6.495e-06 -1.668e-05 4.216e-06 3.759e-06
scaleCoeff1 -2.175e-03 6.866e-03 -1.239e-02 4.672e-03
scaleCoeff2 2.010e-04 -6.830e-04 1.235e-03 -4.669e-04
scaleCoeff3 -5.428e-06 1.561e-05 -2.785e-05 1.046e-05
scaleCoeff4 -2.363e-06 2.174e-05 -4.703e-05 1.902e-05
scaleCoeff5 -5.140e-06 -1.534e-05 4.632e-05 -2.079e-05
scaleCoeff6 3.654e-05 -7.899e-05 1.164e-04 -3.919e-05
scaleCoeff7 -1.730e-05 3.991e-05 -5.957e-05 2.023e-05
shapeCoeff1 -9.002e-05 1.995e-04 -3.297e-04 1.201e-04
shapeCoeff2 9.920e-06 -1.937e-05 3.672e-05 -1.420e-05
shapeCoeff3 -1.607e-06 4.357e-06 -6.935e-06 2.470e-06
shapeCoeff4 2.792e-06 -7.817e-06 1.114e-05 -3.730e-06
shapeCoeff5 -7.817e-06 2.399e-05 -3.436e-05 1.152e-05
shapeCoeff6 1.114e-05 -3.436e-05 5.173e-05 -1.787e-05
shapeCoeff7 -3.730e-06 1.152e-05 -1.787e-05 6.287e-06
Optimization Information
Convergence: successful
Function Evaluations: 221
Gradient Evaluations: 40
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