| Title: | Spatial Deformation and Dimension Expansion Gaussian Processes |
|---|---|
| Description: | Methods for fitting nonstationary Gaussian process models by spatial deformation, as introduced by Sampson and Guttorp (1992) <doi:10.1080/01621459.1992.10475181>, and by dimension expansion, as introduced by Bornn et al. (2012) <doi:10.1080/01621459.2011.646919>. Low-rank thin-plate regression splines, as developed in Wood, S.N. (2003) <doi:10.1111/1467-9868.00374>, are used to either transform co-ordinates or create new latent dimensions. |
| Authors: | Ben Youngman [aut, cre]
|
| Maintainer: | Ben Youngman <[email protected]> |
| License: | GPL-3 |
| Version: | 1.0.0 |
| Built: | 2024-11-12 06:46:23 UTC |
| Source: | CRAN |
Function aniso fits a conventional 2-dimensional anisotropic Gaussian
process, i.e. just with scalings in the x and y coordinates.
aniso(x, z, n, correlation = FALSE, cosine = FALSE, standardise = "together")aniso(x, z, n, correlation = FALSE, cosine = FALSE, standardise = "together")
x |
a 2-column matrix comprising x and y coordinates column-wise, respectively, or a list; see Details for the latter |
z |
a variance-covariance matrix |
n |
an integer number of data |
correlation |
a logical defining whether |
cosine |
a logical defining whether the powered exponential covariance function should be multiplied by the cosine of scaled distances, i.e. giving a damped oscillation; defaults to |
standardise |
a character string that governs whether dimensions are scaled by a common ( |
If x is a list, then it wants elements "x", "z" and "n" as described above.
An object of class deform and then of class anisotropic
Sampson, P. D. and Guttorp, P. (1992) Nonparametric Estimation of Nonstationary Spatial Covariance Structure, Journal of the American Statistical Association, 87:417, 108-119, doi:10.1080/01621459.1992.10475181'
data(solar) aniso(solar$x, solar$z, solar$n) # equivalent to aniso(solar)data(solar) aniso(solar$x, solar$z, solar$n) # equivalent to aniso(solar)
Correlation and covariance matrices from censored data
cencov(x, u) cencor(x, u)cencov(x, u) cencor(x, u)
x |
a numeric matrix |
u |
a numeric matrix giving corresponding points of left-censoring |
For cencov() a covariance matrix is returned and for
cencor() a correlation matrix is returned. Note that cencov()
calls cencor(). Estimates are based on assuming values are from a
multivariate Gaussian distribution.
a matrix
cov and cor for uncensored estimates.
# generate some correlated data n <- 1e2 x <- rnorm(n) y <- 0.25 * x + sqrt(0.75) * rnorm(n) xy <- cbind(x, y) # threshold of zero for left-censoring u <- matrix(0, n, 2) # left-censored correlation matrix cencor(xy, u) # could check with cor(xy) # left-censored covariance matrix cencov(xy, u)# generate some correlated data n <- 1e2 x <- rnorm(n) y <- 0.25 * x + sqrt(0.75) * rnorm(n) xy <- cbind(x, y) # threshold of zero for left-censoring u <- matrix(0, n, 2) # left-censored correlation matrix cencor(xy, u) # could check with cor(xy) # left-censored covariance matrix cencov(xy, u)
Function deform fits a 2-dimensional deformation model, where typically
x and y coordinates in geographic (G-) space will be provided and then deformed
to give new coordinates in deformed (D-) space in which isotropy of a Gaussian
process is optimally achieved.
deform( x, z, n, k = c(10, 10), lambda = c(-1, -1), lambda0 = rep(exp(3), length(k)), correlation = FALSE, cosine = FALSE, bijective = FALSE, bijective.args = NULL, trace = 0, standardise = "together" )deform( x, z, n, k = c(10, 10), lambda = c(-1, -1), lambda0 = rep(exp(3), length(k)), correlation = FALSE, cosine = FALSE, bijective = FALSE, bijective.args = NULL, trace = 0, standardise = "together" )
x |
a 2-column matrix comprising x and y coordinates column-wise, respectively, or a list; see Details for the latter |
z |
a variance-covariance matrix |
n |
an integer number of data |
k |
an integer vector of ranks |
lambda |
specified lambda values; see Details |
lambda0 |
initial lambda values |
correlation |
a logical defining whether |
cosine |
a logical defining whether the powered exponential covariance function should be multiplied by the cosine of scaled distances, i.e. giving a damped oscillation; defaults to |
bijective |
a logical for whether a bijective deformation should be imposed; defaults to FALSE |
bijective.args |
a list specifying quantities to ensure bijectivity, if bijective == TRUE; see Details |
trace |
an integer specifying the amount to report on optimisation (0, default, is nothing; 1 gives a bit) |
standardise |
a character string that governs whether dimensions are scaled by a common ( |
If x is a list, then it wants elements "x", "z" and "n" as described above.
Values of lambda multiply the penalties placed on the wiggliness of the
smooths that form the deformations. Larger values make things less wiggly. Values
of lambda0 specify initial values for lambda, which are still optimised.
bijective.args() is a 4-element list: "mult" is a penalty placed on
the numerical approximation to identifying non-bijectivity, where larger values
impose bijectivity more strictly; "scl" is a scaling placed on the grid
used to numerically identify non-bijectivity, where smaller values will typically
impose bijectivity more strictly; "nx" and "ny" specify the x and y
dimensions of the grid used to numerically identify bijectivity. Defaults are
mult = 1e3, scl = 1, nx = 40 and ny = 40. It is advisable
to use "mult" and not "scl" to control bijectivity, in the first instance.
An object of class deform and then of class deformation
Sampson, P. D. and Guttorp, P. (1992) Nonparametric Estimation of Nonstationary Spatial Covariance Structure, Journal of the American Statistical Association, 87:417, 108-119, doi:10.1080/01621459.1992.10475181
Wood, S.N. (2003), Thin plate regression splines. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65: 95-114. doi:10.1111/1467-9868.00374
data(solar) deform(solar$x, solar$z, solar$n) # equivalent to deform(solar) # bijective deformation deform(solar, bijective = TRUE) # deformation with specified rank deform(solar, k = c(10, 8))data(solar) deform(solar$x, solar$z, solar$n) # equivalent to deform(solar) # bijective deformation deform(solar, bijective = TRUE) # deformation with specified rank deform(solar, k = c(10, 8))
Function exapnd fits a multi-dimensional dimension expansion model, where typically
x and y coordinates in geographic (G-) space will be provided and then scaled and
combined with new latent dimensions (that a functions of x and y) to give new coordinates
in deformed (D-) space in which isotropy of a Gaussian process is optimally achieved.
expand( x, z, n, k = 10, lambda = rep(-1, length(k)), lambda0 = rep(exp(3), length(k)), correlation = FALSE, cosine = FALSE, trace = 0, z0 = NULL, standardise = "together" )expand( x, z, n, k = 10, lambda = rep(-1, length(k)), lambda0 = rep(exp(3), length(k)), correlation = FALSE, cosine = FALSE, trace = 0, z0 = NULL, standardise = "together" )
x |
a 2-column matrix comprising x and y coordinates column-wise, respectively, or a list; see Details for the latter |
z |
a variance-covariance matrix |
n |
an integer number of data |
k |
an integer vector of ranks |
lambda |
specified lambda values |
lambda0 |
initial lambda values |
correlation |
a logical defining whether |
cosine |
a logical defining whether the powered exponential covariance function should be multiplied by the cosine of scaled distances, i.e. giving a damped oscillation; defaults to |
trace |
an integer specifying the amount to report on optimisation (0, default, is nothing; 1 gives a bit) |
z0 |
a scalar giving initial values (which alternate |
standardise |
a character string that governs whether dimensions are scaled by a common ( |
If x is a list, then it wants elements "x", "z" and "n" as described above.
An object of class deform and then of class expansion
Bornn, L., Shaddick, G., & Zidek, J. V. (2012). Modeling nonstationary processes through dimension expansion. Journal of the American Statistical Association, 107(497), 281-289. doi:10.1080/01621459.2011.646919.
# one-dimensional expansion data(solar) expand(solar$x, solar$z, solar$n) # equivalent to expand(solar) # two-dimensional expansion with rank-8 and rank-5 dimensions expand(solar$x, solar$z, solar$n, c(8, 5))# one-dimensional expansion data(solar) expand(solar$x, solar$z, solar$n) # equivalent to expand(solar) # two-dimensional expansion with rank-8 and rank-5 dimensions expand(solar$x, solar$z, solar$n, c(8, 5))
deform objectPlot a fitted deform object
## S3 method for class 'deform' plot( x, start = 1, graphics = "base", breaks = NULL, pal = function(n) hcl.colors(n, "YlOrRd", rev = TRUE), onepage = FALSE, nx = 10, ny = 10, xp = NULL, yp = NULL, xlab = NULL, ylab = NULL, ... )## S3 method for class 'deform' plot( x, start = 1, graphics = "base", breaks = NULL, pal = function(n) hcl.colors(n, "YlOrRd", rev = TRUE), onepage = FALSE, nx = 10, ny = 10, xp = NULL, yp = NULL, xlab = NULL, ylab = NULL, ... )
x |
a fitted |
start |
an integer giving the starting dimension of plots of dimension expansion models; defaults to 1 |
graphics |
a character string that is either |
breaks |
an integer, vector or list; see Details |
pal |
a function specifying the colour palette to use for plots; defaults to |
onepage |
a logical specifying whether all plots should be put on one page; defaults to |
nx |
number of x points to use for plotting grid |
ny |
number of y points to use for plotting grid |
xp |
x points to use for plotting grid |
yp |
y points to use for plotting grid |
xlab |
x-axis label |
ylab |
y-axis label |
... |
extra arguments to pass to |
If breaks is an integer then it specifies the number of breaks to use for colour scales; if it's a vector, then it's the breaks themselves; and if it's a list then it's different breaks for each dimension.
Plots representing all one- or two-dimensional smooths
# deformations data(solar) m0 <- deform(solar$x, solar$z, solar$n) # plot representation of deformation plot(m0) # as above with specified x and y grid xvals <- seq(-123.3, -122.25, by = .05) yvals <- seq(49, 49.4, by = .05) plot(m0, xp = xvals, yp = yvals) # one-dimensional expansion data(solar) m1 <- expand(solar$x, solar$z, solar$n) # plot its three dimensions op <- par(mfrow = c(1, 3)) plot(m1) par(op) # or plot using lattice::levelplot plot(m1, graphics = 'lattice') # or as above, but on one page plot(m1, graphics = 'lattice', onepage = TRUE) # two-dimensional expansion m2 <- expand(solar$x, solar$z, solar$n, c(8, 5)) # plot of its third and fourth dimensions for given x and y values op <- par(mfrow = c(1, 2)) plot(m2, start = 3, xp = xvals, yp = yvals) par(op) # using lattice::levelplot with common breaks across dimensions with # a palette that gives latent dimensions in white where near zero plot(m2, onepage = TRUE, graphics = 'lattice', breaks = seq(-0.35, 0.35, by = 0.1), pal = function(n) hcl.colors(n, 'Blue-Red 3'))# deformations data(solar) m0 <- deform(solar$x, solar$z, solar$n) # plot representation of deformation plot(m0) # as above with specified x and y grid xvals <- seq(-123.3, -122.25, by = .05) yvals <- seq(49, 49.4, by = .05) plot(m0, xp = xvals, yp = yvals) # one-dimensional expansion data(solar) m1 <- expand(solar$x, solar$z, solar$n) # plot its three dimensions op <- par(mfrow = c(1, 3)) plot(m1) par(op) # or plot using lattice::levelplot plot(m1, graphics = 'lattice') # or as above, but on one page plot(m1, graphics = 'lattice', onepage = TRUE) # two-dimensional expansion m2 <- expand(solar$x, solar$z, solar$n, c(8, 5)) # plot of its third and fourth dimensions for given x and y values op <- par(mfrow = c(1, 2)) plot(m2, start = 3, xp = xvals, yp = yvals) par(op) # using lattice::levelplot with common breaks across dimensions with # a palette that gives latent dimensions in white where near zero plot(m2, onepage = TRUE, graphics = 'lattice', breaks = seq(-0.35, 0.35, by = 0.1), pal = function(n) hcl.colors(n, 'Blue-Red 3'))
deform objectPredict from a fitted deform object
## S3 method for class 'deform' predict(object, newdata = NULL, ...)## S3 method for class 'deform' predict(object, newdata = NULL, ...)
object |
a fitted |
newdata |
a 2-column matrix of x and y coordinates |
... |
currently just a placeholder |
A 2-column matrix of predicted x and y points for deformations and a (2 + q)-column matrix for q-dimensional expansions.
# fit a deformation model data(solar) m0 <- deform(solar$x, solar$z, solar$n) # predict D-space points for original locations predict(m0) # predictions for one-dimensional expansion model with specified locations # and standard error estimates data(solar) m1 <- expand(solar$x, solar$z, solar$n) xvals <- seq(-123.3, -122.2, by = .1) yvals <- seq(49, 49.4, by = .1) xyvals <- expand.grid(xvals, yvals) predict(m1, xyvals, se.fit = TRUE)# fit a deformation model data(solar) m0 <- deform(solar$x, solar$z, solar$n) # predict D-space points for original locations predict(m0) # predictions for one-dimensional expansion model with specified locations # and standard error estimates data(solar) m1 <- expand(solar$x, solar$z, solar$n) xvals <- seq(-123.3, -122.2, by = .1) yvals <- seq(49, 49.4, by = .1) xyvals <- expand.grid(xvals, yvals) predict(m1, xyvals, se.fit = TRUE)
deform objectSimulate from a fitted deform object
## S3 method for class 'deform' simulate(object, nsim = 1, seed = NULL, newdata = NULL, ...)## S3 method for class 'deform' simulate(object, nsim = 1, seed = NULL, newdata = NULL, ...)
object |
a fitted |
nsim |
an integer giving the number of simulations |
seed |
an integer giving the seed for simulations |
newdata |
a 2-column matrix of x and y coordinates |
... |
extra arguments to pass to |
Plots representing all one- or two-dimensional smooths
# deformations data(solar) m0 <- deform(solar$x, solar$z, solar$n) # Gaussian process simulations based on fitted deformation model simulate(m0) # one-dimensional expansion model with five simulations and specified locations data(solar) m1 <- expand(solar$x, solar$z, solar$n) xvals <- seq(-123.3, -122.25, by = .05) yvals <- seq(49, 49.4, by = .05) xyvals <- expand.grid(xvals, yvals) simulate(m1, 5, newdata = xyvals)# deformations data(solar) m0 <- deform(solar$x, solar$z, solar$n) # Gaussian process simulations based on fitted deformation model simulate(m0) # one-dimensional expansion model with five simulations and specified locations data(solar) m1 <- expand(solar$x, solar$z, solar$n) xvals <- seq(-123.3, -122.25, by = .05) yvals <- seq(49, 49.4, by = .05) xyvals <- expand.grid(xvals, yvals) simulate(m1, 5, newdata = xyvals)
Variance-covariance matrix for British Columbia solar radiation data
A list with three elements, which are:
a 12-row 2-column matrix of longitude-latitude coordinates for 12 stations
a 12-row 12-column variance-covariance matrix
an integer giving the original sample size
These data were kindly provided by Alexandra Schmidt. They were originally published in Hay (1983) and then used in Sampson and Guttorp's (1992) pioneering deformation paper.
Hay, J. E. (1983) Solar energy system design: The impact of mesoscale variations in solar radiation, Atmosphere-Ocean, 21:2, 138-157, doi:10.1080/07055900.1983.9649161
Sampson, P. D. and Guttorp, P. (1992) Nonparametric Estimation of Nonstationary Spatial Covariance Structure, Journal of the American Statistical Association, 87:417, 108-119, doi:10.1080/01621459.1992.10475181
deform objectPlot the variogram for a fitted deform object
variogram(object, bins = 20, bin.function = "pretty", trim = 0, ...)variogram(object, bins = 20, bin.function = "pretty", trim = 0, ...)
object |
a fitted |
bins |
an integer specifying the number of bins for plotting |
bin.function |
a character specifying a function to use to calculate bins; defaults to |
trim |
a scalar in [0, 0.5], which is passed to |
... |
extra arguments to pass to |
Plot of variogram
# deformations data(solar) m0 <- deform(solar$x, solar$z, solar$n) # empirical versus model-based variogram estimates against distance, # where distance is based on D-space variogram(m0) # which is the default with approximately 20 bins, i.e. variogram(m0, bins = 20) # variogram for one-dimensional expansion without binning data(solar) m1 <- expand(solar$x, solar$z, solar$n) variogram(m1, bins = 0)# deformations data(solar) m0 <- deform(solar$x, solar$z, solar$n) # empirical versus model-based variogram estimates against distance, # where distance is based on D-space variogram(m0) # which is the default with approximately 20 bins, i.e. variogram(m0, bins = 20) # variogram for one-dimensional expansion without binning data(solar) m1 <- expand(solar$x, solar$z, solar$n) variogram(m1, bins = 0)