| Title: | Projected Spatial Gaussian Process Methods |
|---|---|
| Description: | Implements projected sparse Gaussian process Kriging (Ingram 'et. al.', 2008, <doi:10.1007/s00477-007-0163-9>) as an additional method for the 'intamap' package. More details on implementation (Barillec 'et. al.', 2010, <doi:10.1016/j.cageo.2010.05.008>). |
| Authors: | Ben Ingram <[email protected]>, Remi Barillec <[email protected]>, Jon Olav Skoien <[email protected]> |
| Maintainer: | Ben Ingram <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.3-21 |
| Built: | 2025-01-21 06:57:52 UTC |
| Source: | CRAN |
buildMetadata builds a metadata table of likelihood model descriptors in the PSGP framework. This is an internal function and it should not be used directly.
buildMetadata(observations)buildMetadata(observations)
observations |
an observations data frame containing a vector of observation variances ( |
buildMetadata builds a metadata table of likelihood model descriptors in the PSGP framework. The likelihood models are assumed Gaussian with variances specified in the vector observations$oevar (the bias is assumed to be zero). Optionally, biases can be specified in the observations$oebias vector. However, biases are not taken into account in the current version of the PSGP package (they will be in a future release).
Remi Barillec
learnParameters,
makePrediction,
estimateParameters,
spatialPredict,
## Load our favourite dataset data(meuse) obs <- meuse ## Number of observations nobs <- length(obs$y) ## Indicate which likelihood model should be used for each observation obs$oeid <- seq(1:nobs) ## Use random variances for the sake of the example obs$oevar <- rnorm( max(obs$oeid) ) ## Generate metadata table and print it out metadata <- buildMetadata(obs) print(metadata)## Load our favourite dataset data(meuse) obs <- meuse ## Number of observations nobs <- length(obs$y) ## Indicate which likelihood model should be used for each observation obs$oeid <- seq(1:nobs) ## Use random variances for the sake of the example obs$oevar <- rnorm( max(obs$oeid) ) ## Generate metadata table and print it out metadata <- buildMetadata(obs) print(metadata)
This overloads the estimateParameters routine
from the intamap package for interpolation using the PSGP method.
estimateParameters(object, ...)estimateParameters(object, ...)
object |
a list object of Intamap type. Most arguments necessary for interpolation are passed through this object. See intamap-package for further description of the necessary content of this variable. |
... |
other parameters for the generic method, not used for this method |
See psgp-package and learnParameters for
further details.
Remi Barillec, Ben Ingram
learnParameters,
estimateParameters,
makePrediction,
createIntamapObject
# load our favourite dataset data(meuse) coordinates(meuse) = ~x+y meuse$value = log(meuse$zinc) data(meuse.grid) gridded(meuse.grid) = ~x+y proj4string(meuse) = CRS("epsg:28992") proj4string(meuse.grid) = CRS("epsg:28992") # the following two steps are only needed if one wishes to # include observation errors # indicate which likelihood model should be used for each observation # in this case we use a different model for each observation nobs = length(meuse$value) # Number of observations meuse$oeid <- seq(1:nobs) # the variances for the error models are random in this example # in real examples they will come from actual measurements # characteristics meuse$oevar <- abs( rnorm( max(meuse$oeid) ) ) # set up intamap object: obj = createIntamapObject( observations = meuse, predictionLocations = meuse.grid, targetCRS = "epsg:3035", class = "psgp" # Use PSGP for parameter estimation/interpolation ) # do interpolation step: obj = conformProjections(obj) obj = estimateParameters(obj)# load our favourite dataset data(meuse) coordinates(meuse) = ~x+y meuse$value = log(meuse$zinc) data(meuse.grid) gridded(meuse.grid) = ~x+y proj4string(meuse) = CRS("epsg:28992") proj4string(meuse.grid) = CRS("epsg:28992") # the following two steps are only needed if one wishes to # include observation errors # indicate which likelihood model should be used for each observation # in this case we use a different model for each observation nobs = length(meuse$value) # Number of observations meuse$oeid <- seq(1:nobs) # the variances for the error models are random in this example # in real examples they will come from actual measurements # characteristics meuse$oevar <- abs( rnorm( max(meuse$oeid) ) ) # set up intamap object: obj = createIntamapObject( observations = meuse, predictionLocations = meuse.grid, targetCRS = "epsg:3035", class = "psgp" # Use PSGP for parameter estimation/interpolation ) # do interpolation step: obj = conformProjections(obj) obj = estimateParameters(obj)
learnParameters performs maximum likelihood parameter estimation in the PSGP framework.
learnParameters(object)learnParameters(object)
object |
a list object of intamap type. Most arguments necessary for
interpolation are passed through this object.
See |
The Projected Spatial Gaussian Process (PSGP) framework provides an approximation to the full Gaussian process in which the observations are projected sequentially onto an optimal subset of 'active' observations. Spatial interpolation is done using a mixture of covariance kernels (exponential and Matern 5/2).
The function learnParameters is an internal function for estimating the
parameters of the covariance function given the data, using a maximum likelihood
approach. A valid intamap object must be passed in.
PSGP is able to also take the measurement characteristics (i.e. errors) into
account using possibly many error models. For each error model, assumed Gaussian, the
error variance can be specified. The vector
object$observations$oevar contains all variances for the error models (one
value per error model).
Which error model is used for a given observation is determined by the
object$observations$oeid vector of indices, which specifies the index of the
model to be used for each observation.
It is advised to use the intamap wrapper estimateParameters rather than calling this method directly.
Ben Ingram, Remi Barillec
Csato and Opper, 2002; Ingram et al., 2008
makePrediction,
learnParameters,
estimateParameters,
createIntamapObject
# see example in estimateParameters# see example in estimateParameters
makePrediction performs prediction/interpolation within the PSGP package.
makePrediction(object, vario)makePrediction(object, vario)
object |
a list object of intamap type. Most arguments necessary for interpolation are passed through this object. See intamap-package for further description of the necessary content of this variable. Additional meta data about the measurement process is included in this object. In particular, see learnParameters for a way to specify measurement error variances. |
vario |
Log-parameters of the covariance function. For compatibility with the
intamap package, the log-parameters of the PSGP covariance function are stored within
a variogram array object (see |
The Projected Spatial Gaussian Process (PSGP) framework provides an approximation to the full Gaussian process in which the observations are projected sequentially onto an optimal subset of 'active' observations. Spatial interpolation is done using a mixture of covariance kernels (Exponential and Matern 5/2).
The function makePrediction is a function for making predictions at a set
of unobserved inputs (or locations).
Measurement characteristics (i.e. observation error) can be specified if needed. See learnParameters for a description of how to specify measurement error models with given variances.
It is advised to use the intamap wrapper spatialPredict rather than calling this method directly.
Ben Ingram, Remi Barillec
L. Csato and M. Opper. Sparse online Gaussian processes. Neural Computation, 14(3): 641-669, 2002.
B. Ingram, D. Cornford, and D. Evans. Fast algorithms for automatic mapping with space- limited covariance functions. Stochastic Environmental Research and Risk Assessment, 22 (5):661-670, 2008.
learnParameters
spatialPredict,
createIntamapObject
# see example in spatialPredict# see example in spatialPredict
The psgp package provides a spatial interpolation method based on Projected Sequential Gaussian Processes (PSGP) for the intamap package. The PSGP (Csato, 2002) is an approximation to the standard Gaussian process (Rasmussen and Williams, 1996) whereby the observations are projected onto a subset of optimal "active" observations, thus reducing possible redundancy in the data and allowing for faster, memory efficient, interpolation. The projection is done in a sequential manner, that is one observation is projected onto the active subset at a time. This allows for larger datasets to be processed, and overcomes the limitations of standard Gaussian processes related to storing the full covariance matrix (which can be unfeasible for really large datasets).
This implementation of PSGP for spatial interpolation uses a mixture of covariance kernels, namely an Exponential kernel and a Matern kernel with roughness parameter 5/2. The covariance function also includes a nugget term (white noise) and bias term.
The method estimateParameters looks for
a set of covariance function parameters (kernels, white noise and bias) which maximise
the likelihood of the observation.
The method spatialPredict computes predictions (including variance) at a set of unobserved locations.
It is possible to include measurement errors if these are available. These will be taken into account when estimating parameters and making predictions. See learnParameters for more information.
the PSGP package is written in C++ and uses the Armadillo library for all linear algebra routines.
Ben Ingram, Remi Barillec
Csato, L., Gaussian Processes - Iterative Sparse Approximations, Ph.D Thesis, NCRG, Aston University, UK, 2002.
Rasmussen, C. E. and Williams, C. K., Gaussian Processes for Machine Learning, The MIT Press, Cambridge, Massachusetts, 1996.
This overloads the spatialPredict routine
from the intamap package for interpolation using the PSGP method.
spatialPredict(object, ...)spatialPredict(object, ...)
object |
a list object of type PSGP. Most arguments necessary for
interpolation are passed through this object. See |
... |
optional extra arguments (these are only used for debugging purposes) |
See psgp-package and makePrediction for
further detail.
Ben Ingram, Remi Barillec
psgp-package,
estimateParameters,
makePrediction
createIntamapObject
data(meuse) meuse = meuse[1:100,] coordinates(meuse) = ~x+y meuse$value = log(meuse$zinc) data(meuse.grid) gridded(meuse.grid) = ~x+y proj4string(meuse) = CRS("epsg:28992") proj4string(meuse.grid) = CRS("epsg:28992") # Specify a different observation error model for each observation nobs = length(meuse$value) # Number of observations meuse$oeid = seq(1:nobs) # One error model per observation # Indicate the variance for each of these error models meuse$oevar <- abs( rnorm( max(meuse$oeid) ) ) # Set up intamap object obj = createIntamapObject( observations = meuse, predictionLocations = meuse.grid, targetCRS = "epsg:3035", class = "psgp" ) # Estimate parameters and predict at new locations (interpolation) obj = conformProjections(obj) obj = estimateParameters(obj) obj = spatialPredict(obj) # Plot results plotIntamap(obj)data(meuse) meuse = meuse[1:100,] coordinates(meuse) = ~x+y meuse$value = log(meuse$zinc) data(meuse.grid) gridded(meuse.grid) = ~x+y proj4string(meuse) = CRS("epsg:28992") proj4string(meuse.grid) = CRS("epsg:28992") # Specify a different observation error model for each observation nobs = length(meuse$value) # Number of observations meuse$oeid = seq(1:nobs) # One error model per observation # Indicate the variance for each of these error models meuse$oevar <- abs( rnorm( max(meuse$oeid) ) ) # Set up intamap object obj = createIntamapObject( observations = meuse, predictionLocations = meuse.grid, targetCRS = "epsg:3035", class = "psgp" ) # Estimate parameters and predict at new locations (interpolation) obj = conformProjections(obj) obj = estimateParameters(obj) obj = spatialPredict(obj) # Plot results plotIntamap(obj)