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GpGp: Fast Gaussian Process Computing.

Description

Vecchia's (1988)Gaussian process approximation has emerged among its competitorsas a leader in computational scalability and accuracy. This package includesimplementations of the original approximation, as well as severalupdates to it, including the reordered and grouped versions of the approximation outlined in Guinness (2018) and the Fisher scoring algorithmdescribed in Guinness (2019). The package supports spatialmodels, spatial-temporal models, models on spheres, and some nonstationary models.

Details

The main functions of the package arefit_model, andpredictions.fit_model returns estimates of covariance parametersand linear mean parameters. The user is expected to select a covariance functionand specify it with a string. Currently supported covariance functions are

• matern_isotropic

• exponential_isotropic

• matern_anisotropic2D

• exponential_anisotropic2D

• matern_anisotropic3D

• exponential_anisotropic3D

• matern_anisotropic3D_alt

• matern15_isotropic

• matern25_isotropic

• matern35_isotropic

• matern45_isotropic

• matern_scaledim

• exponential_scaledim

• matern15_scaledim

• matern25_scaledim

• matern35_scaledim

• matern45_scaledim

• matern_spacetime

• exponential_spacetime

• matern_nonstat_var

• exponential_nonstat_var

• matern_sphere

• exponential_sphere

• matern_spheretime

• exponential_spheretime

• matern_sphere_warp

• exponential_sphere_warp

• matern_spheretime_warp

• exponential_spheretime_warp

If there are covariates, they can be expressed via a design matrixX, each row containingthe covariates corresponding to the same row inlocs.

Forpredictions, the user should specify prediction locationslocs_pred and a prediction design matrixX_pred.

The vignettes are intended to be helpful for getting a sense of how these functions work.

For Gaussian process researchers, the package also provides access to functions forcomputing the likelihood, gradient, and Fisher information with respectto covariance parameters; reordering functions, nearest neighbor-finding functions, grouping (partitioning) functions, and approximate simulation functions.

Author(s)

Maintainer: Joseph Guinnessjoeguinness@gmail.com

Authors:

• Matthias Katzfusskatzfuss@gmail.com

• Youssef Fahmyyf297@cornell.edu

Multiply approximate Cholesky by a vector

Description

Vecchia's approximation implies a sparse approximation to theinverse Cholesky factor of the covariance matrix. This functionreturns the result of multiplying the inverse of that matrix by a vector(i.e. an approximation to the Cholesky factor).

Usage

L_mult(Linv, z, NNarray)

Arguments

Value

the product of the Cholesky factor with a vector

Examples

n <- 2000locs <- matrix( runif(2*n), n, 2 )covparms <- c(2, 0.2, 0.75, 0.1)ord <- order_maxmin(locs)NNarray <- find_ordered_nn(locs,20)Linv <- vecchia_Linv( covparms, "matern_isotropic", locs, NNarray )z <- rnorm(n)y1 <- fast_Gp_sim_Linv(Linv,NNarray,z)y2 <- L_mult(Linv, z, NNarray)print( sum( (y1-y2)^2 ) )

Multiply transpose of approximate Cholesky by a vector

Description

Vecchia's approximation implies a sparse approximation to theinverse Cholesky factor of the covariance matrix. This functionreturns the result of multiplying the transpose of theinverse of that matrix by a vector(i.e. an approximation to the transpose of the Cholesky factor).

Usage

L_t_mult(Linv, z, NNarray)

Arguments

Value

the product of the transpose of the Cholesky factor with a vector

Examples

n <- 2000locs <- matrix( runif(2*n), n, 2 )covparms <- c(2, 0.2, 0.75, 0.1)NNarray <- find_ordered_nn(locs,20)Linv <- vecchia_Linv( covparms, "matern_isotropic", locs, NNarray )z1 <- rnorm(n)z2 <- L_t_mult(Linv, z1, NNarray)

Multiply approximate inverse Cholesky by a vector

Description

Vecchia's approximation implies a sparse approximation to theinverse Cholesky factor of the covariance matrix. This functionreturns the result of multiplying that matrix by a vector.

Usage

Linv_mult(Linv, z, NNarray)

Arguments

Value

the product of the sparse inverse Cholesky factor with a vector

Examples

n <- 2000locs <- matrix( runif(2*n), n, 2 )covparms <- c(2, 0.2, 0.75, 0.1)ord <- order_maxmin(locs)NNarray <- find_ordered_nn(locs,20)Linv <- vecchia_Linv( covparms, "matern_isotropic", locs, NNarray )z1 <- rnorm(n)y <- fast_Gp_sim_Linv(Linv,NNarray,z1)z2 <- Linv_mult(Linv, y, NNarray)print( sum( (z1-z2)^2 ) )

Multiply transpose of approximate inverse Cholesky by a vector

Description

Vecchia's approximation implies a sparse approximation to theinverse Cholesky factor of the covariance matrix. This functionreturns the result of multiplying the transpose of that matrix by a vector.

Usage

Linv_t_mult(Linv, z, NNarray)

Arguments

Value

the product of the transpose of the sparse inverse Cholesky factor with a vector

Examples

n <- 2000locs <- matrix( runif(2*n), n, 2 )covparms <- c(2, 0.2, 0.75, 0.1)NNarray <- find_ordered_nn(locs,20)Linv <- vecchia_Linv( covparms, "matern_isotropic", locs, NNarray )z1 <- rnorm(n)z2 <- Linv_t_mult(Linv, z1, NNarray)

Ocean temperatures from Argo profiling floats

Description

A dataset containing ocean temperature measurements fromthree pressure levels (depths), measured by profilingfloats from the Argo program. Data collected in Jan, Feb,and March of 2016.

Usage

argo2016

Format

A data frame with 32436 rows and 6 columns

lon

longitude in degrees between 0 and 360

lat

latitude in degrees between -90 and 90

day

time in days

temp100

Temperature at 100 dbars (roughly 100 meters)

temp150

Temperature at 150 dbars (roughly 150 meters)

temp200

Temperature at 200 dbars (roughly 200 meters)

Source

Mikael Kuusela. Argo program:https://argo.ucsd.edu/

Conditional Simulation using Vecchia's approximation

Description

With the prediction locations ordered after the observation locations,an approximation for the inverse Cholesky of the covariance matrixis computed, and standard formulas are applied to obtaina conditional simulation.

Usage

cond_sim(  fit = NULL,  locs_pred,  X_pred,  y_obs = fit$y,  locs_obs = fit$locs,  X_obs = fit$X,  beta = fit$betahat,  covparms = fit$covparms,  covfun_name = fit$covfun_name,  m = 60,  reorder = TRUE,  st_scale = NULL,  nsims = 1)

Arguments

Details

We can specify either a GpGp_fit object (the result offit_model), OR manually enter the covariance function andparameters, the observations, observation locations, and design matrix. We must specify the prediction locations and the prediction design matrix.

compute condition number of matrix

Description

compute condition number of matrix

Usage

condition_number(info)

Arguments

expit function and integral of expit function

Description

expit function and integral of expit function

Usage

expit(x)intexpit(x)

Arguments

Geometrically anisotropic exponential covariance function (two dimensions)

Description

From a matrix of locations and covariance parameters of the form(variance, L11, L21, L22, nugget), return the square matrix ofall pairwise covariances.

Usage

exponential_anisotropic2D(covparms, locs)d_exponential_anisotropic2D(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_exponential_anisotropic2D(): Derivatives of anisotropic exponential covariance

Parameterization

The covariance parameter vector is (variance, L11, L21, L22, nugget)where L11, L21, L22, are the three non-zero entries of a lower-triangularmatrix L. The covariances are

M(x,y) = \sigma^2 exp(-|| L x - L y || )

This means that L11 is interpreted as an inverse range parameter in thefirst dimension.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Geometrically anisotropic exponential covariance function (three dimensions)

Description

From a matrix of locations and covariance parameters of the form(variance, L11, L21, L22, L31, L32, L33, nugget), return the square matrix ofall pairwise covariances.

Usage

exponential_anisotropic3D(covparms, locs)d_exponential_anisotropic3D(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_exponential_anisotropic3D(): Derivatives of anisotropic exponential covariance

Parameterization

The covariance parameter vector is (variance, L11, L21, L22, L31, L32, L33, nugget)where L11, L21, L22, L31, L32, L33 are the six non-zero entries of a lower-triangularmatrix L. The covariances are

M(x,y) = \sigma^2 exp(-|| L x - L y || )

This means that L11 is interpreted as an inverse range parameter in thefirst dimension.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Geometrically anisotropic exponential covariance function (three dimensions, alternate parameterization)

Description

From a matrix of locations and covariance parameters of the form(variance, B11, B12, B13, B22, B23, B33, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

exponential_anisotropic3D_alt(covparms, locs)d_exponential_anisotropic3D_alt(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_exponential_anisotropic3D_alt(): Derivatives of anisotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, B11, B12, B13, B22, B23, B33, smoothness, nugget)where B11, B12, B13, B22, B23, B33, transform the three coordinates as

u_1 = B11[ x_1 + B12 x_2 + (B13 + B12 B23) x_3]

u_2 = B22[ x_2 + B23 x_3]

u_3 = B33[ x_3 ]

(B13,B23) can be interpreted as a drift vector in space over timeif first two dimensions are space and third is time.Assuming x is transformed to u and y transformed to v, the covariances are

M(x,y) = \sigma^2 exp( - || u - v || )

The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic exponential covariance function

Description

From a matrix of locations and covariance parameters of the form(variance, range, nugget), return the square matrix ofall pairwise covariances.

Usage

exponential_isotropic(covparms, locs)d_exponential_isotropic(covparms, locs)d_matern15_isotropic(covparms, locs)d_matern25_isotropic(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_exponential_isotropic(): Derivatives of isotropic exponential covariance

• d_matern15_isotropic(): Derivatives of isotropic matern covariance with smoothness 1.5

• d_matern25_isotropic(): Derivatives of isotropicmatern covariance function with smoothness 2.5

Parameterization

The covariance parameter vector is (variance, range, nugget)=(\sigma^2,\alpha,\tau^2), and the covariance function is parameterizedas

M(x,y) = \sigma^2 exp( - || x - y ||/ \alpha )

The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic exponential covariance function, nonstationary variances

Description

From a matrix of locations and covariance parameters of the form(variance, range, nugget, <nonstat variance parameters>), return the square matrix of all pairwise covariances.

Usage

exponential_nonstat_var(covparms, Z)d_exponential_nonstat_var(covparms, Z)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_exponential_nonstat_var(): Derivatives with respect to parameters

Parameterization

This covariance function multiplies the isotropic exponential covarianceby a nonstationary variance function. The form of the covariance is

C(x,y) = exp( \phi(x) + \phi(y) ) M(x,y)

where M(x,y) is the isotropic exponential covariance, and

\phi(x) = c_1 \phi_1(x) + ... + c_p \phi_p(x)

where\phi_1,...,\phi_p are the spatial basis functionscontained in the lastp columns ofZ, andc_1,...,c_p are the nonstationary variance parameters.

Exponential covariance function, different range parameter for each dimension

Description

From a matrix of locations and covariance parameters of the form(variance, range_1, ..., range_d, nugget), return the square matrix ofall pairwise covariances.

Usage

exponential_scaledim(covparms, locs)d_exponential_scaledim(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_exponential_scaledim(): Derivatives with respect to parameters

Parameterization

The covariance parameter vector is (variance, range_1, ..., range_d, nugget).The covariance function is parameterized as

M(x,y) = \sigma^2 exp( - || D^{-1}(x - y) || )

where D is a diagonal matrix with (range_1, ..., range_d) on the diagonals.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Spatial-Temporal exponential covariance function

Description

From a matrix of locations and covariance parameters of the form(variance, range_1, range_2, nugget), return the square matrix ofall pairwise covariances.

Usage

exponential_spacetime(covparms, locs)d_exponential_spacetime(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_exponential_spacetime(): Derivatives with respect to parameters

Parameterization

The covariance parameter vector is (variance, range_1, range_2, nugget).The covariance function is parameterized as

M(x,y) = \sigma^2 exp( - || D^{-1}(x - y) || )

where D is a diagonal matrix with (range_1, ..., range_1, range_2) on the diagonals.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic exponential covariance function on sphere

Description

From a matrix of longitudes and latitudes and a vector covariance parameters of the form(variance, range, nugget), return the square matrix ofall pairwise covariances.

Usage

exponential_sphere(covparms, lonlat)d_exponential_sphere(covparms, lonlat)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlonlat[i,] andlonlat[j,].

Functions

• d_exponential_sphere(): Derivatives with respect to parameters

Covariances on spheres

The function first calculates the (x,y,z) 3D coordinates, and then inputsthe resulting locations intoexponential_isotropic. This means that we constructcovariances on the sphere by embedding the sphere in a 3D space. There has been someconcern expressed in the literature that such embeddings may produce distortions.The source and nature of such distortions has never been articulated,and to date, no such distortions have been documented. Guinness andFuentes (2016) argue that 3D embeddings produce reasonable models for data on spheres.

Deformed exponential covariance function on sphere

Description

From a matrix of longitudes and latitudes and a vector covariance parameters of the form(variance, range, nugget, <5 warping parameters>), return the square matrix ofall pairwise covariances.

Usage

exponential_sphere_warp(covparms, lonlat)d_exponential_sphere_warp(covparms, lonlat)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlonlat[i,] andlonlat[j,].

Functions

• d_exponential_sphere_warp(): Derivatives with respect to parameters

Warpings

The function first calculates the (x,y,z) 3D coordinates, and then "warps"the locations to(x,y,z) + \Phi(x,y,z), where\Phi is a warpingfunction composed of gradients of spherical harmonic functions of degree 2.See Guinness (2019, "Gaussian Process Learning via Fisher Scoring of Vecchia's Approximation") for details.The warped locations are input intoexponential_isotropic.

Exponential covariance function on sphere x time

Description

From a matrix of longitudes, latitudes, and times, and a vector covariance parameters of the form(variance, range_1, range_2, nugget), return the square matrix ofall pairwise covariances.

Usage

exponential_spheretime(covparms, lonlattime)d_exponential_spheretime(covparms, lonlattime)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlonlattime[i,] andlonlattime[j,].

Functions

• d_exponential_spheretime(): Derivatives with respect to parameters.

Covariances on spheres

The function first calculates the (x,y,z) 3D coordinates, and then inputsthe resulting locations intoexponential_spacetime. This means that we constructcovariances on the sphere by embedding the sphere in a 3D space. There has been someconcern expressed in the literature that such embeddings may produce distortions.The source and nature of such distortions has never been articulated,and to date, no such distortions have been documented. Guinness andFuentes (2016) argue that 3D embeddings produce reasonable models for data on spheres.

Deformed exponential covariance function on sphere

Description

From a matrix of longitudes, latitudes, times, and a vector covariance parameters of the form(variance, range_1, range_2, nugget, <5 warping parameters>), return the square matrix ofall pairwise covariances.

Usage

exponential_spheretime_warp(covparms, lonlattime)d_exponential_spheretime_warp(covparms, lonlattime)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlonlat[i,] andlonlat[j,].

Functions

• d_exponential_spheretime_warp(): Derivatives with respect to parameters

Warpings

The function first calculates the (x,y,z) 3D coordinates, and then "warps"the locations to(x,y,z) + \Phi(x,y,z), where\Phi is a warpingfunction composed of gradients of spherical harmonic functions of degree 2.See Guinness (2019, "Gaussian Process Learning via Fisher Scoring of Vecchia's Approximation") for details.The warped locations are input intoexponential_spacetime. The functiondoes not do temporal warping.

Approximate GP simulation

Description

Calculates an approximation to the inverse Choleskyfactor of the covariance matrix using Vecchia's approximation,then the simulation is produced by solving a linear systemwith a vector of uncorrelated standard normals

Usage

fast_Gp_sim(covparms, covfun_name = "matern_isotropic", locs, m = 30)

Arguments

Value

vector of simulated values

Examples

locs <- as.matrix( expand.grid( (1:50)/50, (1:50)/50 ) )y <- fast_Gp_sim(c(4,0.2,0.5,0), "matern_isotropic",  locs, 30 )fields::image.plot( matrix(y,50,50) )

Approximate GP simulation with specified Linverse

Description

In situations where we want to do many gaussian processsimulations from the same model, we can compute Linverseonce and reuse it, rather than recomputing for each identical simulation.This function also allows the user to input the vector of standard normalsz.

Usage

fast_Gp_sim_Linv(Linv, NNarray, z = NULL)

Arguments

Value

vector of simulated values

Examples

locs <- as.matrix( expand.grid( (1:50)/50, (1:50)/50 ) )ord <- order_maxmin(locs)locsord <- locs[ord,]m <- 10NNarray <- find_ordered_nn(locsord,m)covparms <- c(2, 0.2, 1, 0)Linv <- vecchia_Linv( covparms, "matern_isotropic", locsord, NNarray )y <- fast_Gp_sim_Linv(Linv,NNarray)y[ord] <- yfields::image.plot( matrix(y,50,50) )

Find ordered nearest neighbors.

Description

Given a matrix of locations, find them nearest neighborsto each location, subject to the neighbors comingpreviously in the ordering. The algorithm uses the kdtreealgorithm in the FNN package, adapted to the settingwhere the nearest neighbors must come from previousin the ordering.

Usage

find_ordered_nn(locs, m, lonlat = FALSE, st_scale = NULL)

Arguments

Value

An matrix containing the indices of the neighbors. Rowi of thereturned matrix contains the indices of the nearestmlocations to thei'th location. Indices are ordered within arow to be increasing in distance. By convention, we consider a locationto neighbor itself, so the first entry of rowi isi, thesecond entry is the index of the nearest location, and so on. Because eachlocation neighbors itself, the returned matrix hasm+1 columns.

Examples

locs <- as.matrix( expand.grid( (1:40)/40, (1:40)/40 ) )     ord <- order_maxmin(locs)        # calculate an orderinglocsord <- locs[ord,]            # reorder locationsm <- 20NNarray <- find_ordered_nn(locsord,20)  # find ordered nearest 20 neighborsind <- 100# plot all locations in gray, first ind locations in black,# ind location with magenta circle, m neighhbors with blue circleplot( locs[,1], locs[,2], pch = 16, col = "gray" )points( locsord[1:ind,1], locsord[1:ind,2], pch = 16 )points( locsord[ind,1], locsord[ind,2], col = "magenta", cex = 1.5 )points( locsord[NNarray[ind,2:(m+1)],1],     locsord[NNarray[ind,2:(m+1)],2], col = "blue", cex = 1.5 )

Naive brute force nearest neighbor finder

Description

Naive brute force nearest neighbor finder

Usage

find_ordered_nn_brute(locs, m)

Arguments

Value

An matrix containing the indices of the neighbors. Rowi of thereturned matrix contains the indices of the nearestmlocations to thei'th location. Indices are ordered within arow to be increasing in distance. By convention, we consider a locationto neighbor itself, so the first entry of rowi isi, thesecond entry is the index of the nearest location, and so on. Because eachlocation neighbors itself, the returned matrix hasm+1 columns.

Fisher scoring algorithm

Description

Fisher scoring algorithm

Usage

fisher_scoring(  likfun,  start_parms,  link,  silent = FALSE,  convtol = 1e-04,  max_iter = 40)

Arguments

Estimate mean and covariance parameters

Description

Given a response, set of locations, (optionally) a design matrix,and a specified covariance function, return the maximumVecchia likelihood estimates, obtained with a Fisher scoring algorithm.

Usage

fit_model(  y,  locs,  X = NULL,  covfun_name = "matern_isotropic",  NNarray = NULL,  start_parms = NULL,  reorder = TRUE,  group = TRUE,  m_seq = c(10, 30),  max_iter = 40,  fixed_parms = NULL,  silent = FALSE,  st_scale = NULL,  convtol = 1e-04)

Arguments

Details

fit_model is a user-friendly model fitting functionthat automatically performs many of the auxiliary tasks needed forusing Vecchia's approximation, including reordering, computingnearest neighbors, grouping, and optimization. The likelihoods use a smallpenalty on small nuggets, large spatial variances, and small smoothness parameter.

The Jason-3 windspeed vignette and the Argo temperature vignette are useful sources for ause-cases of thefit_model function for data on sphere. The example below shows a very small example with a simulated dataset in 2d.

Value

An object of classGpGp_fit, which is a list containingcovariance parameter estimates, regression coefficients,covariance matrix for mean parameter estimates, as well as some otherinformation relevant to the model fit.

Examples

n1 <- 20n2 <- 20n <- n1*n2locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )covparms <- c(2,0.1,1/2,0)y <- 7 + fast_Gp_sim(covparms, "matern_isotropic", locs)X <- as.matrix( rep(1,n) )## not run# fit <- fit_model(y, locs, X, "matern_isotropic")# fit

get link function, whether locations are lonlat and space time

Description

get link function, whether locations are lonlat and space time

Usage

get_linkfun(covfun_name)

Arguments

get penalty function

Description

get penalty function

Usage

get_penalty(y, X, locs, covfun_name)

Arguments

get default starting values of covariance parameters

Description

get default starting values of covariance parameters

Usage

get_start_parms(y, X, locs, covfun_name)

Arguments

Automatic grouping (partitioning) of locations

Description

Take in an array of nearest neighbors, and automatically partitionthe array into groups that share neighbors.This is helpful to speed the computations and improve their accuracy.The function returns a list, with each list element containing one orseveral rows of NNarray. The algorithm attempts to find groupings such thatobservations within a group share many common neighbors.

Usage

group_obs(NNarray, exponent = 2)

Arguments

Value

A list with elements defining the grouping. The list entries are:

• all_inds: vector of all indices of all blocks.

• last_ind_of_block: Theith entry tells us thelocation inall_inds of the last index of theith block. Thus the lengthoflast_ind_of_block is the number of blocks, andlast_ind_of_block canbe used to chopall_inds up into blocks.

• global_resp_inds: Theith entry tells us theindex of theith response, as ordered inall_inds.

• local_resp_inds: Theith entry tells us the location within the block of the response index.

• last_resp_of_block: Theith entry tells us thelocation withinlocal_resp_inds andglobal_resp_inds of the lastindex of theith block.last_resp_of_block is toglobal_resp_inds andlocal_resp_indsaslast_ind_of_block is toall_inds.

Examples

locs <- matrix( runif(200), 100, 2 )   # generate random locationsord <- order_maxmin(locs)              # calculate an orderinglocsord <- locs[ord,]                  # reorder locationsm <- 10NNarray <- find_ordered_nn(locsord,m)  # m nearest neighbor indicesNNlist2 <- group_obs(NNarray)          # join blocks if joining reduces squaresNNlist3 <- group_obs(NNarray,3)        # join blocks if joining reduces cubesobject.size(NNarray)object.size(NNlist2)object.size(NNlist3)mean( NNlist2[["local_resp_inds"]] - 1 )   # average number of neighbors (exponent 2)mean( NNlist3[["local_resp_inds"]] - 1 )   # average number of neighbors (exponent 3)all_inds <- NNlist2$all_indslast_ind_of_block <- NNlist2$last_ind_of_blockinds_of_block_2 <- all_inds[ (last_ind_of_block[1] + 1):last_ind_of_block[2] ]local_resp_inds <- NNlist2$local_resp_indsglobal_resp_inds <- NNlist2$global_resp_indslast_resp_of_block <- NNlist2$last_resp_of_blocklocal_resp_of_block_2 <-     local_resp_inds[(last_resp_of_block[1]+1):last_resp_of_block[2]]global_resp_of_block_2 <-     global_resp_inds[(last_resp_of_block[1]+1):last_resp_of_block[2]]inds_of_block_2[local_resp_of_block_2]# these last two should be the same

Windspeed measurements from Jason-3 Satellite

Description

A dataset containing lightly preprocessed windspeed valuesfrom the Jason-3 satellite. Observations near clouds and icehave been removed, and the data have been aggregated (averaged) over 10 second intervals. Jason-3 reportswindspeeds over the ocean only. The data are from a sixday period between August 4 and 9 of 2016.

Usage

jason3

Format

A data frame with 18973 rows and 4 columns

windspeed

wind speed, in maters per second

lon

longitude in degrees between 0 and 360

lat

latitude in degrees between -90 and 90

time

time in seconds from midnight August 4

Source

https://www.ncei.noaa.gov/products/jason-satellite-products

Isotropic Matern covariance function, smoothness = 1.5

Description

From a matrix of locations and covariance parameters of the form(variance, range, nugget), return the square matrix ofall pairwise covariances.

Usage

matern15_isotropic(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Parameterization

The covariance parameter vector is (variance, range, nugget)=(\sigma^2,\alpha,\tau^2), and the covariance function is parameterizedas

M(x,y) = \sigma^2 (1 + || x - y || ) exp( - || x - y ||/ \alpha )

The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Matern covariance function, smoothess = 1.5, different range parameter for each dimension

Description

From a matrix of locations and covariance parameters of the form(variance, range_1, ..., range_d, nugget), return the square matrix ofall pairwise covariances.

Usage

matern15_scaledim(covparms, locs)d_matern15_scaledim(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern15_scaledim(): Derivatives with respect to parameters

Parameterization

The covariance parameter vector is (variance, range_1, ..., range_d, nugget).The covariance function is parameterized as

M(x,y) = \sigma^2 (1 + || D^{-1}(x - y) || ) exp( - || D^{-1}(x - y) || )

where D is a diagonal matrix with (range_1, ..., range_d) on the diagonals.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic Matern covariance function, smoothness = 2.5

Description

From a matrix of locations and covariance parameters of the form(variance, range, nugget), return the square matrix ofall pairwise covariances.

Usage

matern25_isotropic(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Parameterization

The covariance parameter vector is (variance, range, nugget)=(\sigma^2,\alpha,\tau^2), and the covariance function is parameterizedas

M(x,y) = \sigma^2 (1 + || x - y ||/ \alpha + || x - y ||^2/3\alpha^2 ) exp( - || x - y ||/ \alpha )

The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Matern covariance function, smoothess = 2.5, different range parameter for each dimension

Description

From a matrix of locations and covariance parameters of the form(variance, range_1, ..., range_d, nugget), return the square matrix ofall pairwise covariances.

Usage

matern25_scaledim(covparms, locs)d_matern25_scaledim(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern25_scaledim(): Derivatives with respect to parameters

Parameterization

The covariance parameter vector is (variance, range_1, ..., range_d, nugget).The covariance function is parameterized as

M(x,y) = \sigma^2 (1 + || D^{-1}(x - y) || + || D^{-1}(x - y) ||^2/3.0) exp( - || D^{-1}(x - y) || )

where D is a diagonal matrix with (range_1, ..., range_d) on the diagonals.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic Matern covariance function, smoothness = 3.5

Description

From a matrix of locations and covariance parameters of the form(variance, range, nugget), return the square matrix ofall pairwise covariances.

Usage

matern35_isotropic(covparms, locs)d_matern35_isotropic(covparms, locs)d_matern45_isotropic(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern35_isotropic(): Derivatives of isotropicmatern covariance function with smoothness 3.5

• d_matern45_isotropic(): Derivatives of isotropicmatern covariance function with smoothness 3.5

Parameterization

The covariance parameter vector is (variance, range, nugget)=(\sigma^2,\alpha,\tau^2), and the covariance function is parameterizedas

M(x,y) = \sigma^2 ( \sum_{j=0}^3 c_j || x - y ||^j/ \alpha^j ) exp( - || x - y ||/ \alpha )

where c_0 = 1, c_1 = 1, c_2 = 2/5, c_3 = 1/15. The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Matern covariance function, smoothess = 3.5, different range parameter for each dimension

Description

From a matrix of locations and covariance parameters of the form(variance, range_1, ..., range_d, nugget), return the square matrix ofall pairwise covariances.

Usage

matern35_scaledim(covparms, locs)d_matern35_scaledim(covparms, locs)d_matern45_scaledim(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern35_scaledim(): Derivatives with respect to parameters

• d_matern45_scaledim(): Derivatives with respect to parameters

Parameterization

The covariance parameter vector is (variance, range_1, ..., range_d, nugget).The covariance function is parameterized as

M(x,y) = \sigma^2 ( \sum_{j=0}^3 c_j || D^{-1}(x - y) ||^j ) exp( - || D^{-1}(x - y) || )

where c_0 = 1, c_1 = 1, c_2 = 2/5, c_3 = 1/15.where D is a diagonal matrix with (range_1, ..., range_d) on the diagonals.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic Matern covariance function, smoothness = 4.5

Description

From a matrix of locations and covariance parameters of the form(variance, range, nugget), return the square matrix ofall pairwise covariances.

Usage

matern45_isotropic(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Parameterization

The covariance parameter vector is (variance, range, nugget)=(\sigma^2,\alpha,\tau^2), and the covariance function is parameterizedas

M(x,y) = \sigma^2 ( \sum_{j=0}^4 c_j || x - y ||^j/ \alpha^j ) exp( - || x - y ||/ \alpha )

where c_0 = 1, c_1 = 1, c_2 = 3/7, c_3 = 2/21, c_4 = 1/105. The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Matern covariance function, smoothess = 3.5, different range parameter for each dimension

Description

From a matrix of locations and covariance parameters of the form(variance, range_1, ..., range_d, nugget), return the square matrix ofall pairwise covariances.

Usage

matern45_scaledim(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Parameterization

The covariance parameter vector is (variance, range_1, ..., range_d, nugget).The covariance function is parameterized as

M(x,y) = \sigma^2 ( \sum_{j=0}^4 c_j || D^{-1}(x - y) ||^j ) exp( - || D^{-1}(x - y) || )

where c_0 = 1, c_1 = 1, c_2 = 3/7, c_3 = 2/21, c_4 = 1/105. where D is a diagonal matrix with (range_1, ..., range_d) on the diagonals.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Geometrically anisotropic Matern covariance function (two dimensions)

Description

From a matrix of locations and covariance parameters of the form(variance, L11, L21, L22, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_anisotropic2D(covparms, locs)d_matern_anisotropic2D(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_anisotropic2D(): Derivatives of anisotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, L11, L21, L22, smoothness, nugget)where L11, L21, L22, are the three non-zero entries of a lower-triangularmatrix L. The covariances are

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (|| L x - L y || )^\nu K_\nu(|| L x - L y ||)

This means that L11 is interpreted as an inverse range parameter in thefirst dimension.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Geometrically anisotropic Matern covariance function (three dimensions)

Description

From a matrix of locations and covariance parameters of the form(variance, L11, L21, L22, L31, L32, L33, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_anisotropic3D(covparms, locs)d_matern_anisotropic3D(covparms, locs)d_matern_anisotropic3D_alt(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_anisotropic3D(): Derivatives of anisotropic Matern covariance

• d_matern_anisotropic3D_alt(): Derivatives of anisotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, L11, L21, L22, L31, L32, L33, smoothness, nugget)where L11, L21, L22, L31, L32, L33 are the six non-zero entries of a lower-triangularmatrix L. The covariances are

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (|| L x - L y || )^\nu K_\nu(|| L x - L y ||)

This means that L11 is interpreted as an inverse range parameter in thefirst dimension.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Geometrically anisotropic Matern covariance function (three dimensions, alternate parameterization)

Description

From a matrix of locations and covariance parameters of the form(variance, B11, B12, B13, B22, B23, B33, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_anisotropic3D_alt(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Parameterization

The covariance parameter vector is (variance, B11, B12, B13, B22, B23, B33, smoothness, nugget)where B11, B12, B13, B22, B23, B33, transform the three coordinates as

u_1 = B11[ x_1 + B12 x_2 + (B13 + B12 B23) x_3]

u_2 = B22[ x_2 + B23 x_3]

u_3 = B33[ x_3 ]

NOTE: the u_1 transformation is different from previous versions of this function.NOTE: now (B13,B23) can be interpreted as a drift vector in space over time.Assuming x is transformed to u and y transformed to v, the covariances are

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (|| u - v || )^\nu K_\nu(|| u - v ||)

The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic Matern covariance function with random effects for categories

Description

From a matrix of locations and covariance parameters of the form(variance, range, smoothness, category variance, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_categorical(covparms, locs)d_matern_categorical(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_categorical(): Derivatives of isotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, range, smoothness, category variance, nugget)=(\sigma^2,\alpha,\nu,c^2,\tau^2), and the covariance function is parameterizedas

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (|| x - y ||/\alpha )^\nu K_\nu(|| x - y ||/\alpha )

The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.The category variancec^2 is added if two observation from same categoryNOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic Matern covariance function

Description

From a matrix of locations and covariance parameters of the form(variance, range, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_isotropic(covparms, locs)d_matern_isotropic(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_isotropic(): Derivatives of isotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, range, smoothness, nugget)=(\sigma^2,\alpha,\nu,\tau^2), and the covariance function is parameterizedas

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (|| x - y ||/\alpha )^\nu K_\nu(|| x - y ||/\alpha )

The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic Matern covariance function, nonstationary variances

Description

From a matrix of locations and covariance parameters of the form(variance, range, smoothness, nugget, <nonstat variance parameters>), return the square matrix of all pairwise covariances.

Usage

matern_nonstat_var(covparms, Z)d_matern_nonstat_var(covparms, Z)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_nonstat_var(): Derivatives with respect to parameters

Parameterization

This covariance function multiplies the isotropic Matern covarianceby a nonstationary variance function. The form of the covariance is

C(x,y) = exp( \phi(x) + \phi(y) ) M(x,y)

where M(x,y) is the isotropic Matern covariance, and

\phi(x) = c_1 \phi_1(x) + ... + c_p \phi_p(x)

where\phi_1,...,\phi_p are the spatial basis functionscontained in the lastp columns ofZ, andc_1,...,c_p are the nonstationary variance parameters.

Matern covariance function, different range parameter for each dimension

Description

From a matrix of locations and covariance parameters of the form(variance, range_1, ..., range_d, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_scaledim(covparms, locs)d_matern_scaledim(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_scaledim(): Derivatives with respect to parameters

Parameterization

The covariance parameter vector is (variance, range_1, ..., range_d, smoothness, nugget).The covariance function is parameterized as

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (|| D^{-1}(x - y) || )^\nu K_\nu(|| D^{-1}(x - y) || )

where D is a diagonal matrix with (range_1, ..., range_d) on the diagonals.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Spatial-Temporal Matern covariance function

Description

From a matrix of locations and covariance parameters of the form(variance, range_1, range_2, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_spacetime(covparms, locs)d_matern_spacetime(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_spacetime(): Derivatives with respect to parameters

Parameterization

The covariance parameter vector is (variance, range_1, range_2, smoothness, nugget).The covariance function is parameterized as

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (|| D^{-1}(x - y) || )^\nu K_\nu(|| D^{-1}(x - y) || )

where D is a diagonal matrix with (range_1, ..., range_1, range_2) on the diagonals.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.NOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Space-Time Matern covariance function with random effects for categories

Description

From a matrix of locations and covariance parameters of the form(variance, spatial range, temporal range, smoothness, category, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_spacetime_categorical(covparms, locs)d_matern_spacetime_categorical(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_spacetime_categorical(): Derivatives of isotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, range, smoothness, category, nugget)=(\sigma^2,\alpha_1,\alpha_2,\nu,c^2,\tau^2), and the covariance function is parameterizedas

d = ( || x - y ||^2/\alpha_1 + |s-t|^2/\alpha_2^2 )^{1/2}

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (d)^\nu K_\nu(d)

(x,s) and (y,t) are the space-time locations of a pair of observations.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.The category variancec^2 is added if two observation from same categoryNOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Space-Time Matern covariance function with local random effects for categories

Description

From a matrix of locations and covariance parameters of the form(variance, spatial range, temporal range, smoothness, cat variance, cat spatial range, cat temporal range, cat smoothness, nugget),return the square matrix ofall pairwise covariances.This is the covariance for the following model for data from cateogory k

Y_k(x_i,t_i) = Z_0(x_i,t_i) + Z_k(x_i,t_i) + e_i

where Z_0 is Matern with parameters (variance,spatial range,temporal range,smoothness)and Z_1,...,Z_K are independent Materns with parameters(cat variance, cat spatial range, cat temporal range, cat smoothness),and e_1, ..., e_n are independent normals with variance (variance * nugget)

Usage

matern_spacetime_categorical_local(covparms, locs)d_matern_spacetime_categorical_local(covparms, locs)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlocs[i,] andlocs[j,].

Functions

• d_matern_spacetime_categorical_local(): Derivatives of isotropic Matern covariance

Parameterization

The covariance parameter vector is (variance, range, smoothness, category, nugget)=(\sigma^2,\alpha_1,\alpha_2,\nu,c^2,\tau^2), and the covariance function is parameterizedas

d = ( || x - y ||^2/\alpha_1 + |s-t|^2/\alpha_2^2 )^{1/2}

M(x,y) = \sigma^2 2^{1-\nu}/\Gamma(\nu) (d)^\nu K_\nu(d)

(x,s) and (y,t) are the space-time locations of a pair of observations.The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix.The category variancec^2 is added if two observation from same categoryNOTE: the nugget is \sigma^2 \tau^2, not \tau^2.

Isotropic Matern covariance function on sphere

Description

From a matrix of longitudes and latitudes and a vector covariance parameters of the form(variance, range, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_sphere(covparms, lonlat)d_matern_sphere(covparms, lonlat)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlonlat[i,] andlonlat[j,].

Functions

• d_matern_sphere(): Derivatives with respect to parameters

Matern on Sphere Domain

The function first calculates the (x,y,z) 3D coordinates, and then inputsthe resulting locations intomatern_isotropic. This means that we constructcovariances on the sphere by embedding the sphere in a 3D space. There has been someconcern expressed in the literature that such embeddings may produce distortions.The source and nature of such distortions has never been articulated,and to date, no such distortions have been documented. Guinness andFuentes (2016) argue that 3D embeddings produce reasonable models for data on spheres.

Deformed Matern covariance function on sphere

Description

From a matrix of longitudes and latitudes and a vector covariance parameters of the form(variance, range, smoothness, nugget, <5 warping parameters>), return the square matrix ofall pairwise covariances.

Usage

matern_sphere_warp(covparms, lonlat)d_matern_sphere_warp(covparms, lonlat)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlonlat[i,] andlonlat[j,].

Functions

• d_matern_sphere_warp(): Derivatives with respect to parameters.

Warpings

The function first calculates the (x,y,z) 3D coordinates, and then "warps"the locations to(x,y,z) + \Phi(x,y,z), where\Phi is a warpingfunction composed of gradients of spherical harmonic functions of degree 2.See Guinness (2019, "Gaussian Process Learning via Fisher Scoring of Vecchia's Approximation") for details.The warped locations are input intomatern_isotropic.

Matern covariance function on sphere x time

Description

From a matrix of longitudes, latitudes, and times, and a vector covariance parameters of the form(variance, range_1, range_2, smoothness, nugget), return the square matrix ofall pairwise covariances.

Usage

matern_spheretime(covparms, lonlattime)d_matern_spheretime(covparms, lonlattime)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlonlattime[i,] andlonlattime[j,].

Functions

• d_matern_spheretime(): Derivatives with respect to parameters

Covariances on spheres

The function first calculates the (x,y,z) 3D coordinates, and then inputsthe resulting locations intomatern_spacetime. This means that we constructcovariances on the sphere by embedding the sphere in a 3D space. There has been someconcern expressed in the literature that such embeddings may produce distortions.The source and nature of such distortions has never been articulated,and to date, no such distortions have been documented. Guinness andFuentes (2016) argue that 3D embeddings produce reasonable models for data on spheres.

Deformed Matern covariance function on sphere

Description

From a matrix of longitudes, latitudes, times, and a vector covariance parameters of the form(variance, range_1, range_2, smoothness, nugget, <5 warping parameters>), return the square matrix ofall pairwise covariances.

Usage

matern_spheretime_warp(covparms, lonlattime)d_matern_spheretime_warp(covparms, lonlattime)

Arguments

Value

A matrix withn rows andn columns, with the i,j entrycontaining the covariance between observations atlonlat[i,] andlonlat[j,].

Functions

• d_matern_spheretime_warp(): Derivatives with respect to parameters

Warpings

The function first calculates the (x,y,z) 3D coordinates, and then "warps"the locations to(x,y,z) + \Phi(x,y,z), where\Phi is a warpingfunction composed of gradients of spherical harmonic functions of degree 2.See Guinness (2019, "Gaussian Process Learning via Fisher Scoring of Vecchia's Approximation") for details.The warped locations are input intomatern_spacetime. The functiondoes not do temporal warping.

Sorted coordinate ordering

Description

Return the ordering of locations sorted along one of thecoordinates or the sum of multiple coordinates

Usage

order_coordinate(locs, coordinate)

Arguments

Value

A vector of indices giving the ordering, i.e. the first element of this vector is the index of the first location.

Examples

n <- 100             # Number of locationsd <- 2               # dimension of domainlocs <- matrix( runif(n*d), n, d )ord1 <- order_coordinate(locs, 1 )ord12 <- order_coordinate(locs, c(1,2) )

Distance to specified point ordering

Description

Return the ordering of locations increasing in theirdistance to some specified location

Usage

order_dist_to_point(locs, loc0, lonlat = FALSE)

Arguments

Value

A vector of indices giving the ordering, i.e. the first element of this vector is the index of the location nearest toloc0.

Examples

n <- 100             # Number of locationsd <- 2               # dimension of domainlocs <- matrix( runif(n*d), n, d )loc0 <- c(1/2,1/2)ord <- order_dist_to_point(locs,loc0)

Maximum minimum distance ordering

Description

Return the indices of an approximation to the maximum minimum distance ordering.A point in the center is chosen first, and then each successive pointis chosen to maximize the minimum distance to previously selected points

Usage

order_maxmin(locs, lonlat = FALSE, space_time = FALSE, st_scale = NULL)

Arguments

Value

A vector of indices giving the ordering, i.e. the first element of this vector is the index of the first location.

Examples

# planar coordinatesnvec <- c(50,50)locs <- as.matrix( expand.grid( 1:nvec[1]/nvec[1], 1:nvec[2]/nvec[2] ) )ord <- order_maxmin(locs)par(mfrow=c(1,3))plot( locs[ord[1:100],1], locs[ord[1:100],2], xlim = c(0,1), ylim = c(0,1) )plot( locs[ord[1:300],1], locs[ord[1:300],2], xlim = c(0,1), ylim = c(0,1) )plot( locs[ord[1:900],1], locs[ord[1:900],2], xlim = c(0,1), ylim = c(0,1) )# longitude/latitude coordinates (sphere)latvals <- seq(-80, 80, length.out = 40 )lonvals <- seq( 0, 360, length.out = 81 )[1:80]locs <- as.matrix( expand.grid( lonvals, latvals ) )ord <- order_maxmin(locs, lonlat = TRUE)par(mfrow=c(1,3))plot( locs[ord[1:100],1], locs[ord[1:100],2], xlim = c(0,360), ylim = c(-90,90) )plot( locs[ord[1:300],1], locs[ord[1:300],2], xlim = c(0,360), ylim = c(-90,90) )plot( locs[ord[1:900],1], locs[ord[1:900],2], xlim = c(0,360), ylim = c(-90,90) )

Middle-out ordering

Description

Return the ordering of locations increasing in theirdistance to the average location

Usage

order_middleout(locs, lonlat = FALSE)

Arguments

Value

A vector of indices giving the ordering, i.e. the first element of this vector is the index of the location nearest the center.

Examples

n <- 100             # Number of locationsd <- 2               # dimension of domainlocs <- matrix( runif(n*d), n, d )ord <- order_middleout(locs)

penalize large values of parameter: penalty, 1st deriative, 2nd derivative

Description

penalize large values of parameter: penalty, 1st deriative, 2nd derivative

Usage

pen_hi(x, tt, aa)dpen_hi(x, tt, aa)ddpen_hi(x, tt, aa)

Arguments

penalize small values of parameter: penalty, 1st deriative, 2nd derivative

Description

penalize small values of parameter: penalty, 1st deriative, 2nd derivative

Usage

pen_lo(x, tt, aa)dpen_lo(x, tt, aa)ddpen_lo(x, tt, aa)

Arguments

penalize small values of log parameter: penalty, 1st deriative, 2nd derivative

Description

penalize small values of log parameter: penalty, 1st deriative, 2nd derivative

Usage

pen_loglo(x, tt, aa)dpen_loglo(x, tt, aa)ddpen_loglo(x, tt, aa)

Arguments

Compute Gaussian process predictions using Vecchia's approximations

Description

With the prediction locations ordered after the observation locations,an approximation for the inverse Cholesky of the covariance matrixis computed, and standard formulas are applied to obtainthe conditional expectation.

Usage

predictions(  fit = NULL,  locs_pred,  X_pred,  y_obs = fit$y,  locs_obs = fit$locs,  X_obs = fit$X,  beta = fit$betahat,  covparms = fit$covparms,  covfun_name = fit$covfun_name,  m = 60,  reorder = TRUE,  st_scale = NULL)

Arguments

Details

We can specify either a GpGp_fit object (the result offit_model), OR manually enter the covariance function andparameters, the observations, observation locations, and design matrix. We must specify the prediction locations and the prediction design matrix.

compute gradient of spherical harmonics functions

Description

compute gradient of spherical harmonics functions

Usage

sph_grad_xyz(xyz, Lmax)

Arguments

Print summary of GpGp fit

Description

Print summary of GpGp fit

Usage

## S3 method for class 'GpGp_fit'summary(object, ...)

Arguments

test likelihood object for NA or Inf values

Description

test likelihood object for NA or Inf values

Usage

test_likelihood_object(likobj)

Arguments

Entries of inverse Cholesky approximation

Description

This function returns the entries of the inverse Choleskyfactor of the covariance matrix implied by Vecchia's approximation.For return matrixLinv,Linv[i,] contains the non-zero entries of rowi ofthe inverse Cholesky matrix. The columns of the non-zero entriesare specified inNNarray[i,].

Usage

vecchia_Linv(covparms, covfun_name, locs, NNarray, start_ind = 1L)

Arguments

Value

matrix containing entries of inverse Cholesky

Examples

n1 <- 40n2 <- 40n <- n1*n2locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )covparms <- c(2, 0.2, 0.75, 0)NNarray <- find_ordered_nn(locs,20)Linv <- vecchia_Linv(covparms, "matern_isotropic", locs, NNarray)

Grouped Vecchia approximation to the Gaussian loglikelihood, zero mean

Description

This function returns a grouped version (Guinness, 2018) of Vecchia's (1988) approximation to the Gaussianloglikelihood. The approximation modifies the ordered conditionalspecification of the joint density; rather than each term in the productconditioning on all previous observations, each term conditions ona small subset of previous observations.

Usage

vecchia_grouped_meanzero_loglik(covparms, covfun_name, y, locs, NNlist)

Arguments

Value

a list containing

• loglik: the loglikelihood

Examples

n1 <- 20n2 <- 20n <- n1*n2locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )covparms <- c(2, 0.2, 0.75, 0)y <- fast_Gp_sim(covparms, "matern_isotropic", locs, 50 )ord <- order_maxmin(locs)NNarray <- find_ordered_nn(locs,20)NNlist <- group_obs(NNarray)#loglik <- vecchia_grouped_meanzero_loglik( covparms, "matern_isotropic", y, locs, NNlist )

Grouped Vecchia approximation, profiled regression coefficients

Description

This function returns a grouped version (Guinness, 2018) of Vecchia's (1988) approximation to the Gaussianloglikelihood and the profile likelihood estimate of the regressioncoefficients. The approximation modifies the ordered conditionalspecification of the joint density; rather than each term in the productconditioning on all previous observations, each term conditions ona small subset of previous observations.

Usage

vecchia_grouped_profbeta_loglik(covparms, covfun_name, y, X, locs, NNlist)

Arguments

Value

a list containing

• loglik: the loglikelihood

• betahat: profile likelihood estimate of regression coefficients

• betainfo: information matrix forbetahat.

The covariancematrix for$betahat is the inverse of$betainfo.

Examples

n1 <- 20n2 <- 20n <- n1*n2locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )X <- cbind(rep(1,n),locs[,2])covparms <- c(2, 0.2, 0.75, 0)y <- fast_Gp_sim(covparms, "matern_isotropic", locs, 50 )ord <- order_maxmin(locs)NNarray <- find_ordered_nn(locs,20)NNlist <- group_obs(NNarray)#loglik <- vecchia_grouped_profbeta_loglik( #    covparms, "matern_isotropic", y, X, locs, NNlist )

Grouped Vecchia loglikelihood, gradient, Fisher information

Description

This function returns a grouped version (Guinness, 2018) of Vecchia's (1988) approximation to the Gaussianloglikelihood, the gradient, and Fisher information, and the profile likelihood estimate of the regressioncoefficients. The approximation modifies the ordered conditionalspecification of the joint density; rather than each term in the productconditioning on all previous observations, each term conditions ona small subset of previous observations.

Usage

vecchia_grouped_profbeta_loglik_grad_info(  covparms,  covfun_name,  y,  X,  locs,  NNlist)

Arguments

Value

a list containing

• loglik: the loglikelihood

• grad: gradient with respect to covariance parameters

• info: Fisher information for covariance parameters

• betahat: profile likelihood estimate of regression coefs

• betainfo: information matrix forbetahat.

The covariancematrix for$betahat is the inverse of$betainfo.

Examples

n1 <- 20n2 <- 20n <- n1*n2locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )X <- cbind(rep(1,n),locs[,2])covparms <- c(2, 0.2, 0.75, 0)y <- fast_Gp_sim(covparms, "matern_isotropic", locs, 50 )ord <- order_maxmin(locs)NNarray <- find_ordered_nn(locs,20)NNlist <- group_obs(NNarray)#loglik <- vecchia_grouped_profbeta_loglik_grad_info( #    covparms, "matern_isotropic", y, X, locs, NNlist )

Vecchia's approximation to the Gaussian loglikelihood, zero mean

Description

This function returns Vecchia's (1988) approximation to the Gaussianloglikelihood. The approximation modifies the ordered conditionalspecification of the joint density; rather than each term in the productconditioning on all previous observations, each term conditions ona small subset of previous observations.

Usage

vecchia_meanzero_loglik(covparms, covfun_name, y, locs, NNarray)

Arguments

Value

a list containing

• loglik: the loglikelihood

Examples

n1 <- 20n2 <- 20n <- n1*n2locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )covparms <- c(2, 0.2, 0.75, 0)y <- fast_Gp_sim(covparms, "matern_isotropic", locs, 50 )ord <- order_maxmin(locs)NNarray <- find_ordered_nn(locs,20)#loglik <- vecchia_meanzero_loglik( covparms, "matern_isotropic", y, locs, NNarray )

Vecchia's approximation to the Gaussian loglikelihood, with profiled regression coefficients.

Description

This function returns Vecchia's (1988) approximation to the Gaussianloglikelihood, profiling out the regression coefficients. The approximation modifies the ordered conditionalspecification of the joint density; rather than each term in the productconditioning on all previous observations, each term conditions ona small subset of previous observations.

Usage

vecchia_profbeta_loglik(covparms, covfun_name, y, X, locs, NNarray)

Arguments

Value

a list containing

• loglik: the loglikelihood

• betahat: profile likelihood estimate of regression coefficients

• betainfo: information matrix forbetahat.

The covariancematrix for$betahat is the inverse of$betainfo.

Examples

n1 <- 20n2 <- 20n <- n1*n2locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )X <- cbind(rep(1,n),locs[,2])covparms <- c(2, 0.2, 0.75, 0)y <- X %*% c(1,2) + fast_Gp_sim(covparms, "matern_isotropic", locs, 50 )ord <- order_maxmin(locs)NNarray <- find_ordered_nn(locs,20)#loglik <- vecchia_profbeta_loglik( covparms, "matern_isotropic", y, X, locs, NNarray )

Vecchia's loglikelihood, gradient, and Fisher information

Description

This function returns Vecchia's (1988) approximation to the Gaussianloglikelihood, profiling out the regression coefficients, and returningthe gradient and Fisher information. Vecchia's approximation modifies the ordered conditionalspecification of the joint density; rather than each term in the productconditioning on all previous observations, each term conditions ona small subset of previous observations.

Usage

vecchia_profbeta_loglik_grad_info(covparms, covfun_name, y, X, locs, NNarray)

Arguments

Value

A list containing

• loglik: the loglikelihood

• grad: gradient with respect to covariance parameters

• info: Fisher information for covariance parameters

• betahat: profile likelihood estimate of regression coefs

• betainfo: information matrix forbetahat.

The covariance matrix for$betahat is the inverse of$betainfo.

Examples

n1 <- 20n2 <- 20n <- n1*n2locs <- as.matrix( expand.grid( (1:n1)/n1, (1:n2)/n2 ) )X <- cbind(rep(1,n),locs[,2])covparms <- c(2, 0.2, 0.75, 0)y <- X %*% c(1,2) + fast_Gp_sim(covparms, "matern_isotropic", locs, 50 )ord <- order_maxmin(locs)NNarray <- find_ordered_nn(locs,20)#loglik <- vecchia_profbeta_loglik_grad_info( covparms, "matern_isotropic", #    y, X, locs, NNarray )