Title: Spatial Misalignment: Interpolation, Linkage, and Estimation
Version: 1.1.0
Date: 2025-09-11
Description: Provides functions to estimate, predict and interpolate areal data. For estimation and prediction we assume areal data is an average of an underlying continuous spatial process as in Moraga et al. (2017) <doi:10.1016/j.spasta.2017.04.006>, Johnson et al. (2020) <doi:10.1186/s12942-020-00200-w>, and Wilson and Wakefield (2020) <doi:10.1093/biostatistics/kxy041>. The interpolation methodology is (mostly) based on Goodchild and Lam (1980, ISSN:01652273).
License: GPL-3
Encoding: UTF-8
LazyData: true
RoxygenNote: 7.3.2
SystemRequirements: GDAL (>= 2.0.1), GEOS (>= 3.4.0), PROJ (>= 4.8.0)
LinkingTo: Rcpp, RcppEigen
Imports: numDeriv, Rcpp, sf, mvtnorm, stats, parallel, Matrix
Depends: R (≥ 4.1.0)
URL: https://lcgodoy.me/smile/, https://github.com/lcgodoy/smile/
BugReports: https://github.com/lcgodoy/smile/issues/
Suggests: knitr, rmarkdown, ggplot2, graphics
VignetteBuilder: knitr
Language: en-US
NeedsCompilation: yes
Packaged: 2025-09-22 21:34:59 UTC; lcgodoy
Author: Lucas da Cunha Godoy ORCID iD [aut, cre]
Maintainer: Lucas da Cunha Godoy <lcgodoy@duck.com>
Repository: CRAN
Date/Publication: 2025-09-22 22:00:02 UTC

Areal Interpolation

Description

This function estimates variables observed at a "source" region into a "target" region. "Source" and "target" regions represent two different ways to divide a city, for example. For more details, see https://lcgodoy.me/smile/articles/sai.html.

Usage

ai(source, target, vars)

ai_var(source, target, vars, vars_var, sc_vars = FALSE, var_method = "CS")

Arguments

source

a sf object - source spatial data.

target

a sf object - target spatial data.

vars

a character representing the variables (observed at the source) to be estimated at the target data.

vars_var

a scalar of type character representing the name of the variable in the source dataset that stores the variances of the variable to be estimated at the target data.

sc_vars

boolean indicating whether vars should be scaled by its observed variance (if available).

var_method

a character representing the method to approximate the variance of the AI estimates. Possible values are "CS" (Cauchy-Schwartz) or "MI" (Moran's I).

Value

the target (of type sf) with estimates of the variables observed at the source data.

Internal use only

Description

aggregate.sf taken from sf.

Usage

aggregate_aux(
  x,
  by,
  FUN,
  ...,
  do_union = TRUE,
  simplify = TRUE,
  join = st_intersects
)

Arguments

x

internal use

by

internal use

FUN

internal use

...

internal use

do_union

internal use

simplify

internal use

join

internal use

Pairwise distances for a list of matrices (Internal use)

Description

Pairwise distances for a list of matrices (Internal use)

Cross-distances for two lists of matrices (Eigen version)

Prediction cross-distances (Eigen version)

Usage

single_dists(mat_list)

mult_dists(mat_list1, mat_list2, return_single)

pred_cdist(mat_list, pred_mat)

get_grid_list(x_to_list, by)

dist_from_grids(y_grid, by)

mult_dist_from_grids(y_grid, x_grid, by)

Arguments

mat_list

internal use

mat_list1

internal use

mat_list2

internal use

return_single

internal use

pred_mat

internal use

x_to_list

internal use

by

internal use

y_grid

internal use

x_grid

internal use

Pairwise distances between two matrices (Eigen version)

Description

Computes Euclidean distance between rows of m1 and rows of m2.

Usage

crossdist(m1, m2)

Arguments

m1

A matrix where each row is a 2D coordinate.

m2

A matrix where each row is a 2D coordinate.

Creating a symmetric distance matrix (Eigen version)

Description

Computes the Euclidean distance between all pairs of rows in a matrix.

Usage

distmat(my_mat)

Arguments

my_mat

A matrix where each row is a 2D coordinate.

MLEs for fixed V.

Description

MLEs for fixed V.

Usage

est_mle(y, Vinv)

Arguments

y

response variable.

Vinv

inverse of V

Details

internal use.

Find phi parameter for the Exponential spatial auto-correlation function

Description

Function designed to find the phi parameter such that the correlation between points within a given distance d is at most a given value.

Usage

find_phi(
  d,
  nu,
  kappa,
  mu2,
  family = "matern",
  range = c(1e-04, 1000),
  cut = 0.05
)

Arguments

d

maximum distance for spatial dependence equal to cut.

nu

smoothness parameter associated with the Matern cov. function.

kappa

one of the smoothness parameters associated with the Generalized Wendland covariance function

mu2

one of the smoothness parameters associated with the Generalized Wendland covariance function

family

covariance function family, the options are c("matern", "gw", "cs", "spher", "pexp", "gaussian").

range

Minimum and maximum distance to be considered. The default is range = c(1e-04, 1000).

cut

desired spatial correlation at a distance d, the default is cut = .05.

Value

a numeric value indicating the range parameter such that the spatial correlation between two points at distance d is cut.

Fitting an underlying continuous process to areal data

Description

This function fits a spatial process model to areal data by estimating the parameters of an underlying continuous process. It leverages the stats::optim function for Maximum Likelihood estimation, providing flexibility in optimization algorithms and control parameters.

Usage

fit_spm(x, ...)

## S3 method for class 'spm'
fit_spm(
  x,
  model,
  theta_st,
  nu = NULL,
  kappa = 1,
  mu2 = 1.5,
  apply_exp = FALSE,
  opt_method = "Nelder-Mead",
  control_opt = list(),
  comp_hess = TRUE,
  ...
)

fit_spm2(
  x,
  model,
  nu,
  kappa = 1,
  mu2 = 1.5,
  comp_hess = TRUE,
  phi_min,
  phi_max,
  nphi = 10,
  cores = getOption("mc.cores", 1L)
)

Arguments

x

An object of type spm. The dimension of theta_st depends on the number of variables analyzed and whether the input is an spm object.

...

Additional parameters passed to stats::optim.

model

A character scalar indicating the covariance function family. Options are c("matern", "pexp", "gaussian", "spherical", "gw").

theta_st

A named numeric vector containing initial parameter values.

nu

A numeric value specifying the \nu parameter for the Matern or Powered Exponential covariance functions. If model is "matern" and nu is not provided, it defaults to 0.5. If model is "pexp" and nu is not provided, it defaults to 1. In both cases, this results in the exponential covariance function.

kappa

\kappa \in \{0, \ldots, 3 \} parameter for the GW covariance function.

mu2

The smoothness parameter \mu for the GW function.

apply_exp

A logical scalar indicating whether to exponentiate non-negative parameters.

opt_method

A character scalar specifying the optimization algorithm (see stats::optim).

control_opt

A named list of control arguments for the optimization algorithm (see stats::optim).

comp_hess

A logical indicating whether to compute the Hessian matrix.

phi_min

A numeric scalar representing the minimum phi value for grid search.

phi_max

A numeric scalar representing the maximum phi value for grid search.

nphi

A numeric scalar indicating the number of phi values for grid search.

cores

An integer scalar indicating the number of cores to use. Defaults to getOption("mc.cores"). No effect on Windows.

Details

The function utilizes optimization algorithms from stats::optim to determine Maximum Likelihood estimators (MLEs) and their standard errors for a model adapted for areal data. Users can customize the optimization process by providing control parameters through the control_opt argument (a named list, see stats::optim for details), specifying lower and upper bounds for parameters, and selecting the desired optimization algorithm via opt_method (must be available in stats::optim).

Additionally, the function supports various covariance functions, including Matern, Exponential, Powered Exponential, Gaussian, and Spherical. The apply_exp argument, a logical scalar, allows for exponentiation of non-negative parameters, enabling optimization over the entire real line.

The underlying model assumes a point-level process:

Y(\mathbf{s}) = \mu + S(\mathbf{s})

where S ~ GP(0, \sigma^2 C(\lVert \mathbf{s} - \mathbf{s}_2 \rVert; \theta)). The observed areal data is then assumed to behave as the average of the process over each area:

Y(B) = \lvert B \rvert^{-1} \int_{B} Y(\mathbf{s}) \, \textrm{d} \mathbf{s}

.

Value

An spm_fit object containing the estimated model parameters.

Examples

data(liv_lsoa) ## loading the LSOA data

msoa_spm <- sf_to_spm(sf_obj = liv_msoa, n_pts = 500,
                     type = "regular", by_polygon = FALSE,
                     poly_ids = "msoa11cd",
                     var_ids = "leb_est")
## fitting model
theta_st_msoa <- c("phi" = 1) # initial value for the range parameter

fit_msoa <-
  fit_spm(x = msoa_spm,
          theta_st = theta_st_msoa,
          model = "matern",
          nu = .5,
          apply_exp  = TRUE,
          opt_method = "L-BFGS-B",
          control    = list(maxit = 500))

AIC(fit_msoa)

summary_spm_fit(fit_msoa, sig = .05)

Akaike's (and Bayesian) An Information Criterion for spm_fit objects.

Description

Akaike's (and Bayesian) An Information Criterion for spm_fit objects.

Usage

## S3 method for class 'spm_fit'
AIC(object, ..., k = 2)

## S3 method for class 'spm_fit'
BIC(object, ...)

Arguments

object

a spm_fit object.

...

optionally more fitted model objects.

k

numeric, the penalty per parameter to be used; the default 'k = 2' is the classical AIC. (for compatibility with stats::AIC.

Value

a numeric scalar corresponding to the goodness of fit measure.

Liverpool Lower Super Output Area.

Description

A dataset containing containing the LSOA's for Liverpool along with estimates for Index of Multiple Deprivation. Data taken from Johnson et al. 2020

Usage

liv_lsoa

Format

A sf data frame with 298 rows and 6 variables:

lsoa11cd

LSOA code

lsoa11cd

LSOA name

male

Male population

female

Female population

imdscore

Index of Multiple Deprivation

area

LMSOA area, in km^2

Details

The data was projected to EPSG 27700 and units changed to km

Source

https://ij-healthgeographics.biomedcentral.com/articles/10.1186/s12942-020-00200-w

Liverpool Middle Super Output Area.

Description

A dataset containing containing the MSOA's for Liverpool along with estimates for Life Expectancy at Birth. Data taken from Johnson et al. 2020

Usage

liv_msoa

Format

A sf data frame with 61 rows and 4 variables:

msoa11cd

MSOA code

msoa11cd

MSOA name

lev_est

Estimated life expectancy at birth, in years

area

MSOA area, in km^2

Details

The data was projected to EPSG 27700 and units changed to km

Source

https://ij-healthgeographics.biomedcentral.com/articles/10.1186/s12942-020-00200-w

Matern covariance function for a given distance matrix.

Description

Computing the Matern covariance function for a matrix of distances.

Computing the Matern covariance function between polygons.

Usage

mat_cov(dists, sigsq, phi, nu)

comp_mat_cov(cross_dists, n, n2, sigsq, phi, nu)

pexp_cov(dists, sigsq, phi, nu)

comp_pexp_cov(cross_dists, n, n2, sigsq, phi, nu)

gauss_cov(dists, sigsq, phi)

comp_gauss_cov(cross_dists, n, n2, sigsq, phi)

spher_cov(dists, sigsq, phi)

comp_spher_cov(cross_dists, n, n2, sigsq, phi)

cs_cov(dists, sigsq, phi)

comp_cs_cov(cross_dists, n, n2, sigsq, phi)

gw_cov(dists, sigsq, phi, kappa, mu)

comp_gw_cov(cross_dists, n, n2, sigsq, phi, kappa, mu)

Arguments

dists

a numeric matrix representing the distance between spatial entities.

sigsq

the \sigma^2 parameter from the Matern covariance function.

phi

the \phi parameter from the Matern covariance function, controls the range of the spatial dependence.

nu

the \nu parameter from the Matern covariance function, controls the differentiability of the process.

cross_dists

a list such that each position contains the cross distances between points within different polygons.

Value

The matern covariance function (for a stationary and isotropic process) associated with the provided distances (dists) and the given set of parameters.

The matern covariance matrix associated with a set of polygons.

See Also

single_exp, single_matern

single_exp, single_matern, mat_cov

Calculates the (global) Moran's I

Description

Calculates the (global) Moran's I

Usage

morans_i(sf_dt, variable)

Arguments

sf_dt

a sf (with POLYGON geometry) dataset.

variable

a character representing one of the variables from sf_dt.

Value

a numeric scalar.

Nova Lima census tracts

Description

A dataset containing containing the census tracts for the city of Nova Lima in Minas Gerais - Brazil.

Usage

nl_ct

Format

A sf data frame with 113 rows and 14 variables:

cd_setor

unique identifier

hh_density

average household density

var_hhd

variance of the household density

avg_income

average income per household

var_income

variance of the income per household

pop

population in the census tract

avg_age

average age of the inhabitants in the census tract

var_age

variance of the variable age in the census tract

prop_women

proportion of women

prop_elder

proportion of people with 55 years of age or older

illit_rate

illiteracy rate

prop_white

proportion of self-declared white people

prop_black

proportion of self-declared black people

prop_native

proportion of self-declared native people

Details

The data is project using the SIRGAS 2000.

Prediction over the same or a different set of regions (or points).

Description

Realizes predictions that can be useful when researchers are interested in predict a variable observed in one political division of a city (or state) on another division of the same region.

Usage

predict_spm(x, ...)

## S3 method for class 'spm_fit'
predict_spm(x, .aggregate = TRUE, ...)

## S3 method for class 'sf'
predict_spm(x, spm_obj, n_pts, type, outer_poly = NULL, id_var, ...)

Arguments

x

a sf object such that its geometries are either points or polygons.

...

additional parameters

.aggregate

logical. Should the predictions be aggregated? In case the input is only a "fit" object, the aggregation is made over the polygons on which the original data was observed. In case the input x is composed by sf POLYGONS, the aggregation is made over this new partition of the study region.

spm_obj

an object of either class spm_fit or mspm_fit

n_pts

a numeric scalar standing for number of points to form a grid over the whole region to make the predictions

type

character type of grid to be generated. See st_sample in the package sf.

outer_poly

(object) sf geometry storing the "outer map" we want to compute the predictions in.

id_var

if x is a set of POLYGONS (areal data) instead of a set of points, the id_var is the name (or index) of the unique identifier associated to each polygon.

Value

a list of size 4 belonging to the class spm_pred. This list contains the predicted values and the mean and covariance matrix associated with the conditional distribution used to compute the predictions.

Examples


data(liv_lsoa) ## loading the LSOA data
data(liv_msoa) ## loading the MSOA data

msoa_spm <- sf_to_spm(sf_obj = liv_msoa, n_pts = 500,
                      type = "regular", by_polygon = FALSE,
                      poly_ids = "msoa11cd",
                      var_ids = "leb_est")
## fitting model
theta_st_msoa <- c("phi" = 1) # initial value for the range parameter

fit_msoa <-
   fit_spm(x = msoa_spm,
           theta_st = theta_st_msoa,
           model = "matern",
           nu = .5,
           apply_exp  = TRUE,
           opt_method = "L-BFGS-B",
           control    = list(maxit = 500))

pred_lsoa <- predict_spm(x = liv_lsoa, spm_obj = fit_msoa, id_var = "lsoa11cd")

Calculate Smallest Eigenvalue for Power Exponential Correlation Matrices

Description

This function computes the smallest eigenvalue of a correlation matrix derived from the power exponential correlation function. It evaluates this across a grid of values for the power parameter (nu) and the practical range parameter (rho), based on a provided distance matrix.

Usage

sev_pexp(range_nu, range_rho, grid_len = 50, dmat)

Arguments

range_nu

A numeric vector of length 2, specifying the minimum and maximum values for the power parameter nu. nu typically ranges between 0 and 2 (e.g., nu = 1 for exponential, nu = 2 for Gaussian).

range_rho

A numeric vector of length 2, specifying the minimum and maximum values for the practical range parameter rho. rho must be positive.

grid_len

An integer specifying the number of points to create for both nu and rho sequences. The total number of grid combinations will be grid_len^2. Default is 50.

dmat

A numeric matrix representing the distance matrix between locations. The distances should be non-negative.

Details

The practical range rho is defined here as the distance at which the correlation is 0.1. The internal scale parameter phi is calculated as phi = rho / (log(10)^(1/nu)). The power exponential correlation function is assumed to be of the form C(h) = exp(-(h/phi)^nu), where h is distance. The function smile:::pexp_cov is used internally to compute the covariance/correlation matrix with a sill of 1.

The function first creates a grid of nu and rho parameters. For each pair of (rho, nu) in the grid: 1. It calculates the scale parameter phi for the power exponential correlation function, where phi = rho / (log(10)^(1/nu)). This definition implies that the correlation is 0.1 at the distance rho. 2. It computes the power exponential correlation matrix using smile:::pexp_cov(dists = dmat, sill = 1, range = phi, smooth = nu). Note the use of an internal function from the smile package. 3. It calculates the eigenvalues of this correlation matrix. 4. The minimum eigenvalue is extracted. The final output is a tibble containing all parameter combinations and their corresponding minimum eigenvalues.

Value

A tibble with three columns:

rho

The practical range parameter value.

nu

The power parameter value.

lambda

The smallest eigenvalue of the power exponential correlation matrix corresponding to the rho and nu pair.

single sf to spm

Description

Transforming a sf into a spm object (Internal use)

Usage

single_sf_to_spm(
  sf_obj,
  n_pts,
  type = "regular",
  by_polygon = FALSE,
  poly_ids = NULL,
  var_ids = NULL
)

sf_to_spm(
  sf_obj,
  n_pts,
  type = "regular",
  by_polygon = FALSE,
  poly_ids = NULL,
  var_ids = NULL
)

Arguments

sf_obj

a sf object s.t. its geometries are polygons.

n_pts

a numeric scalar representing the number of points to create a grid in the study region on which the polygons in sf_obj is observed. Alternatively, it can be a vector of the same length as nrow(sf_obj). In this case, it generates the given number of points for each polygon in sf_obj.

type

a character indicating the type of grid to be generated. The options are c("random", "regular", "hexagonal"). For more details, see st_sample in the sf package.

by_polygon

a logical indicating whether we should generate n_pts by polygon or for the n_pts for the whole study region.

poly_ids

a character vector informing the name of the variable in sf_obj that represents the polygons unique identifiers. In case this is not informed, we assume the id of the polygons are given by their row numbers.

var_ids

a scalar or vector of type character indicating the (numerical) variables that are going to be analyzed.

Value

a named list of size 6 belonging to the class spm. This list stores all the objects necessary to fit models using the fit_spm.

Examples

data(liv_lsoa) # loading the LSOA data

msoa_spm <- sf_to_spm(sf_obj = liv_msoa, n_pts = 1000,
                      type = "regular", by_polygon = FALSE,
                      poly_ids = "msoa11cd",
                      var_ids = "leb_est")

Evaluate log-lik

Description

Evaluate the log-likelihood for a given set of parameters

Usage

singl_ll_nn_hess(
  theta,
  .dt,
  dists,
  npix,
  model,
  nu = NULL,
  kappa = 1,
  mu2 = 1.5,
  apply_exp = FALSE
)

Arguments

theta

a numeric vector of size 3 containing the parameters values associated with \mu, \sigma^2, and \phi, respectively.

.dt

a numeric vector containing the variable Y.

dists

a list of size three. The first containing the distance matrices associated with the regions where Y was measured, the second for the distance matrices associated with X, and the last containing the cross-distance matrices.

npix

a integer vector containing the number of pixels within each polygon. (Ordered by the id variables for the polygons).

model

a character indicating which covariance function to use. Possible values are c("matern", "pexp", "gaussian", "spherical").

nu

\nu parameter. Not necessary if mode is "gaussian" or "spherical"

kappa

\kappa \in \{0, \ldots, 3 \} parameter for the GW cov function.

mu2

the smoothness parameter \mu for the GW function.

apply_exp

a logical indicating whether the exponential transformation should be applied to variance parameters. This facilitates the optimization process.

Details

Internal use.

Value

a scalar representing -log.lik.

Evaluate log-lik

Description

Evaluate the log-likelihood for a given set of parameters

Usage

singl_log_lik(
  theta,
  .dt,
  dists,
  npix,
  model,
  nu = NULL,
  kappa = 1,
  mu2 = 1.5,
  apply_exp = FALSE
)

Arguments

theta

a numeric vector of size 4 (\mu, \sigma^2, \alpha, \phi) containing the parameters associated with the model.

.dt

a numeric vector containing the variable Y.

dists

a list of size distance matrices at the point level.

npix

a integer vector containing the number of pixels within each polygon. (Ordered by the id variables for the polygons).

model

a character indicating which covariance function to use. Possible values are c("matern", "pexp", "gaussian", "spherical", "cs", "gw").

nu

\nu parameter. Not necessary if model is "gaussian" or "spherical"

kappa

\kappa \in \{0, \ldots, 3 \} parameter for the GW cov function.

mu2

the smoothness parameter \mu for the GW function.

apply_exp

a logical indicating whether the exponential transformation should be applied to variance parameters. This facilitates the optimization process.

Details

Internal use.

Value

a scalar representing -log.lik.

Evaluate log-lik

Description

Evaluate the log-likelihood for a given set of parameters - No nugget + profile likelihood

Usage

singl_log_lik_nn(
  theta,
  .dt,
  dists,
  npix,
  model,
  nu = NULL,
  kappa = 1,
  mu2 = 1.5,
  apply_exp = FALSE
)

Arguments

theta

a scalar for the \phi parameter.

.dt

a numeric vector containing the variable Y.

dists

a list of size three. The first containing the distance matrices associated with the regions where Y was measured, the second for the distance matrices associated with X, and the last containing the cross-distance matrices.

npix

a integer vector containing the number of pixels within each polygon. (Ordered by the id variables for the polygons).

model

a character indicating which covariance function to use. Possible values are c("matern", "pexp", "gaussian", "spherical", "cs", "gw", "tapmat").

nu

\nu parameter. Not necessary if mode is "gaussian" or "spherical"

kappa

\kappa \in \{0, \ldots, 3 \} parameter for the GW cov function.

mu2

the smoothness parameter \mu for the GW function.

apply_exp

a logical indicating whether the exponential transformation should be applied to variance parameters. This facilitates the optimization process.

Details

Internal use.

Value

a scalar representing -log.lik.

Evaluate log-lik

Description

Evaluate the log-likelihood for a given set of parameters - New parametrization + profile likelihood

Usage

singl_log_plik(
  theta,
  .dt,
  dists,
  npix,
  model,
  nu = NULL,
  kappa = 1,
  mu2 = 1.5,
  apply_exp = FALSE
)

Arguments

theta

a vector of size 2 containing the parameters associated with the model. These parameters are \nu and \phi, respectively.

.dt

a numeric vector containing the variable Y.

dists

a list of size three. The first containing the distance matrices associated with the regions where Y was measured, the second for the distance matrices associated with X, and the last containing the cross-distance matrices.

npix

a integer vector containing the number of pixels within each polygon. (Ordered by the id variables for the polygons).

model

a character indicating which covariance function to use. Possible values are c("matern", "pexp", "gaussian", "spherical", "cs", "gw", "tapmat").

nu

\nu parameter. Not necessary if mode is "gaussian" or "spherical"

kappa

\kappa \in \{0, \ldots, 3 \} parameter for the GW cov function.

mu2

the smoothness parameter \mu for the GW function.

apply_exp

a logical indicating whether the exponential transformation should be applied to variance parameters. This facilitates the optimization process.

Details

Internal use.

Value

a scalar representing -log.lik.

Cubic spline covariance function (scalar)

Description

Computing the Spherical covariance function for a scalar distance.

Usage

single_cs(d, sigsq, phi)

Arguments

d

a scalar representing the distance on which it is desired to evaluate the covariance function.

sigsq

the \sigma^2 parameter from the Spherical covariance. function.

phi

the \phi parameter from the Spherical covariance function, controls the range of the spatial dependence.

Value

a scalar representing the (gaussian) covariance between two observations d apart of each other.

See Also

single_exp, single_matern, single_matern3, single_matern5, mat_cov

Matern covariance function (scalar - generic)

Description

Computing the Matern covariance function for a scalar distance, adapted from geoR.

Usage

single_exp(d, sigsq, phi)

single_matern3(d, sigsq, phi)

single_matern5(d, sigsq, phi)

single_matern(d, sigsq, phi, nu)

single_gauss(d, sigsq, phi)

Arguments

d

a scalar representing the distance on which it is desired to evaluate the covariance function.

sigsq

the \sigma^2 parameter from the Matern covariance function.

phi

the \phi parameter from the Matern covariance function, controls the range of the spatial dependence.

nu

the \nu parameter from the Matern covariance function, controls the differentiability of the process.

Details

single_matern3 and single_matern5 are optimized for when \nu is 1.5 or 2.5, respectively. Similarly, single_exp and single_gauss represent the cases where \nu = 0.5 or \nu \to \infty. In other words, they are the exponential and Gaussian covariance functions.

Value

a scalar representing the (matern) covariance between two observations d apart of each other.

See Also

single_matern3, single_matern5 single_exp, mat_cov

Matern Generalized Wendland (GW) covariance function (scalar - generic)

Description

adapted from Bevilacqua et al. 2019.

Usage

single_gw0(d, sigsq, phi, mu)

single_gw1(d, sigsq, phi, mu)

single_gw2(d, sigsq, phi, mu)

single_gw3(d, sigsq, phi, mu)

single_gw(d, sigsq, phi, kappa, mu)

Arguments

d

a scalar representing the distance on which it is desired to evaluate the covariance function.

sigsq

the \sigma^2 parameter from the Matern covariance function.

phi

the \phi parameter from the Matern covariance function, controls the range of the spatial dependence.

mu

a parameter that controls the smoothness of the covariance function. Note that, \mu \geq 1.

kappa

\kappa \in \{0, \ldots, 3 \}.

Value

a scalar representing the GW covariance between two observations d apart of each other.

Powered Exponential covariance function (scalar)

Description

Computing the Powered Exponential covariance function for a scalar distance.

Usage

single_pexp(d, sigsq, phi, nu)

Arguments

d

a scalar representing the distance on which it is desired to evaluate the covariance function.

sigsq

the \sigma^2 parameter from the Exponential covariance function.

phi

the \phi parameter from the Exponential covariance function, controls the range of the spatial dependence.

nu

the \nu \in (0, 2] parameter representing the "power"

Value

a scalar representing the (exponential) covariance between two observations d apart of each other.

See Also

single_exp, single_matern, single_matern3, single_matern5, mat_cov

Spherical covariance function (scalar)

Description

Computing the Spherical covariance function for a scalar distance.

Usage

single_spher(d, sigsq, phi)

Arguments

d

a scalar representing the distance on which it is desired to evaluate the covariance function.

sigsq

the \sigma^2 parameter from the Spherical covariance. function.

phi

the \phi parameter from the Spherical covariance function, controls the range of the spatial dependence.

Value

a scalar representing the (gaussian) covariance between two observations d apart of each other.

See Also

single_exp, single_matern, single_matern3, single_matern5, mat_cov

smile: Spatial MIsaLignment Estimation

Description

smile: Spatial MIsaLignment Estimation

Remove holes from a sfc POLYGON

Description

internal use. Taken from https://github.com/michaeldorman/nngeo/

Usage

st_remove_holes(x)

Arguments

x

a sf or sfc polygon.

Value

a sf or sfc polygon.

Summarizing spm_fit

Description

Provides a data.frame with point estimates and confidence intervals for the parameters of the model fitted using the spm_fit function.

Usage

summary_spm_fit(x, sig = 0.05)

Arguments

x

a spm_fit object.

sig

a real number between 0 and 1 indicating significance level to be used to compute the confidence intervals for the parameter estimates.

Value

a data.frame summarizing the parameters estimated by the fit_spm function.

Transform method for sf objects

Description

Internal usage.

Usage

## S3 method for class 'sf'
transform(`_data`, ...)

Arguments

_data

a sf object

...

additional options

Value

an sf object.

Voronoi Data Linkage

Description

Reminder, have to create an example. This will be exported after we submit the paper for publication.

Usage

vdl(coords_sf, areal_sf, vars, buff)

Arguments

coords_sf

sf POINT target dataset.

areal_sf

sf POLYGON source dataset.

vars

a character representing the variables (observed at the source - polygon) to be estimated at the target data.

buff

scalar numeric. Mostly for internal use.

Value

a sf object for the coords_sf spatial data set.

Voronoi Data Linkage - Single variable and variance

Description

Reminder, have to create an example. This will be exported after we submit the paper for publication.

Usage

vdl_var(coords_sf, areal_sf, res_var, variance, var_method = "CS", buff)

Arguments

coords_sf

sf POINT target dataset.

areal_sf

sf POLYGON source dataset.

res_var

a character - the name of the variable in the areal_sf to be estimated in the coords_sf.

variance

a character - the name of the variable variance in the areal_sf to be estimated in the coords_sf.

var_method

a character representing the method to approximate the variance of the AI estimates. Possible values are "CS" (Cauchy-Schwartz) or "MI" (Moran's I).

buff

scalar numeric. Mostly for internal use.

Value

a sf object, containing the id_coords variable and the list_vars for the coords_sf spatial data set.

Voronoi Tessellation inside a polygon

Description

voronoi tessellation of a given a set of points inside a polygon. This is an internal use function.

Usage

vor_build(points_sf, poly_sf)

Arguments

points_sf

sf data frame containing the points' coordinates

poly_sf

a sf polygon

Value

a sf object containing the polygons associated with the voronoi tessellation of points_sf the polygon poly_sf

Building weight matrix W for Areal Interpolation

Description

internal use. W_{ij} = | A_i \, cap \, B_j |.

Usage

w_col(source_unit, target)

build_w(source, target)

est_w(W, source_dt, target)

var_w(W, var_vec, target, method = "CS", rho_mi)

Arguments

source_unit

a single geometry from the source dataset.

target

a sf object - target spatial data.

source

a sf object - source spatial data.

W

the weight matrix.

source_dt

a data.frame object representing the source dataset but excluding the geometry, i.e. the spatial information, column.

var_vec

a numeric vector with variances observed at the source data.

method

a character representing the method to approximate the variance of the AI estimates. Possible values are "CS" (Cauchy-Schwartz) or "MI" (Moran's I).

rho_mi

numeric calculated Moran's I.

Value

A n \times m numeric matrix. Where n is the number of observations in the target and m is the sample size in the source dataset.