fdaPDEdeprecated  R Documentation 
Only executed when smooth.FEM.basis
is run with the option CPP_CODE
= FALSE
. It computes the mass matrix. The element (i,j) of this matrix contains the integral over the domain of the product between the ith and kth element
of the Finite Element basis. As common practise in Finite Element Analysis, this quantities are computed iterating over all the mesh triangles.
Only executed when smooth.FEM.basis
is run with the option CPP_CODE
= FALSE
. It computes the stifness matrix. The element (i,j) of this matrix contains the integral over the domain of the scalar product between the gradient of the ith and kth element
of the Finite Element basis. As common practise in Finite Element Analysis, this quantities are computed iterating over all the mesh triangles.
Only executed when the function smooth.FEM.basis
is run with the option CPP_CODE
= FALSE
. It evaluates the Finite Element basis functions and their derivatives up to order 2 at the specified set of locations.
This version of the function is implemented using only R code. It is called by R_smooth.FEM.basis.
Only executed when the function smooth.FEM.basis
is run with the option CPP_CODE
= FALSE
. It evaluates a FEM object at the specified set of locations.
This function implements a spatial regression model with differential regularization; isotropic and stationary case. In particular, the regularizing term involves the Laplacian of the spatial field. Spacevarying covariates can be included in the model. The technique accurately handle data distributed over irregularly shaped domains. Moreover, various conditions can be imposed at the domain boundaries.
This function implements a spatial regression model with differential regularization; anysotropic case. In particular, the regularizing term involves a second order elliptic PDE, that models the spacevariation of the phenomenon. Spacevarying covariates can be included in the model. The technique accurately handle data distributed over irregularly shaped domains. Moreover, various conditions can be imposed at the domain boundaries.
This function implements a spatial regression model with differential regularization; anysotropic and nonstationary case. In particular, the regularizing term involves a second order elliptic PDE with spacevarying coefficients, that models the spacevariation of the phenomenon. Spacevarying covariates can be included in the model. The technique accurately handle data distributed over irregularly shaped domains. Moreover, various conditions can be imposed at the domain boundaries.
This function is a wrapper of the Triangle library (http://www.cs.cmu.edu/~quake/triangle.html). It can be used
to create a triangulation of the domain of interest starting from a list of points, to be used as triangles' vertices, and a list of segments, that define the domain boundary. The resulting
mesh is a Constrained Delaunay triangulation. This is constructed in a way to preserve segments provided in the input segments
without splitting them. This imput can be used to define the boundaries
of the domain. If this imput is NULL, it generates a triangulation over the
convex hull of the points.
This function refines a Constrained Delaunay triangulation into a Conforming Delaunay triangulation. This is a wrapper of the Triangle library (http://www.cs.cmu.edu/~quake/triangle.html). It can be used to
refine a mesh created previously with create.MESH.2D. The algorithm can add Steiner points (points through which the segments
are splitted)
in order to meet the imposed refinement conditions.
Plot a mesh MESH2D object, generated by create.MESH.2D
or refine.MESH.2D
. Circles indicate the mesh nodes.
R_mass(FEMbasis) R_stiff(FEMbasis) R_smooth.FEM.basis( locations, observations, FEMbasis, lambda, covariates = NULL, GCV ) R_eval.FEM.basis(FEMbasis, locations, nderivs = matrix(0, 1, 2)) R_eval.FEM(FEM, locations) smooth.FEM.basis( locations = NULL, observations, FEMbasis, lambda, covariates = NULL, BC = NULL, GCV = FALSE, CPP_CODE = TRUE ) smooth.FEM.PDE.basis( locations = NULL, observations, FEMbasis, lambda, PDE_parameters, covariates = NULL, BC = NULL, GCV = FALSE, CPP_CODE = TRUE ) smooth.FEM.PDE.sv.basis( locations = NULL, observations, FEMbasis, lambda, PDE_parameters, covariates = NULL, BC = NULL, GCV = FALSE, CPP_CODE = TRUE ) create.MESH.2D(nodes, nodesattributes = NA, segments = NA, holes = NA, triangles = NA, order = 1, verbosity = 0) refine.MESH.2D(mesh, minimum_angle, maximum_area, delaunay, verbosity) ## S3 method for class 'MESH2D' plot(x, ...)
FEMbasis 
A 
locations 
A #observationsby2 matrix where each row specifies the spatial coordinates 
observations 
A vector of length #observations with the observed data values over the domain.
The locations of the observations can be specified with the 
lambda 
A scalar or vector of smoothing parameters. 
covariates 
A #observationsby#covariates matrix where each row represents the covariates associated with the corresponding observed data value in 
GCV 
Boolean. If 
nderivs 
A vector of lenght 2 specifying the order of the partial derivatives of the bases to be evaluated. The vectors' entries can be 0,1 or 2, where 0 indicates that only the basis functions, and not their derivatives, should be evaluated. 
FEM 
A 
BC 
A list with two vectors:

CPP_CODE 
Boolean. If 
PDE_parameters 
A list specifying the spacevarying parameters of the elliptic PDE in the regularizing term: 
nodes 
A #nodesby2 matrix containing the x and y coordinates of the mesh nodes. 
nodesattributes 
A matrix with #nodes rows containing nodes' attributes. These are passed unchanged to the output. If a node is added during the triangulation process or mesh refinement, its attributes are computed by linear interpolation using the attributes of neighboring nodes. This functionality is for instance used to compute the value of a Dirichlet boundary condition at boundary nodes added during the triangulation process. 
segments 
A #segmentsby2 matrix. Each row contains the row's indices in 
holes 
A #holesby2 matrix containing the x and y coordinates of a point internal to each hole of the mesh. These points are used to carve holes in the triangulation, when the domain has holes. 
triangles 
A #trianglesby3 (when 
order 
Either '1' or '2'. It specifies wether each mesh triangle should be represented by 3 nodes (the triangle' vertices) or by 6 nodes (the triangle's vertices and midpoints).
These are
respectively used for linear (order = 1) and quadratic (order = 2) Finite Elements. Default is 
verbosity 
This can be '0', '1' or '2'. It indicates the level of verbosity in the triangulation process. 
mesh 
A MESH2D object representing the triangular mesh, created by create.MESH.2D. 
minimum_angle 
A scalar specifying a minimun value for the triangles angles. 
maximum_area 
A scalar specifying a maximum value for the triangles areas. 
delaunay 
A boolean parameter indicating whether or not the output mesh should satisfy the Delaunay condition. 
x 
A MESH2D object defining the triangular mesh, as generated by 
... 
Arguments representing graphical options to be passed to par. 
These functions are Deprecated in this release of fdaPDE, they will be marked as Defunct and removed in a future version.
A square matrix with the integrals of all the basis' functions pairwise products. The dimension of the matrix is equal to the number of the nodes of the mesh.
A square matrix with the integrals of all the basis functions' gradients pairwise dot products. The dimension of the matrix is equal to the number of the nodes of the mesh.
A list with the following quantities:

A 

A 

If covariates is not 

If GCV is 

If GCV is 

If GCV is 
A matrix of basis function values. Each row indicates the location where the evaluation has been taken, the column indicates the basis function evaluated
A matrix of numeric evaluations of the FEM
object. Each row indicates the location where the evaluation has been taken, the column indicates the
function evaluated.
A list with the following variables:
fit.FEM
A FEM
object that represents the fitted spatial field.
PDEmisfit.FEM
A FEM
object that represents the Laplacian of the estimated spatial field.
solution
A list, note that all terms are matrices or row vectors: the j
th column represents the vector of related to lambda[j]
if lambda.selection.criterion="grid"
and lambda.selection.lossfunction="unused"
.
In all the other cases is returned just the column related to the best penalization parameter
f
Matrix, estimate of function f, first half of solution vector
g
Matrix, second half of solution vector
z_hat
Matrix, prediction of the output in the locations
beta
If covariates
is not NULL
, a matrix with number of rows equal to the number of covariates and number of columns equal to length of lambda. It is the regression coefficients estimate
rmse
Estimate of the root mean square error in the locations
estimated_sd
Estiimate of the standard deviation of the error
optimization
A detailed list of optimization related data:
lambda_solution
numerical value of best lambda acording to lambda.selection.lossfunction
, 1 if lambda.selection.lossfunction="unused"
lambda_position
integer, postion in lambda_vector
of best lambda acording to lambda.selection.lossfunction
, 1 if lambda.selection.lossfunction="unused"
GCV
numeric value of GCV in correspondence of the optimum
optimization_details
list containing further information about the optimization method used and the nature of its termination, eventual number of iterations
dof
numeric vector, value of dof for all the penalizations it has been computed, empty if not computed
lambda_vector
numeric value of the penalization factors passed by the user or found in the iterations of the optimization method
GCV_vector
numeric vector, value of GCV for all the penalizations it has been computed
time
Duration of the entire optimization computation
bary.locations
A barycenter information of the given locations if the locations are not mesh nodes.
A list with the following variables:
fit.FEM
A FEM
object that represents the fitted spatial field.
PDEmisfit.FEM
A FEM
object that represents the Laplacian of the estimated spatial field.
solution
A list, note that all terms are matrices or row vectors: the j
th column represents the vector of related to lambda[j]
if lambda.selection.criterion="grid"
and lambda.selection.lossfunction="unused"
.
In all the other cases is returned just the column related to the best penalization parameter
f
Matrix, estimate of function f, first half of solution vector
g
Matrix, second half of solution vector
z_hat
Matrix, prediction of the output in the locations
beta
If covariates
is not NULL
, a matrix with number of rows equal to the number of covariates and number of columns equal to length of lambda. It is the regression coefficients estimate
rmse
Estimate of the root mean square error in the locations
estimated_sd
Estiimate of the standard deviation of the error
optimization
A detailed list of optimization related data:
lambda_solution
numerical value of best lambda acording to lambda.selection.lossfunction
, 1 if lambda.selection.lossfunction="unused"
lambda_position
integer, postion in lambda_vector
of best lambda acording to lambda.selection.lossfunction
, 1 if lambda.selection.lossfunction="unused"
GCV
numeric value of GCV in correspondence of the optimum
optimization_details
list containing further information about the optimization method used and the nature of its termination, eventual number of iterations
dof
numeric vector, value of dof for all the penalizations it has been computed, empty if not computed
lambda_vector
numeric value of the penalization factors passed by the user or found in the iterations of the optimization method
GCV_vector
numeric vector, value of GCV for all the penalizations it has been computed
time
Duration of the entire optimization computation
bary.locations
A barycenter information of the given locations if the locations are not mesh nodes.
A list with the following variables:
fit.FEM
A FEM
object that represents the fitted spatial field.
PDEmisfit.FEM
A FEM
object that represents the Laplacian of the estimated spatial field.
solution
A list, note that all terms are matrices or row vectors: the j
th column represents the vector of related to lambda[j]
if lambda.selection.criterion="grid"
and lambda.selection.lossfunction="unused"
.
In all the other cases is returned just the column related to the best penalization parameter
f
Matrix, estimate of function f, first half of solution vector
g
Matrix, second half of solution vector
z_hat
Matrix, prediction of the output in the locations
beta
If covariates
is not NULL
, a matrix with number of rows equal to the number of covariates and number of columns equal to length of lambda. It is the regression coefficients estimate
rmse
Estimate of the root mean square error in the locations
estimated_sd
Estiimate of the standard deviation of the error
optimization
A detailed list of optimization related data:
lambda_solution
numerical value of best lambda acording to lambda.selection.lossfunction
, 1 if lambda.selection.lossfunction="unused"
lambda_position
integer, postion in lambda_vector
of best lambda acording to lambda.selection.lossfunction
, 1 if lambda.selection.lossfunction="unused"
GCV
numeric value of GCV in correspondence of the optimum
optimization_details
list containing further information about the optimization method used and the nature of its termination, eventual number of iterations
dof
numeric vector, value of dof for all the penalizations it has been computed, empty if not computed
lambda_vector
numeric value of the penalization factors passed by the user or found in the iterations of the optimization method
GCV_vector
numeric vector, value of GCV for all the penalizations it has been computed
time
Duration of the entire optimization computation
bary.locations
A barycenter information of the given locations if the locations are not mesh nodes.
An object of the class MESH2D with the following output:

A #nodesby2 matrix containing the x and y coordinates of the mesh nodes. 

A vector of length #nodes, with entries either '1' or '0'. An entry '1' indicates that the corresponding node is a boundary node; an entry '0' indicates that the corresponding node is not a boundary node. 

nodesattributes A matrix with #nodes rows containing nodes' attributes. These are passed unchanged to the output. If a node is added during the triangulation process or mesh refinement, its attributes are computed by linear interpolation using the attributes of neighboring nodes. This functionality is for instance used to compute the value of a Dirichlet boundary condition at boundary nodes added during the triangulation process. 

A #trianglesby3 (when 

A vector of length #segments with entries either '1' or '0'. An entry '1' indicates that the corresponding element in 

A #edgesby2 matrix containing all the edges of the triangles in the output triangulation. Each row contains the row's indices in 

A vector of lenght #edges with entries either '1' or '0'. An entry '1' indicates that the corresponding element in 

A #trianglesby3 matrix. Each row contains the indices of the three neighbouring triangles. An entry '1' indicates that one edge of the triangle is a boundary edge. 

A #holesby2 matrix containing the x and y coordinates of a point internal to each hole of the mesh. These points are used to carve holes in the triangulation, when the domain has holes. 

Either '1' or '2'. It specifies wether each mesh triangle should be represented by 3 nodes (the triangle' vertices) or by 6 nodes (the triangle's vertices and midpoints).
These are respectively used for linear (order = 1) and quadratic (order = 2) Finite Elements. Default is 
A MESH2D object representing the refined triangular mesh, with the following output:

A #nodesby2 matrix containing the x and y coordinates of the mesh nodes. 

A vector of length #nodes, with entries either '1' or '0'. An entry '1' indicates that the corresponding node is a boundary node; an entry '0' indicates that the corresponding node is not a boundary node. 

nodesattributes A matrix with #nodes rows containing nodes' attributes. These are passed unchanged to the output. If a node is added during the triangulation process or mesh refinement, its attributes are computed by linear interpolation using the attributes of neighboring nodes. This functionality is for instance used to compute the value of a Dirichlet boundary condition at boundary nodes added during the triangulation process. 

A #trianglesby3 (when 

A #edgesby2 matrix. Each row contains the row's indices of the nodes where the edge starts from and ends to. 

A vector of lenght #edges with entries either '1' or '0'. An entry '1' indicates that the corresponding element in 

A #trianglesby3 matrix. Each row contains the indices of the three neighbouring triangles. An entry '1' indicates that one edge of the triangle is a boundary edge. 

A #holesby2 matrix containing the x and y coordinates of a point internal to each hole of the mesh. These points are used to carve holes in the triangulation, when the domain has holes. 

Either '1' or '2'. It specifies wether each mesh triangle should be represented by 3 nodes (the triangle' vertices) or by 6 nodes (the triangle's vertices and midpoints).
These are respectively used for linear (order = 1) and quadratic (order = 2) Finite Elements. Default is 
refine.MESH.2D
, create.FEM.basis
create.MESH.2D
, create.FEM.basis
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