Spatial regression with differential regularization: anysotropic and nonstationary case (elliptic PDE with spacevarying coefficients)
Description
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.
Usage
Arguments
locations 
A #observationsby2 matrix where each row specifies the spatial coordinates x and y of the corresponding observations in the vector observations .
This parameter can be NULL . In this case the spatial coordinates of the corresponding observations are assigned as specified in observations .

observations 
A vector of length #observations with the observed data values over the domain.
The locations of the observations can be specified with the locations argument.
Otherwise if only the vector of observations is given, these are consider to be located in the corresponding node in the table
nodes of the mesh. In this last case, an NA value in the observations vector indicates that there is no observation associated to the corresponding
node.

FEMbasis 
A FEMbasis object describing the Finite Element basis, as created by create.FEM.basis .

lambda 
A scalar or vector of smoothing parameters.

PDE_parameters 
A list specifying the spacevarying parameters of the elliptic PDE in the regularizing term: K , a function that for each spatial location in the spatial domain
(indicated by the vector of the 2 spatial coordinates) returns a 2by2 matrix of diffusion coefficients. This induces an anisotropic
smoothing with a local preferential direction that corresponds to the first eigenvector of the diffusion matrix K.The function must support recycling for efficiency reasons, thus if the input parameter is a #pointby2 matrix, the output should be
an array with dimensions 2by2by#points.b , a function that for each spatial location in the spatial domain returns
a vector of length 2 of transport coefficients. This induces a local smoothing only in the direction specified by the vector b . The function must support recycling for efficiency reasons, thus if the input parameter is a #pointby2 matrix, the output should be
a matrix with dimensions 2by#points; c , a function that for each spatial location in the spatial domain returns a scalar reaction coefficient.
c induces a shrinkage of the surface to zero. The function must support recycling for efficiency reasons, thus if the input parameter is a #pointby2 matrix, the output should be
a vector with length #points; u , a function that for each spatial location in the spatial domain returns a scalar reaction coefficient.
u induces a reaction effect. The function must support recycling for efficiency reasons, thus if the input parameter is a #pointby2 matrix, the output should be
a vector with length #points.

covariates 
A #observationsby#covariates matrix where each row represents the covariates associated with the corresponding observed data value in observations .

BC 
A list with two vectors:
BC_indices , a vector with the indices in nodes of boundary nodes where a Dirichlet Boundary Condition should be applied;
BC_values , a vector with the values that the spatial field must take at the nodes indicated in BC_indices .

GCV 
Boolean. If TRUE the following quantities are computed: the trace of the smoothing matrix, the estimated error standard deviation, and
the Generalized Cross Validation criterion, for each value of the smoothing parameter specified in lambda .

CPP_CODE 
Boolean. If TRUE the computation relies on the C++ implementation of the algorithm. This usually ensures a much faster computation.

Value
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 PDE misfit for the estimated spatial field.

beta

If covariates is not NULL , a matrix with number of rows equal to the number of covariates and numer of columns equal to length of lambda. The j th column represents the vector of regression coefficients when
the smoothing parameter is equal to lambda[j] .

edf

If GCV is TRUE , a scalar or vector with the trace of the smoothing matrix for each value of the smoothing parameter specified in lambda .

stderr

If GCV is TRUE , a scalar or vector with the estimate of the standard deviation of the error for each value of the smoothing parameter specified in lambda .

GCV

If GCV is TRUE , a scalar or vector with the value of the GCV criterion for each value of the smoothing parameter specified in lambda .

References
Azzimonti, L., Sangalli, L.M., Secchi, P., Domanin, M., and Nobile, F., 2014. Blood flow velocity field estimation via spatial regression with PDE penalization Blood flow velocity field estimation via spatial regression with PDE penalization. DOI. 10.1080/01621459.2014.946036.
Azzimonti, L., Nobile, F., Sangalli, L.M., and Secchi, P., 2014. Mixed Finite Elements for Spatial Regression with PDE Penalization. SIAM/ASA Journal on Uncertainty Quantification, 2(1), pp.305335.
See Also
smooth.FEM.basis
, smooth.FEM.PDE.basis
Examples
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42  # Loading the mesh
data(mesh.2D.rectangular)
# Create the FEM basis object
FEMbasis = create.FEM.basis(mesh.2D.rectangular)
# Create a vector of noisy samples of an underlying spatial field,
# located over the nodes of the mesh
observations = sin(0.2*pi*mesh.2D.rectangular$nodes[,1]) +
rnorm(n = nrow(mesh.2D.rectangular$nodes), sd = 0.1)
# Set the smoothing coefficient
lambda = c(10^2)
#Set the space vriant coefficients of the penalizying PDE
K_func<function(points)
{
mat<c(0.01,0,0,1)
output = array(0, c(2, 2, nrow(points)))
for (i in 1:nrow(points))
output[,,i] = 0.5*mat %*% t(points[i,1]^2)
output
}
b_func<function(points)
{
output = array(0, c(2, nrow(points)))
for (i in 1:nrow(points))
output[,i] = 0
output
}
c_func<function(points)
{
rep(c(0), nrow(points))
}
u_func<function(points)
{
rep(c(0), nrow(points))
}
# Assemble the parameters in one object
PDE_parameters = list(K = K_func, b = b_func, c = c_func, u = u_func)
# Estimate the underlying spatial field and plot these
FEM_CPP_PDE = smooth.FEM.PDE.sv.basis(observations = observations,
FEMbasis = FEMbasis, lambda = lambda, PDE_parameters = PDE_parameters)
plot(FEM_CPP_PDE$fit.FEM)
