# smooth.FEM: Spatial regression with differential regularization In fdaPDE: Functional Data Analysis and Partial Differential Equations (PDE); Statistical Analysis of Functional and Spatial Data, Based on Regression with PDE Regularization

 smooth.FEM R Documentation

## Spatial regression with differential regularization

### Description

This function implements a spatial regression model with differential regularization. The regularizing term involves a Partial Differential Equation (PDE). In the simplest case the PDE involves only the Laplacian of the spatial field, that induces an isotropic smoothing. When prior information about the anisotropy or non-stationarity is available the PDE involves a general second order linear differential operator with possibly space-varying coefficients. The technique accurately handle data distributed over irregularly shaped domains. Moreover, various conditions can be imposed at the domain boundaries

### Usage

```smooth.FEM(locations = NULL, observations, FEMbasis,
covariates = NULL, PDE_parameters = NULL, BC = NULL,
incidence_matrix = NULL, areal.data.avg = TRUE,
search = "tree", bary.locations = NULL,
family = "gaussian", mu0 = NULL, scale.param = NULL, threshold.FPIRLS = 0.0002020,
max.steps.FPIRLS = 15, lambda.selection.criterion = "grid", DOF.evaluation = NULL,
lambda.selection.lossfunction = NULL, lambda = NULL, DOF.stochastic.realizations = 100,
DOF.stochastic.seed = 0, DOF.matrix = NULL, GCV.inflation.factor = 1,
lambda.optimization.tolerance = 0.05,
inference.data.object=NULL)
```

### Arguments

 `locations` A #observations-by-2 matrix in the 2D case and #observations-by-3 matrix in the 2.5D and 3D case, where each row specifies the spatial coordinates `x` and `y` (and `z` in 2.5D and 3D) of the corresponding observation in the vector `observations`. If the locations of the observations coincide with (or are a subset of) the nodes of the mesh in the `FEMbasis`, leave the parameter `locations = NULL` for a faster implementation. `observations` A vector of length #observations with the observed data values over the domain. If the `locations` argument is left NULL the vector of the observations have to be of length #nodes of the mesh in the FEMbasis. In this case, each observation is associated to the corresponding node in the mesh. If the observations are observed only on a subset of the mesh nodes, fill with `NA` the values of the vector `observations` in correspondence of unobserved data. `FEMbasis` A `FEMbasis` object describing the Finite Element basis, as created by `create.FEM.basis`. `covariates` A #observations-by-#covariates matrix where each row represents the covariates associated with the corresponding observed data value in `observations` and each column is a different covariate. `PDE_parameters` A list specifying the parameters of the PDE in the regularizing term. Default is NULL, i.e. regularization is by means of the Laplacian (stationary, isotropic case). If the coefficients of the PDE are constant over the domain `PDE_parameters` must contain: `K`, a 2-by-2 matrix of diffusion coefficients. This induces an anisotropic smoothing with a preferential direction that corresponds to the first eigenvector of the diffusion matrix K; `b`, a vector of length 2 of advection coefficients. This induces a smoothing only in the direction specified by the vector `b`; `c`, a scalar reaction coefficient. `c` induces a shrinkage of the surface to zero. If the coefficients of the PDE are space-varying `PDE_parameters` must contain: `K`, a function that for each spatial location in the spatial domain (indicated by the vector of the 2 spatial coordinates) returns a 2-by-2 matrix of diffusion coefficients. The function must support recycling for efficiency reasons, thus if the input parameter is a #point-by-2 matrix, the output should be an array with dimensions 2-by-2-by-#points. `b`, a function that for each spatial location in the spatial domain returns a vector of length 2 of transport coefficients. The function must support recycling for efficiency reasons, thus if the input parameter is a #point-by-2 matrix, the output should be a matrix with dimensions 2-by-#points; `c`, a function that for each spatial location in the spatial domain returns a scalar reaction coefficient. The function must support recycling for efficiency reasons, thus if the input parameter is a #point-by-2 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 #point-by-2 matrix, the output should be a vector with length #points. For 2.5D and 3D, only the Laplacian is available (`PDE_parameters=NULL`). `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`. `incidence_matrix` A #regions-by-#triangles/tetrahedrons matrix where the element (i,j) equals 1 if the j-th triangle/tetrahedron is in the i-th region and 0 otherwise. This is needed only for areal data. In case of pointwise data, this parameter is set to `NULL`. `areal.data.avg` Boolean. It involves the computation of Areal Data. If `TRUE` the areal data are averaged, otherwise not. `search` a flag to decide the search algorithm type (tree or naive or walking search algorithm). `bary.locations` A list with three vectors: `locations`, location points which are same as the given locations options. (checks whether both locations are the same); `element ids`, a vector of element id of the points from the mesh where they are located; `barycenters`, a vector of barycenter of points from the located element. `family` This parameter specify the distibution within exponential family used for GLM model. The following distribution are implemented: "binomial", "exponential", "gamma", "poisson", "gaussian", "invgaussian". The default link function for binomial is `logit` if you want either `probit` or `clogloc` set `family = "probit"`, `family = "cloglog"`. `mu0` This parameter is a vector that set the starting point for FPIRLS algorithm. It represent an initial guess of the location parameter. Default is set to observation for non binary distribution while equal to `0.5(observations + 0.5)` for binary data. `scale.param` Dispersion parameter of the chosen distribution. This is only required for "gamma", "gaussian", "invgaussian". User may specify the parameter as a positive real number. If the parameter is not supplied, it is estimated from data according to Wilhelm Sangalli 2016. `threshold.FPIRLS` This parameter is used for arresting algorithm iterations. Algorithm stops when two successive iterations lead to improvement in penalized log-likelihood smaller than threshold.FPIRLS. Default value `threshold.FPIRLS = 0.0002020`. `max.steps.FPIRLS` This parameter is used to limit the maximum number of iteration. Default value `max.steps.FPIRLS=15`. `lambda.selection.criterion` This parameter is used to select the optimization method for the smoothing parameter `lambda`. The following methods are implemented: 'grid', 'newton', 'newton_fd'. The former is a pure evaluation method. A test vector of `lambda` must be provided. The remaining two are optimization methods that automatically select the best penalization according to `lambda.selection.lossfunction` criterion. They implement respectively a pure Newton method and a finite differences Newton method. Default value `lambda.selection.criterion='grid'` `DOF.evaluation` This parameter is used to identify if and how to perform degrees of freedom computation. The following possibilities are allowed: NULL, 'exact' and 'stochastic' In the former case no degree of freedom is computed, while the other two methods enable computation. Stochastic computation of DOFs may be slightly less accurate than its deterministic counterpart, but it is fairly less time consuming. Stochastic evaluation is highly suggested for meshes with more than 5000 nodes. Default value `DOF.evaluation=NULL` `lambda.selection.lossfunction` This parameter is used to determine if some loss function has to be evaluated. The following possibilities are allowed: NULL and 'GCV' (generalized cross validation) If NULL is selected, `lambda.selection.criterion='grid'` is required. 'GCV' is employed for both `lambda.selection.criterion='grid'` and optimization methods. Default value `lambda.selection.lossfunction=NULL` `lambda` a vector of spatial smoothing parameters provided if `lambda.selection.criterion='grid'`. An optional initialization otherwise. `DOF.stochastic.realizations` This positive integer is considered only when `DOF.evaluation = 'stochastic'`. It is the number of uniform random variables used in stochastic DOF evaluation. Default value `DOF.stochastic.realizations=100`. `DOF.stochastic.seed` This positive integer is considered only when `DOF.evaluation = 'stochastic'`. It is a user defined seed employed in stochastic DOF evaluation. Default value `DOF.stochastic.seed = 0` means random. `DOF.matrix` Matrix of degrees of freedom. This parameter can be used if the DOF.matrix corresponding to `lambda` is available from precedent computation. This allows to save time, since the computation of the DOFs is the most time consuming part of GCV evaluation. `GCV.inflation.factor` Tuning parameter used for the estimation of GCV. Default value `GCV.inflation.factor = 1.0` or `1.8` in GAM. It is advised to set `GCV.inflation.factor` larger than 1 to avoid overfitting. `lambda.optimization.tolerance` Tolerance parameter, a double between 0 and 1 that fixes how much precision is required by the optimization method: the smaller the parameter, the higher the accuracy. Used only if `lambda.selection.criterion='newton'` or `lambda.selection.criterion='newton_fd'`. Default value `lambda.optimization.tolerance=0.05`. `inference.data.object` An `inferenceDataObject` that stores all the information regarding inference over the linear parameters of the model. This parameter needs to be consistent with `covariates`, otherwise will be discarded. If set and well defined, the function will have in output the inference results. It is suggested to create this object via `inferenceDataObjectBuilder` function, so that the object is guaranteed to be well defined.

### 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 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 related to `lambda[j]` if `lambda.selection.criterion="grid"` and `lambda.selection.lossfunction=NULL`. In all the other cases, only the column related to the best smoothing parameter is returned.

`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`Estimate of the standard deviation of the error.

• `optimization`A detailed list of optimization related data:

`lambda_solution`numerical value of best lambda according to `lambda.selection.lossfunction`, -1 if `lambda.selection.lossfunction=NULL`.

`lambda_position`integer, position in `lambda_vector` of best lambda according to `lambda.selection.lossfunction`, -1 if `lambda.selection.lossfunction=NULL`.

`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`vector of positive numbers, DOFs for all the lambdas in `lambda_vector`, empty or invalid if not computed.

`lambda_vector`vector of positive numbers: penalizations either passed by the user or found in the iterations of the optimization method.

`GCV_vector`vector of positive numbers, GCV values for all the lambdas in `lambda_vector`

• `time`Duration of the entire optimization computation.

• `bary.locations`Barycenter information of the given locations, if the locations are not mesh nodes.

• `GAM_output`A list of GAM related data:

`fn_hat`A matrix with number of rows equal to number of locations and number of columns equal to length of lambda. Each column contains the evaluaton of the spatial field in the location points.

`J_minima`A vector of the same length of lambda, containing the reached minima for each value of the smoothing parameter.

`variance.est`A vector which returns the variance estimates for the Generative Additive Models.

• `inference`A list set only if a well defined [inferenceDataObject] is passed as parameter to the function; contains all inference outputs required:

`p_values`list of lists set only if at least one p-value is required; contains the p-values divided by implementation:

`wald`list containing all the Wald p-values required, in the same order of the `type` list in `inference.data.object`. If one-at-the-time tests are required, the corresponding item is a vector of p values ordered as the rows of `coeff` matrix in `inference.data.object`.

`speckman`list containing all the Speckman p-values required, in the same order of the `type` list in `inference.data.object`. If one-at-the-time tests are required, the corresponding item is a vector of p values ordered as the rows of `coeff` matrix in `inference.data.object`.

`eigen_sign_flip`list containing all the Eigen-Sign-Flip p-values required, in the same order of the `type` list in `inference.data.object`. If one-at-the-time tests are required, the corresponding item is a vector of p values ordered as the rows of `coeff` matrix in `inference.data.object`.

`CI`list of lists set only if at least one confidence interval is required; contains the confidence intervals divided by implementation:

`wald`list containing all the Wald confidence intervals required, in the same order of the `type` list in `inference.data.object`. Each item is a matrix with 3 columns and p rows, p being the number of rows of `coeff` matrix in `inference.data.object`; each row is the CI for the corresponding row of `coeff` matrix.

`speckman`list containing all the Speckman confidence intervals required, in the same order of the `type` list in `inference.data.object`. Each item is a matrix with 3 columns and p rows, p being the number of rows of `coeff` matrix in `inference.data.object`; each row is the CI for the corresponding row of `coeff` matrix.

### References

• Sangalli, L. M., Ramsay, J. O., Ramsay, T. O. (2013). Spatial spline regression models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(4), 681-703.

• Azzimonti, L., Sangalli, L. M., Secchi, P., Domanin, M., Nobile, F. (2015). Blood flow velocity field estimation via spatial regression with PDE penalization. Journal of the American Statistical Association, 110(511), 1057-1071.

• Matthieu Wilhelm & Laura M. Sangalli (2016). Generalized spatial regression with differential regularization. Journal of Statistical Computation and Simulation, 86:13, 2497-2518.

• Federico Ferraccioli, Laura M. Sangalli & Livio Finos (2021). Some first inferential tools for spatial regression with differential regularization. Journal of Multivariate Analysis, to appear

### Examples

```library(fdaPDE)

#### No prior information about anysotropy/non-stationarity (laplacian smoothing) ####
data(horseshoe2D)
boundary_nodes = horseshoe2D\$boundary_nodes
boundary_segments = horseshoe2D\$boundary_segments
locations = horseshoe2D\$locations

mesh = create.mesh.2D(nodes = rbind(boundary_nodes, locations), segments = boundary_segments)
FEMbasis = create.FEM.basis(mesh)
lambda = 10^-1
# no covariate
data = fs.test(mesh\$nodes[,1], mesh\$nodes[,2]) + rnorm(nrow(mesh\$nodes), sd = 0.5)

solution = smooth.FEM(observations = data, FEMbasis = FEMbasis, lambda = lambda)
plot(solution\$fit.FEM)

# with covariates
covariate = covs.test(mesh\$nodes[,1], mesh\$nodes[,2])
data = fs.test(mesh\$nodes[,1], mesh\$nodes[,2]) + 2*covariate + rnorm(nrow(mesh\$nodes), sd = 0.5)

#Inferential tests and confidence intervals
inference.data.object = inferenceDataObjectBuilder(test = 'oat', type = 'w', dim = 2, n_cov = 1)

solution = smooth.FEM(observations = data, covariates = covariate,
FEMbasis = FEMbasis, lambda = lambda,
inference.data.object=inference.data.object)

# beta estimate:
solution\$solution\$beta
# tests over beta estimates p-values:
solution\$inference\$beta\$p_values
# confidence intervals for beta estimates:
solution\$inference\$beta\$CI
# non-parametric estimate:
plot(solution\$fit.FEM)

# Choose lambda with GCV - stochastic grid evaluation:
lambda = 10^(-2:0)
solution = smooth.FEM(observations = data,
covariates = covariate,
FEMbasis = FEMbasis,
lambda = lambda, DOF.evaluation = 'stochastic',
lambda.selection.lossfunction = 'GCV')
bestLambda = solution\$optimization\$lambda_solution
# Choose lambda with GCV - Newton finite differences stochastic evaluation -:
solution = smooth.FEM(observations = data,
covariates = covariate,
FEMbasis = FEMbasis,
DOF.evaluation = 'stochastic', lambda.selection.lossfunction = 'GCV')
bestLambda = solution\$optimization\$lambda_solution

#### Smoothing with prior information about anysotropy/non-stationarity and boundary conditions ####
# See Azzimonti et al. for reference to the current example
data(quasicircle2D)
boundary_nodes = quasicircle2D\$boundary_nodes
boundary_segments = quasicircle2D\$boundary_segments
locations = quasicircle2D\$locations
data = quasicircle2D\$data

mesh = create.mesh.2D(nodes = rbind(boundary_nodes, locations), segments = boundary_segments)
FEMbasis = create.FEM.basis(mesh)
lambda = 10^-2

# Set the PDE parameters
R = 2.8
K1 = 0.1
K2 = 0.2
beta = 0.5
K_func<-function(points)
{
output = array(0, c(2, 2, nrow(points)))
for (i in 1:nrow(points))
output[,,i]=10*rbind(c(points[i,2]^2+K1*points[i,1]^2+K2*(R^2-points[i,1]^2-points[i,2]^2),
(K1-1)*points[i,1]*points[i,2]),
c((K1-1)*points[i,1]*points[i,2],
points[i,1]^2+K1*points[i,2]^2+K2*(R^2-points[i,1]^2-points[i,2]^2)))
output
}

b_func<-function(points)
{
output = array(0, c(2, nrow(points)))
for (i in 1:nrow(points))
output[,i] = 10*beta*c(points[i,1],points[i,2])
output
}

c_func<-function(points)
{
rep(c(0), nrow(points))
}

u_func<-function(points)
{
rep(c(0), nrow(points))
}
PDE_parameters = list(K = K_func, b = b_func, c = c_func, u = u_func)

# Set the boundary conditions
BC = NULL
BC\$BC_indices = which(mesh\$nodesmarkers == 1) # b.c. on the complete boundary
BC\$BC_values = rep(0,length(BC\$BC_indices)) # homogeneus b.c.

# Since the data locations are a subset of the mesh nodes for a faster solution use:
dataNA = rep(NA, FEMbasis\$nbasis)
dataNA[mesh\$nodesmarkers == 0] = data
#grid evaluation
solution = smooth.FEM(observations = dataNA,
FEMbasis = FEMbasis,
lambda = lambda,
PDE_parameters = PDE_parameters,
BC = BC)
plot(solution\$fit.FEM)
image(solution\$fit.FEM)
# Newton's method
solution = smooth.FEM(observations = dataNA,
FEMbasis = FEMbasis,
PDE_parameters = PDE_parameters,
BC = BC)
plot(solution\$fit.FEM)
image(solution\$fit.FEM)

#### Smoothing with areal data ####
# See Azzimonti et al. for reference to the current exemple
data(quasicircle2Dareal)
incidence_matrix = quasicircle2Dareal\$incidence_matrix
data = quasicircle2Dareal\$data
mesh = quasicircle2Dareal\$mesh

FEMbasis = create.FEM.basis(mesh)
lambda = 10^-4

# Set the PDE parameters
R = 2.8
K1 = 0.1
K2 = 0.2
beta = 0.5
K_func<-function(points)
{
output = array(0, c(2, 2, nrow(points)))
for (i in 1:nrow(points))
output[,,i]=10*rbind(c(points[i,2]^2+K1*points[i,1]^2+K2*(R^2-points[i,1]^2-points[i,2]^2),
(K1-1)*points[i,1]*points[i,2]),
c((K1-1)*points[i,1]*points[i,2],
points[i,1]^2+K1*points[i,2]^2+K2*(R^2-points[i,1]^2-points[i,2]^2)))
output
}

b_func<-function(points)
{
output = array(0, c(2, nrow(points)))
for (i in 1:nrow(points))
output[,i] = 10*beta*c(points[i,1],points[i,2])
output
}

c_func<-function(points)
{
rep(c(0), nrow(points))
}

u_func<-function(points)
{
rep(c(0), nrow(points))
}
PDE_parameters = list(K = K_func, b = b_func, c = c_func, u = u_func)

# Set the boundary conditions
BC = NULL
BC\$BC_indices = which(mesh\$nodesmarkers == 1) # b.c. on the complete boundary
BC\$BC_values = rep(0,length(BC\$BC_indices)) # homogeneus b.c.
#grid evaluation
solution = smooth.FEM(observations = data,
incidence_matrix = incidence_matrix,
FEMbasis = FEMbasis,
lambda = lambda,
PDE_parameters = PDE_parameters,
BC = BC)
plot(solution\$fit.FEM)
image(solution\$fit.FEM)
#Newton's method
solution = smooth.FEM(observations = data,
incidence_matrix = incidence_matrix,
FEMbasis = FEMbasis,
PDE_parameters = PDE_parameters,
BC = BC)
plot(solution\$fit.FEM)
image(solution\$fit.FEM)

```

fdaPDE documentation built on Nov. 10, 2022, 5:06 p.m.