statFEM_analysis package

statFEM_analysis.oneDim module

statFEM_analysis.oneDim.BigPhiMat(J, grid)

Function to compute the \(\Phi\) matrix.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J\))

• grid (array) – grid in the definition of \(\Phi\)

Returns

\(\Phi\) matrix returned as a sparse csr_matrix.

Return type

csr_matrix

class statFEM_analysis.oneDim.MyExpression(*args, **kwargs)

Bases: dolfin.function.expression.UserExpression

Class to allow users to user their own functions to create a FEniCS UserExpression.

eval(value, x)

value_shape()

statFEM_analysis.oneDim.c_post(x, y, c, Y, B)

This function evaluates the posterior covariance at \((x,y)\)

Parameters

• x (float) – one of two points to evaluate the covariance at

• y (float) – one of two points to evaluate the covariance at

• c (function) – function which returns the prior covariance at any given pair \((x,y)\)

• Y (array) – vector of sensor locations

• B (array) – the matrix \(\epsilon^{2}I+C_{Y}\) to be inverted in order to obtain the posterior

Returns

posterior covariance evaluated at \((x,y)\)

Return type

float

statFEM_analysis.oneDim.c_post_fem_assembler(J, k_f, grid, Y, ε, par, trans)

Function to assemble the statFEM posterior cov mat on a reference grid specified by grid.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J\))

• k_f (function) – the covariance function for the forcing \(f\)

• grid (array) – the fixed reference grid \(\{x_{i}\}_{i=1}^{N}\) on which to assemble the posterior cov mat

• Y (array) – vector of sensor locations

• par (bool) – argument indicating whether the computation of the forcing cov mat should be done in parallel

• trans (bool) – argument indicating whether the computation of the forcing cov mat should be computed assuming k_f is translation invariant or not

Returns

statFEM posterior cov matrix evaluated on grid

Return type

array

statFEM_analysis.oneDim.cov_assembler(J, k_f, grid, parallel, translation_inv)

Function to assemble the approximate FEM covariance matrix on the reference grid.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J\))

• k_f (function) – the covariance function for the forcing \(f\)

• grid (array) – the reference grid where the FEM cov matrix should be computed on

• parallel (bool) – argument indicating whether the intermediate computation of \(C_f\) should be done in parallel

• translation_inv (bool) – argument indicating whether the intermediate computation of \(C_f\) should be computed assuming k_f is translation invariant or not

Returns

approximate FEM covariance matrix on the reference grid

Return type

array

statFEM_analysis.oneDim.fem_cov_assembler_post(J, k_f, Y, parallel, translation_inv)

Function to create the matrix \(C_{Y,h}\) and the vector function \(c^{(h)}\) required for the statFEM posterior mean.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J\))

• k_f (function) – the covariance function for the forcing \(f\)

• Y (array) – vector of sensor locations

• parallel (bool) – argument indicating whether the computation of the forcing cov mat should be done in parallel

• translation_inv (bool) – argument indicating whether the computation of the forcing cov mat should be computed assuming k_f is translation invariant or not

Returns

returns tuple of matrix \(C_{Y,h}\) and the vector function \(c^{(h)}\)

Return type

tuple

statFEM_analysis.oneDim.gen_sensor(ε, m, k, Y, u_quad, grid, par=False, trans=True, tol=1e-09, maxiter=50, require_f=False)

Function to generate noisy sensor observations of the solution u on a sensor grid Y.

Parameters

• ϵ (float) – controls the amount of sensor noise

• m (function) – mean function for the forcing \(f\)

• k (function) – cov function for the forcing \(f\)

• Y (array) – vector of sensor locations

• u_quad (function) – function to accurately compute the solution \(u\) given a realisation of the forcing \(f\)

• grid (array) – grid where forcing \(f\) is sampled on

• par (bool, optional) – argument indicating whether the computation of the forcing cov matrix should be done in parallel, by default False

• trans (bool, optional) – argument indicating whether the computation of the forcing cov matrix should be computed assuming k is translation invariant or not, by default True

• tol (_type_, optional) – controls the size of the tiny diagonal perturbation added to forcing cov matrix to ensure it is strictly positive definite, by default 1e-9

• maxiter (int, optional) – parameter which controls the accuracy of the quadrature used in u_quad, by default 50

• require_f (bool, optional) – argument indicating whether or not to also return the realisation of the forcing \(f\), by default False

Returns

If require_f is set to True returns a tuple of two arrays containing the generated noisy sensor observations and the realisation of the forcing. If require_f is set to False returns just the generated noisy sensor observations.

Return type

tuple or array

statFEM_analysis.oneDim.kernMat(k, grid, parallel=True, translation_inv=False)

Function to compute the covariance matrix \(K\) corresponding to the covariance kernel \(k\) on a grid.

Parameters

• k (function) – forcing covariance function

• grid (array) – grid on which to evaluate the covariance function

• parallel (bool, optional) – specifies whether or not the cov kernel is translation invariant, by default True

• translation_inv (bool, optional) – specifies whether or not the cov kernel is translation invariant, by default False

Returns

Covariance matrix K with \(ij-\) where \(x_i\) is the \(i\)-th point of the grid.

Return type

array

statFEM_analysis.oneDim.m_post(x, m, c, v, Y, B)

This function evalutes the posterior mean at the point \(x\).

Parameters

• x (float) – point where the posterior mean will be evaluated

• m (function) – function which computes the prior mean at a given point y

• c (function) – function which returns the vector (c(x,y)) for y in Y (note: c is the prior covariance function)

• v (array) – vector of noisy sensor readings

• Y (array) – vector of sensor locations

• B (array) – the matrix \(\epsilon^{2}I+C_Y\) to be inverted in order to obtain the posterior

Returns

posterior mean at the point \(x\)

Return type

float

statFEM_analysis.oneDim.m_post_fem_assembler(J, f_bar, k_f, ε, Y, v_dat, par=False, trans=True)

Function to assemble the statFEM posterior mean function.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J\))

• f_bar (function) – the mean function for the forcing \(f\)

• k_f (function) – the covariance function for the forcing \(f\)

• ϵ (float) – controls the amount of sensor noise

• Y (array) – vector of sensor locations

• v_dat (array) – vector of noisy sensor observations

• par (bool, optional) – argument passed to fem_cov_assembler_post’s argument parallel, by default False

• trans (bool, optional) – argument passed to fem_cov_assembler_post’s argument translation_inv, by default True

Returns

statFEM posterior mean function

Return type

function

statFEM_analysis.oneDim.mean_assembler(h, f_bar)

This function assembles the mean for the statFEM prior for our 1-D problem.

Parameters

• h (float) – mesh size h

• f_bar (FEniCS function/expression/constant) – mean function for the forcing

Returns

mean of the approximate statFEM prior

Return type

FEniCS function

statFEM_analysis.oneDim.post_fem_cov_assembler(J, k_f, grid, Y, parallel, translation_inv)

Function which assembles the matrices \(\Sigma_X,\Sigma_{XY}\), and \(\Sigma_Y\) required for the statFEM posterior covariance.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J\))

• k_f (function) – the covariance function for the forcing \(f\)

• grid (array) – the fixed reference grid \(\{x_{i}\}_{i=1}^{N}\) on which to assemble the posterior cov mat

• Y (array) – vector of sensor locations

• parallel (bool) – argument indicating whether the computation of the forcing cov mat should be done in parallel

• translation_inv (bool) – argument indicating whether the computation of the forcing cov mat should be computed assuming k_f is translation invariant or not

Returns

tuple of the matrices \(\Sigma_X,\Sigma_{XY}\), and \(\Sigma_Y\)

Return type

tuple

statFEM_analysis.oneDim.sample_gp(n_sim, m, k, grid, par=False, trans=True, tol=1e-09)

Function to sample a GP with mean \(m\) and cov \(k\) on a grid.

Parameters

• n_sim (int) – number of trajectories to be sampled

• m (function) – mean function for the GP

• k (function) – cov function for the GP

• grid (array) – grid of points on which to sample the GP

• par (bool, optional) – argument indicating whether the computation of the cov matrix should be done in parallel, by default False

• trans (bool, optional) – argument indicating whether the computation of the cov matrix should be computed assuming k is translation invariant or not, by default True

• tol (_type_, optional) – controls the size of the tiny diagonal perturbation added to cov matrix to ensure it is strictly positive definite, by default 1e-9

Returns

array containing n_sim samples from a GP with mean \(m\) and cov \(k\) evaluated on the grid grid

Return type

array

statFEM_analysis.twoDim module

statFEM_analysis.twoDim.BigPhiMat(J, grid)

Function to compute the \(\Phi\) matrix.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J^{2}\))

• grid (array) – grid in the definition of \(\Phi\)

Returns

\(\Phi\) matrix returned as a sparse csr_matrix.

Return type

csr_matrix

class statFEM_analysis.twoDim.MyExpression(*args, **kwargs)

Bases: dolfin.function.expression.UserExpression

Class to allow users to user their own functions to create a FEniCS UserExpression.

eval(value, x)

value_shape()

statFEM_analysis.twoDim.c_post(x, y, c, Y, B)

This function evaluates the posterior covariance at \((x,y)\)

Parameters

• x (float) – one of two points to evaluate the covariance at

• y (float) – one of two points to evaluate the covariance at

• c (function) – function which returns the prior covariance at any given pair \((x,y)\)

• Y (array) – vector of sensor locations

• B (array) – the matrix \(\epsilon^{2}I+C_{Y}\) to be inverted in order to obtain the posterior

Returns

posterior covariance evaluated at \((x,y)\)

Return type

float

statFEM_analysis.twoDim.c_post_fem_assembler(J, k_f, grid, Y, ε, par, trans)

Function to assemble the statFEM posterior cov mat on a reference grid specified by grid.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J^{2}\))

• k_f (function) – the covariance function for the forcing \(f\)

• grid (array) – the fixed reference grid \(\{x_{i}\}_{i=1}^{N}\) on which to assemble the posterior cov mat

• Y (array) – vector of sensor locations

• par (bool) – argument indicating whether the computation of the forcing cov mat should be done in parallel

• trans (bool) – argument indicating whether the computation of the forcing cov mat should be computed assuming k_f is translation invariant or not

Returns

statFEM posterior cov matrix evaluated on grid

Return type

array

statFEM_analysis.twoDim.cov_assembler(J, k_f, grid, parallel, translation_inv)

Function to assemble the approximate FEM covariance matrix on the reference grid.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J^{2}\))

• k_f (function) – the covariance function for the forcing \(f\)

• grid (array) – the reference grid where the FEM cov matrix should be computed on

• parallel (bool) – argument indicating whether the intermediate computation of \(C_f\) should be done in parallel

• translation_inv (bool) – argument indicating whether the intermediate computation of \(C_f\) should be computed assuming k_f is translation invariant or not

Returns

approximate FEM covariance matrix on the reference grid

Return type

array

statFEM_analysis.twoDim.fem_cov_assembler_post(J, k_f, Y, parallel, translation_inv)

Function to create the matrix \(C_{Y,h}\) and the vector function \(c^{(h)}\) required for the statFEM posterior mean.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J^{2}\))

• k_f (function) – the covariance function for the forcing \(f\)

• Y (array) – vector of sensor locations

• parallel (bool) – argument indicating whether the computation of the forcing cov mat should be done in parallel

• translation_inv (bool) – argument indicating whether the computation of the forcing cov mat should be computed assuming k_f is translation invariant or not

Returns

returns tuple of matrix \(C_{Y,h}\) and the vector function \(c^{(h)}\)

Return type

tuple

statFEM_analysis.twoDim.gen_sensor(ε, m, k, Y, J, par, trans, tol=1e-09, require=False)

Function to generate noisy sensor observations of the solution u on a sensor grid Y.

Parameters

• ϵ (float) – controls the amount of sensor noise

• m (function) – mean function for the forcing \(f\)

• k (function) – cov function for the forcing \(f\)

• Y (array) – vector of sensor locations

• J (int) – controls the FE mesh size (\(h=1/J^{2}\))

• par (bool, optional) – argument indicating whether the computation of the forcing cov matrix should be done in parallel, by default False

• trans (bool, optional) – argument indicating whether the computation of the forcing cov matrix should be computed assuming k is translation invariant or not, by default True

• tol (_type_, optional) – controls the size of the tiny diagonal perturbation added to forcing cov matrix to ensure it is strictly positive definite, by default 1e-9

• require_f (bool, optional) – argument indicating whether or not to also return the realisation of the forcing \(f\), by default False

Returns

If require_f is set to True returns a tuple of two arrays containing the generated noisy sensor observations and the realisation of the forcing. If require_f is set to False returns just the generated noisy sensor observations.

Return type

tuple or array

statFEM_analysis.twoDim.kernMat(k, grid, parallel=True, translation_inv=False)

Function to compute the covariance matrix \(K\) corresponding to the covariance kernel \(k\) on a grid.

Parameters

• k (function) – forcing covariance function

• grid (array) – grid on which to evaluate the covariance function

• parallel (bool, optional) – specifies whether or not the cov kernel is translation invariant, by default True

• translation_inv (bool, optional) – specifies whether or not the cov kernel is translation invariant, by default False

Returns

Covariance matrix K with \(ij-\) where \(x_i\) is the \(i\)-th point of the grid.

Return type

array

statFEM_analysis.twoDim.m_post(x, m, c, v, Y, B)

This function evalutes the posterior mean at the point \(x\).

Parameters

• x (float) – point where the posterior mean will be evaluated

• m (function) – function which computes the prior mean at a given point y

• c (function) – function which returns the vector (c(x,y)) for y in Y (note: c is the prior covariance function)

• v (array) – vector of noisy sensor readings

• Y (array) – vector of sensor locations

• B (array) – the matrix \(\epsilon^{2}I+C_Y\) to be inverted in order to obtain the posterior

Returns

posterior mean at the point \(x\)

Return type

float

statFEM_analysis.twoDim.m_post_fem_assembler(J, f_bar, k_f, ε, Y, v_dat, par=False, trans=True)

Function to assemble the statFEM posterior mean function.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J^{2}\))

• f_bar (function) – the mean function for the forcing \(f\)

• k_f (function) – the covariance function for the forcing \(f\)

• ϵ (float) – controls the amount of sensor noise

• Y (array) – vector of sensor locations

• v_dat (array) – vector of noisy sensor observations

• par (bool, optional) – argument passed to fem_cov_assembler_post’s argument parallel, by default False

• trans (bool, optional) – argument passed to fem_cov_assembler_post’s argument translation_inv, by default True

Returns

statFEM posterior mean function

Return type

function

statFEM_analysis.twoDim.mean_assembler(h, f_bar)

This function assembles the mean for the statFEM prior for our 1-D problem.

Parameters

• h (float) – mesh size h

• f_bar (FEniCS function/expression/constant) – mean function for the forcing

Returns

mean of the approximate statFEM prior

Return type

FEniCS function

statFEM_analysis.twoDim.post_fem_cov_assembler(J, k_f, grid, Y, parallel, translation_inv)

Function which assembles the matrices \(\Sigma_X,\Sigma_{XY}\), and \(\Sigma_Y\) required for the statFEM posterior covariance.

Parameters

• J (int) – controls the FE mesh size (\(h=1/J^{2}\))

• k_f (function) – the covariance function for the forcing \(f\)

• grid (array) – the fixed reference grid \(\{x_{i}\}_{i=1}^{N}\) on which to assemble the posterior cov mat

• Y (array) – vector of sensor locations

• parallel (bool) – argument indicating whether the computation of the forcing cov mat should be done in parallel

• translation_inv (bool) – argument indicating whether the computation of the forcing cov mat should be computed assuming k_f is translation invariant or not

Returns

tuple of the matrices \(\Sigma_X,\Sigma_{XY}\), and \(\Sigma_Y\)

Return type

tuple

statFEM_analysis.twoDim.sample_gp(n_sim, m, k, grid, par, trans, tol=1e-09)

Function to sample a GP with mean \(m\) and cov \(k\) on a grid.

Parameters

• n_sim (int) – number of trajectories to be sampled

• m (function) – mean function for the GP

• k (function) – cov function for the GP

• grid (array) – grid of points on which to sample the GP

• par (bool, optional) – argument indicating whether the computation of the cov matrix should be done in parallel, by default False

• trans (bool, optional) – argument indicating whether the computation of the cov matrix should be computed assuming k is translation invariant or not, by default True

• tol (_type_, optional) – controls the size of the tiny diagonal perturbation added to cov matrix to ensure it is strictly positive definite, by default 1e-9

Returns

array containing n_sim samples from a GP with mean \(m\) and cov \(k\) evaluated on the grid grid

Return type

array

statFEM_analysis.maxDist module

statFEM_analysis.maxDist.wass(a, b, n_bins)

This function computes an approximation of the 2-Wasserstein distance between two datasets.

Parameters

• a (array) – dataset

• b (array) – dataset

• n_bins (int) – controls number of bins used to create the histograms for the dataset.

Returns

estimate of the 2-Wasserstein distance between the two data-sets a and b

Return type

float