Likelihood Evaluations Based on Fast Fourier Transforms

Likelihood Evaluations Based on Fast Fourier Transforms#

The module implements Fourier Methods for one-dimensional signals and two-dimensional signals. The functions of immediate use to practitioners are gp_rfft_lpdf to evaluate the likelihood of a one-dimensional signal and gp_inv_rfft to construct a non-centered parameterization using white noise. The two-dimensional analogues are gp_rfft2_lpdf and gp_inv_rfft2. The module also provides functions to evaluate kernels directly in the Fourier domain [1], including gp_periodic_exp_quad_cov_rfft and gp_periodic_matern_cov_rfft.

Utility Functions#

The following functions, and their two-dimensional analogues, primarily exist as utility functions but may be useful for more complex models.

  • gp_rfft transforms Gaussian process realizations to the Fourier domain and scales the coefficients such that they are white noise under the Gaussian process prior.

  • gp_rfft_log_abs_det_jac evaluates the log absolute determinant of the Jacobian associated with the transformations gp_rfft.

Together, these two functions are used by gp_rfft_lpdf to evaluate the likelihood.

  • gp_unpack_rfft unpacks complex Fourier coefficients of size n %/% 2 + 1 to a real vector of size n for easier manipulation.

  • gp_pack_rfft is, unsurprisingly, the inverse of gp_unpack_rfft and packs a real vector of size n into a complex vector of size n %/% 2 + 1 such that the inverse RFFT can be applied.

  • gp_evaluate_rfft_scale evaluates the expected standard deviation of Fourier coefficients obtained by transforming a Gaussian process. The values are arranged to match the output of gp_unpack_rfft.

Function Reference#

One-dimensional Signals#

vector gp_evaluate_rfft_scale(vector cov_rfft, int n)#

Evaluate the scale of Fourier coefficients.

Parameters:

  • cov_rfft – Precomputed real fast Fourier transform of the kernel with shape (..., n %/% 2 + 1).

  • n – Size of the real signal. Necessary because the size cannot be inferred from cov_rfft.

Returns:

Scale of Fourier coefficients with shape (..., n %/% 2 + 1).

vector gp_unpack_rfft(complex_vector x, int n)#

Unpack the Fourier coefficients of a real Fourier transform with n %/% 2 + 1 elements to a vector of n elements.

Parameters:

  • z – Real Fourier transform coefficients.

  • n – Size of the real signal. Necessary because the size cannot be inferred from rfft.

Returns:

Unpacked vector of n elements comprising the n %/% 2 + 1 real parts of the zero frequency term, complex terms, and Nyqvist frequency term (for even n). The subsequent (n - 1) %/% 2 elements are the imaginary parts of complex coefficients.

vector gp_rfft(vector y, vector loc, vector cov_rfft)#

Transform a Gaussian process realization to white noise in the Fourier domain.

Parameters:

  • y – Realization of the Gaussian process with shape (..., n).

  • loc – Mean of the Gaussian process with shape (..., n).

  • cov_rfft – Precomputed real fast Fourier transform of the kernel with shape (..., n // 2 + 1).

Returns:

Fourier-domain white noise with shape (..., n). See gp_unpack_rfft for details on the data structure.

real gp_rfft_log_abs_det_jac(vector cov_rfft, int n)#

Evaluate the log absolute determinant of the Jacobian associated with gp_rfft.

Parameters:

  • cov_rfft – Precomputed real fast Fourier transform of the kernel with shape (..., n %/% 2 + 1).

  • n – Size of the real signal. Necessary because the size cannot be inferred from rfft.

Returns:

Log absolute determinant of the Jacobian.

real gp_rfft_log_abs_det_jacobian(vector cov_rfft, int n)#
real gp_rfft_lpdf(vector y, vector loc, vector cov_rfft)#

Evaluate the log probability of a one-dimensional Gaussian process realization in Fourier space.

Parameters:

  • y – Realization of a Gaussian process with n grid points.

  • loc – Mean of the Gaussian process of size n.

  • cov_rfft – Precomputed real fast Fourier transform of the kernel of size n %/% 2 + 1.

Returns:

Log probability of the Gaussian process realization.

complex_vector gp_pack_rfft(vector z)#

Transform a real vector with n elements to a vector of complex Fourier coefficients with n elements ready for inverse real fast Fourier transformation.

vector gp_inv_rfft(vector z, vector loc, vector cov_rfft)#

Transform white noise in the Fourier domain to a Gaussian process realization, i.e., a non-centered* parameterization in the Fourier domain.

The n real white noise variables must be assembled into a length-n %/% 2 + 1 complex vector with structure expected by the fast Fourier transform. The input vector z comprises

  • the real zero frequency term,

  • (n - 1) %/% 2 real parts of positive frequency terms,

  • the real Nyqvist frequency term if n is even,

  • and (n - 1) %/% 2 imaginary parts of positive frequency terms.

Parameters:

  • z – Fourier-domain white noise comprising n elements.

  • loc – Mean of the Gaussian process.

  • cov_rfft – Real fast Fourier transform of the covariance kernel.

Returns:

Realization of the Gaussian process with n elements.

vector gp_periodic_exp_quad_cov_rfft(int n, real sigma, real length_scale, real period)#

Evaluate the real fast Fourier transform of the periodic squared exponential kernel.

Parameters:

  • n – Number of grid points.

  • sigma – Scale of the covariance.

  • length_scale – Correlation length.

  • period – Period for circular boundary conditions.

Returns:

Fourier transform of the squared exponential kernel of size n %/% 2 + 1.

vector gp_periodic_matern_cov_rfft(real nu, int n, real sigma, real length_scale, real period)#

Evaluate the real fast Fourier transform of the periodic Matern kernel.

Parameters:

  • nu – Smoothness parameter (1 / 2, 3 / 2, and 5 / 2 are typical values).

  • n – Number of grid points.

  • sigma – Scale of the covariance.

  • length_scale – Correlation length.

  • period – Period for circular boundary conditions.

Returns:

Fourier transform of the squared exponential kernel of size n %/% 2 + 1.

Two-dimensional Signals#

matrix gp_evaluate_rfft2_scale(matrix cov_rfft2, int width)#

Evaluate the scale of Fourier coefficients.

Parameters:

  • cov_rfft2 – Precomputed real fast Fourier transform of the kernel with shape (height, width %/% 2 + 1).

  • width – Number of columns of the signal (cannot be inferred from the Fourier coefficients).

Returns:

Scale of Fourier coefficients with shape (height, width %/% 2 + 1).

matrix gp_unpack_rfft2(complex_matrix z, int m)#

Unpack the complex Fourier coefficients of a two-dimensional real Fourier transform with shape to a real matrix with shape (height, width).

TODO: add details on packing structure.

Parameters:

  • z – Two-dimensional real Fourier transform coefficients with shape (height, width %/% 2 + 1).

  • width – Number of columns of the signal (cannot be inferred from the Fourier coefficients z).

Returns:

Unpacked matrix with shape (height, width).

complex_matrix gp_pack_rfft2(matrix z)#

Transform a real matrix with shape (height, width) to a matrix of complex Fourier coefficients with shape (height, width %/% 2 + 1) ready for inverse real fast Fourier transformation in two dimensions.

Parameters:

z – Unpacked matrices with shape (height, width).

Returns:

Two-dimensional real Fourier transform coefficients.

matrix gp_rfft2(matrix y, matrix loc, matrix cov_rfft2)#

Transform a Gaussian process realization to white noise in the Fourier domain.

Parameters:

  • y – Realization of the Gaussian process with shape (height, width).

  • loc – Mean of the Gaussian process with shape (height, width).

  • cov_rfft2 – Precomputed real fast Fourier transform of the kernel with shape (height, width %/% 2 + 1).

Returns:

Unpacked matrix with shape (height, width).

matrix gp_inv_rfft2(matrix z, matrix loc, matrix cov_rfft2)#

Transform white noise in the Fourier domain to a Gaussian process realization.

Parameters:

  • z – Unpacked matrix with shape (height, width).

  • loc – Mean of the Gaussian process with shape (height, width).

  • cov_rfft2 – Precomputed real fast Fourier transform of the kernel with shape (height, width %/% 2 + 1).

Returns:

Realization of the Gaussian process.

real gp_rfft2_log_abs_det_jac(matrix cov_rfft2, int width)#

Evaluate the log absolute determinant of the Jacobian associated with gp_rfft2.

Parameters:

  • cov_rfft2 – Precomputed real fast Fourier transform of the kernel with shape (height, width %/% 2 + 1).

  • width – Number of columns of the signal (cannot be inferred from the precomputed kernel Fourier transform cov_rfft2).

Returns:

Log absolute determinant of the Jacobian.

real gp_rfft2_log_abs_det_jacobian(matrix cov_rfft2, int width)#
real gp_rfft2_lpdf(matrix y, matrix loc, matrix cov_rfft2)#

Evaluate the log probability of a two-dimensional Gaussian process realization in Fourier space.

Parameters:

  • y – Realization of a Gaussian process with shape (height, width), where height is the number of rows, and width is the number of columns.

  • loc – Mean of the Gaussian process with shape (height, width).

  • cov_rfft2 – Precomputed real fast Fourier transform of the kernel with shape (height, width %/% 2 + 1).

Returns:

Log probability of the Gaussian process realization.

matrix gp_periodic_exp_quad_cov_rfft2(int height, int width, real sigma, vector length_scale, vector period)#

Evaluate the two-dimensional real fast Fourier transform of the periodic squared exponential kernel.

Parameters:

  • height – Number of grid rows.

  • width – Number of grid columns.

  • sigma – Scale of the covariance.

  • length_scale – Correlation lengths in each dimension.

  • period – Periods for circular boundary conditions in each dimension.

Returns:

Fourier transform of the periodic squared exponential kernel with shape (height, width %/% 2 + 1).

matrix gp_periodic_matern_cov_rfft2(real nu, int height, int width, real sigma, vector length_scale, vector period)#

Evaluate the two-dimensional real fast Fourier transform of the periodic Matern kernel.

Parameters:

  • nu – Smoothness parameter (1 / 2, 3 / 2, and 5 / 2 are typical values).

  • height – Number of grid rows.

  • width – Number of grid columns.

  • sigma – Scale of the covariance.

  • length_scale – Correlation lengths in each dimension.

  • period – Periods for circular boundary conditions in each dimension.

Returns:

Fourier transform of the periodic Matern kernel with shape (height, width %/% 2 + 1).

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