Building the covariance model#

ML-GPR represents the covariance of the data as a sum of named components. A model can contain smooth intrinsic foreground emission, chromatic mode mixing, a 21-cm signal, and heteroscedastic thermal noise.

Configuration example#

This extract defines two foreground components, one 21-cm component, and the noise-amplitude parameter. Settings not shown here are inherited from the packaged defaults by load_with_defaults().

kern.uv_bins_du = 20
kern.uv_bins_n_uni = 0
kern.fg = ["intrinsic", "mixing"]
kern.eor = ["signal"]

[intrinsic]
type = "MRBF"
variance.prior = "Uniform(0.8, 1.2)"
lengthscale.prior = "Uniform(10, 100)"
var_alpha.prior = "Fixed(0)"
ls_alpha.prior = "Fixed(0)"

[mixing]
type = "MMat32"
variance.prior = "Log10Uniform(-3, -0.5)"
variance.log_scale = true
lengthscale.prior = "Uniform(2, 60)"
var_alpha.prior = "Fixed(0)"
ls_alpha.prior = "Uniform(0, 1.5)"

[signal]
type = "MExponential"
variance.prior = "Log10Uniform(-5, -1)"
variance.log_scale = true
lengthscale.prior = "Uniform(0.2, 2)"
var_alpha.prior = "Fixed(0)"
ls_alpha.prior = "Fixed(0)"

[kern.noise]
alpha.prior = "Uniform(0.9, 1.6)"

Component names and composition#

Sections listed under kern.fg and kern.eor become components prefixed with fg_ and eor_. The example therefore produces fg_intrinsic, fg_mixing, eor_signal, and noise. These prefixes let posterior methods select groups with patterns such as "fg*" and "eor*".

Listed sections are additive by default. Place "x" or "*" between two section names to multiply their kernels into one component:

kern.fg = ["frequency_structure", "x", "second_structure", "foreground_2"]

This creates the components fg_frequency_structure (the product of the first two kernels) and fg_foreground_2.

Available frequency kernels#

Stationary configuration types construct a UVScaledKernel with the corresponding family:

TOML `type`	Kernel family	Frequency covariance	Additional parameter
`MRBF`	RBF	Infinitely differentiable, very smooth covariance.	—
`MMat32`	Matérn \(\nu=3/2\)	Once-differentiable covariance.	—
`MMat52`	Matérn \(\nu=5/2\)	Twice-differentiable covariance.	—
`MExponential`	Matérn \(\nu=1/2\)	Exponential covariance with a sharp loss of correlation.	—
`MCosine`	Cosine	Periodic covariance; `lengthscale` is interpreted as its period.	—
`MRatQuad` or `MRatQuad2`	Rational quadratic	A scale mixture of RBF covariances. Both names map to the same current implementation.	`power`
`VAEKernTorch`	VAE 21-cm covariance	Covariance derived from the power spectrum decoded at latent coordinates `x1`, `x2`, and any further trained dimensions.	`fitter_filename`, `x1` … `xN`

Every stationary family has variance and either lengthscale / ls_alpha or the wedge parameters described below. The VAE kernel has a variance parameter in addition to its latent coordinates.

The noise component is constructed automatically as WhiteHeteroscedasticKernel; its alpha parameter scales the measured per-frequency, per-UV-bin noise variance. Multi-epoch frequency kernels are wrapped by TimeKron; the available time covariance models are listed in Multi-epoch and Cross-GPR models.

Available priors#

Each candidate hyperparameter has a prior entry. The accepted configuration expressions are:

Expression	Arguments	Meaning
`Uniform(lower, upper)`	Natural-value lower and upper bounds.	Constant density over the closed interval. Implemented by `UniformPrior`.
`Log10Uniform(lower, upper)`	Lower and upper \(\log_{10}\) bounds.	Log-uniform density over positive natural values. For example, `Log10Uniform(-5, -1)` spans \(10^{-5}\) to \(10^{-1}\). Implemented by `Log10UniformPrior`.
`Gaussian(mu, sigma)`	Natural-value mean and standard deviation.	Unbounded normal density. Implemented by `GaussianPrior`.
`Fixed(value)`	One natural value.	Removes the parameter from the sampled vector and keeps it at the given value.

Priors are defined on natural kernel values. The optional parameter.log_scale = true setting is separate from the prior: it makes the inference methods store and explore log10(parameter) while retaining the stated natural-value prior and the corresponding transformation Jacobian.

UV-dependent parameterizations#

UVScaledKernel evaluates one covariance per UV bin. Its amplitude and frequency coherence can vary with the mean baseline length \(u\):

Parameterization	UV dependence	Configuration
Constant	The same variance or lengthscale is used in every UV bin.	`var_alpha.prior = "Fixed(0)"` and `ls_alpha.prior = "Fixed(0)"`
Variance power law	\(\mathrm{variance}(u) \propto (u/u_\mathrm{min})^{\mathrm{var\_alpha}}\), normalized across the fitted UV bins.	Set a free or fixed `var_alpha` prior and leave `use_uv_ps = false`.
Measured angular-power scaling	The variance shape follows the angular power measured from the input data and is normalized across UV bins.	`use_uv_ps = true`. In this mode `var_alpha` is not a kernel parameter.
Lengthscale evolution	\(\ell(u) = \ell_0 / [1 + 10^{-3}\,\mathrm{ls\_alpha}\,\ell_0 (u-u_\mathrm{min})]\).	Set a free or fixed `ls_alpha` prior and leave `wedge_parametrization = false`.
Wedge/delay evolution	The frequency scale is the inverse of a baseline-dependent delay built from a wedge angle and an additive delay buffer.	`wedge_parametrization = true` with priors for `theta_rad` and `delay_buffer_us`. `lengthscale` and `ls_alpha` are then not sampled parameters.

use_uv_ps controls only the variance law and can be combined with either lengthscale parameterization. uv_min and uv_max restrict a component to a baseline interval; l_max and uv_break control the cap applied to derived lengthscales.

Units and normalization#

The input is divided into UV bins and normalized before fitting. Model parameters use the following conventions:

stationary frequency lengthscales are in MHz;
UV coordinates and UV limits are in wavelengths;
theta_rad is in radians;
delay_buffer_us is in microseconds;
kernel variances refer to the internally normalized data.

Noise model#

The noise input can be a measured noise realization, a NoiseStdCube, or a scalar variance when constructing MultiData directly. In the high-level interface, process_noise_cube() applies the differencing, SEFD estimation, baseline scaling, and noise simulation selected in the configuration.

Thermal noise is independent between frequencies in the likelihood. Its measured variance sets the diagonal covariance, and kern.noise.alpha optionally rescales that level during inference.