Diagnostics¶

Tools enabled by end-to-end differentiability of the forward model. These provide Fisher information analysis, gradient SEDs (saliency maps), and Green’s function time-sensitivity analysis.

Fisher Information¶

tengri.analysis.diagnostics.fisher.compute_jacobian(predict_fn, params, param_keys)[source]¶

Compute the Jacobian ∂m/∂θ of model predictions w.r.t. parameters.

Parameters:

predict_fn (callable) – Function mapping a flat parameter array to model predictions.
params (array, shape (n_params,)) – Parameter values.
param_keys (tuple of str) – Parameter names (for static hashing).

Returns:

Jacobian matrix.

Return type:

array, shape (n_data, n_params)

tengri.analysis.diagnostics.fisher.compute_fisher_matrix(forward_model, params, noise, data_type='photometry', param_names=None)[source]¶

Compute the Fisher Information Matrix.

F_ij = sum_k (1/sigma_k^2) * (dm_k/dtheta_i) * (dm_k/dtheta_j)

Parameters:

forward_model (ForwardModel) – Configured forward model.
params (dict) – Parameter values at which to evaluate the FIM.
noise (array) – 1-sigma uncertainties on the data.
data_type (str) – “photometry” or “spectroscopy”.
param_names (list of str, optional) – Which parameters to include. Defaults to all physical params (excludes xi — the GP latent is high-dimensional).

Returns:

fim (array, shape (n_params, n_params)) – Fisher Information Matrix.
names (list of str) – Parameter names corresponding to FIM rows/columns.

tengri.analysis.diagnostics.fisher.fisher_parameter_errors(fim)[source]¶

Compute 1-sigma parameter uncertainties from the FIM.

sigma_i = sqrt((F^{-1})_ii)

Parameters:: fim (array, shape (n_params, n_params)) – Fisher Information Matrix.
Returns:: 1-sigma marginal uncertainties.
Return type:: array, shape (n_params,)

tengri.analysis.diagnostics.fisher.fisher_correlation_matrix(fim)[source]¶

Compute parameter correlation matrix from the FIM.

Parameters:: fim (array, shape (n_params, n_params)) – Fisher Information Matrix.
Returns:: Correlation matrix (values in [-1, 1]).
Return type:: array, shape (n_params, n_params)

Saliency (Gradient SEDs)¶

tengri.analysis.diagnostics.saliency.compute_gradient_sed(forward_model, params, param_name)[source]¶

Compute ∂SED(λ)/∂θ — the gradient of the SED w.r.t. one parameter.

Parameters:

forward_model (ForwardModel) – Configured forward model.
params (dict) – Parameter values.
param_name (str) – Which parameter to differentiate with respect to.

Returns:

gradient_sed (array, shape (n_wave,)) – ∂SED/∂θ at each wavelength.
wavelength (array, shape (n_wave,)) – Rest-frame wavelengths (Angstrom).

tengri.analysis.diagnostics.saliency.compute_all_gradient_seds(forward_model, params, param_names=None)[source]¶

Compute gradient SEDs for all physical parameters.

Parameters:

forward_model (ForwardModel) – Configured forward model.
params (dict) – Parameter values.
param_names (list of str, optional) – Parameters to compute gradients for.

Returns:

gradients (dict of {param_name: array (n_wave,)}) – Gradient SED per parameter.
wavelength (array, shape (n_wave,)) – Rest-frame wavelengths.

tengri.analysis.diagnostics.saliency.compute_photometry_sensitivity(forward_model, params, param_names=None)[source]¶

Compute ∂flux_band/∂θ for each filter and parameter.

Returns a matrix showing how sensitive each photometric band is to each parameter — useful for understanding degeneracies and planning filter sets.

Parameters:

forward_model (ForwardModel) – Configured forward model (must have filters set).
params (dict) – Parameter values.
param_names (list of str, optional) – Parameters to include.

Returns:

sensitivity (array, shape (n_filters, n_params)) – ∂flux_band/∂θ_param matrix.
param_names (list of str) – Parameter names (columns).

Green’s Functions¶

tengri.analysis.diagnostics.green_functions.compute_green_function(ssp_flux_at_z, ssp_wave, filter_wave=None, filter_trans=None, wave_target=None)[source]¶

Compute Green’s function G(t_age) for a filter or wavelength.

G(t_age) = flux contribution per unit stellar mass at age t_age.

For photometry: G = int L_SSP(lambda|t_age) * T(lambda) * lambda dlambda: / int T(lambda) * lambda dlambda

For a single wavelength: G = L_SSP(wave_target | t_age)

Parameters:

ssp_flux_at_z (array, shape (n_age, n_wave)) – SSP spectra at fixed metallicity.
ssp_wave (array, shape (n_wave,)) – Wavelength grid (Angstrom).
filter_wave (array, optional) – Filter wavelength grid.
filter_trans (array, optional) – Filter transmission.
wave_target (float, optional) – Single wavelength (Angstrom). Used if no filter provided.

Returns:

Green’s function G(t_age).

Return type:

array, shape (n_age,)

tengri.analysis.diagnostics.green_functions.compute_window_function(green_fn, mean_sfr_on_ages)[source]¶

Compute window function W(t_age) = G(t_age) * <SFR(t_age)>.

The window function tells you which lookback times actually contribute to the observed flux, given the galaxy’s mean SFH.

Parameters:

green_fn (array, shape (n_age,)) – Green’s function G(t_age).
mean_sfr_on_ages (array, shape (n_age,)) – Mean SFR evaluated at the SSP age grid (Msun/yr).

Returns:

Window function W(t_age) (unnormalized).

Return type:

array, shape (n_age,)

tengri.analysis.diagnostics.green_functions.compute_window_function_fourier(window_fn, ssp_ages_yr)[source]¶

Compute Fourier transform of window function.

|W_tilde(omega)|^2 tells you the PSD sensitivity: how much power at frequency omega contributes to the variance of this observable.

From Munoz+2026 Eq. 11:: sigma^2_L = int |W_tilde(omega)|^2 P(omega) d_omega / (2*pi)

Parameters:

window_fn (array, shape (n_age,)) – Window function W(t_age).
ssp_ages_yr (array, shape (n_age,)) – SSP ages in years (for frequency axis).

Returns:

power_transfer (array, shape (n_freq,)) – |W_tilde(omega)|^2 — the PSD-to-observable transfer function.
omega (array, shape (n_freq,)) – Angular frequencies (rad/yr).

tengri.analysis.diagnostics.green_functions.compute_time_sensitivity_matrix(ssp_flux_at_z, ssp_wave, wavelengths_target)[source]¶

Compute sensitivity of multiple wavelengths to different ages.

Returns a matrix G(wavelength, age) showing which wavelengths are sensitive to which stellar population ages.

Parameters:

ssp_flux_at_z (array, shape (n_age, n_wave)) – SSP spectra at fixed metallicity.
ssp_wave (array, shape (n_wave,)) – Wavelength grid (Angstrom).
wavelengths_target (array, shape (n_target,)) – Target wavelengths to evaluate (Angstrom). E.g., [1500, 2500, 4000, 5500, 6563, 8000, 16000] for FUV, NUV, Balmer break, V-band, H-alpha, I-band, H-band.

Returns:

sensitivity – G(wavelength, age) matrix. Each row is the Green’s function at that wavelength.

Return type:

array, shape (n_target, n_age)