estimation op• Data kinds: signal × signal → table
• Call: import fullseye as fs; fs.ledger.crlb_gaussian(x, theta, sigma: 'float' = 1.0, *, model: 'str' = 'gaussian_peak') -> 'dict' (to call the implementation directly, import mathestimation; mathestimation.crlb_gaussian(x, theta, sigma: 'float' = 1.0, *, model: 'str' = 'gaussian_peak') -> 'dict'; from the registry, opsmath.get("crlb_gaussian"))
Cramér–Rao lower bound under white Gaussian noise (standard deviation sigma).
For y_i = f(x_i; θ) + ε_i, ε ~ N(0, σ²), the Fisher information matrix is I = JᵀJ / σ² (J = ∂f/∂θ, central differences); the covariance of any unbiased estimator is at least I⁻¹. std is the lower bound √diag(I⁻¹) of each parameter's standard deviation. model: constant (c) / line (a + bx) / gaussian_peak (A, μ, s) / exp_decay (A, k) / sinusoid (A, f, φ).
Gate: for the mean of a constant, σ²/n. The Monte Carlo variance of line least squares (an efficient estimator) matches the lower bound.
Every mathops op validates its input before computing (nothing slips through silently):
• complex input raises ValueError — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• masked arrays with masked elements raise ValueError — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• NaN/Inf raises ValueError on every input (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• estimation_tour — py -3.11 examples/estimation_tour.py
table as input)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
estimation)assign_hungarian · hist_distance · stat_ttest_paired · stat_ttest_welch · stat_ks_test · stat_chi2_gof · kalman_smooth · mat_expm
*Provenance: mathestimation.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.