estimation op• Data kinds: matrix × matrix × matrix × matrix × matrix × signal × matrix → table
• Call: import fullseye as fs; fs.ledger.kalman_smooth(z, F, H, Q, R, x0, P0) -> 'dict' (to call the implementation directly, import mathestimation; mathestimation.kalman_smooth(z, F, H, Q, R, x0, P0) -> 'dict'; from the registry, opsmath.get("kalman_smooth"))
Kalman filter (forward) and Rauch–Tung–Striebel smoother (backward) for the linear Gaussian state-space model x_{k+1} = F x_k + w (w ~ N(0,Q)), z_k = H x_k + v (v ~ N(0,R)).
z is the (T, m) observations (rows with NaN are treated as missing and the update is skipped). Returns {"x_filt", "P_filt", "x_smooth", "P_smooth", "nll"}.
Gate: the smoothing result agrees exactly with least squares using all observations and the prior at once (batch MAP).
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 · crlb_gaussian · 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.