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Tracking filters: EKF and IMM

Source: packages/classical/src/ekf.ts (ekfPredict), packages/classical/src/imm.ts (immPredict), CLI tma-track. Requirements: SRD Annex B §B.3 (as amended 2026-10-28), spec 010.

Phase 3's target can turn once. Batch least squares fits a single constant-velocity track, so it cannot follow a turn. G4 needs a reference that can, and SRD Annex B names a tracking filter.

Common ground​

Both filters:

  • Start where classical does. They are initialised at zig_index, the first sample on leg 2, from the batch solution on samples up to it. The initial covariance is the batch fit's own: σ²(JᵀJ)⁻¹, propagated to that time.
  • Run causally. They update once per bearing, so the row at sample t uses bearings up to t. They emit one row per sample from zig_index + 1, the same rows as tma-solve --per-timestep, so the estimators pair row for row.
  • Use the wrapped innovation. The innovation is z − h wrapped to (−π, π], with a Joseph-form covariance update.
  • Take measurement noise from the scenario. That is the scenario's σ, floored at 10⁻⁴ rad so that noise-free data keep the innovation variance positive.
  • Report divergence. A state outside the plausibility bounds (over 25 m/s or over 100 km) is reported as converged = false. The filter then re-initialises from the batch solution on the samples so far.

EKF (constant velocity)​

The state is (x, y, vx, vy). Process noise is white-noise acceleration σ_a (m/s²), and σ_a is the only parameter.

The single-model EKF is reported in phase 3 but is not the gated reference. Spec 010's first tests showed why:

  • Before the turn it is exact. On noise-free data it holds the truth to about 10⁻⁶ in range.
  • After the turn it usually drifts. In the first hand-built case (a 90° starboard turn followed by an ownship zig) its range error grows to about 50% twenty minutes after the turn at every ratio σ_a/σ tried, while a batch fit on the post-turn bearings alone recovers the truth exactly. It is not hopeless everywhere: the G0 panel found the mirror-image port case ends at 0.33% with σ_a/σ = 0.32.

The mechanism (G0 panel, numerics, reproduced with an independent EKF and a UKF): a constant-velocity model carried through the turn, then covariance collapse. While ownship holds course, every scaled track own + k(target − own) explains the bearings equally well, and the filter drifts along that family. It is related to, and looser than, the Cartesian instability Aidala described (1979).

IMM (constant velocity + coordinated turn)​

The interacting-multiple-model tracker (Blom and Bar-Shalom) runs two EKFs on the state (x, y, vx, vy, ω):

  • CV: constant velocity, white-noise acceleration σ_cv. It carries ω but does not use it.
  • CT: coordinated turn at rate ω (positive is clockwise, as the generator's turns), with white-noise acceleration σ_ct and a turn-rate random walk σ_ω.

Each step:

  1. Mix. Each model's prior blends both estimates through a two-state Markov chain whose switch probability is switch per second.
  2. Predict. Each model predicts, using a numerical Jacobian of its motion.
  3. Update. Each model takes the bearing.
  4. Reweight. The innovation likelihoods reweight the model probabilities.
  5. Report. The output is the probability-weighted mixture.

The prior on ω at initialisation has a standard deviation of 0.05 rad/s, and the initial model probabilities are 0.9 (CV) and 0.1 (CT).

The four parameters (σ_cv, σ_ct, σ_ω, switch) are tuned on the phase-3 validation split, frozen in the decision record, and then used unchanged at G4 (spec 010 FR-004).

What it does​

  • Noise-free, with an ownship zig after the target's turn. With G0's settings it ends within 0.2% range on the hand-built 90° turn at 1°/s, in both directions; this is gated at G0 (check tracker-noise-free) and shows the implementation follows a turn. It is not a general re-convergence result: over 30–120° and 0.5–3°/s the same geometry is reported, not gated, and some fast turns stay far off; and other settings (for example the viewer's) fail the gated case itself.
  • At σ = 1°, before the turn. With no CV process noise and rare switching (σ_cv 0, switch 10⁻⁵/s) its median range error at the last sample before the turn is about that of per-timestep batch least squares (3.1% against 3.4% on 120 scenarios). G0 gates the ratio at 1.25 (check tracker-noisy-sanity). The check is one-sided: it catches an under-stated measurement variance (σ × 0.1 gives 2.1 ×), not an over-stated one (σ × 10 gives 1.002 ×), because with no CV process noise the filter is close to recursive least squares whatever R is. With G0's noise-free settings (switch 10⁻³/s) it is 1.8 × batch least squares: frequent switching leaks the turn model's noise into straight legs, so the tuning grid must include low switch rates. Turn-ready settings are about 1.7–1.8 × batch least squares on straight legs at the true σ, and telling them σ × 10 brings that to about 1.1 ×: a scale on R is a tuning lever too (G0 panel round 2).
  • Noise-free, before the turn. Every row from zig + 60 s to the turn is within G0's tolerances (0.1% range, 0.1° course, 0.01 kt).
  • At σ = 1°, five minutes after the turn. On 150 validation-like scenarios with untuned settings, its median range error is about 15.6%. The best single-model EKF reaches 14.8%, sliding-window least squares 42–100%, and batch least squares over the whole track 28%.

Generator 0.5.0 (phase 4)​

G0 on 0.5.0 (experiments/20261030-g0-p4-r2/) adds three tracker checks:

  • Re-convergence after every manoeuvre kind. G0's IMM setting (σ_ct 0.01) cannot follow a deceleration: 0.93 range error on a hand-built −2 m/s case. With σ_ct 0.1, every hand-built case ends within 0.6% when ownship zigs afterwards: two turns either side, ±2 m/s, and turn, speed change, turn. A single coordinated-turn model with a signed ω handles S-turns poorly at small σ_ω.
  • A paired straight-leg ratio. The median per-scenario ratio to batch least squares is 1.05 over 1,000 scenarios, against a limit of 1.15. The ratio of medians, 1.17–1.26, was noisy on 200 scenarios.
  • NIS. The median per-scenario mean NIS, through immPredict, is 1.06 with the scenario's σ, 0.71 with σ² × 1.5 and 1.38 with σ² × 0.75. The band is [0.75, 1.3], so a variance mis-scaled either way fails. The ratio check cannot see an over-stated variance, and on 0.5.0 it cannot see a mis-stated σ at all.

Reading it critically​

  • The IMM's advantage shows only with exact data. At σ = 1°, five minutes after a turn, the best single EKF is level with or better than the best IMM tried so far (panel: medians 16.6% against 17.8%, P90 39.6% against 51.6%, on 200 scenarios). The decision record must settle whether G4 gates against the tuned IMM only or the better of the two tuned trackers, and widen the IMM grid (including a scale on the initial covariance). There the bearings, not the filter, limit what is knowable. That is the regime G4 judges, so a network that beats the IMM must extract information the filter leaves on the table: for example, a prior over how targets turn.
  • Re-convergence needs geometry. When the target turns after ownship's last zig (about half of the turning scenarios), a filter with process noise can observe range again only through the continuity of position across the turn. A batch or smoothing fit that knows the speed is constant through the turn could do better: that prior is exactly what a network can learn. G0 reports the noise-free median error at manoeuvre + 5 min on generated scenarios but does not gate it.
  • The IMM is not the strongest possible reference. A smoother, a particle filter, or an IMM in modified polar coordinates could do better. The owner chose the IMM on 2026-10-28 as the textbook tracker for manoeuvring targets (SRD Annex B, decision D3 revisited).
  • Re-initialisation is crude. After an implausible state the filter restarts from a constant-velocity batch over every sample so far, turn included.
  • Initialisation after an instantaneous zig is slow. With iteration 1's instantaneous zig, the batch solution the filter starts from is nearly unobservable, and the IMM needs more than 60 s to settle (worst 1% range). Phase-3 data have rate-limited turns, where it settles within 60 s. G0 gates the tracker on generator-0.4.0 data only.

Known limitation: re-initialisation can give an indefinite covariance (spec 015 R5). The re-initialisation fits the batch solution over every bearing so far. After target manoeuvres that constant-velocity fit can be degenerate: its track passes through ownship's position at an earlier sample.

  • At that sample, one Jacobian row dominates JᵀJ, whose condition number reaches about 10¹⁶.
  • The initial covariance σ²(JᵀJ)⁻¹ is then not positive definite.
  • On the seed-214 test split this happens in 43 of 40,000 scenarios: 87 of 44,343 (re-)initialisations.
  • The frozen reference is not changed.

Guarded initialisation (spec 016, opt-in: -g1, -g2). --guarded-init 1 on tma-track (or guardedInit in code) refuses a fit that has either fault:

  • its track comes within 100 m of ownship at any fitted sample;
  • its QR factor's conditioning estimate is above 10⁷.

The tracker then retries every 6 samples. Level 1 also builds P₀ in square-root form, σ² · p0 · R⁻¹R⁻ᵀ, from a Householder QR of the Jacobian at tᵢ. That is positive definite by construction, and equal to the frozen P₀ to rounding on a healthy fit.

Level 2 (-g2) does the same, but a re-initialisation first fits only the last 60 samples (10 min), falling back to all.

Every frozen reference is level 0 and unchanged.

On seed-211 validation (specs/016-imm-guarded-init/research.md):

  • Level 1 is accuracy-neutral for the IMM, the EKF and the fused tracker. Every G5-row ratio to the frozen tracker is 1.000 ± 0.003.
  • Level 1 removes every non-positive-definite IMM covariance. The frozen IMM has 21 such steps, in 6 of 10,000 scenarios; level 1 has none. It reports no estimate on 0.07% more rows.
  • Level 2 is worse. It is 2–9% worse on range, course and speed, and still has non-PD steps: a 10-min window is too short to observe range after a manoeuvre.

Running it​

pnpm tma-track --in <split.f32pack> --out <pred.ndjson> --filter imm \
  --sigma-cv 0.003 --sigma-ct 0.03 --sigma-omega 0.003 --switch 0.001
pnpm tma-track --in <split.f32pack> --out <pred.ndjson> --filter ekf --sigma-a 0.03

The estimator name records the settings (for example imm-cv0.003-ct0.03-w0.003-s0.001). Predictions are written as they are made.

Tuned for phase 3​

On the seed-208 validation split at σ = 1° (spec 010 R2; experiments/20261028-p3-research/), by median range error five minutes after the turn:

  • IMM: 13.6% (σ_cv 0.003, σ_ct 0.1, σ_ω 0.001, switch 0.001/s, R × 2).
  • Single EKF: 15.3% (σ_a 0.1, P0 × 4, R × 4).
  • Batch least squares: 29.7%.

The IMM is therefore G4's frozen reference.

  • Held course. Both trackers' straight tunings are within 0.3 points of batch least squares (3.2–3.3% against 3.0%).
  • Under noise, after the turn. Where ownship zigs after the turn, the reference reaches 12.5%, against 31.5% for a batch fit on post-turn bearings alone. It uses the pre-turn track, as it should.

Re-tuned for phase 4​

On the seed-211 validation split (generator 0.5.0: 0–3 course changes, one speed change in 30%), by median range error over all G5 rows (σ = 1°, zig + 5/10/15/20 min; spec 012 R2, experiments/20261030-p4-research/):

  • IMM: 9.24% (σ_cv 0.01, σ_ct 0.03, σ_ω 0.01, switch 0.001/s, P0 × 4, R × 8): G5's frozen reference, imm-cv0.01-ct0.03-w0.01-s0.001-p4-r8.
  • Single EKF: 9.72% (σ_a 0.03, P0 × 4, R × 8).
  • Batch least squares: 12.2% overall, but 6.7% on settled rows, where the IMM reaches 7.2%. A batch fit over a long straight segment beats a recursive filter there.

Two declared grids were run, the second because the first's optimum sat on its R and P0 edges. The surface is flat in σ_cv and σ_ct: the top four settings are within 0.05 points.