TRACKING / ASSOCIATION / UNCERTAINTY

One return.
Two plausible targets.

Joint probabilistic data association (JPDA) weighs complete, feasible assignments before updating each target. Explore what changes when measurements can be clutter, targets can be missed, and nobody gets to claim the same return twice.

Based on Fortmann, Bar-Shalom & Scheffe (1983) and Bar-Shalom, Daum & Huang (2009).

INTERACTIVE / ONE SCAN

Reason jointly. Update softly.

Drag a measurement, or use the position controls. Watch the joint events, marginal probabilities, and updated target estimates respond.

3 feasible joint eventsΣ event probabilities = 1.000
Prediction JPDA mean Independent PDA mean Measurement

Dashed ellipses: 95% innovation contours, not gates. Filled ellipses: 95% contours of the moment-matched posterior Gaussians. Every displayed measurement remains eligible.

SUM OVER EVENTS

Marginal association probabilities

Each target row sums to 1. A measurement’s target probabilities plus its clutter probability sum to 1.

NORMALIZED JOINT WEIGHTS

Feasible assignments

Select an event to inspect its lines. The JPDA update still uses all events.

MOMENT-MATCH THE MIXTURE

Updated state & uncertainty

The spread term preserves uncertainty between association alternatives; a mean between returns is not a selected measurement.

01 / MODEL

A small, explicit association problem

This is a synthetic Cartesian teaching example with two established point targets and a fixed measurement set. Target states and measurements are 2D positions; H = I. Each target independently generates at most one return with probability PD. Clutter is an independent homogeneous Poisson process with spatial intensity λ. There are no births, deaths, track management, or temporal trajectory simulation.

All candidate measurements are considered: PG = 1 is the large-gate simplification used in the 1983 paper’s Section III and footnote 5. The drawing window is only a viewport. Its boundary and the dashed contours do not gate measurements or change the likelihood.

02 / JOINT EVENTS

Normalize complete explanations of the scan

Let at = 0 denote a missed detection and at = j denote target t generating measurement j. Positive assignments must be distinct. Any unassigned measurement is clutter. With predicted measurement mean μt and innovation covariance St = Pt− + R:

\[\ell_{tj}=\frac{P_{D,t}\,\mathcal N(z_j;\mu_t,S_t)}{\lambda},\qquad w(a)=\prod_{t:a_t=0}(1-P_{D,t})\prod_{t:a_t>0}\ell_{t,a_t},\qquad p(a\mid Z)=\frac{w(a)}{\sum_{a'\in\mathcal A}w(a')} .\]

The likelihood ratio comes from the 2009 paper’s (38); the joint weight is the parametric Poisson-clutter form in 1983 (3.18) and 2009 (47), after removing a common λm factor. No additional clutter-count factorial is inserted. The demo enumerates all feasible events and normalizes their weights in log space.

03 / MARGINALS & UPDATE

Average the branches—and keep their spread

\[\beta_{tj}=\sum_{a\in\mathcal A:a_t=j}p(a\mid Z),\qquad \hat x_t^+=\sum_{j=0}^{m}\beta_{tj}\hat x_{tj},\qquad P_t^+=\sum_{j=0}^{m}\beta_{tj}\left[P_{tj}+(\hat x_{tj}-\hat x_t^+)(\hat x_{tj}-\hat x_t^+)^\top\right].\]

The missed-detection branch retains the prior. Each measurement branch uses its ordinary Kalman correction. These mixture moments are equivalent to the innovation and covariance formulas in 2009 (39)–(44). Marginalization follows 1983 (3.19)–(3.20) and 2009 (51).

The lab computes exact association sums for this scan under its independent Gaussian prediction model, then shows the decoupled JPDAF Gaussian update. A recursive JPDA filter approximates the resulting mixture; this is not an exact multitarget posterior over all time. Separate Gaussian tracks can move together under persistent ambiguity (track coalescence). Maintaining cross-covariances, as in JPDACF, is beyond this lab.

PRIMARY REFERENCES

Read the source papers

  1. Sonar Tracking of Multiple Targets Using Joint Probabilistic Data Association
    Thomas E. Fortmann, Yaakov Bar-Shalom & Molly Scheffe. IEEE Journal of Oceanic Engineering, OE-8(3), pp. 173–184, July 1983.
    Joint-event constraints, Poisson-clutter weights, and marginal probabilities: Section III, especially (3.18)–(3.20), printed p. 177.
  2. The Probabilistic Data Association Filter
    Yaakov Bar-Shalom, Fred Daum & Jim Huang. IEEE Control Systems Magazine, pp. 82–100, December 2009. DOI: 10.1109/MCS.2009.934469.
    Likelihood ratio (38), PDA moment update (39)–(44), and JPDA event weights and marginals (45)–(51), printed pp. 90–93.

Figures and numerical examples on this page are generated from the teaching model above; they do not reproduce the papers’ sonar experiments.

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