Tier V.B — Point process model (TMTPP)

Generative model: events arrive in time according to an intensity $\lambda(t)$, each carries a mark $Q \sim f(Q)$, and each is observed with probability $P_d(Q; \text{satellite})$. The “thinned marked temporal point process” (TMTPP) is the mathematical object on which everything else at Tier V is built — see [daley2003,daleyVereJones2008] for foundations.

The full mathematical derivation lives in the TMTPP theory page. This page gives the architectural view: the components, their interfaces, and where they plug into the rest of plumax.

The three components¶

Temporal — $\lambda(t)$ (events / second)¶

The intensity function tells you how rapidly events arrive at time $t$. Examples from the catalogue in 02_intensity_zoo.md:

13 deterministic kernels currently implemented in plumax.population.intensity (catalogued in the intensity zoo); LGCP is the v1.5 next kernel — it’s the natural model when clustering is environmental rather than self-exciting.

Each kernel is an equinox.Module exposing the same __call__(t) → λ and sample_priors() interface. Adding a new kernel is a one-file PR.

Marks — $f(Q)$ (probability density on kg/s)¶

The mark distribution gives the size of an event conditional on it happening.

Mark families — form and operational fit.

$Q_\text{min}$ (power-law) and $Q_\text{break}$ (lognormal-Pareto) are themselves parameters.

The mark distribution is what Tier V actually wants to recover — it’s the population-scale answer to “how big are the leaks at this kind of facility?”

Detection thinning — $P_d(Q; \text{satellite})$ (probability)¶

Not every event is observed. Each satellite has a probability of detection that depends on the leak size, viewing geometry, surface, and atmospheric state.

The operational form in the methane literature is the Hill function (Cusworth 2021, Sherwin 2024):

where $Q_{50}$ is the leak size at which detection probability is 0.5 and $k$ controls the steepness.

POD calibration uncertainty — hierarchical prior¶

Per-instrument controlled-release campaigns (Sherwin et al. 2024) deliver a posterior on $(Q_{50}, k)$, not a point. v1 default: hierarchical prior carrying calibration-campaign uncertainty:

This is the middle ground between (a) hard-coding published values (biased when those are uncertain) and (b) full joint inference (cleanest but identifiability concern with $\lambda$). v2 promotes to joint inference when basin data warrants.

10 POD models currently in plumax.population.pod, described visually in the POD gallery. Variants:

• Hill — operational standard.

• Varying-coefficient Hill — $Q_{50} = g(\text{albedo}, \text{SZA}, \text{scene class})$.

• Spectral-aware — explicitly carries the SWIR retrieval noise floor as a function of column XCH₄.

• Full GLM — generalised linear model with multiple scene covariates.

TMTPP likelihood — canonical form¶

For a set of detected events with per-event posteriors $\{p(Q \mid \text{observation}_i)\}_{i=1}^{n}$ and detection times $\{t_i\}$ over a window $[0, T]$:

The first sum scores each detected event under: (a) the temporal intensity at the detection time, and (b) the integrated mark contribution that combines the per-event likelihood with the population mark distribution and the satellite POD. The second integral is the expected number of events that would have been detected under the model — subtracts the right amount so the posterior is consistent.

Practical evaluation¶

The mark integral $\int P_d(Q)\, L_i(Q)\, f(Q)\, \mathrm{d}Q$ is computed via the importance-weighted Monte Carlo estimator from 06a § Mark likelihood:

with samples $Q_i^{(s)} \sim p(Q \mid \text{observation}_i)$ and $\pi_\text{per-event}(Q)$ the per-event prior used at Tier I–IV. The $1 / \pi_\text{per-event}$ factor is the importance weight; without it the population fit double-counts the per-event prior.

Point-regime simplification¶

When per-event posteriors are tightly concentrated ($\operatorname{CV} < 20\%$) and $f$ is smooth on that scale, the importance-weighted MC reduces to the Point regime of 06a:

This is the form implemented in plumax.population.fitting.pod_powerlaw_model (to be rebuilt on xtremax.point_processes.ThinningProcess over a MarkedTemporalPointProcess under #106). It’s the simplification, not the canonical form — explicit regime selection per 06a § Regime selection rule decides when it’s safe to use.

Numerical stability of the integrated thinned-rate term¶

The integral $\int P_d(Q)\, f(Q)\, \mathrm{d}Q$ over heavy-tailed $f$ (power-law tail, Pareto) and saturating $P_d$ (Hill) must not be evaluated by naive quadrature in linear $Q$ — the heavy tail underflows.

Where it plugs into plumax¶

TMTPP inputs.

TMTPP outputs.

Population vs. per-source — the v1 commitment¶

The TMTPP fits aggregate the population. Two distinct framings:

• Across-population (v1 default for inventory accounting): fit one $(\lambda, f, P_d)$ over all sources of a class within a basin/region. $Q$ means “size of an event drawn from this class”.

• Per-source longitudinal (v1 for persistency forecasting on a known facility): fit one $(\lambda_\text{facility}(t), f_\text{facility}(Q), P_d)$ per facility, with hierarchical shrinkage to the population. $Q$ means “size of an event from this specific facility”.

Both have library support; the choice is driven by the scientific question, not by the methodology. Inventory totals (06d) use across-population; dispatch decisions for a known leak history (06c) use per-source.

Module layout¶

Tier V.B module layout — concern, target module, status.

Validation strategy¶

• Likelihood gradient. jax.grad matches finite differences within tolerance. Cheap unit test.

• Synthetic recovery — Point regime. Already in 06_stationary_numpyro_mcmc for power-law mark.

• Synthetic recovery — Full regime with importance correction. Generate per-event posteriors with a known $\pi_\text{per-event}$, fit population, recover $(\lambda^{*}, f^{*}, P_d^{*})$ within reported posterior. Mirrors the importance-correction round trip from 06a § Validation.

• SBC — point. Across 1000 simulated populations, per-parameter rank statistics uniform.

• SBC — soft observation. Same SBC but with the soft-observation layer (per-event posteriors as input). Validates the cross-tier inference end-to-end.

• Identifiability stress test. Generate data where $\lambda$ is high but $P_d$ is low (vs. the opposite). Quantitative target: posterior correlation $\operatorname{corr}(\lambda_0, Q_{50}) > 0.5$ when confounded; $< 0.2$ when well-separated. Confirms the model knows what it can’t disentangle.

• Log-space integration test. Compare log-space Gauss–Hermite to Pareto importance sampling for $\int \mathrm{Hill}(Q) \cdot \mathrm{LN\text{-}Pareto}(Q)\, \mathrm{d}Q$ across $\alpha \in [1.5, 4]$. Linear quadrature should fail loudly past $\alpha > 2$. Catches the silent thinned-rate underflow bug.

• Hierarchical POD coverage. With known controlled-release calibration injected as $(Q_{50,\text{pub}}, \sigma_{Q_{50},\text{pub}})$, the hierarchical POD prior should produce $Q_{50}$ posteriors that contain the true value at 95% CI ~95% of the time across simulated basins.

Open questions¶