Temporal and Stochastic Processes

mixle.process collects temporal and stochastic-process families that are otherwise easy to miss in the broader distribution tree. These models are for event-time data, self-excitation, renewal structure, birth-death trajectories, and random partition processes.

The public namespace re-exports:

• HawkesProcessDistribution;

• PowerLawHawkesDistribution;

• MultivariateHawkesProcessDistribution;

• InhomogeneousPoissonProcessDistribution;

• RenewalProcessDistribution;

• BirthDeathSamplingDistribution;

• ContinuousTimeMarkovChainDistribution;

• ChineseRestaurantProcessDistribution.

When to Use a Process Model

Use a process model when the observation is not just a row, but an event history or trajectory. The timing is part of the likelihood.

Keep the observation window with the data. Event-history likelihoods depend on start time, stop time, censoring, and exposure; dropping that context can make the same timestamp sequence mean something different.

Process	Use when	Key idea
Hawkes	Events trigger more events	Intensity rises after recent arrivals and decays over time.
Power-law Hawkes	Excitation has long memory	Triggering decays with a heavy-tailed kernel.
Multivariate Hawkes	Event types excite each other	A matrix controls cross-type excitation.
Inhomogeneous Poisson	Rate changes over time but events do not self-excite	Intensity is time-varying and exogenous.
Renewal	Waiting times are iid or family-modeled	Interarrival distribution drives the process.
Birth-death sampling	Counts evolve through arrivals and removals	Trajectory likelihood depends on birth and death rates.
Continuous-time Markov chain	State trajectories are fully observed with dwell times	Transition rates are estimated from jump counts and state exposure.
Chinese restaurant process	Partitions grow sequentially	New clusters appear with concentration-controlled probability.

Point Processes

Hawkes and Poisson process models are appropriate for timestamped event sequences. Depending on the fitted family, the model may expose intensity, expected count, sampling, and sequence log-density behavior.

Use Hawkes processes for:

• incident cascades;

• user activity bursts;

• aftershock-like phenomena;

• alert streams where one event increases near-future risk.

Use inhomogeneous Poisson processes for:

• seasonality or time-of-day rates;

• scheduled operational load;

• background arrivals without self-excitation.

Use multivariate Hawkes processes when event types influence each other, such as support categories, market event classes, operational alerts, or social interaction types.

Compare self-exciting models against a non-self-exciting baseline before interpreting triggering. Apparent excitation can come from seasonality, batching, missing exposure, or changing observation windows.

Renewal Processes

RenewalProcessDistribution models sequences through the distribution of interarrival times. It is useful when the waiting-time law is the main scientific question and there is no self-exciting feedback.

Renewal models compose naturally with scalar interarrival distributions. For example, a Gamma or log-normal interarrival family can be fit as part of a larger event model.

For renewal workflows, inspect residual waiting times and censoring behavior. A fitted interarrival distribution can look acceptable in aggregate while systematically missing early or late arrivals.

Birth-Death Processes

BirthDeathSamplingDistribution models trajectories where a population, queue, active set, or count evolves by births and deaths. Use it when both increments and decrements are meaningful, and the path itself is observed.

Examples include:

• queue size traces;

• active session counts;

• population dynamics;

• open/closed case counts.

Check whether zero, absorbing, or capacity states need explicit support. A birth-death fit should not hide impossible negative counts or unmodeled capacity limits in preprocessing.

Continuous-Time Markov Chains

ContinuousTimeMarkovChainDistribution models fully observed trajectories with an initial state and a sequence of (dwell_time, next_state) jumps. The generator matrix has off-diagonal rates q_ij and diagonal entries derived from total exit rates.

from mixle.inference import optimize
from mixle.stats.processes.ctmc import ContinuousTimeMarkovChainEstimator

trajectories = [
    (0, [(0.8, 1), (1.2, 0), (0.5, 2)]),
    (1, [(1.0, 0), (0.7, 2)]),
]

est = ContinuousTimeMarkovChainEstimator(num_states=3)
ctmc = optimize(trajectories, est, max_its=1, out=None)
print(ctmc.generator)

For fully observed trajectories, the MLE is closed form: q_ij = n_ij / T_i, where n_ij is the observed transition count and T_i is total dwell time in state i. mixle.inference.certify reports this family as GLOBAL_UNIQUE.

That certificate depends on the fully observed trajectory assumption. If jumps, dwell times, or state labels are censored or inferred upstream, record that source of uncertainty separately instead of treating the CTMC fit as a fully observed closed-form result.

Chinese Restaurant Processes

ChineseRestaurantProcessDistribution models a sequence of cluster assignments where new clusters can appear over time. It is useful as a prior or standalone distribution for partition-valued data.

For fitted finite approximations and variational mixtures, see the Dirichlet-process material in Model Families.

Composing Process Models

Process models are often only one field in a heterogeneous observation. A single application might combine:

• a Hawkes process over event times;

• a categorical distribution over event type;

• a positive continuous family over severity, amount, or duration;

• a calibrated rule that escalates low-confidence records for review.

That is the Mixle modeling shape: timing, labels, magnitudes, and decisions can remain separate components while sharing one scoring and inference story.

Diagnostics

For process models, inspect more than aggregate likelihood:

• compare observed and simulated event counts;

• check residual waiting times;

• inspect intensity around bursts;

• hold out contiguous time ranges;

• compare a self-exciting model against an inhomogeneous Poisson baseline;

• verify calibration of predicted counts or intervals;

• for CTMCs, compare simulated dwell times and transition counts against held-out trajectories.

Certification

Process families now participate in estimation certificates:

• inhomogeneous Poisson, birth-death, and CTMC fits are classified as GLOBAL_UNIQUE when their closed-form count/exposure MLE applies;

• Hawkes variants are classified as STATIONARY because branching EM or ML over self-excitation is non-convex;

• renewal-process certificates inherit the guarantee of the interarrival family used in the M-step.

Use Analysis Utilities for extreme-value and spatial diagnostics, Inference for proper scoring and model comparison, and Production Workflows for drift monitoring.

Release Evidence

For process models, preserve:

• observation-window definitions and exposure assumptions;

• timestamp units, time zones, and censoring policy;

• baseline comparisons such as renewal or inhomogeneous Poisson alternatives;

• residual, simulation, or count-calibration diagnostics;

• certificate assumptions for closed-form fits; and

• blocked or missing optional dependencies for accelerated process examples.

This evidence keeps temporal structure from being mistaken for ordinary row modeling.