Poisson processes
PoissonProcess
PoissonProcess represents a homogeneous Poisson process with constant rate \(\lambda>0\).
Constructor
PoissonProcess(rate)
Properties
| Property | Meaning |
|---|---|
rate |
Constant intensity \(\lambda\). |
lambda_ |
Python-safe alias for rate. |
Methods
count_probability(n, t)— compute \(\mathbb P(N(t)=n)\).increment_probability(n, s, t)— compute the increment law.interarrival_samples(n, rng=None)— sample exponential inter-arrival times.arrival_times(n, rng=None)— sample cumulative arrival times.conditional_first_arrival_cdf(y, s)— conditional first-arrival distribution.conditional_arrival_times(k, s)— ordered arrival times conditioned on the count.superpose(other)— superpose two homogeneous Poisson processes.split(probability)— split the process by Bernoulli thinning.simulate(t_max, rng=None)— generate event times up to a horizon.
Example
from stochx.stochastic import PoissonProcess
process = PoissonProcess(rate=3.0)
print(process.count_probability(2, 1.0))
print(process.arrival_times(5))
NonHomogeneousPoissonProcess
NonHomogeneousPoissonProcess represents a Poisson process with intensity \(\lambda(t)\) and mean function
\[m(t)=\int_0^t\lambda(x)\,dx.\]
Constructor
NonHomogeneousPoissonProcess(intensity_function, mean_function=None)
Properties
| Property | Meaning |
|---|---|
intensity_function |
Callable intensity \(\lambda(t)\). |
mean_function |
Callable mean function \(m(t)\) when supplied or constructed. |
Methods
mean(t)— compute \(m(t)\).count_probability(n, t)— count probability using the mean function.increment_probability(n, s, t)— increment law on \([s,t]\).simulate(t_max, rng=None)— simulate event times under the intensity.