Implementing Poisson Process Using MATLAB

The Poisson process is a mathematical model describing random events occurring at fixed time or space intervals, widely used in communications, finance, biology, and other fields. Implementing Poisson process simulation and parameter estimation in MATLAB can be accomplished through the following steps.

### Poisson Process Simulation The key characteristics of Poisson process are event independence and constant average occurrence rate. Assuming the event rate (λ) is known, we can simulate event occurrence times by generating inter-arrival intervals following exponential distribution.

Generating inter-event intervals: In Poisson process, the time between consecutive events follows exponential distribution with parameter λ. In MATLAB, the `exprnd` function can generate these random time intervals using syntax: `intervals = exprnd(1/lambda, [1, nEvents])` where nEvents represents the number of events to simulate. Calculating event times: By cumulative summation of generated intervals using `cumsum` function, we obtain specific event occurrence times, forming the simulated path of Poisson process: `eventTimes = cumsum(intervals)`.

### Parameter Estimation In practical applications, λ may be unknown and requires estimation from observed data. MATLAB provides multiple statistical methods for parameter estimation:

Maximum Likelihood Estimation (MLE): For Poisson process, the MLE of λ equals the average number of observed events per unit time. Using `fitdist` function with 'Exponential' distribution type: `pd = fitdist(observedIntervals', 'Exponential')` where the estimated rate parameter can be extracted as `lambda_hat = 1/pd.mu`. Confidence interval calculation: Utilizing MATLAB's Statistical Toolbox, confidence intervals for λ can be computed using `paramci(pd)` to evaluate estimation reliability.

### Extended Applications Non-homogeneous Poisson process: When event rate varies with time, simulation can be achieved by modifying λ as a time-dependent function, implementing time-varying rate using function handles. Multi-event Poisson processes: In finance or biology fields, simulating multiple independent Poisson processes can be efficiently implemented through matrix operations in MATLAB, using multidimensional array processing for parallel simulation.

Through MATLAB's statistical computing and random number generation capabilities, Poisson process simulation and parameter estimation become efficient and intuitive, suitable for various random event analysis and prediction scenarios. The implementation leverages core functions like `exprnd`, `cumsum`, and `fitdist` with proper parameter handling for accurate modeling.