## Task 1: Mathematical model on how inhomogeneous Poisson becomes homogeneous Poisson using hierarchical Bayesian theory.

We can use hierarchical Bayesian theory to explain the effect of separating sources as decomposing observations whose rate lambda is first sampled from a Gamma distribution (three types: signal delay, dwell delay, overflow delay). Then, the delay rate of each type (lambda1, lambda2, lambda3) is used to generate a Poisson process.

Let's define the model:

1. Observational model for the data:
$$(Y_i \mid \lambda_i) \sim \text{Poisson}(\lambda_i), \quad i=1, \ldots, k$$
where $Y_i$ represents the observed delay for the $i$-th type (signal delay, dwell delay, or overflow delay), and $\lambda_i$ is the corresponding rate parameter.

2. Structural model for the parameters of the likelihood:
$$(\lambda_i \mid \alpha, \beta) \sim \text{Gamma}(\alpha, \beta), \quad i=1, \ldots, k$$
where $\alpha$ and $\beta$ are the shape and rate parameters of the Gamma distribution, respectively.

3. Hyperparameter model for the parameters of the structural model:
$$\alpha \sim \text{Uniform}(0, 0.5)$$
$$\beta \sim \text{Gamma}(0.1, 1)$$

By separating the sources of delay into signal delay, dwell delay, and overflow delay, we are essentially decomposing the inhomogeneous Poisson process into multiple homogeneous Poisson processes, each with its own rate parameter $\lambda_i$. The hierarchical Bayesian model allows us to account for the uncertainty in the rate parameters by assigning them a Gamma prior distribution, which is a conjugate prior for the Poisson likelihood. The hyperparameters $\alpha$ and $\beta$ are given their own prior distributions to complete the hierarchical structure.

This hierarchical Bayesian approach helps to explain how separating the sources of delay can lead to a more homogeneous Poisson process for each source, as the rate parameters are modeled as being drawn from a common Gamma distribution. The posterior distribution of the rate parameters can be inferred using Bayesian inference techniques, such as Markov Chain Monte Carlo (MCMC) methods.

## Task 2: Five representative references that assume a Poisson process (exponential interarrival time).

Five papers that assume a Poisson process (exponential interarrival time) and are relevant to transportation:

1. "[Impact of Variability of Interarrival and Service Times](https://link.springer.com/chapter/10.1007/978-3-031-36763-2_4.)"¹. This paper analyzes the influence of the distributions of some input parameters of the models on the Response times. It applies different distributions to the Interarrival times of requests keeping the distribution of Service times fixed.

2. "[15.1: Introduction - Statistics LibreTexts](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/15%3A_Renewal_Processes/15.01%3A_Introduction)"². This literature discusses the sequence of interarrival times X = (X1, X2, …) as an independent, identically distributed sequence of random variables. It assumes that X takes values in [0, ∞) and P(X > 0) > 0, so that the interarrival times are nonnegative, but not identically 0.

3. "[Interarrival Time - an overview | ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/interarrival-time)"³. From the viewpoint of mathematical tractability, the negative-exponential interarrival or service-time distribution is the simplest, and fortunately these distributions serve as a good fit in many practical problems.

Two papers that resolve interarrival exponential time through hierarchical Bayesian:

4. "[A hierarchical Bayesian entry time realignment method to study the long-term natural history of diseases](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8941125/)"⁴⁵. This paper developed BETR (Bayesian entry time realignment), a hierarchical Bayesian method for investigating the long-term natural history of diseases using data from patients followed over short durations.

5. "[Spatiotemporal Prediction Using Hierarchical Bayesian Modeling](https://2020.iccspa.org/wp-content/uploads/2021/11/1570631439.pdf)"⁶. This paper leverages the hierarchy of Bayesian models using the Gaussian process to predict long-term traffic status in urban settings.

Source: Conversation with Bing, 4/4/2024