Let be the the number of people who arrive at the bus stop i between time t and time . We assume that the count process follows an inhomogeneous Poisson process with rate parameter . We then model the rate parameter on the logarithmic scale as a linear model. That is,
If we only include these terms, the model can be seen as a first order Fourier expansion of the function . This simplistic model neglects key terms that we need to include in the model that address certain problems discussed in the Currently Working On section. First, there is a dependency among observations along the same route. We can include this via a AR(1) process term, where the number of people who arrive at the bus stop i between t and depends on the number of people who arrive at the bus stop i between t and . We can include these dependencies by updating the above model to include the log rate parameter at stop i-1 on the right hand side, . The coefficient will depend highly on spatial relations between these points. Therefore, we can express this term as:
Moreover, we'd like to include fixed terms with respect to day of the week , month , and any other temporal trends that we may find.
Zero Inflated Poisson Regression and Mixture Models
Estimation - An Introduction to MCMC, Gibbs Sampling, and Preferential Sampling