We begin by describing the simplest version of our model. Later on, we will introduce several modifications, in order to deal with time dependent parameters.
where is a fixed parameter, Poisson(), i.i.d., independent of are the innovations, and is the binomial thinning operator: with i.i.d Bernoulli() random variables. Later on, we shall introduce time dependence, and hence we will be considering that is a funtion of .
The INAR(1) process is a homogeneous Markov chain with transition probabilities
The expectation and variance of the binomial thinning operator are
A simple under-reporting scheme
where and represent the frequency and intensity of the under-reporting process, respectively. We will eventually be interested in considering that is time dependent: . That is, for each , we observe with probability , and a -thinning of with probability , independently of the past .
Hence, what we observe (the reported counts) are
Properties of the model
The auto-covariance function of the observed process is
Hence, the auto-correlation function of is a multiple of :
When the distribution of the observed process is a zero-inflated Poisson distribution.
From the mixture we derive initial estimations for , , and , to be used in a maximum likelihood estimation procedure.
The likelihood function of is quite cumbersome to compute,
hence the forward algorithm (Lystig and Hughes (2002)), used in the context of HMC is a suitable option.
Consider the forward probabilities
Then, the likelihood function is
and are the so-called emission and transition probabilities.
Transition probabilities are computed as
While emission probabilities are given by
From this computations, a nonlinear optimization program computes the MLE estimates of the parameters.
Reconstructing the hidden chain
In order to reconstruct the hidden series , the Viterbi algorithm (Viterbi, 1967) is used.
The idea is to provide the latent chain that maximizes the likelyhood of the latent process given the observed series, assuming all the parameters are known.
Let be the likelihood function of the model, then
Since does not depend on , it is enough to maximise the probability .
The hidden series is reconstructed as:
Having observed , we are interested in predicting , for , and in evaluating the uncertainty of these predictions.
From equation 3, we have that , so that, if we have a good estimate for , then we can predict by means of its expectation .
From (1), assuming that the expectation of the innovations depends on , that is, the noise is Poisson() it is straightforward to see that
T.C. Lystig, J.P. Hughes (2002), Exact computation of the observed information matrix for hidden Markov models, Jr of Comp.and Graph. Stat.
Viterbi, A.J. (1967), Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Transactions on Information Theory, 13, 260-269