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# The model

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.

Consider a hidden process (the non-observable actual number of counts in the phenomenon under study) with Po-INAR(1) structure:

(1)

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

The under-reported phenomenon is modeled by assuming that the observed counts are

(2)

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 mean and the variance of a stationary INAR(1) process with Poisson() innovations are .
Its auto-covariance and auto-correlation functions are and respectively.
Hence,

(3)

The auto-covariance function of the observed process is

Hence, the auto-correlation function of is a multiple of :

Parameter estimation

The marginal probability distribution of is a \textbf{mixture of two Poisson distributions}

(4)

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

(5)

with .

Then, the likelihood function is

and are the so-called emission and transition probabilities.

Transition probabilities are computed as

(6)

While emission probabilities are given by

(7)

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:

Predictions

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

(8)

The easiest way to estimate is by substituting by in (3), to get
, and then in 8 to get

(9)

Bibliography

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