NDARMA(1,1) formulation

As a choice model

An observed time series

$X_t$ with marginal distribution

$X_t\sim \mathrm{\Pi }$, in which

$\mathrm{\Pi }$ can be anything, say,

$Categorical(\lambda )$ or

$Poisson(\lambda )$, NDARMA(1,1) is

$$X_t=\{\begin{array}{ll}{X}_{t-1}& \text{with probability }{\alpha }_1\\ U_t& \text{with probability }{\beta }_0\\ U_{t-1}& \text{with probability }{\beta }_1\end{array},$$

in which

$U_t$ is a latent process with

$U_t\sim \mathrm{\Pi }$ and

${\alpha }_1+{\beta }_0+{\beta }_1=1$.

With ARMA(1,1)-type formulation

For a latent innovation process

$U_t$ with

$U_t\sim \mathrm{\Pi }$, an observed time series

$X_t$ with marginal distribution

$X_t\sim \mathrm{\Pi }$ is an NDARMA(1,1) process/dice with and is given by,

$$X_t=a_{1,t}\cdot X_{t-1}+b_{0,t}\cdot U_t+b_{1,t}\cdot U_{t-1}$$

in which

$U_t$ is a latent innovation process/dice

$U_t$ with

$U_t\sim \mathrm{\Pi }$ and

$D_t=[a_{1,t},b_{0,t},b_{1,t}]$ a latent decision variable (dice)

$D_t\sim Multinomial(1;ab)$ with decision probabilities

$ab=[{\alpha }_1,{\beta }_0,{\beta }_1]$

The Log-likelihood of this process (if I'm not mistaken) may be written as

$$\begin{array}{rl}\log \mathcal{L}& =\log Pr(X_t=x|\lambda ,{\alpha }_1,{\beta }_0,{\beta }_1;X_{t-1},\dots )\\ & =\sum _{t=1}^N[{\alpha }_{1}Pr(X_{t-1}=x|\dots )+{\beta }_{0}Pr(U_t=x|\dots )+{\beta }_{1}Pr(U_{t-1}=x|\dots )]\end{array}$$

Some Stan attempts

data {
  int<lower=1> N; // Number of observations
  int<lower=1> k; // Number of categories in X and U
  int<lower=1, upper=k> X_t[N]; // Observed data X_t
}

parameters {
  simplex[k] lambda; // Parameters of the multinomial distribution Pi
  simplex[3] ab; // Probabilities for dice (alpha1, beta0, beta1)
}

model {

  int U_t[N];
  array[3,N] int D_t; 

  // Priors
  lambda ~ dirichlet(rep_vector(2.0, k));
  ab ~ dirichlet(rep_vector(2.0, 3));

  for (t in 2:N) {
    vector[3] contributions;
    // array[3] real D_t;
    // D_t ~ multinomial(ab);

    contributions[1] = D_t[1, t] + X_t[t-1];
    contributions[2] = D_t[2, t] + U_t[t-1];
    contributions[3] = D_t[3, t] + U_t[t];

    target += log_sum_exp(contributions);
  }
}

---

NO

The conditional probability of the mixture should be

$$\begin{array}{rl}& Pr(X_t=x|\lambda ,{\alpha }_1,{\beta }_0,{\beta }_1;X_{t-1},\dots )=\\ & {\alpha }_{1}Pr(X_{t-1}=x|\dots )+{\beta }_{0}Pr(U_t=x|\dots )+{\beta }_{1}Pr(U_{t-1}=x|\dots )\end{array}$$

So the log-likelihood would be

$$\begin{array}{rl}\log \mathcal{L}& =\log Pr(X_t=x|\lambda ,{\alpha }_1,{\beta }_0,{\beta }_1;X_{t-1},\dots )\\ & =\sum _{t=1}^N[{\alpha }_{1}Pr(X_{t-1}=x|\dots )+{\beta }_0\underset{{\lambda }_x}{\underbrace{Pr(U_t=x|\dots )}}+{\beta }_1\underset{{\lambda }_x}{\underbrace{Pr(U_{t-1}=x|\dots )}}\\ & =\sum _{t=1}^N[{\alpha }_{1}Pr(X_{t-1}=x|\dots )+{\lambda }_x({\beta }_0+{\beta }_1)]\\ & =\sum _{t=1}^N[{\alpha }_{1}Pr(X_{t-1}=x|\dots )+{\lambda }_x(1-{\alpha }_1)]\end{array}$$

$P(X_t,X_{t-1};{ϵ}_t,{ϵ}_{t-1})=P(X_t,{ϵ}_t|X_{t-1},{ϵ}_{t-1})\ast P(X_{t-1},{ϵ}_{t-1})$

[ P(X_t = i_0, X_{t-1} = i_1, U_t = j_0, U_{t-1} = j_1) ]

[ = p_{j_0} \cdot \left( (\alpha_0 \delta_{i_0 j_0} + \alpha_1 \delta_{i_0 j_1} + \alpha_1 \delta_{i_0 i_1}) \cdot P(X_{t-1} = i_1, U_{t-1} = j_1) \right. ]

[

• \beta_2 \cdot P(X_{t-1} = i_1, U_{t-1} = j_1, X_{t-2} = i_0, U_{t-1} = i_0)
• \beta_2 \cdot P(X_{t-1} = i_1, X_{t-2} = i_0, U_{t-1} = j_1) \Bigg) ]

$p_{j_0}\cdot (({\beta }_0{\delta }_{i_0{j}_0}+{\beta }_1{\delta }_{i_0{j}_1}+{\alpha }_1{\delta }_{i_0{i}_1})\cdot p_{j_1}\cdot ((1-{\beta }_0)p_{i_1}+{\beta }_0{\delta }_{i_1{j}_1}))$

$p_{j_0}{p}_{j_1}\cdot ({\beta }_0{\delta }_{i_0{j}_0}+{\beta }_1{\delta }_{i_0{j}_1}+{\alpha }_1{\delta }_{i_0{i}_1})((1-{\beta }_0)p_{i_1}+{\beta }_0{\delta }_{i_1{j}_1})$