Copula-based Markov zero-inflated count time series models with application

ABSTRACT

Count time series data with excess zeros are observed in several applied disciplines. When these zero-inflated counts are sequentially recorded, they might result in serial dependence. Ignoring the zero-inflation and the serial dependence might produce inaccurate results. In this paper, Markov zero-inflated count time series models based on a joint distribution on consecutive observations are proposed. The joint distribution function of the consecutive observations is constructed through copula functions. First- and second-order Markov chains are considered with the univariate margins of zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), or zero-inflated Conway–Maxwell–Poisson (ZICMP) distributions. Under the Markov models, bivariate copula functions such as the bivariate Gaussian, Frank, and Gumbel are chosen to construct a bivariate distribution of two consecutive observations. Moreover, the trivariate Gaussian and max-infinitely divisible copula functions are considered to build the joint distribution of three consecutive observations. Likelihood-based inference is performed and asymptotic properties are studied. To evaluate the estimation method and the asymptotic results, simulated examples are studied. The proposed class of models are applied to sandstorm counts example. The results suggest that the proposed models have some advantages over some of the models in the literature for modeling zero-inflated count time series data.

1. Introduction

Zero-inflated counts time series can be found in several fields such as environmental sciences, public health, and economics. For example, in monthly counts of sandstorms in some areas, in rare diseases with low infection rates, and in crimes such as arson. In these cases, the observed counts may include a considerable frequency of zeros. However, during certain seasons, these counts could result in larger values. Additionally, these zero-inflated counts are usually autocorrelated when the data is collected over time. Overlooking the frequent occurrence of zeros and the serial correlation could lead to false inference. In many real-life time series examples, the series are not stationary and observe some sort of trend and seasonal features. In Figure 1, an example of such time series is shown. It displays the monthly counts of strong sandstorms recorded by the AQI airport station in Eastern Province, Saudi Arabia. One can see the frequent occurrence of zeros in the distribution of the counts, serial dependence, decreasing trend, and seasonality in the series. Standard time series models fail to account for such problems. Motivated by these problems, we propose and develop a class of copula-based Markov time series models for zero-inflated counts with the presence of covariates.

Figure 1.

Time series plot of monthly count of sandstorms, bar-plot of distribution of sandstorm counts, autocorrelation function, and the circular plot.

Zero-inflated regression models for independent counts were first proposed by Lambert [16] via a generalized linear model (GLM), assuming the counts follow zero-inflated Poisson (ZIP) distribution. Later on, other distributional assumptions were proposed, such as the zero-inflated negative binomial distribution (ZINB) in [23] and the zero-inflated Conway–Maxwell–Poisson (ZICMP) distribution in [24]. Recently, modeling serially dependent zero-inflated counts has gained more attention. Authors in [7] proposed a regression ZIP model to analyze air pollution-related emergency room visit counts. Under the integer-valued generalized autoregressive conditional heteroscedatic (INGARCH) class of models, Zhu [30] and Gonçalves et al. [4] proposed zero-inflated INGARCH models with different distributional assumptions. Yang et al. [28] proposed state-space models to handle a time series of zero-inflated counts with both ZIP and ZINB distributions. Assuming the sequence of zero-inflated counts followed a Markov process, Wang [27] introduced Poisson regression models to fit daily numbers of phone calls. Yang et al. [29] also fit Markov models to public health surveillance for syphilis using a partial likelihood approach.

In this paper, copula-based models are proposed. Copulas are multivariate distributions with uniform margins on the unit interval. There is ample research on how they have been used to capture the correlation structures of continuous time series data (for example, see [6,9,13,21]). Due to computational complexity, there is not as much literature for count time series data as there is for continuous time series data. Joe [14] reviewed Markov models with copula-based transition probabilities for time series of counts. Masarotto and Varin [18] introduced marginal regression models for count time series data with the serial dependence being captured by a Gaussian copula. Jia et al. [11] and Lennon and Yuan [17] applied the same models but suggested different estimation methods. Alqawba et al. [2] extended the models proposed by Masarotto and Varin [18] to include a class of models that accommodates for zero-inflation in the time series of counts.

Here, we extend the work done by Joe [14] on constructing the count time series models through copula-based joint distributions of consecutive observations by including a class of models that accounts for zero-inflation in the count time series. An important advantage of copula-based Markov models is that they help avoid some strict distributional assumptions on the marginals, such as the infinite divisibility condition [14]. The latter condition is not necessarily satisfied when we assume the counts follow a zero-inflated distribution. In addition, the copula-based models extend nicely to non-stationary processes through time-varying parameters in the univariate margins of the ZIP, ZINB, and ZICMP distributions proposed in this paper.

This paper is organized as follows. In Section 2, we provide background on the theories needed to construct the proposed class of Markov models. We briefly describe the zero-inflated regression models following the ZIP, ZINB, and ZICMP distributions, respectively. We then define copula functions. In Section 3, we present the proposed copula-based Markov models and discuss in detail the first- and second-order Markov models. In Section 4, we describe the statistical inference method applied via likelihood inference and provide asymptotic results of the maximum likelihood estimates. Section 5 presents simulation studies. Then, the proposed models are used to analyze the Sandstorm counts in Section 6. We end the paper with a summary in Section 7.

2. Background

A time series is a sequence of random variables, $\left\{Y_t\right\}$, taken over an equally spaced discrete time, which is $t=1,\dots ,n$ for some fixed n. The observed values, $\left\{y_t\right\}$, are usually referred to as the realization of the stochastic process $\left\{Y_t\right\}$. An important feature of a time series is that consecutive observations are serially dependent. Hence, time series analysis is concerned with accounting for the serial dependence. A complete description of a time series process is the joint multivariate distribution of the observed values, which is

$$F(y_1,\dots ,y_n)=\mathrm{P}\mathrm{r}(Y_1\le y_1,\dots ,Y_n\le y_n).$$ (1)

However, constructing multivariate distributions is quite challenging, especially when the time series is discrete and observes an unusual behavior, such as zero inflation. As mentioned in the introduction, the main objective of this paper is to build a multivariate distribution as given in (1) to describe the zero-inflated count time series through a class of copula-based Markov models. Next, we discuss appropriate margins of these zero-inflated counts and their multivariate distributions via copula theories.

2.1. Zero-inflated count regression models

In this section, we will revisit the zero inflated regression models under different distributional assumptions. First, we will examine the ZIP regression model with independent counts [16]. Then we will consider other distributional assumptions, such as the ZINB distribution [23] and ZICMP [24], which can accommodate for over-dispersion and under-dispersion when choosing the latter.

Suppose $Y_t$ denotes a random count at time t with the probability mass function (pmf) and the cumulative distribution function (cdf) given as $f_t$ and $F_t$, respectively.

1. ZIP: ${\omega }_t$ zero-inflation parameter, ${\lambda }_t$ intensity parameter, and

   $$f_t\left(y_t\right)={\omega }_t{I}_{\{y_t=0\}}+(1-{\omega }_t)\frac{e^{-{\lambda }_t}{\lambda }_t^{y_t}}{y_t!},$$

   where $I_{\{y_t=0\}}$ is the indicator function, ${\omega }_t\in [0,1]$ and ${\lambda }_t>0$. If ${\omega }_t\to 0$, the baseline Poisson distribution is obtained.
2. ZINB: ${\omega }_t$ zero-inflation parameter, ${\lambda }_t$ intensity parameter, ${\kappa }_t$ dispersion parameter, and

   $$f_t\left(y_t\right)={\omega }_t{I}_{\{y_t=0\}}+(1-{\omega }_t)\frac{\mathrm{\Gamma }({\kappa }_t+y_t)}{\mathrm{\Gamma }\left({\kappa }_t\right)y_t!}{\left(\frac{{\kappa }_t}{{\kappa }_t+{\lambda }_t}\right)}^{{\kappa }_t}{\left(\frac{{\lambda }_t}{{\kappa }_t+{\lambda }_t}\right)}^{y_t},$$

   where $I_{\{y_t=0\}}$ is the indicator function, ${\omega }_t\in [0,1]$, ${\lambda }_t>0$ and ${\kappa }_t\ge 0$. If ${\omega }_t\to 0$, the baseline NB distribution is obtained.
3. ZICMP: ${\omega }_t$ zero-inflation parameter, ${\lambda }_t$ intensity parameter, ${\kappa }_t$ dispersion parameter, and

   $$f_{Y_t}\left(y_t\right)={\omega }_t{I}_{\{y_t=0\}}+(1-{\omega }_t)\frac{{\lambda }_t^{y_t}}{(y_t!{)}^{{\kappa }_t}Z({\lambda }_t,{\kappa }_t)},$$

   where $I_{\{y_t=0\}}$ is the indicator function, ${\omega }_t\in [0,1]$, ${\lambda }_t>0$ and ${\kappa }_t\ge 0$. If ${\omega }_t\to 0$, the baseline CMP distribution is obtained, and if ${\kappa }_t=1$, the ZIP distribution is obtained.

Following GLM [19], we will simultaneously fit two GLMs for the ZIP parameters and three GLM's for the ZINB and ZICMP parameters, which is: (1) the intensity parameter ${\lambda }_t$, with the logarithmic link function; (2) the zero-inflation parameter ${\omega }_t$ for $t=1,\dots ,n$ with the logit link function; and finally, in the case of ZINB and ZICMP distribution, (3) the dispersion parameter ${\kappa }_t$, with the logarithmic link function just like the intensity parameter. That is,

$$\begin{array}{rl}& \log \left({\lambda }_t\right)={\mathrm{\text{β}}}^{\prime }{\mathbf{x}}_t,\end{array}$$ (2)

$$\begin{array}{rl}& \mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\omega }_t\right)=\log \left(\frac{{\omega }_t}{1-{\omega }_t}\right)={\mathit{\gamma }}^{\prime }{\mathbf{z}}_t,\end{array}$$ (3)

and

$$\mathrm{l}\mathrm{o}\mathrm{g}\left({\kappa }_t\right)={\mathit{\alpha }}^{\prime }{\mathbf{w}}_t,$$ (4)

where ${\mathbf{x}}_t=(x_{1t},\dots ,x_{kt}{)}^{\prime }$, ${\mathbf{z}}_t=(z_{1t},\dots ,z_{lt}{)}^{\prime }$, and ${\mathbf{w}}_t=(w_{1t},\dots ,w_{mt}{)}^{\prime }$ are the associated covariates that affect the intensity parameter ${\lambda }_t$, the zero-inflation parameter ${\omega }_t$, and the dispersion parameter ${\kappa }_t$, respectively. In addition, $\mathrm{\text{β}}=({\mathrm{\text{β}}}_1,\dots ,{\mathrm{\text{β}}}_k{)}^{\prime }$, $\mathit{\gamma }=({\gamma }_1,\dots ,{\gamma }_l{)}^{\prime }$, and $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_m{)}^{\prime }$ are the regression coefficients for the log-linear model given in (2), the logit model given in (3), and the log-linear model given in (4), respectively.

2.2. Copula

The multivariate normal distribution is a natural extension of the univariate normal distribution to higher dimensions, and it plays a central role in statistical theory. However, there is not one but several multivariate extensions of univariate discrete distributions to higher dimensions. And copulas facilitate such extensions and constructions of multivariate distributions with given continuous or discrete marginals, which model various types of dependence. An extensive and detailed discussion of copulas is contained in [13]. A d dimensional copula $C(a_1,a_2,\dots ,a_d):[0,1{]}^d\to [0,1]$ is simply a multivariate cdf with all d univariate marginals uniform on the unit interval. A multivariate cdf for $\mathbf{Y}=(Y_1,Y_2,\dots ,Y_d)$ is given as

$$F(y_1,y_2,\dots ,y_d)=C\left(F_1\right(y_1),F_2(y_2),\dots ,F_d(y_d\left)\right).$$

If all the margins are integer valued as given in [13] on page 27, the multivariate probability mass function can be obtained as

$$\begin{array}{rl}f(y_1,y_2,\dots ,y_d)& =P(Y_1=y_1,Y_2=y_2,\dots ,Y_d=y_d)\\ & =\sum _{j_1=1}^2\sum _{j_2=1}^2\dots \sum _{j_d=1}^2(-1{)}^{j_1+j_2+\cdots +j_d}C(u_{1j_1},u_{2j_2}\dots ,u_{d{j}_d}),\end{array}$$

where $u_{t1}=F_t\left(y_t\right)$ and $u_{t2}=F_t\left(y_t^{-}\right)$. The term $F_t\left(y_t^{-}\right)$ is the left-hand limit of $F_t$ at $y_t$, which is equal to $F_t(y_t-1)$ when $Y_t$ is integer valued random variable. A comprehensive list of copulas can be found in [13].

3. Copula-based Markov chain models

A general form of p order Markov models with copula-based transition probabilities as defined in [13] is given as

$$Y_t=g({ϵ}_t;Y_{t-1},\dots ,Y_{t-p}),$$

where $\left\{{ϵ}_t\right\}$ is an i.i.d stochastic continuous latent process and $g(.)$ is assumed to be an increasing function in ${ϵ}_t$ for $t=1,\dots ,n$. Thus, the observed value $Y_t$ depends on the past through only $Y_{t-1},\dots ,Y_{t-p}$. If the process $\left\{Y_t\right\}$ is continuous, then there exists a simple stochastic representation for the Markov model. However, for discrete process, as in the case here, there is no simple stochastic representation for the model [14]. Next, we discuss the first-order Markov models and their immediate extensions to second-order Markov models.

3.1. First- order Markov models

Suppose $\left\{Y_t\right\}$ is a zero-inflated count time series following a first-order Markov chain with one of the distributions introduced in Section 2. Then, taking advantage of the chain rule of probability and the Markov property, the multivariate joint density distribution of $Y_1,\dots ,Y_n$ is given as

$$\mathrm{P}\mathrm{r}(Y_1=y_1,\dots ,Y_n=y_n)=\mathrm{P}\mathrm{r}(Y_1=y_1)\prod _{t=2}^n\mathrm{P}\mathrm{r}(Y_t=y_t|Y_{t-1}=y_{t-1}).$$ (5)

The transition probability, i.e. conditional probability in the right hand of (5), depends on the joint density function of $Y_t,Y_{t-1}$ and can be found using the copula functions as shown in Section 2. That is, let

$$F_{12}(y_t,y_{t-1})=C\left(F_t\right(y_t|{\mathbf{X}}_t;\mathit{\theta }),F_{t-1}(y_{t-1}|{\mathbf{X}}_{t-1};\mathit{\theta });\mathit{\delta }),$$ (6)

where $C(.;\mathit{\delta })$ is a bivariate copula function with parameter vector $\mathit{\delta }$. The covariates ${\mathbf{X}}_t=({\mathbf{x}}_t,{\mathbf{z}}_t,{\mathbf{w}}_t)$, for $t=1,\dots ,n$, are the covariates corresponding to the intensity (mean) parameter ${\lambda }_t$, the zero-inflation parameter ${\omega }_t$ and the dispersion parameter ${\kappa }_t$, if existed, respectively. Notice that in some cases, these parameters, or part of them, may be constant across time when the covariates are not significant and dropped from the model. The parameter vector $\mathit{\theta }=({\mathrm{\text{β}}}^{\prime },{\mathit{\gamma }}^{\prime },{\mathit{\alpha }}^{\prime }{)}^{\prime }$ is the unknown marginal regression coefficient. Hence, the transition probability is given as

$$\mathrm{P}\mathrm{r}(Y_t=y_t|Y_{t-1}=y_{t-1})=\frac{\mathrm{P}\mathrm{r}(Y_t=y_t,Y_{t-1}=y_{t-1})}{f_{t-1}(y_{t-1}|{\mathbf{X}}_{t-1};\mathit{\theta })},$$ (7)

where

$$\mathrm{P}\mathrm{r}(Y_t=y_t,Y_{t-1}=y_{t-1})=F_{12}(y_t,y_{t-1})-F_{12}(y_t^{-},y_{t-1})-F_{12}(y_t,y_{t-1}^{-})+F_{12}(y_t^{-},y_{t-1}^{-}),$$

and $y_t^{-}=y_t-1$ since $Y_t$ is a discrete random variable for all t.

Several choices of the bivariate copula function $C(.;\mathit{\delta })$ can be selected depending on the degree and the sign of the dependence and the tail behavior of the copula. Table 1 shows examples of copula choices considered in this paper. For example, if there is symmetry, the Gaussian copula is recommended. However, Gumbel or reflected Gumbel copulas perform better than Gaussian copula with the existence of tail dependence [14]. Moreover, the Frank and the Gaussian copulas are both reflection symmetric and allow for negative dependence (see [13] for more details).

Table 1. Bivariate Copula functions.

Copula	Copula function
Gaussian	$C(u1,u2;\delta )={\mathrm{\Phi }}_{\delta }\left({\mathrm{\Phi }}^{-1}\right(u_1),{\mathrm{\Phi }}^{-1}(u_2\left)\right),\delta \in [-1,1]$
Frank	$C(u_1,u_2;\delta )=-\frac{1}{\delta }\log \left[1+\frac{(e^{-\delta u_1}-1)(e^{-\delta u_2}-1)}{e^{-\delta -1}}\right],\delta \in \mathbb{R}\left\{0\right\}$
Gumbel	$C(u_1,u_2;\delta )=\exp [-((-\log \left(u_1\right){)}^{\delta }+(-\log \left(u_2\right){)}^{\delta }{)}^{1/\delta }],\delta \ge 1$
reflected Gumbel	$C(u_1,u_2;\delta )=u_1+u_2-1+\exp [-((-\log \left(u_1\right){)}^{\delta }+(-\log \left(u_2\right){)}^{\delta }{)}^{1/\delta }],\delta \ge 1$
Plackett	$C(u_1,u_2;\delta )=\frac{[1+(\delta -1\left)\right(u_1+u_2\left)\right]-\sqrt{[1+(\delta -1\left)\right(u_1+u_2){]}^2-4u_1{u}_2\delta (\delta -1)}}{2(\delta -1)},\delta \ge 0$

Next, we will discuss the immediate extension of the first-order Markov models. That is, the second-order Markov models where the zero-inflated count $Y_t$ depended on the past two counts.

3.2. Second order Markov models

Suppose $\left\{Y_t\right\}$ is a zero-inflated count time series of second-order Markov chains following one of the distributions introduced in Section 2. Now, the multivariate joint distribution of $Y_1,\dots ,Y_n$ is given by

$$\begin{array}{rl}\mathrm{P}\mathrm{r}(Y_1=y_1,\dots ,Y_n=y_n)& =\mathrm{P}\mathrm{r}(Y_1=y_1,Y_2=y_2)\\ & \times \prod _{t=3}^n\mathrm{P}\mathrm{r}(Y_t=y_t|Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2}),\end{array}$$ (8)

where the conditional probability of $Y_2$ given $Y_1=y_1$ can be evaluated using (7). However, for the second-order transition probabilities of $Y_t$ given $Y_{t-1}=y_{t-1}\mathrm{a}\mathrm{n}\mathrm{d}{Y}_{t-2}=y_{t-2}$, we need to fit an appropriate trivariate copula function for the joint distribution of $Y_t,Y_{t-1}$ and $Y_{t-2}$ for $t=3,\dots ,n$.

The most popular choice is the trivariate Gaussian copula. Using the joint multivariate distribution with discrete margin given in (5) and the Gaussian copula function, the transition probabilities of $Y_t$ given $Y_{t-1}=y_{t-1}\mathrm{a}\mathrm{n}\mathrm{d}{Y}_{t-2}=y_{t-2}$ for $t=3,\dots ,n$ is given by

$$\mathrm{P}\mathrm{r}(Y_t=y_t|Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2})=\frac{\mathrm{P}\mathrm{r}(Y_t=y_t,Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2})}{\mathrm{P}\mathrm{r}(Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2})},$$

where the joint distribution is given by

$$\begin{array}{rl}& \mathrm{P}(Y_t=y_t,Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2})\\ & =\sum _{j_1=0}^1\sum _{j_2=0}^1\sum _{j_3=0}^1(-1{)}^{j_1+j_2+j_3}{F}_{123}(y_t-j_1,y_{t-1}-j_2,y_{t-2}-j_3),\end{array}$$ (9)

where the function $F_{123}(.)$ is given by

$$\begin{array}{rl}& F_{123}(y_t,y_{t-1},y_{t-2})\\ & ={\mathrm{\Phi }}_{R\left(\mathit{\delta }\right)}\left(F_t\right(y_t|{\mathbf{X}}_t;\mathit{\theta }),F_{t-1}(y_{t-1}|{\mathbf{X}}_{t-1};\mathit{\theta }),F_{t-2}(y_{t-2}|{\mathbf{X}}_{t-2};\mathit{\theta }\left)\right),\end{array}$$

for $t=3,\dots ,n$ where $R\left(\mathit{\delta }\right)$ is a $3\times 3$ correlation matrix, with $\mathit{\delta }=({\delta }_1,{\delta }_2{)}^{\prime }$ as a vector of the dependence structure parameters. The bivariate copula margins $F_{12}$ and $F_{23}$ of the trivariate copula function $F_{123}$ are assumed to be the same and, where case, given by the bivariate Gaussian copula.

Another way of evaluating the trivariate joint density function, when the Gaussian copula function is chosen, can be via integrating over rectangle probability [20]. That is

$$\begin{array}{rl}& \mathrm{P}\mathrm{r}(Y_t=y_t,Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2})\\ & ={\int }_{{\mathcal{D}}_t(y_t;\mathit{\theta })}{\int }_{{\mathcal{D}}_{t-1}(y_{t-1};\mathit{\theta })}{\int }_{{\mathcal{D}}_{t-2}(y_{t-2};\mathit{\theta })}{\phi }_{R\left(\mathit{\delta }\right)}(z_{t-2},z_{t-1},z_t)\mathrm{d}{z}_{t-2}\mathrm{d}{z}_{t-1}\mathrm{d}{z}_t,\end{array}$$ (10)

where

$${\mathcal{D}}_t(y_t;\mathit{\theta })=\left[{\mathrm{\Phi }}^{-1}\right\{F_t(y_t^{-}|{\mathbf{X}}_t;\mathit{\theta })\},{\mathrm{\Phi }}^{-1}\{F_t(y_t|{\mathbf{X}}_t;\mathit{\theta })\left\}\right].$$ (11)

Although the Gaussian copula function in (10) has no closed-form, there are several accurate deterministic approximations of the function when the dimension is low, such as the case here with the trivariate Gaussian or the bivariate Gaussian.

Another way of calculating the trivariate joint distribution, if the closed copula function is desired, can be found by employing the Laplace transform (LT) of a non-negative random variable through a max-infinite divisible (max-id) copula as introduced in [15]. They stated that when the copula functions pair copula function were chosen as $C_{12}=C_{23}=H$, where H is a permutation symmetric max-id bivariate copula function, and $C_{13}$ is the independent copula function, then the model would be appropriate for generating a second-order Markov chain.

Hence, the function $F_{123}(.)$ in (9) becomes the following trivariate max-id copula

$$\begin{array}{rl}& F_{123}(y_t,y_{t-1},y_{t-2})\\ & =\psi \left(\sum _{j\in \{t,t-2\}}\left[-\log H(e^{-0.5{\psi }^{-1}(F_j;{\delta }_1)},e^{-0.5{\psi }^{-1}(F_{t-1};{\delta }_1)};{\delta }_2)\right.\right.\\ & \left.\left.+\frac{1}{2}{\psi }^{-1}(F_j;{\delta }_1)\right];{\delta }_1\right),\end{array}$$ (12)

where $F_j=F_j(y_j|{\mathbf{X}}_j;\mathit{\theta })$ for j = t, t−2 and $t=3,\dots ,n$. The function $\psi (.;{\delta }_1)$ is the LT with ${\delta }_1$ describing the minimal dependence, and $H(.;{\delta }_2)$ is a permutation symmetric max-id bivariate copula function with ${\delta }_2$ describing the stronger pairwise dependence. The bivariate margins of (12) are given by

$$\begin{array}{rl}& F_{i2}(y_j,y_{t-1})\\ & =\psi \left(-\log H(e^{-0.5{\psi }^{-1}(F_j;{\delta }_1)},e^{-0.5{\psi }^{-1}(F_{t-1};{\delta }_1)};{\delta }_2)\right.\\ & \left.+\frac{1}{2}{\psi }^{-1}(F_j;{\delta }_1)+\frac{1}{2}{\psi }^{-1}(F_{t-1};{\delta }_1);{\delta }_1\right),\end{array}$$ (13)

for i = 1, 3, $j=t,t-2$, and

$$F_{13}(y_t,y_{t-2})=\psi \left({\psi }^{-1}\right(F_t;{\delta }_1)+{\psi }^{-1}(F_{t-2};{\delta }_1);{\delta }_1).$$

The use of the above trivariate max-id copula is suggested when there is stronger dependence for measurements at nearer time points [14]. He also stated that in the case of large value clustering (such as when the time series observes seasonality) a good choice for $H(.;{\delta }_2)$ is the bivariate Gumbel copula and $\psi (.;{\delta }_1)$ is the positive stable Laplace transform, which results in the function $F_{123}(.)$ in (12) to be a trivariate extreme value copula. The Gaussian copula and the max-id copula can be extended to fit Markov models of order greater than 2.

4. Statistical inference

4.1. Log-likelihood functions

The inference or estimation method performed for the Markov models' parameter vector $\mathit{\vartheta }=({\mathit{\theta }}^{\prime },{\mathit{\delta }}^{\prime }{)}^{\prime }$ presented in this chapter is the maximum likelihood estimation (MLE) method. As stated previously, likelihood inference method is easily applied when the chosen copula family has simple form. In addition, likelihood inference gives us the advantage of performing hypothesis testing through the likelihood ratio statistics and model selection using the log-likelihood function. Next, we give a detailed description of the likelihood functions of the two models presented earlier.

For the first-order Markov models, the likelihood function is given by (5), and the log-likelihood is given as

$$\begin{array}{rl}l(\mathit{\vartheta };\mathbf{y})=\log f_1(y_1|{\mathbf{X}}_1;\mathit{\theta })& +\sum _{t=2}^n\left[\log \left\{\sum _{j_1=0}^1\sum _{j_2=0}^1{F}_{12}(y_t-j_1,y_{t-1}-j_2)\right\}\right.\\ & \left.\phantom{\sum _{t=2}^n}-\log f_{t-1}(y_{t-1}|{\mathbf{X}}_{t-1};\mathit{\theta })\right],\end{array}$$ (14)

where $F_{12}(y_t-,y_{t-1})$ is given by (6), and $\mathit{\theta }$ and $\mathit{\delta }$ are the parameter vectors of the marginals and the dependence structure, respectively. The log-likelihood function in (14) has closed-form if the copula family chosen to define $C(.;\mathit{\delta })$ has a closed-form.

For second-order Markov models with Gaussian copula function, the log-likelihood function is given as

$$\begin{array}{rl}l(\mathit{\vartheta };\mathbf{y})& =\log \left\{{\int }_{{\mathcal{D}}_1(y_1;\mathit{\theta })}{\int }_{{\mathcal{D}}_2(y_2;\mathit{\theta })}{\phi }_{R\left(\mathit{\delta }\right)}(z_2,z_1)\mathrm{d}{z}_2\mathrm{d}{z}_1\right\}\\ & +\sum _{t=3}^n\left[\log \left\{{\int }_{{\mathcal{D}}_t(y_t;\mathit{\theta })}{\int }_{{\mathcal{D}}_{t-1}(y_{t-1};\mathit{\theta })}{\int }_{{\mathcal{D}}_{t-2}(y_{t-2};\mathit{\theta })}{\phi }_{R\left(\mathit{\delta }\right)}(z_{t-2},z_{t-1},z_t)\mathrm{d}{z}_{t-2}\mathrm{d}{z}_{t-1}\mathrm{d}{z}_t\right\}\right.\\ & \left.-\log \left\{{\int }_{{\mathcal{D}}_{t-1}(y_{t-1};\mathit{\theta })}{\int }_{{\mathcal{D}}_{t-2}(y_{t-2};\mathit{\theta })}{\phi }_{R\left(\mathit{\delta }\right)}(z_{t-2},z_{t-1})\mathrm{d}{z}_{t-2}\mathrm{d}{z}_{t-1}\right\}\right],\end{array}$$ (15)

where ${\mathcal{D}}_t(y_t;\mathit{\theta })$ is given by (11). The expression in (15) is not in a closed form, and approximations are needed for the rectangle probabilities. However, for the trivariate max-id copula, the log-likelihood function can take a closed form and is given by

$$\begin{array}{rl}l(\mathit{\vartheta };\mathbf{y})& =\log \left\{\sum _{j_1=0}^1\sum _{j_2=0}^1{F}_{12}(y_1-j_1,y_2-j_2)\right\}\\ & +\sum _{t=3}^n\left[\log \left\{\sum _{j_1=0}^1\sum _{j_2=0}^1\sum _{j_3=0}^1(-1{)}^{j_1+j_2+j_3}{F}_{123}(y_t-j_1,y_{t-1}-j_2,y_{t-2}-j_3)\right\}\right.\\ & \left.-\log \left\{\sum _{j_1=0}^1\sum _{j_2=0}^1{F}_{12}(y_{t-1}-j_1,y_{t-2}-j_2)\right\}\right],\end{array}$$ (16)

where $F_{12}(y_{t-1},y_{t-2})$ and $F_{123}(y_t,y_{t-1},y_{t-2})$ are given by (13) and (12), respectively.

Hence, the maximum likelihood estimates of $\mathit{\vartheta }=({\mathit{\theta }}^{\prime },{\mathit{\delta }}^{\prime }{)}^{\prime }$ can be obtained by:

$$\hat{\mathit{\vartheta }}=\underset{\mathit{\vartheta }}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}l(\mathit{\vartheta };\mathbf{y}).$$

This optimization will produce a Hessian Matrix that yields the observed Fisher information matrix. To get the standard errors of the ML estimates of $\mathit{\vartheta }$, one can take the inverse of the Fisher information matrix. In the next section, asymptotic results are derived to prove that the inverse of the Fisher information matrix of the above log-likelihood functions evaluated at the MLE of $\mathit{\vartheta }$ can be used as an estimated covariance of matrix of $\hat{\mathit{\vartheta }}$.

4.2. Asymptotic properties

To draw some inference on Markov processes, Billingsley [3] gave important results that basically state that, under certain regularity conditions, the asymptotic likelihood theory and numerical maximum likelihood from the i.i.d case can be extended to hold with dependent data following a Markov process. First, we will consider the first-order Markov model with the corresponding log-likelihood given in (14). As listed in [12] on page 318, the regularity conditions needed in order to use the results from [3] for the first-order Markov process are verified. Assume that we have $\left\{Y_t\right\}$ for $t=1,\dots ,n$ a first-order Markov chain with state space $\mathcal{S}$, and $\mathrm{P}\mathrm{r}(Y_t=y_t|Y_{t-1}=y_{t-1};\mathit{\vartheta })$ is a family of transition densities with respect to a counting measure and with column vector parameter $\mathit{\vartheta }$ of dimension r in the parameter space $\mathbf{\Theta }$. In addition, for asymptotic analysis, rewrite the log-likelihood function given in (14) as

$$l_n\left(\mathit{\vartheta }\right)=\sum _{t=2}^{n}p(\mathit{\vartheta };y_{t-1},y_t),$$ (17)

where $p(\mathit{\vartheta };y_{t-1},y_t)=\log \mathrm{P}\mathrm{r}(Y_t=y_t|Y_{t-1}=y_{t-1};\mathit{\vartheta }),$ for $t=2,\dots ,n$. Note that the first probability, $\mathrm{P}\mathrm{r}(Y_1=y_1;\mathit{\theta })$, is omitted from the function since the first observation, $y_1$, is asymptotically insignificant.

Now, given the regularity conditions in [12], the following asymptotic results from the i.i.d case hold for our Markov processes. In particular, there exists a root ${\hat{\mathit{\vartheta }}}_n$ of $\mathrm{\partial }/\mathrm{\partial }\mathit{\vartheta }{l}_n\left(\mathit{\vartheta }\right)=\mathbf{0}$ such that

1. The ML estimator ${\hat{\mathit{\vartheta }}}_n$ of $\mathit{\vartheta }=({\mathit{\theta }}^{\prime },{\mathit{\delta }}^{\prime }{)}^{\prime }$ converges in probability to the true value of $\mathit{\vartheta }$, say ${\mathit{\vartheta }}^0$. That is, ${\hat{\mathit{\vartheta }}}_n$ is a consistent estimator of $\mathit{\vartheta }$.

2. The ML estimator ${\hat{\mathit{\vartheta }}}_n$ is asymptotically normal. That is, $n^{1/2}({\hat{\mathit{\vartheta }}}_n-{\mathit{\vartheta }}^0){\displaystyle \stackrel{d}{⟶}{N}_n(\mathbf{0},{\mathrm{\Sigma }}^{-1}({\mathit{\vartheta }}^0\left)\right),}$ where $\mathrm{\Sigma }\left({\mathit{\vartheta }}^0\right)=\mathrm{V}\mathrm{a}\mathrm{r}\left({\mathit{\vartheta }}^0\right).$

3. The log-likelihood ratios for hypotheses involving nested models for the parameter $\mathit{\vartheta }$ have asymptotic chi-square distribution. That is, $2\left[\underset{\mathit{\vartheta }}{\mathrm{m}\mathrm{a}\mathrm{x}}{l}_n\right(\mathit{\vartheta })-l({\mathit{\vartheta }}^0\left)\right]{\displaystyle {⟶}^d{\chi }_r^2.}$

Hence, as in the i.i.d case, the numerical maximization of (17) yields the observed Fisher information matrix, which can be used as an estimated covariance matrix of $\hat{\mathit{\vartheta }}$. That is, for large n, $n^{-1}{\mathrm{\Sigma }}^{-1}\left(\hat{\mathit{\vartheta }}\right)\approx I_n^{-1}\left(\hat{\mathit{\vartheta }}\right),$ where

$$I_n^{-1}\left(\hat{\mathit{\vartheta }}\right)=-{\left[\frac{{\mathrm{\partial }}^2{l}_n\left(\mathit{\vartheta }\right)}{\mathrm{\partial }\mathit{\vartheta }\mathrm{\partial }{\mathit{\vartheta }}^{\prime }}{|}_{\hat{\mathit{\vartheta }}}\right]}^{-1}=-{\left[\frac{{\mathrm{\partial }}^2\sum _{t=2}^{n}p(\mathit{\vartheta };y_{t-1},y_t)}{\mathrm{\partial }\mathit{\vartheta }\mathrm{\partial }{\mathit{\vartheta }}^{\prime }}{|}_{\hat{\mathit{\vartheta }}}\right]}^{-1}.$$

Joe [12] argued that the theory in [3] still applies for higher-order Markov processes, assuming the order is known. He also stated that extension of the asymptotic theory to a case where the transition probabilities depend on covariates should be possible.

5. Simulated examples

To evaluate the performance of the proposed method and confirm the asymptotic results, a comprehensive simulation study was conducted. We carried out the simulation in the statistical software R [22]. Out of the several processes to choose from, we simulated first-order stationary Markov processes with joint distribution of consecutive observations following the bivariate Gaussian and Frank copulas. The marginal distributions were chosen to be the ZIP, ZINB, and ZICMP distributions. Since we assumed the process is stationary, we set the marginal distributions' parameters, $\mathit{\theta }$, to be constant across time. For Gaussian copula, the marginal parameters were chosen as (1) ZIP with $\mathit{\theta }=(\lambda =3,\omega =0.3{)}^{\prime }$, (2) ZINB with $\mathit{\theta }=(\lambda =3,\omega =0.3,\kappa =5{)}^{\prime }$, and (3) ZICMP with $\mathit{\theta }=(\lambda =3,\omega =0.2,\kappa =0.5{)}^{\prime }$. The dependence parameter for the bivariate Gaussian copula was chosen as $\delta =0.5$ across all three models.

We generated 500 simulated datasets for each of the above models with the sample sizes, n = 100, 200 and 500. The evaluation criterion was chosen to be the mean absolute deviation error (MADE), which is given by:

$$\frac{1}{m}\sum _{i=1}^m|{\hat{\vartheta }}_i-\vartheta |,$$

where m is the number of replications, i.e. m = 500.

The parameter estimates were obtained after constructing the log-likelihood function given in (14) for the ZIP, ZINB, and ZICMP distributions. A summary of the simulation results are shown in Table 2, which represents the count time series ZIP, ZINB, and ZICMP models with joint distribution of consecutive observations following the bivariate Gaussian copula. The results indicate that the proposed estimation method produces reasonable estimates and relatively small MADEs. In addition, as the sample size increases, the parameter estimates appeared to converge to the true parameter values. To assess the approximate normality of the estimates, Q–Q plots of the ML estimates for the 500 ZIP, ZINB, and ZICMP replicates of length n = 500 are shown in Figure 2. These plots agree with the asymptotic results given in Section 4.2.

Table 2. Mean of estimates, MADEs (within parentheses) for Markov zero-inflated models with Gaussian copula.

Model	n	λ	ω	κ	δ
ZIP	100	2.9992(0.2773)	0.2949(0.0591)		0.4840(0.0748)
	200	3.0056(0.1698)	0.2961(0.0402)		0.4903(0.0542)
	500	2.9980(0.1051)	0.2977(0.0249)		0.4933(0.0342)
ZINB	100	3.0078(0.3225)	0.2961(0.0661)	4.9338(1.6244)	0.4809(0.0801)
	200	3.0109(0.2252)	0.2968(0.0581)	5.2743(1.4477)	0.4860(0.0581)
	500	2.9990(0.1430)	0.2980(0.0290)	4.9858(1.0384)	0.4913(0.0362)
ZICMP	100	3.4689(0.6813)	0.2008(0.0488)	0.5516(0.0860)	0.4771(0.0747)
	200	3.3383(0.4619)	0.2016(0.0332)	0.5404(0.0611)	0.4847(0.0540)
	500	3.2545(0.2996)	0.2031(0.0210)	0.5326(0.0404)	0.4885(0.0336)

Figure 2.

Q–Q plots of the ML estimates for the 500 ZIP, ZINB and ZICMP processes of length n = 500.

Next, we simulate a stationary Markov process with dependence structure following the Frank copula. The marginal parameters were chosen as (1) ZIP with $\mathit{\theta }=(\lambda =3,\omega =0.3{)}^{\prime }$, (2) ZINB with $\mathit{\theta }=(\lambda =4.1,\omega =0.25,\kappa =5{)}^{\prime }$, and (3) ZICMP with $\mathit{\theta }=(\lambda =4.1,\omega =0.25,\kappa =0.5{)}^{\prime }$. The dependence parameter for the bivariate Frank copula is chosen to be $\delta =3.2$ across all three models. A summary of the simulation results are shown in Table 3, which represents the count time series ZIP, ZINB, and ZICMP models with joint distribution of consecutive observations following the bivariate Frank copula. Similar to the Gaussian copula, the proposed method yields robust performance.

Table 3. Mean of estimates, MADEs (within parentheses) for Markov zero-inflated models with Frank copula.

Model	n	λ	ω	κ	δ
ZIP	100	2.9897(0.2513)	0.2935(0.0556)		3.1585(0.5723)
	200	3.0272(0.1916)	0.2902(0.0388)		3.1809(0.5316)
	500	3.0409(0.1051)	0.2965(0.0249)		3.2226(0.3563)
ZINB	100	4.1530(1.1330)	0.2555(0.1434)	0.6780(0.2990)	3.0515(0.7628)
	200	4.0739(0.8759)	0.2471(0.1177)	0.5779(0.2390)	3.1333(0.6089)
	500	4.1200(0.5459)	0.2528(0.0768)	0.5307(0.1348)	3.1394(0.3645)
ZICMP	100	4.3895(0.5621)	0.2493(0.0509)	0.5167(0.0719)	3.2445(0.5745)
	200	4.1753(0.4619)	0.2500(0.0355)	0.5082(0.0490)	3.2312(0.4222)
	500	4.1159(0.3890)	0.2486(0.0203)	0.5052(0.0339)	3.2431(0.2785)

6. Application to sandstorm counts

The data set used in this example consists of the monthly count of strong sandstorms recorded by the AQI airport station in Eastern Province, Saudi Arabia. These data are a subset of the data studied in [1]. The station happens to be located in one of the major dust producing regions in the world [10]. Sandstorms are a weather event that result from strong wind releasing dust from the ground and transfering it long distances [5]. Sandstorms can cause many environmental and human-related hazards. For example, sandstorms impact the air quality, disturb daily activities, and disrupt transportation. Hence, studying and accurately analyzing the behavior of these phenomena is important to successfully forecast such events.

The monthly counts studied here are characterized as strong sandstorms by the AQI airport station. Tao et al. [26] stated that a strong sandstorm reduces the level of visibility to less than 500 m and with average wind speeds of 17.2 to 24.4 m/s. The counts of these events contain zero inflation. Several works have been applied on handling rare events such as strong sandstorms (e.g. see [25] and [8]). Here we apply the proposed Markov zero-inflated count time series models with copula-based transition probabilities.

The data set consists of 348 monthly counts of strong sandstorms, starting from January 1978 to December 2013. The main objective was to apply the proposed models and investigate if there were any significant seasonal and trend components. Additionally, we investigated if there were any other predictors that affected the frequency of sandstorms, such as the monthly counts of dust haze events, maximum wind speed, temperature, and relative humidity.

Figure 1 shows the sandstorms series plot, the autocorrelation function, bar-plot of the distribution of sandstorm counts, and circular plot of the monthly mean count of sandstorms. From the time series plot and the bar-plot, we could see that the distribution of the sandstorm counts had more zeros relative to a Poisson distribution with the same empirical mean. These zeros represented about 59% of the sample. Decreasing trend could also be observed from the time series plot. Additionally, seasonality was also seen from the autocorrelation function and circular plot. In fact, from the circular plot, we concluded that most sandstorms occurred during spring time, i.e. March, April, and May. Thus, trend and seasonal covariates were added to the models.

Hence, we fitted several models with different marginal distributions and dependence structures. The marginal distributions were chosen to be the ZIP, ZINB, and ZICMP distributions with the log-linear function of the intensity parameter given as

$$\log \left({\lambda }_t\right)={\mathrm{\text{β}}}_0+{\mathrm{\text{β}}}_1(t\times {10}^{-3})+{\mathrm{\text{β}}}_2{x}_{1t}+{\mathrm{\text{β}}}_3{x}_{2t}+{\mathrm{\text{β}}}_4{x}_{3t},$$

and the logit function for the zero-inflation parameter given as

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\omega }_t\right)={\gamma }_0+{\gamma }_1{z}_{1t}+{\gamma }_2{z}_{2t}+{\gamma }_3{z}_{3t},$$

for $t=1,\dots ,n$, where $x_{1t}=z_{1t}=\cos \left(2\pi t/12\right)$, $x_{2t}=z_{2t}=\sin \left(2\pi t/12\right)$, and $x_{3t}=z_{3t}$ are the monthly count of dust haze events. The log-function of the dispersion parameter (if it existed) was given by $\log \left(\kappa \right)=\alpha$, i.e. it was chosen to be constant across time. We considered both first-order and second-order dependence structures. In the first-order Markov models, we fitted the bivariate Gaussian, Frank, Gumbel, reflected Gumbel, and Plackett copula functions for the joint distribution of two consecutive observations. For the second-order Markov models, we fitted the trivariate Gaussian and max-id copula functions for the joint distribution of three consecutive observations. In the trivariate max-id copula function, we chose the Laplace transform function, $\psi (.)$, to be either the positive staple Laplace transform (PSLT) or the log series Laplace transform (LSLT) with $H(.;\mathit{\delta })$ chosen to be either the bivariate Frank or Gumbel copulas. Out of these models, we selected two models, each with different marginals, based on the model selection criteria Akaike information criterion ( $\mathrm{A}\mathrm{I}\mathrm{C}$), Bayesian information criterion (BIC), and root mean square prediction errors (RMSPE).

Table 4 shows comparisons of the ZIP, ZINB, and ZICMP Markov models with the different dependence structures. The increase of the dependence order improves the models. The second-order Markov models outperformed the first-order Markov models in terms of the $\mathrm{A}\mathrm{I}\mathrm{C}$, BIC, and RMSPE values. However, the second-order parameters of the trivariate Gaussian copula and the Laplace transform parameters, in the trivariate max-id copula, are not always significant and can be dropped if necessary. Additionally, having the same dependence structure, the Markov models with ZINB margins seem to fit the sandstorm data better than the models with ZIP and ZICMP margins. Within the ZIP and ZINB margins, the bivariate reflected Gumbel and Frank copula function are chosen to model the dependence structures. The trivariate Gaussian copula and the trivariate max-id copula with PSLT and bivariate Gumbel copula function for the ZICMP margin. Hence, we want to show how each dependence structure can be interpreted in term of the autocorrelation. The ordinary Poisson and NB Markov models are also shown in the table as a comparison with the best models from the ZIP and ZINB Markov models. The results show that the zero-inflated models outperform the ordinary models.

Table 4. Comparisons of the ZIP, ZINB, and ZICMP models with different dependence structures.

Marginal	Copula	Order	$\mathrm{A}\mathrm{I}\mathrm{C}$a	BICb	RMSPEc
ZIP	Gaussian	1	435.55	945.76	1.59
	Frank	1	434.11	942.88	1.58
	Gumbel	1	438.32	951.32	1.60
	ref.Gumbel	1	433.87	942.41	1.57
	Plackett	1	435.21	945.09	1.58
	Gaussian	2	433.85	950.22	1.56
	PSLT/Frank	2	431.25	948.87	1.56
	LSLT/Frank	2	433.77	953.90	1.55
	PSLT/Gumbel	2	418.46	923.28	1.62
ZINB	Gaussian	1	425.86	938.08	1.59
	Frank	1	424.54	935.45	1.59
	Gumbel	1	428.20	942.77	1.60
	ref.Gumbel	1	424.15	934.67	1.58
	Plackett	1	425.5	937.38	1.59
	Gaussian	2	424.06	942.34	1.56
	PSLT/Frank	2	417.00	928.23	1.57
	LSLT/Frank	2	423.67	941.56	1.57
	PSLT/Gumbel	2	413.05	920.32	1.63
ZICMP	Gaussian	1	477.10	1040.56	1.91
	Frank	1	475.06	1036.49	1.88
	Gumbel	1	473.49	1033.3	1.79
	ref.Gumbel	1	473.73	1033.84	1.87
	Plackett	1	469.06	1024.48	1.77
	Gaussian	2	449.97	994.16	1.72
	PSLT/Frank	2	457.44	1009.11	1.80
	LSLT/Frank	2	450.91	996.04	1.77
	PSLT/Gumbel	2	455.12	1004.47	1.83
Poisson	Frank	1	495.90	1038.9	1.76
	ref.Gumbel	1	488.16	1023.43	1.75
NB	Frank	1	446.28	947.52	2.32
	ref.Gumbel	1	445.69	946.35	2.29

^a AIC $=l(\hat{\mathit{\vartheta }};\mathbf{y})-r$,

^b BIC $=-2l(\hat{\mathit{\vartheta }};\mathbf{y})+r\log n$.

^c $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{P}\mathrm{E}=\left\{1/n-p\sum _{t=1+p}^n\right[y_t-\hat{\mathrm{E}}(Y_t|Y_{t-1}=y_{t-1},\dots ,Y_{t-p}=y_{t-p};\hat{\mathit{\vartheta }}){]}^2{\}}^{1/2}.$

Table 5 shows that the zero-inflated Markov models are capable of accounting for first-order dependence. However, only the models with ZICMP margins account for the second-order dependence. The autocorrelation coefficients are similar when the dependence structure is the same for the models with ZIP and ZINB margins. For the marginal parameters, $\mathit{\theta }$, the estimates are quite similar between the ZIP and ZINB and slightly different from the ZICMP. All models suggest significant decreasing trend in the number of strong sandstorms since ${\mathrm{\text{β}}}_1<0$. Seasonality is also significant at annual frequencies since ${\mathrm{\text{β}}}_2,{\mathrm{\text{β}}}_3,{\gamma }_1$ and ${\gamma }_2$ are significantly different from zero. Finally, the affect of dust haze is significant since both ${\mathrm{\text{β}}}_4$ and ${\gamma }_3$ are significantly different from zero. To compare between the Markov models in term of the dependence, we consider the Kendall's tau. The Kendall's tau, when the chosen copula function is the reflected Gumbel, is given by ${\tau }_K=1-{\delta }_1^{-1},$ so for the ZIP margin, it equals to ${\tau }_K=0.1908$, which is similar to the one corresponding to the ZINB margin, ${\tau }_K=0.2156$. When the copula function is the Frank, the Kendall's tau is then given as ${\tau }_K=1-41-D_1\left({\delta }_1\right)/{\delta }_1,$ where $D_1(.)$ is the Debye function $D_1\left(x\right)=1/x{\int }_0^{x}t/e^t-1\mathrm{d}t.$ Thus, for the ZIP margin it equals to ${\tau }_K=0.1664$, which is similar to the one corresponding to the ZINB margin, ${\tau }_K=0.1870$. In both cases, the ZINB distribution provides slightly stronger dependence than the ZIP distribution.

Table 5. Parameter estimates (standard errors) for the copula-based Markov models fit to the sandstorms count series.

	ZIP		ZINB		ZICMP
Parameter	ref.Gumbel	Frank	ref.Gumbel	Frank	Gaussian	PSLT/Gumbel
${\mathrm{\text{β}}}_0$	0.9578(0.1190)	0.9705(0.1172)	0.9960(0.1564)	0.9391(0.1593)	0.8139(0.0577)	0.7701(0.1938)
${\mathrm{\text{β}}}_1$	−4.2579(0.6098)	−4.0715(0.6049)	−4.9031(0.8052)	−4.7335(0.8168)	−2.3656(0.0302)	−2.4943(0.6999)
${\mathrm{\text{β}}}_2$	−0.2184(0.0890)	−0.1789(0.0879)	−0.1956(0.1233)	−0.1836(0.1253)	−0.0921(0.0107)	−0.1248(0.0860)
${\mathrm{\text{β}}}_3$	0.3722(0.0957)	0.3650(0.0950)	0.4371(0.1255)	0.4379(0.1283)	0.2203(0.0554)	0.2466(0.0827)
${\mathrm{\text{β}}}_4$	0.0656(0.0088)	0.0638(0.0089)	0.0635(0.0121)	0.0636(0.0124)	0.0452(0.0024)	0.0426(0.0112)
${\gamma }_0$	0.6627(0.2717)	0.7264(0.2677)	0.5357(0.3077)	0.5926(0.3086)	1.1825(0.0481)	1.1292(0.2221)
${\gamma }_1$	0.6236(0.2561)	0.6615(0.2514)	0.6706(0.2991)	0.6607(0.3006)	0.6252(0.0717)	0.6266(0.2014)
${\gamma }_2$	−0.9051(0.2507)	−0.8798(0.2427)	−0.8819(0.2814)	−0.7799(0.2799)	−0.9827(0.0456)	−0.9824(0.2070)
${\gamma }_3$	−0.1407(0.0444)	−0.1498(0.0448)	−0.1596(0.0538)	−0.1664(0.0545)	−0.1565(0.0181)	−0.1467(0.0335)
α			1.4876(0.5709)	1.4303(0.5431)	0.8083(0.0186)	0.7540(0.1400)
${\delta }_1$	1.2358(0.0765)	1.5326(0.4672)	1.2748(0.0896)	1.7328(0.5034)	0.2779(0.0444)	1.0242(0.0885)
${\delta }_2$					0.1818(0.0238)	1.1922(0.1057)
${\tau }_K$	0.1908	0.1664	0.2156	0.1870	0.1793	0.1612

Figure 3 displays the predicted values, which were the conditional expectations of $Y_t$ given $Y_{t-1}$ for the ZIP and ZINB Markov models and $Y_t$ given $Y_{t-1}$ and $Y_{t-2}$ for ZICMP Markov models for $t=1,\dots ,n$. The ZIP and ZINB models perform better than the ZICMP, especially with the first hundred observations where non-zero counts are more frequent. Within each margin, the reflected Gumbel and Frank copulas are very similar with the ZIP and ZINB margins. However, with the ZICMP margin, the Gaussian copula is better than the max-id copula.

Figure 3.

Predicted values using the conditional expectations of the Markov models fit to the sandstorm count series. Dots represent the observed counts.

Finally, we compare our proposed class of zero-inflated Markov models to some zero-inflated time series of counts in the literature. We first fit the zero-inflated integer-valued autoregressive (ZIINAR) models with ZIP and ZINB distributed counts presented in [29] to the Sandstorm counts. Also, we fit the class of zero-inflated nonlinear state-space (ZINLSS) time series counts given in [2] with ZIP, ZINB, and ZICMP distributions. Table 6 shows the parameter estimates and their standard errors for the five models along sides the AIC of each models. The estimates of the marginal parameters are quite similar to our Markov models. However, the serial dependence parameters coming from ZIINAR indicate stronger serial dependence. The standard errors of our Markov models' parameters are smaller than both methods in Table 6. The AIC values are also similar across the classes of models, but the Markov ones provide slightly smaller values. An important advantage of the models proposed in this paper is the straight forward estimation method which utilizes the MLE. In contrast, Yang et al. [29] maximized the partial likelihood function to estimate the parameters of the ZIINAR class. This could lead to loss of some information. In addition, the estimation method in [2] uses sequential importance sampling that requires intense computations. Also, the proposed class of Markov models provides a variety dependence structures that we can choose from.

Table 6. Parameter estimates (standard errors) for the ZIINAR [29] and ZINLSS [2] models fit to the sandstorms count series.

	ZIINAR		ZINLSS
Parameter	ZIP	ZINB	ZIP	ZINB	ZICMP
${\mathrm{\text{β}}}_0$	0.7314(0.1424)	0.6898(0.1813)	0.9977(0.1175)	0.9709(0.1570)	0.7978(0.1888)
${\mathrm{\text{β}}}_1$	$-4.1779\left(0.5435\right)$	$-4.6293\left(0.6839\right)$	$-4.1493\left(0.6065\right)$	$-4.7477\left(0.7976\right)$	$-2.4523\left(0.5772\right)$
${\mathrm{\text{β}}}_2$	$-0.0647\left(0.0964\right)$	$-0.0369\left(0.1321\right)$	$-0.2004\left(0.0885\right)$	$-0.1813\left(0.1243\right)$	$-0.1089\left(0.0723\right)$
${\mathrm{\text{β}}}_3$	0.3722(0.0957)	0.4146(0.1236)	0.3461(0.0938)	0.4231(0.1239)	0.2093(0.0786)
${\mathrm{\text{β}}}_4$	0.3295(0.0936)	0.0676(0.0119)	0.0627(0.0088)	0.0645(0.0123)	0.0435(0.0094)
${\gamma }_0$	0.6685(0.2647)	0.5157(0.3114)	0.7647(0.2622)	0.5656(0.3047)	0.6629(0.2119)
${\gamma }_1$	0.5891(0.2356)	0.6894(0.2923)	0.6163(0.2460)	0.6648(0.2925)	$-1.0047\left(0.2132\right)$
${\gamma }_2$	$-0.8578\left(0.2338\right)$	$-0.7940\left(0.2740\right)$	$-0.8931\left(0.2401\right)$	$-0.8363\left(0.2736\right)$	$-0.1496\left(0.0344\right)$
${\gamma }_3$	$-0.1542\left(0.0439\right)$	$-0.1754\left(0.0566\right)$	$-0.1489\left(0.0424\right)$	$-0.1659\left(0.0524\right)$	$-0.2466\left(0.1613\right)$
α		1.6104		0.6400(0.2437)	1.1733(0.2230)
δ	0.4110(0.1344)	0.4182(0.1669)	0.2580(0.0623)	0.2503(0.0724)	0.2870(0.0780)
AIC	435.44	427.06	435.45	425.81	430.8

7. Summary

Many scientific investigations or natural phenomena result in data that consist of counts measured sequentially at different time points. And it is common to see a large number of zeros in the count time series data. Applying ordinary Poisson and NB distributions to these time series of counts might not be appropriate due to the frequent occurrence of zeros. In this paper, we have extended the work done by Joe [14] to include a class of models that accounts for zero inflation. We used the marginal ZIP, ZINB, and ZICMP distributions to build Markov zero-inflated count time series models. The serial dependence was captured through constructing bivariate and trivariate joint distributions of the consecutive observations. The joint distribution function of the consecutive observations is constructed through copula functions such as the Gaussian, Frank, Gumbel copula functions. Simulation studies were conducted to evaluate the maximum likelihood estimation method. The studies showed that the estimated parameters are consistent and normally distributed. The proposed Markov models are applied on the sandstorm counts. The models prove to be reliable on handling different zero-inflated count time series data. Future direction is to consider pair copula construction introduced in [20] to fit second-order Markov models.

Disclosure statement

No potential conflict of interest was reported by the author(s).