*Article* **A New Overdispersed Integer-Valued Moving Average Model with Dependent Counting Series**

**Kaizhi Yu and Huiqiao Wang \***

School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130, China<EMAIL_ADDRESS>
**\*** Correspondence<EMAIL_ADDRESS>
**Abstract:** A new integer-valued moving average model is introduced. The assumption of independent counting series in the model is relaxed to allow dependence between them, leading to the overdispersion in the model. Statistical properties were established for this new integer-valued moving average model with dependent counting series. The Yule–Walker method was applied to estimate the model parameters. The estimator's performance was evaluated using simulations, and the overdispersion test of the INMA(1) process was applied to examine the dependence between counting series.

**Keywords:** integer-valued moving average model; counting series; dispersion test

#### **1. Introduction**

Integer-valued time series can be encountered in numerous fields, such as epidemiology, insurance, and intraday stock transitions. The most widely used model is the integer-valued autoregressive (INAR) model, a recursive model first introduced by Alzaid and Al-Osh [1] and is similar to the traditional autoregressive (AR) model. Du and Li [2] generalized the model to the *p*-th order (which was called INAR(*p*) model) and proved the ergodic and Markov properties of the model. Similar to the continuous-valued moving average model, the *q*-th order integer-valued moving average model INMA(*q*) was introduced by Al-Osh and Alzaid [3], which is a slightly different form proposed by McKenzie [4].

Many researchers generalize the INAR model to deal with different real-life situations. Weiß [5] presented a new INAR(*p*) model showing possible marginal distributions of the DSD family. This model overcomes the difficulty of choosing the appropriate marginal distribution. Monteiro and Scotto [6] defined the periodic integer-valued autoregressive model, driven by a periodic sequence of independent Poisson-distributed random variables. Weiß [7] proposed the extended Poisson INAR(1) model, where the innovations are assumed to be serially dependent. Zhu [8] introduced a negative binomial INGARCH model to handle integer-valued time series with overdispersion and potential extreme observations. The study by Weiß [9] discussed threshold models for integer-valued time series with infinite range and briefly discussed new models for counting data time series with a finite range. Kang and Wang [10] generalized the mixture INAR(1) model based on mixing Pegram and binomial thinning operator. Li and Wang [11] proposed the first-order mixed integer-valued autoregressive process with zero-inflated generalized power series innovations, which contains the commonly used zero-inflated Poisson and geometric distributions. To handle the non-stationary integer-valued time series with a large dispersion, Kim and Park [12] introduced an integer-valued autoregressive process with a signed binomial thinning operator (INARS(*p*)).

Various modified thinning operators have been proposed to capture the specificity of real data, and many new INAR-type models have been defined. For example, Zheng and Basawa [13] introduced the random coefficient thinning operator while Risti´c and

**Citation:** Yu, K.; Wang, H. A New Overdispersed Integer-Valued Moving Average Model with Dependent Counting Series. *Entropy* **2021**, *23*, 706. https://doi.org/ 10.3390/e23060706

Academic Editor: Christian H. Weiss

Received: 30 April 2021 Accepted: 31 May 2021 Published: 2 June 2021

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Bakouch [14] proposed the negative binomial thinning operator. The *p*th-order integervalued autoregressive process with signed generalized power series thinning operator was proposed by Zhang et al. [15].

The most significant generalization of the thinning operator was made by Risti´c et al. [16] which they called the dependent Bernoulli thinning operator. They constructed the new sequence of the Bernoulli random variable allowing correlation between counting series. Based on this, Mileti´c Ili´c et al. [17] proposed the new model, based on the mix of regular binomial thinning and dependent thinning operator. For more details of thinning operators, refer to Weiß [18].

MA-type models are very important in time series analysis. The method of moving average is generally popular in statistical and mathematical analyses. Some researchers study forecasting using the moving average method. Winters [19] analyzed the exponentially weighted moving average method for forecasting sales. Cox [20] proposed the weighted moving average method to predict the Markov series. Landauskas et al. [21] introduced an algebraic approach to select the appropriate weight coefficients for weight moving averages performing better than classical moving average predictors. Wind speed prediction using combined time series model and neural network prediction was studied by Nan et al. [22]. Other applied weight moving average methods in control charts. Alevizakos et al. [23] proposed the triple exponentially weighted moving average control chart (TEWMA), which improves the detection ability of the classical control chart. Capizzi and Masarotto [24] proposed the adaptive exponentially weighted moving average control chart (AEWMA), which weights past observations of the monitored process using a suitable function of the current error. Adegoke et al. [25] studied the multivariate homogeneously weighted moving average (MHWMA) control chart for monitoring a process mean vector.

Some researchers constructed MA-type models from the perspective of count time series. Brännäs and Quoreshi [26] used the INMA process to model the number of transactions for intraday stocks and extended the model to include explanatory variables. Brännäs and Hall [27] mainly focused on the estimation in the INMA model. The construction of the thinning operation from these studies is based on independent counting series, which is a strong assumption. Thus, to make the model more flexible in capturing the specificity of different data types, this assumption should be relaxed.

For small counts of intraday transactions in stocks per minute, the decision of buyer or seller could be affected by the public news, which means decisions from different individuals may not be uncorrelated. The change in inventories can sometimes be described as an INMA process. However, during a certain period, the change can be influenced by the same external factor. For the INMA model applying a discrete risk model, the number of claims for certain insurance will be affected by the same factor, such as natural disasters. Thus, these claims are no longer independent. The independence between counting series should be relaxed. Allowing the correlation between counting series is natural, and therefore the INMA model based on dependent counting series can be derived to handle different real data situations.

The rest paper is organized as follows. Section 2 presents the model construction and discussions on some relevant statistical properties, and Section 3 discussed the estimation of unknown parameters. Section 4 shows the numerical simulation results and give dispersion test for dependence between counting series, while the conclusions are given in Section 5.

#### **2. The Model and Basic Properties**

#### *2.1. The Model Construction*

The counting series {*U*}*i*∈N of the integer-valued model is defined as:

$$\mathcal{U}\_{\dot{\imath}} = (1 - V\_{\dot{\imath}})\mathcal{W}\_{\dot{\imath}} + V\_{\dot{\imath}}Z\_{\dot{\imath}}$$

{*W*}*i*∈N is a sequence of *i*.*i*.*d* random variable with *Wi* ∼ *B*(1, *β*), *β* ∈ [0, 1]. {*V*}*i*∈N is a sequence of *i*.*i*.*d* random variable with *Vi* ∼ *B*(1, *θ*), *θ* ∈ [0, 1]. *Z* is a random variable with *Z* ∼ *B*(1, *β*). The operator ◦*θ* is defined by *β* ◦*θ εt* = ∑*εt i*=1 *Ui*, it is the dependent Bernoulli thinning operator, where *εt* is a non-negative integer-valued random variable. Based on this construction, we can easily verify that *E*(*Ui*) = *β*, *Var*(*Ui*) = *β*(1 − *β*), *corr*(*Ui*, *Uj*) = *θ*2, which has promising dependence between the counting series. Now we generalize the dependent count series to the INMA(*q*) process, For convenience, we use ◦ instead of ◦*θ* to simplify the notation.

**Definition 1** (Dependent Counting series Integer-valued Moving Average Model (DCINMA))**.** *The DCINMA(q) model is defined as:*

$$X\_t = \varepsilon\_t + \beta\_1 \circ \varepsilon\_{t-1} + \dots + \beta\_q \circ \varepsilon\_{t-q}$$

*βj* ◦ *εt*−*j*, *j* = 1, 2, ... , *q is the dependent Bernoulli thinning operator, and the following conditions should be satisfied.*

*A1.* {*εt*}*t*∈N *is a sequence of i. i. d non-negative random variables.*

*A2. The counting variable* {*Ui*}*i*∈N *is independent of εt for any i*, *t.*

*A3. βj* ◦ *εt*−*j, for any j* = 1, 2, . . . , *q are mutually independent.*

*2.2. The Numerical Properties for DCINMA(q) Model*

We denote *με* and *σ*2 *ε* as the mean and variance of term *εt*.

**Theorem 1.** *The numerical characteristics of* {*Xt*} *in Definition 1 are as follows:*

$$\begin{aligned} (i) \ E(\mathbf{X}\_t) &= \mu\_t (1 + \sum\_{i=1}^q \beta\_i) \\ (ii) \ Var(\mathbf{X}\_t) &= \sigma\_\varepsilon^2 + \sum\_{i=1}^q \left[ \mu\_\varepsilon \beta\_i + \mu\_\varepsilon^2 \theta^2 \beta\_i (1 - \beta\_i) - \mu\_\varepsilon (\theta^2 \beta\_i - \theta^2 \beta\_i^2 + \beta\_i^2) + \beta\_i^2 \sigma\_\varepsilon^2 \right] \\ (iii) \end{aligned}$$

$$\operatorname{cov}(\mathbf{X}\_t, \mathbf{X}\_{t-k}) = \begin{cases} \sigma\_\varepsilon^2 \sum\_{i=0}^{q-k} \beta\_i \beta\_{i+k} & k = 1, \dots, q \\ 0 & k \ge q + 1 \end{cases}$$

**Proof.** See Appendix A.

**Theorem 2.** {*Xt*} *is the process defined in Definition 1, then* {*Xt*} *is a covariance stationary process.*

**Proof.** It can be seen from the Theorem 1 that the unconditional mean and unconditional variance of *Xt* is a finite constant given the distribution of *εt*. Thus, *Xt* is a stationary process.

**Theorem 3.** {*Xt*} *is the process defined in Definition 1, then* {*Xt*} *is ergodic in mean and autocovariance function.*

**Proof.** See Appendix A.

*2.3. The Probability Generating Functions for DCINMA Model*

Risti´c et al. [16] derived the probability generating function of the *n* ∑ *i*=1 *Ui* as follows:

$$\begin{aligned} \Phi\_{\mathsf{U}} &= E[\mathsf{s}^{(\mathsf{U}\_{1}+\mathsf{U}\_{2}+...+\mathsf{U}\_{n})}] \\ &= (1-\beta)(1-\beta(1-\theta)(1-s))^{n} + \beta(1-(\beta+\theta-\beta\theta)(1-s))^{n} \end{aligned}$$

The above equation implies that the term *n* ∑ *i*=1 *Ui* has a distribution of:

*U*1 + *U*2 + ... + *Un* = *Bin*(*n*, *β*(1 − *θ*)) *w*.*p* 1 − *β Bin*(*n*, *β* + *θ* − *βθ*) *w*.*p β*

Then the probability generating function (PGF) of {*Xt*} is:

$$\Phi\_{X\_{\theta}}(\mathbf{s}) = \left[ (1 - \beta) \cdot \phi\_{\mathbb{E}}(1 - \beta(1 - \theta)(1 - \mathbf{s})) + \beta \cdot \phi\_{\mathbb{E}}(1 - (\beta + \theta - \beta\theta)(1 - \mathbf{s})) \right] \cdot \phi\_{\mathbb{E}}(\mathbf{s})$$

*φε*(*s*) is the probability generating function of the *εt*. Given the distribution of *εt*, the explicit expression of the probability generating function can be derived. Suppose Poisson distribution of *εt*, then

$$\phi\_{X\_0}(s) = \left[ (1 - \beta) \cdot \mathcal{e}^{-\lambda \beta (1 - \theta)(1 - s)} + \beta \mathcal{e}^{-\lambda (\beta + \theta - \beta \theta)(1 - s)} \right] \cdot \mathcal{e}^{(\lambda (s - 1))}$$

The probability generating function is defined by the probabilities. The uniqueness of a power series expansion implies that the probability generating function in turn defines probabilities.

Therefore, we can derive the probability of {*Xt*}. For *Xt* = *j*, the probability mass function of *Xt* is:

$$P(X\_t = j) = [\frac{1}{j!} \frac{d^j \phi(s)}{ds^j}]\_{s=0}$$

The bivariate probability generating function of {*Xt*} is Φ*Xt*,*Xt*−1 (*s*1,*s*2). Thus, deriving the explicit expression of bivariate probability generating function with Poisson innovation is easy for DCINMA(1) process.

$$\begin{aligned} E(s\_1^{X\_t} s\_2^{X\_{t-1}}) &= E(s\_1^{\mathcal{G} \cap \mathfrak{e}\_{t-1} + \mathfrak{e}\_t} \cdot s\_2^{\mathcal{G} \cap \mathfrak{e}\_{t-2} + \mathfrak{e}\_{t-1}}) \\ &= E(s\_1^{\mathcal{G} \cap \mathfrak{e}\_{t-1}} \cdot s\_1^{\mathfrak{e}\_t} \cdot s\_2^{\mathcal{G} \cap \mathfrak{e}\_{t-2}} \cdot s\_2^{\mathfrak{e}\_{t-1}}) \\ &= E(s\_1^{\mathfrak{e}\_t}) \cdot E(s\_1^{\mathcal{G} \cap \mathfrak{e}\_{t-2}}) \cdot E(s\_1^{\mathcal{G} \cap \mathfrak{e}\_{t-2}} \cdot s\_2^{\mathfrak{e}\_{t-1}}) \end{aligned}$$

Given the Poisson distribution of the innovation term, we can obtain:

$$E(s\_1^{\varepsilon\_t}) = e^{\lambda(s\_1 - 1)}, \\ E(s\_2^{\beta \circ \varepsilon\_{t-1}}) = (1 - \beta) \cdot e^{-\lambda \beta (1 - \theta)(1 - s\_2)} + \beta \cdot e^{-\lambda (\beta + \theta - \beta \theta)(1 - s\_2)}$$

$$E(s\_1^{\beta \circ \varepsilon\_{t-1}} \cdot s\_2^{\varepsilon\_{t-1}}) = (1 - \beta) \cdot e^{\lambda s\_2 [1 - \beta(1 - \theta)(1 - s\_2)]} + \beta \cdot e^{\lambda s\_2 [(1 - \beta - \theta - \beta \theta)(1 - s\_1)]}$$

#### *2.4. Compare with the INMA(q) Model*

The mean and the covariance of the *q*-th order integer-valued moving average model has the same expression, and the variance of the INMA(*q*) process is:

$$V\_{im\text{nr}} = \sigma\_\varepsilon^2 + \sum\_{i=1}^q [\sigma\_\varepsilon^2 \beta\_i + \mu\_\varepsilon \beta\_i (1 - \beta\_i)]^2$$

From Theorem 1, the variance of the DCINMA process presents a more complicated expression than the INMA(*q*) process due to the correlation between the counting series of parameter *θ*. For the Poisson innovation, the overdispersion index of the INMA and DCINMA model is as follows:

$$I\_{inma} = 1,\\ I\_{dimma} = \frac{1 + \beta + \beta\lambda\theta^2(1 - \beta)}{1 + \beta}$$

Since the value of *λ*, *β* and *θ* are all non-negative, the term 1 + *βλθ*2(1 − *β*) > 1. When two models (INMA(1) and DCINMA(1)) share the same *λ* and *β*, the *θ* will determine whether there is dependence between counting series (*θ* = 0). Thus, if we want to test *θ* = 0, it is equivalent to evaluating whether the model is overdispersed. If the value of *θ* is 0, the model degenerates to INMA model.

#### *2.5. Compare the Entropy with INMA(1) Model*

Entropy is an important concept in physics, but it can also be applied to other disciplines, including cosmology and economics. Entropy is a measure of the randomness or disorder of a system. In our case, entropy can be seen as a dispersion measure for the model. Thus, we evaluate the model from the perspective of entropy. The definition of Shannon entropy as follows:

$$H(\mathcal{Y}) = -\sum\_{i=1}^{n} p(y\_i) \cdot \ln p(y\_i)$$

where *Y* is a discrete random variable with probability mass function taking values on *y*1, ... , *yn*. We denote the *φdinma*(*s*) and *φinma*(*s*) as probability generating functions of the DCINMA(1) and INMA(1) process, respectively. Suppose the same innovation term for the two models follows the Poisson distribution. We can rewrite the probability generating function of them as:

$$\begin{aligned} \phi\_{\text{dimma}}(s) &= (1 - \beta) \cdot \mathfrak{c}^{(s-1)\cdot\left[\lambda\beta(1-\theta)+\lambda\right]} + \beta \cdot \mathfrak{c}^{(s-1)\cdot\left[\lambda\left(\beta+\theta-\beta\theta+\lambda\right)\right]}\\ \phi\_{\text{dimma}}(s) &= \mathfrak{c}^{(s-1)\cdot(\lambda\beta+\lambda)} \end{aligned}$$

Thus, we can conclude the distribution of DCINMA(1) and INMA(1) process based on the definition of the probability generating function. *Xinma*(1) *t* and *Xdinma*(1) *t* denote the sample from INMA(1) model and DCINMA(1) model.

$$\begin{aligned} X\_t^{\
in \text{max}(1)} &\sim Poi(\lambda \beta + \lambda) \\ X\_t^{\
dim \text{var}(1)} &\sim \begin{cases} Poi(\lambda \beta (1 - \theta) + \lambda) & w.p & 1 - \beta \\ Poi(\lambda (\beta + \theta - \beta \theta) + \lambda) & w.p & \beta \end{cases} \end{aligned}$$

The Shannon entropy for both models can be derived as follows:

$$\begin{split} H(X\_{t}^{\text{imma}(1)}) &= (\lambda\beta + \lambda)[1 - \log((\lambda\beta + \lambda))] + \epsilon^{(\lambda\beta + \lambda)} \cdot \sum\_{k=0}^{\infty} \frac{(\lambda\beta + \lambda)^{k} \log(k!)}{k!} \\ H(X\_{t}^{\text{dimma}(1)}) &= (1 - \beta) \cdot \{ (\lambda\beta(1 - \theta) + \lambda)[1 - \log((\lambda\beta(1 - \theta) + \lambda))] \} \\ &+ \epsilon^{(\lambda\beta(1 - \theta) + \lambda)} \cdot \sum\_{k=0}^{\infty} \frac{(\lambda\beta(1 - \theta) + \lambda)^{k} \log(k!)}{k!} \\ &+ \beta \cdot \{ (\lambda(\beta + \theta - \beta\theta) + \lambda)[1 - \log((\lambda(\beta + \theta - \beta\theta) + \lambda))] \\ &+ \epsilon^{(\lambda(\beta + \theta - \beta\theta) + \lambda)} \cdot \sum\_{k=0}^{\infty} \frac{(\lambda(\beta + \theta - \beta\theta) + \lambda)^{k} \log(k!)}{k!} \end{split}$$

The expression of entropy for the DCINMA(1) model is more complicated than the INMA(1) model due to the additional parameter *θ*.

#### **3. Parameter Estimation**

The estimation of the INMA model is complicated. Brännäs and Hall [27] discussed the Yule–Walker estimator, generalized moment method (GMM) based on the probability generating function (PGF) function, and the conditional least square method. Here, we did not attempt to use maximum likelihood estimation, which requires density functions that are generally not easily obtained in the INMA model, especially for this dependent situation. The results of generalized moments method-based probability generating function (PGF) function estimator are highly correlated with the values of *z*1 and *z*2 in Φ*k*(*z*1, *z*2), which are not stable. On the other hand, for the conditional least square method, the number of estimation equations are less than the number of parameters. This means that there is an additional parameter *θ* to be estimated.

Therefore, we derive the Yule–Walker estimator to obtain the unknown parameters for the Poisson DCINMA(*q*) model. We denote the following symbols: *μX* is the sample mean of {*Xt*}, *με* is the mean of innovation term *εt*. *γ*0 is the sample variance of {*Xt*}, *γ*1, *γ*2, ... , *γq* are the sample covariance of 1-*th*, 2-*th*,. . . , *q*-*th* order. The *β*0 for the equations below is 1.

Then, the unknown parameters in the DCINMA(*q*) model can be solved by equations through the sample moments function, which is as follows:

$$\begin{cases} \mu\_{X} = \mu\_{\varepsilon}(1 + \beta\_{1} + \beta\_{2} + \dots + \beta\_{q}) \\ \gamma\_{0} = \sigma\_{\varepsilon}^{2} + \sum\_{i=1}^{q} [\mu\_{\varepsilon}\beta\_{i} + \mu\_{\varepsilon}^{2}\theta^{2}\beta\_{i}(1 - \beta\_{i}) - \mu\_{\varepsilon}(\theta^{2}\beta\_{i} - \theta^{2}\beta\_{i}^{2} + \beta\_{i}^{2}) + \beta\_{i}^{2}\sigma\_{\varepsilon}^{2}] \\ \gamma\_{1} = \lambda\left(\beta\_{0}\beta\_{1} + \beta\_{1}\beta\_{2} + \dots + \beta\_{q-1}\beta\_{q}\right) \\ \gamma\_{2} = \lambda\left(\beta\_{0}\beta\_{2} + \beta\_{1}\beta\_{3} + \dots + \beta\_{q-2}\beta\_{q}\right) \\ \vdots \\ \gamma\_{q-1} = \lambda\left(\beta\_{0}\beta\_{q-1} + \beta\_{1}\beta\_{q}\right) \\ \gamma\_{q} = \lambda\left(\beta\_{0}\beta\_{q}\right) \end{cases}$$

#### **4. Simulation Study**

*4.1. Estimation of the Model Parameters*

In this section, we present some simulation results to show the performance of the estimator using different sample sizes. (*n* = 100, 300, 700, 1000). We focused on the Poisson DCINMA(1) model. The three group parameter values considered in this model are as follows:

> **Model A**: (*λ* = 1, *θ* = 0.6, *β* = 0.2) **Model B**: (*λ* = 4, *θ* = 0.7, *β* = 0.1) **Model C**: (*λ* = 5, *θ* = 0.5, *β* = 0.1)

We used the above parameter groups to generate the data and applied Yule–Walker method, then computed the bias and standard error based on 10,000 replications for each parameter group. The estimation results and their performance are reported in Table 1.

As shown in Table 1, we obtained nearly convergent estimators in all cases. In all cases, as the sample size increased, the bias decreased.

**Table 1.** Some numerical results of the estimates for true values of the parameters *λ*, *θ* and *β*.

**Table 1.** *Cont*.

#### *4.2. Testing for Dependence between Counting Series*

From Section 2.5, when the two processes have the same *λ* and *β*, the DCINMA model always presents overdispersion due to *θ*. We tested whether the value of *θ* is 0, which is equivalent to assessing whether the DCINMA process presents overdispersion. Aleksandrov and Weiß [28] proposed the diagnostic test for the INMA process. In this section, only the overdispersion test was applied. Under the null hypothesis *H*0 : *θ* = 0 (the process does not present overdispersion), the distribution of index dispersion has the following form:

$$\mathcal{I}\_{disp} \stackrel{d}{\longrightarrow} N(1 - \frac{1}{T} \frac{1 + 3\beta}{1 + \beta}, \frac{1}{T}(2 + 4(\frac{\beta}{1 + \beta})^2))$$

We then analyzed the simulation results to assess the performance of the overdispersion test. The nominal level *α* = 0.05 was employed, for sample sizes, *n* = 100, 150, 250. We did 1000 replications to calculate the size and power of the test. The following parameter groups were considered:

> **Model D:** (*λ* = 3, *θ*=0.4, *β* = 0.5) **Model E:** (*λ* = 5, *θ*=0.4, *β* = 0.3) **Model F:** (*λ* = 6, *θ*=0.3, *β* = 0.4) **Model G:** (*λ* = 7, *θ*=0.7, *β* = 0.8)

From Table 2, given the same values for *λ* and *β*, under the null hypothesis, *θ* = 0, the size of ˆ*Idisp* are close to nominal level. Under the alternative situation, *θ* = 0, for all cases, as *n* increase the power quickly increases to 1.

**Table 2.** Size and power of DCINMA(1) and INMA(1) model.

#### **5. Real Data Example**

We then applied the proposed model to a real dataset. Unlike the integer-valued autoregressive model, since the density function of this DCINMA process is hard to obtain, the Akaike information criterion (AIC) is difficult to use in measuring the fitness of such an INMA-type model. To assess the performance of the model, we adapted the parametric resampling method by Jung et al. [29].

We used crime data available from the Forecasting Principles site. The data consist of 144 observations of monthly larceny counts for the City of Pittsburgh from January of 1990 to December of 2001. It would be highly improbable for criminals to remain in the same place for a long time.

The calculated of sample mean (4.73) and variance (6.40) suggest that the dataset is overdispersed. In addition, the sample autocorrelation function (ACF) in Figure 1 shows that the order of the model should be set to one. We fitted the data set into the DCINMA(1) model and INMA(1) model to evaluate the model performance for overdispersed data. The estimation results for the two models are presented in Table 3.

**Table 3.** Estimation results of DCINMA(1) and INMA(1) model.

The parametric resampling method can be performed in several steps:

**Step 1:** Generate samples with length equal to the original dataset from the fitted model for *R* times.

**Step 2:** Compute the autocorrelation function (ACF) for each sample series. Then derive the empirical sample autocorrelation function (ACF) for the different lag orders.

**Step 3:** Using the results from **Step 2**, compute the 100(1 − *α*/2) and 100(*α*/2) quantiles.

For the given example, *R* was set to 5000, and *α* was set to 0.05. We plotted the acceptance envelope for DCINMA(1) model and INMA(1) model, and the results are shown in Figures 2 and 3. The sample autocorrelations for the real dataset lie within the acceptance envelopes of the DCINMA(1) model except for only one point, while the important second order of the autocorrelation function (ACF) exceed the upper bound of the acceptance envelopes of the INMA(1) model. It is clear that the acceptance envelopes of the DCINMA(1) model performs better than the acceptance envelopes of the INMA(1) model. Thus, the proposed model is suitable for this dataset, adequately representing the autocorrelation for real dataset.

**Figure 1.** Time series plot (top figure) and autocorrelation function (below figure panel) for actual Larceny dataset.

**Figure 2.** Acceptance envelope of the DCINMA(1) model for the autocorrelation function for the larceny dataset.

**Figure 3.** Acceptance envelope of the INMA(1) model for the autocorrelation function for the larceny dataset.

#### **6. Conclusions**

In this paper, we constructed a new integer-valued moving average model with dependent counting series. The statistical properties of the proposed model were discussed and evaluated. The parameter estimation of the proposed model is based on the Yule–Walker method. The new model presents overdispersion due to the dependence parameter *θ*, which means the dependence between counting series can be verified by an overdispersion test. Numerical simulation results were used to evaluate the performance of the estimation and overdispersion test.

Future extensions for this study are as follows. First, we only focused on the stationary DCINMA(1) model in this paper. However, cases with non-stationary series are more common. The switched system is an important model in studying hybrid systems, particularly from the perspective of control science and engineering. Refer to ([30–32]) for more detailed discussion. The mechanism conducts the transformation between different subsystems, providing an approach to generalize the DCINMA process into a non-stationary case. A function can be introduced to control the change for different parameter values of the stationary case, which characterize non-stationary in series. Second, the parameter *θ* in our model provides a probability for overdispersion. In a switched system, the switching signal is a piecewise constant function, depending on time, external signal, output, and its own past value. Thus, the weighted switching signal function with weights *θ* and (1 − *θ*) can be considered, where the weight *θ* gives the probability for switching to different subsystems.

**Author Contributions:** Conceptualization, K.Y. and H.W.; methodology, K.Y.; software, H.W.; validation, K.Y. and H.W.; formal analysis, K.Y.; investigation, H.W.; resources, K.Y.; data curation, K.Y.; writing—original draft preparation, H.W.; writing—review and editing, K.Y.; visualization, H.W.; supervision, K.Y.; project administration, K.Y.; funding acquisition, K.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data used to support the findings of this study are included within the article.

**Acknowledgments:** This research was supported by the National Social Science Fund of China (No. 18BTJ039) and Fundamental Research Funds for the Central Universities (No. JBK2102021). In addition, the authors will be grateful for the editor and reviewers' comments and suggestions, which led to improvements of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

**Proof of Theorem 1.** we give the detailed derivation process for 1-th order, the *q*-th order can be derived in the same manner. term (*i*) and (*ii*) of the Theorem 1 are easy to verify, so we focus on the term (*iii*).

$$\begin{split} Var(\mathbf{X}\_t) &= Var(\boldsymbol{\beta} \circ \boldsymbol{\varepsilon}\_{t-1}) + Var(\boldsymbol{\varepsilon}\_t) \\ &= V[E[\boldsymbol{\beta} \circ \boldsymbol{\varepsilon}\_{t-1} | \boldsymbol{\varepsilon}\_{t-1}]] + E[V[\boldsymbol{\beta} \circ \boldsymbol{\varepsilon}\_{t-1} | \boldsymbol{\varepsilon}\_{t-1}]] + Var(\boldsymbol{\varepsilon}\_t) \end{split}$$

Then, we focus on the term *V*[*β* ◦ *εt*−1|*εt*−1].

$$\begin{split} V[\boldsymbol{\beta}\diamond\boldsymbol{\varepsilon}\_{t-1}|\boldsymbol{\varepsilon}\_{t-1}] &= E[(\sum\_{i=1}^{\varepsilon\_{t-1}}\boldsymbol{U}\_{i})^{2}|\boldsymbol{\varepsilon}\_{t-1}] - [E[\sum\_{i=1}^{\varepsilon\_{t-1}}\boldsymbol{U}\_{i}|\boldsymbol{\varepsilon}\_{t-1}]]^{2} \\ &= \boldsymbol{\varepsilon}\_{t-1}\cdot\boldsymbol{E}[\boldsymbol{U}\_{1}^{2}] + (\boldsymbol{\varepsilon}\_{t-1}-1)\cdot\boldsymbol{\varepsilon}\_{t-1}\cdot\boldsymbol{E}[\boldsymbol{U}\_{1}\cdot\boldsymbol{U}\_{2}] - [\boldsymbol{\varepsilon}\_{t-1}\cdot\boldsymbol{\beta}]^{2} \\ &= \boldsymbol{\varepsilon}\_{t-1}\cdot(\boldsymbol{\beta}\cdot(1-\boldsymbol{\beta})+\boldsymbol{\beta}^{2}) + \boldsymbol{\varepsilon}\_{t-1}\cdot(\boldsymbol{\varepsilon}\_{t-1}-1)\cdot(\boldsymbol{\theta}^{2}\cdot\boldsymbol{\beta}\cdot(1-\boldsymbol{\beta})+\boldsymbol{\beta}^{2}) \\ &- \boldsymbol{\beta}^{2}\cdot\boldsymbol{\varepsilon}\_{t-1}^{2} \end{split}$$

$$\begin{aligned} E\left[V[\boldsymbol{\beta}\circ\boldsymbol{\varepsilon}\_{t-1}|\boldsymbol{\varepsilon}\_{t-1}]\right] &= (\boldsymbol{\beta}\cdot(1-\boldsymbol{\beta})+\boldsymbol{\beta}^{2})\cdot E[\boldsymbol{\varepsilon}\_{t-1}] - E[\boldsymbol{\beta}^{2}\cdot\boldsymbol{\varepsilon}\_{t-1}^{2}] \\ &+ E[\boldsymbol{\varepsilon}\_{t-1}\cdot(\boldsymbol{\varepsilon}\_{t-1}-1)]\cdot(\boldsymbol{\theta}^{2}\cdot\boldsymbol{\beta}\cdot(1-\boldsymbol{\beta})+\boldsymbol{\beta}^{2}) \\ &= \boldsymbol{\lambda}\cdot(\boldsymbol{\beta}\cdot(1-\boldsymbol{\beta})-\boldsymbol{\beta}^{2})-\boldsymbol{\beta}^{2}\cdot(\boldsymbol{\lambda}+\boldsymbol{\lambda}^{2}) \\ &+ (\boldsymbol{\lambda}+\boldsymbol{\lambda}^{2}-\boldsymbol{\lambda})\cdot(\boldsymbol{\theta}^{2}\cdot\boldsymbol{\beta}\cdot(1-\boldsymbol{\beta})+\boldsymbol{\beta}^{2}) \\ &= \boldsymbol{\lambda}^{2}\cdot\boldsymbol{\beta}\cdot(1-\boldsymbol{\beta})\cdot\boldsymbol{\theta}^{2}+\boldsymbol{\beta}\cdot\boldsymbol{\lambda}-\boldsymbol{\beta}^{2}\cdot\boldsymbol{\lambda} \end{aligned}$$

$$\begin{split} Var(\mathbf{X}\_t) &= V[\boldsymbol{\beta} \cdot \boldsymbol{\varepsilon}\_{t-1}] \boldsymbol{\lambda}^2 \cdot \boldsymbol{\beta} \cdot (1 - \boldsymbol{\beta}) \cdot \boldsymbol{\theta}^2 + \boldsymbol{\beta} \cdot \boldsymbol{\lambda} - \boldsymbol{\beta}^2 \cdot \boldsymbol{\lambda} + \boldsymbol{\lambda} \\ &= \boldsymbol{\beta} \cdot \boldsymbol{\lambda} + \boldsymbol{\lambda}^2 \cdot \boldsymbol{\beta} \cdot (1 - \boldsymbol{\beta}) \cdot \boldsymbol{\theta}^2 + \boldsymbol{\lambda} \end{split}$$

**Proof of Theorem 3.** let *Yt* = *Xt* − *μX*, {*Xt*} is the process defined in Section 2. To prove the *Yt* is ergodic in autocovariance function, it is sufficient to show that:

$$\lim\_{T \to \infty} \frac{1}{T} \sum\_{l=0}^{T} \left\{ E\left(\mathbf{Y}\_t \mathbf{Y}\_{t+v} \mathbf{Y}\_{t+l} \mathbf{Y}\_{t+v+l}\right) - R^2(v) \right\} = 0$$

It is obvious *E*(*Yi t*) < ∞. For *i* = 1, 2, 3, 4, let *E*(*Y*2 *t* ) = *c*2, *E*(*Y*4 *t* ) = *c*4. **The case** *v* = 0**:**

$$E(\mathcal{Y}\_t \mathcal{Y}\_{t+\upsilon} \mathcal{Y}\_{t+l} \mathcal{Y}\_{t+\upsilon+l}) = E(\mathcal{Y}\_t^2 \mathcal{Y}\_{t+l}^2)$$

If *l* = 1,

$$E(\mathbf{Y}\_t \mathbf{Y}\_{t+\upsilon} \mathbf{Y}\_{t+l} \mathbf{Y}\_{t+\upsilon+l}) = E(\mathbf{Y}\_t^2 \mathbf{Y}\_{t+l}^2) \le \left( E(|\mathbf{Y}\_t^2|^2 |\mathbf{Y}\_{t+l}^2|^2) \right)^{1/2} = c\_4$$

If *l* > 1, *Yt* and *Yt*+*l* are irrelevant, then

$$E(\mathbf{Y}\_t \mathbf{Y}\_{t+\upsilon} \mathbf{Y}\_{t+l} \mathbf{Y}\_{t+\upsilon+l}) = E(\mathbf{Y}\_t^2 \mathbf{Y}\_{t+l}^2) = c\_2^2 = \gamma\_Y^2(0)$$

$$\frac{1}{T} \sum\_{l=0}^T \{ E(\mathbf{Y}\_t \mathbf{Y}\_{t+\upsilon} \mathbf{Y}\_{t+l} \mathbf{Y}\_{t+\upsilon+l}) - R^2(\upsilon) \} \le \frac{1}{T} \{ \varepsilon\_4 - \gamma\_Y^2(0) \} \to 0$$

$$\begin{array}{c} \text{The case } v \ge 1; \\ \text{If } l \le v+1; \end{array}$$

$$E(\mathbf{Y}\_t \mathbf{Y}\_{t+\upsilon} \mathbf{Y}\_{t+l} \mathbf{Y}\_{t+\upsilon+l}) \le [E(|\mathbf{Y}\_t \mathbf{Y}\_{t+\upsilon}|^2 | \mathbf{Y}\_{t+l} \mathbf{Y}\_{t+l+\upsilon}|^2]^{\frac{1}{2}} \le [E(|\mathbf{Y}\_t|^4 | \mathbf{Y}\_{t+\upsilon}|^4 | \mathbf{Y}\_{t+l}|^4 | \mathbf{Y}\_{t+l+\upsilon}|^4]^{\frac{1}{4}} = c\_4$$

If *l* > *v* + 1, *YtYt*+*l* and *Yt*+*lYt*+*l*+*v* are irrelevant, then

$$E(\mathbf{Y}\_t \mathbf{Y}\_{t+\upsilon} \mathbf{Y}\_{t+l} \mathbf{Y}\_{t+\upsilon+l}) = E(\mathbf{Y}\_t \mathbf{Y}\_{t+\upsilon})E(\mathbf{Y}\_{t+l} \mathbf{Y}\_{t+l+\upsilon}) = R^2(\upsilon)$$

$$\frac{1}{T} \sum\_{l=0}^T \{ E(\mathbf{Y}\_t \mathbf{Y}\_{t+\upsilon} \mathbf{Y}\_{t+l} \mathbf{Y}\_{t+\upsilon+l}) - R^2\_{\upsilon} \} \le \frac{1}{T} \{ (c\_4 - R^2(\upsilon))(\upsilon + 1) \} \to 0$$

We can obtain:

$$\lim\_{T \to \infty} E[(\mathcal{R}\_n^2(v) - \mathcal{R}^2(v))^2] = 0$$

Which implies:

$$
\mathbb{R}^2(v) \xrightarrow{p} \mathbb{R}^2(v).
$$

The above equation is holding for variable *Yt*:

$$\mathcal{Y}\_t = X\_t - \mu\_{X\_{\prime}} \mathring{R}\_Y^2(\upsilon) \stackrel{p}{\to} R\_Y^2(\upsilon),$$

We need to prove that

$$P(|\mathcal{R}\_X^2(v) - \mathcal{R}\_X^2(v)| \ge \epsilon) \to 0$$

Because of *R*2 *X*(*v*) = *R*2 *Y*(*v*), rewriting the above equation as

$$\begin{aligned} P(|\mathring{R}\_X^2(\upsilon) - R\_X^2(\upsilon)| \ge \epsilon) &= P(|\mathring{R}\_X^2(\upsilon) - \mathring{R}\_Y^2(\upsilon) + \mathring{R}\_Y^2(\upsilon) - R\_X^2(\upsilon)| \ge \epsilon) \\ &\le P(|\mathring{R}\_X^2(\upsilon) - \mathring{R}\_Y^2(\upsilon)| \ge \epsilon/2) + P(|\mathring{R}\_Y^2(\upsilon) - R\_Y^2(\upsilon)| \ge \epsilon/2) \end{aligned}$$

*R*ˆ 2 *Y*(*v*) and *R*ˆ 2 *X*(*v*) expressing as

$$\mathcal{R}\_X^2(\upsilon) = \frac{1}{T} \sum\_{t=1}^{T-k} (X\_{t+k} - \mathcal{X})(X\_t - \mathcal{X}),\\\mathcal{R}\_Y^2(\upsilon) = \frac{1}{T} \sum\_{t=1}^{T-k} Y\_{t+k} Y\_t = \frac{1}{T} \sum\_{t=1}^{T-k} (X\_{t+k} - \mu\_X)(X\_t - \mu\_X)$$

Then the following expression can be derived

$$\begin{split}P(|\mathbb{R}\_X^2(v) - \mathbb{R}\_Y^2(v)| \ge \varepsilon/2) &= P((\mathbb{X}^2 - \mu\_X^2) + (\mu\_X - \mathbb{X})(\mathbb{X}\_{l+k} + \mathbb{X}\_T) \ge \varepsilon/2) \\ &\le P(\bar{\mathcal{X}}^2 - \mu\_X^2 \ge \varepsilon/4) + P((\mu\_X - \bar{\mathcal{X}})(\bar{\mathcal{X}}\_{l+k} + \bar{\mathcal{X}}\_T) \ge \varepsilon/4) \\ &\ge \varepsilon/4) \end{split}$$

Since *X*¯ *p* → *μX*, by Slutsky's theorem

$$(\mathbb{X}^2 - \mu\_X^2 \overset{p}{\to} 0, (\mu\_X - \mathbb{X})(\mathbb{X}\_{t+k} + \mathbb{X}\_T) \overset{p}{\to} 0)$$

Consequently,

*P*(|*R*ˆ 2 *X*(*v*) − *R*2 *X*(*v*)| ≥ ) → 0, *T* → ∞

#### **References**


**Laurent Dufoss´e**

Laurent Dufosse has held the position of Professor of Food Science and Biotechnology since ´ 2006, at Reunion Island University, which is located on a volcanic island in the Indian Ocean, near Madagascar and Mauritius. The island is one of France's overseas territories, with almost one million inhabitants, and the university has 19,000 students. Previously, Professor Dufosse was a researcher ´ and senior lecturer at the Universite de Bretagne Occidentale, Quimper, Brittany, France. He attended ´ the University of Burgundy, Dijon, France, where he received his PhD in Food Science in 1993, and has been involved in the field of the biotechnology of food ingredients for more than 30 years. His main research interests over the last 20 years have mainly been the microbial production of pigments and the study of aryl carotenoids, such as isorenieratene, C50 carotenoids, azaphilones and anthraquinones. These studies have relevance for applications in science and technology, in areas such as the supplying of sustainable components in many industrial sectors and the development of biobased pigments and dyeing agents for the textile industry.


*Article* **Effect of Cold-Deformation on Austenite Grain Growth Behavior in Solution-Treated Low Alloy Steel**

**Xianguang Zhang 1,\*, Kiyotaka Matsuura 2 and Munekazu Ohno 2**

Received: 28 October 2018; Accepted: 21 November 2018; Published: 1 December 2018

**Abstract:** The occurrence of abnormal grain growth (AGG) of austenite during annealing is a serious problem in steels with carbide and/or nitride particles, which should be avoided from a viewpoint of mechanical properties. The effects of cold deformation prior to annealing on the occurrence of AGG have been investigated. It was found that the temperature range of the occurrence of AGG is shifted toward a low temperature region by cold deformation, and that the shift increases with the increase of the reduction ratio. The lowered AGG occurrence temperature is attributed to the fine and near-equilibrium AlN particles that are precipitated in the cold-deformed steel, which is readily dissolved during annealing. In contrast, coarse and non-equilibrium AlN particles precipitated in the undeformed steel, which is resistant to dissolution during annealing.

**Keywords:** austenite; abnormal grain growth; cold-deformation; precipitate

#### **1. Introduction**

Microalloying has been used as an important austenite grain refinement technology in steels. Austenite grain growth can be inhibited by microalloying, which causes the precipitation of second-phase particles in austenite [1–5]. Abnormal grain growth (AGG) frequently occurs in micro-alloyed steels during annealing. The austenite grain growth of the micro-alloyed steels usually experiences "normal → abnormal → normal" growth modes in sequence, with an increase of the annealing temperature [5]. AGG occurs within a certain temperature range during annealing, which is referred to as an AGG temperature range. The AGG temperature range of steels is an important factor in designing the heat treatment to achieve fine-grained products.

The austenite grain growth behavior in low-alloy steels is largely dependent on the stability of second-phase particles. At low temperatures, the second-phase particles are stable, and the austenite grain boundaries can be effectively pinned by the particles. The grain growth is thus greatly retarded. However, at relatively high temperatures, the second-phase particles become unstable, and they readily dissolve or coarsen [5]. Then, some of the austenite grains may become unpinned, and they may grow much faster than the matrix grains. This leads to the occurrence of AGG [6]. There is a transition temperature at which the austenite grain growth switches from normal grain growth (NGG) to AGG mode. This transition temperature is referred to as a grain coarsening temperature, *T*c [7]. In addition, at temperatures that are much higher than the *T*c, NGG resumes, due to the almost complete dissolution of the pinning particles, and the grain growth becomes free [5]. Hence, there is another transition temperature, the AGG finishing temperature (*T*f), at which the grain growth switches from AGG to NGG mode. *T*c and *T*f are important parameters for understanding the grain growth behavior of austenite.

Cold-deformation, such as cold-rolling, hammering, or cold-forging, has been widely used in the steel manufacturing process. It was found that hot-deformation could accelerate the nucleation and coarsening of the second-phase particles due to the large amount of dislocation that is introduced into the steels by deformation [8–10]. In addition, previous studies [11–13] have revealed that cold-deformation also has a strong influence on the precipitation and growth of the precipitates during subsequent annealing. It was found that the second-phase particles were finer and distributed more uniformly by cold-deformation [11]. Meanwhile, cold-deformation makes the precipitates resistant to coarsening [11,12].

The grain growth behavior of austenite relies on the presence of second-phase particles. Therefore, it is expected that the cold-deformation should have a strong influence on the AGG temperature range of the low-alloy steels. However, previous works have mainly focused on the effect of cold-deformation on the precipitation or coarsening behaviors of second-phase particles. Rare attention has been paid to its influence on austenite grain growth behavior. Recently, it was reported by the present authors [13,14] that an inhomogeneous distribution of AlN particles in ferrite/pearlite banded steel became homogeneous, due to cold-deformation prior to austenization, which increased the low *T*c of the banded steel significantly. However, to the best of the authors' knowledge, no systematic work has been reported regarding the effects of cold deformation on the AGG temperature range of austenite. The present study has been undertaken to clarify the effects of cold-deformation on austenite grain growth behavior in solution-treated low-alloy steel. It was found that the AGG temperature range of austenite is shifted toward the lower temperature region by applying prior cold-deformation.

#### **2. Materials and Methods**

The composition of the low-alloy steel used in the present study is displayed in Table 1. The solubility of AlN in austenite can be written as [15]:

$$\log[\text{AlI}][\text{N}] = 1.03 - 6770/T \tag{1}$$

Where [Al] and [N] represent the solubilities of aluminum and nitrogen at the absolute temperature *T*. According to Equation (1), the complete dissolution temperature of AlN in the current steel was calculated to be 1453 K. To completely dissolve the AlN, and to eliminate the banded segregation of Mn, the solution treatment temperature and time was designed to be 1573 K for 2 h. After the solution treatment, the specimen was air-cooled to room temperature.

**Table 1.** Composition of the steel used in this study (wt %).

The cold-deformation was carried out by cold-rolling, with reduction ratios ranging from 10 to 70%. Both the solution-treated (ST), and solution-treated and deformed (STD) specimens were annealed at temperatures between 1173–1473 K for various periods, as schematically shown in Figure 1. The heating rate of the specimens was about 5 K/s. After annealing, the specimens were quenched in ice water to freeze the microstructure. The specimens were cut and mechanically polished to obtain a mirror-like surface. To reveal the prior austenite grain boundaries, the specimens were etched in a supersaturated picric acid solution at 333 K. The microstructure was characterized by using an optical microscope (OM). The well-known linear intercept method was used to evaluate the mean austenite grain size. About 450 to 550 intercepts were counted for each sample.

**Figure 1.** Schematic illustrations of the annealing treatment.

The AlN particles were characterized by using transmission electron microscopy (TEM, JEOL JEM-2010, JEOL Ltd., Tokyo, Japan) equipped with energy dispersive spectroscopy (EDS) operating at 200 kV on carbon extraction replicas eroded off the specimens. At least a thousand AlN precipitates were measured for each case to evaluate the volume fraction and particle sizes.

The volume fraction of the AlN precipitates from the carbon extraction replica can be calculated based on the McCall-Boyed method [16] as follows:

$$f = \left(\frac{1.4\pi}{6}\right)\left(\frac{ND^3}{V}\right),\tag{2}$$

where *N* is the number of the precipitates, *D* is the mean diameter of the particles, and *V* is the volume of the matrix from which the precipitates were extracted. The thickness of the matrix can be assumed as the mean diameter of the particles [17]. Thus, Equation (2) can be written as:

$$f = \left(\frac{1.4\pi}{6}\right)\left(\frac{ND^2}{S}\right),\tag{3}$$

where *S* is the area of the corresponding matrix.

#### **3. Results and Discussion**

#### *3.1. Austenite Grain Growth Behavior*

Figure 2 shows the prior austenite grain structures (now martensite after quenching) of the solution-treated (ST) specimens after annealing at different temperatures for various periods of time. Microstructure observation revealed that the austenite grain growth during annealing at 1298 K is very slow and the prior austenite grain size almost has no change. Hence, only the microstructure after annealing at 1298 K for 1800 s is shown Figure 2a. It is clear that fine and uniform prior austenite grain structures were formed, which is attributed to the occurrence of NGG. However, mixed prior austenite grain structures were formed after the specimen was annealed at 1323 K for 60 s, as shown in Figure 2b, where abnormal coarse grain (as indicated by the red arrow) mixed with fine matrix grain structures (as indicated by the black arrow) were observed. After a prolonged holding time of up to 1800 s, extremely coarse grain structures were developed (Figure 2c). Hence, typical AGG occurred at this temperature. After the sample was annealed at 1423 K for 1800 s, slightly coarse but uniform grain structures were formed (Figure 2d). This indicates that the grain growth switched to NGG mode again.

**Figure 2.** Prior austenite grain structures of the solution-treated (ST) specimens after annealing at (**a**) 1298 K for 1800 s, (**b**) 1323 K for 60 s, (**c**) 1323 K for 1800 s, and (**d**) 1423 K for 1800 s. The red and black arrows in (**b**) to indicate the abnormal coarse and fine matrix grains, respectively.

According to the above results, the occurrence AGG led to extremely coarse grain structures, while the NGG led to relatively fine grain structures after annealing. Therefore, the mean austenite grain size of the ST specimens after annealed at various temperatures for 1800 s were plotted in Figure 3a to show the austenite grain growth behavior during annealing. The average austenite grain sizes after annealing at 1298 K (Figure 2a), 1323 K (Figure 2c), and 1423 K (Figure 2d) are indicated by a black solid, red solid, and open arrows, correspondingly. From Figure 3a, it can be seen that the prior austenite grain size is fine, and that it has almost no change below 1298 K. This indicates that the austenite grain growth was greatly retarded, and it grew under the NGG mode below 1298 K. The coarse grains that were formed by AGG appeared at intermediate temperatures. Slightly coarse grains that were formed by NGG appeared at high temperatures. Therefore, the ST specimens experienced "normal → abnormal → normal" grain growth modes that were usually observed in the micro-alloyed steels [5]. The grain growth modes and the transition temperatures of *T*c and *T*f are indicated in Figure 3a. It is necessary to point out that the grain growth behaviors were examined carefully by changing the annealing temperatures within 25 K intervals, and that only partial results are shown in Figure 3a.

To understand the effects of cold-deformation on the austenite grain growth behavior, the average austenite grain sizes at various temperatures in the solution-treated and cold-deformed by 50% (STD-50%) specimens were examined. The measured austenite grain size as a function of annealing temperature is summarized in Figure 3b. A similar grain structure changing tendency was observed as in that of the ST specimens; viz., a fine grain structure at low temperatures, a coarse grain structure at intermediate temperatures, and slightly coarse and uniform grain structures at high temperatures. Therefore, the STD-50% specimen experienced "normal → abnormal → normal" grain growth modes as well. The prior austenite grain structure evolution process was similar to that of the ST specimens, but at different temperatures; therefore, the microstructures are not shown here again. Importantly, comparing Figure 3a,b shows that the AGG temperature range was shifted toward a low temperature by cold-deformation.

**Figure 3.** Average prior austenite grain sizes of the (**a**) ST and (**b**) solution-treated and cold-deformed by 50% (STD-50%) specimens after annealing at various temperatures for 1800 s.

To gain an overall image of the effect of cold-deformation on austenite grain growth behavior, the austenite grain growth modes against annealing temperatures in the specimens deformed with various reduction ratios are summarized in Figure 4. The "normal → abnormal → normal" grain growth modes occurred in all the specimens. Importantly, the AGG temperature range was gradually depressed toward low temperatures with the increase in the reduction ratio, which was almost saturated up to 70%. The decreased *T*c and *T*f temperatures, induced by the cold-deformation, can be explained by its influences on the evolution of the AlN precipitates, which is discussed in the following section.

**Figure 4.** Austenite grain growth behavior against annealing temperatures in the specimens deformed at various reduction ratios.

#### *3.2. The Evolution of AlN Precipitates*

As reviewed in the Introduction, the stability of the second-phase particles during annealing plays a key role in the austenite grain growth behavior. To understand the reasons for the suppression of the AGG temperature range by the cold-deformation, the evolution of the AlN particles were carefully studied by TEM observations. The volume fractions and sizes of the AlN particles in both the ST and STD-50% specimens quenched from various temperatures from 1073 to 1353 K are summarized in Figure 5a,b, respectively. For comparison, the equilibrium volume fraction of the AlN particles calculated from Equation (1) is also plotted in Figure 5a, as shown by the blue broken line.

**Figure 5.** (**a**) Volume fraction and (**b**) size of the AlN particles in the ST and STD-50% specimens after annealing at various temperatures for 1800 s. TEM images of the AlN particles (indicated by the black arrows) in (**c**) ST and (**d**) STD-50% specimens after annealing at 1173 K for 1800 s.

It should be pointed out that after the solution treatment, the AlN precipitates should have been fully dissolved into the austenite. It was reported that the precipitation of AlN is difficult during the cooling process, and that it is sensitive to the cooling rate [18–20]. The precipitation of AlN can be entirely suppressed at a cooling rate larger than 1 K/s [21], which is approximately equivalent to air cooling a 50 mm diameter steel bar [22]. Therefore, the AlN precipitation during the air-cooling process is negligible in this study. The AlN precipitates should mainly be re-precipitated during the reheating process in both the ST and STD specimens.

According to the precipitation-time-temperature (PTT) diagram of AlN [23], the AlN precipitation should have been completed below 1073 K in the current steel during the annealing process. Therefore, the AlN particles within the specimens quenched from 1073 K (ferrite + austenite two phase region) were examined. According to Figure 5a,b, fine- and near-equilibrium AlN particles precipitated in the STD-50% specimen at 1073 K. Meanwhile, a much lower volume fraction of the AlN particles than the equilibrium was precipitated in the ST specimen. These indicate that the AlN particles have been almost fully precipitated in the STD-50% specimen, while the AlN particles were only partially been precipitated in the ST specimen. The near-equilibrium precipitation in the STD-50% specimen should result from the dislocation-induced precipitation of AlN [11]. It needs to point out that the slightly overestimated AlN volume fraction in the STD-50% specimen compared to the

equilibrium prediction should be caused by the differences between the metallographic observation and theoretical calculation.

It is necessary to point out that as the annealing temperature was raised up to 1173 K (just after the ferrite-to-austenite transformation), the AlN particles in the STD-50% specimen were less coarsened than that of the ST specimen (Figure 5b). This could be more clearly seen from the TEM images of the AlN particles in both the undeformed and cold-deformed specimens after annealing at 1173 K for 1800 s, as shown in Figure 5c,d, respectively. Smaller AlN particles were formed in the STD-50% specimen, than that of the ST specimen. Therefore, the AlN particles in the prior cold-deformed specimen had sluggish growth kinetics. This may strongly influence the dissolution behavior of the precipitates at high temperatures. Similar phenomena have also been reported by Furubayashi et al. [11] and Kesternich [12], while the mechanism(s) for this are still not very clear.

Regarding the pinning effect of the second-phase particles, there is a maximum radius of the precipitates, *r*crit, for pinning the grain growth [6]. That is, only the fine second phase particles with a radius smaller than *r*crit, can effectively pin the grain boundaries. The *r*crit can be expressed as [6]:

$$\sigma\_{\rm crit} = 3 \frac{\mathcal{R}\_0 f}{\pi} \left( 3 - 4 / Z \right)^{-1} \,, \tag{4}$$

where *R*0 is the average grain size, *Z* is a parameter that is used to represents the inhomogeneity of the initial grains, and *f* is the volume fraction of second-phase particles. It is obvious that for a given uniform initial grain structure, *r*crit linearly decreases with the decrease of *f* [6].

According to Figure 5a, the fraction of the AlN particles is stable in the STD-50% specimen below 1223 K (as indicated by symbol 1 in Figure 5a). Therefore, both the *r*crit and the pinning force, which is proportional to *f*/*r* [6] (here, *r* is the average radius of the pinning particles), may have no obvious change, and the grain boundaries can be well-pinned. This results in the sluggish austenite grain growth kinetics in the STD-50% specimens below 1223 K, as shown in Figure 3b. However, when the annealing temperature was increased up to 1248 K (position 2 in Figure 5a), the fraction of the AlN particles in the STD-50% specimen decreased obviously. This may be caused by the accelerated diffusion rates of the elements at elevated temperatures, the increased solubility of the AlN in austenite, and the easy dissolution of the fine AlN particles in the STD-50% specimen. Hence, the *r*crit may decrease, according to Equation (4). Then, the volume faction of the AlN precipitates available to pin the grain boundaries may decrease steeply at 1248 K. This may cause the occurrence of AGG at this temperature.

As for the ST specimen, the fraction of AlN particles were almost constant until 1298 K (position 3 in Figure 5a). Therefore, both *r*crit and the pinning force were stable, and thus the grain growth could be effectively retarded. This agrees with the observed sluggish grain growth kinetics below 1298 K in Figure 3a. However, as the annealing temperature was raised up to 1323 K (position 4 in Figure 5a), the fractions of the AlN particles were decreased, and the *r*crit may decrease accordingly. This may result in the occurrence of AGG in the ST specimen, as shown in Figure 2b,c.

According to the discussion above, it can be concluded that the change in the volume fraction of the AlN particles during annealing plays a key role in the austenite grain growth behavior. The rapid decrease in the AlN volume fraction may greatly change the *r*crit, and then the effective volume fraction of the second phase particles to pin the gain boundaries may decrease dramatically, which may result in the occurrence of AGG.

As reviewed in the Introduction, the transition from AGG to NGG mode occurs at high temperatures when the AlN precipitates are nearly completely dissolved, and the grain growth becomes free. According to Figure 5a, the fractions of the AlN particles in both the deformed and undeformed specimens were low at 1353 K. Experimental results revealed that AGG occurred within 60 s when the ST specimen was annealed at 1353 K, while NGG occurred in the STD-50% specimen, even after a longer holding time (Figure 3b). The average diameters of the AlN particles in ST and STD-50% specimens after annealing at 1353 K for 0 s were evaluated to be around 60 and 30 nm, correspondingly. It was reported that the dissolution time and the size of the precipitates had a parabolic relationship [23]. Accordingly, the dissolution time of the AlN particles in the ST specimen was around four times that in the STD-50% specimen. Therefore, it was possible that the fine AlN precipitates in the STD-50% specimen dissolved very quickly during annealing, which made the grain growth become free and change to the NGG mode. However, since the dissolution of the coarse precipitates in the ST specimen took a longer time, AGG could occur. To quickly dissolve the relatively large AlN precipitates in the ST specimen, this required higher temperatures. This may promote the AGG-to-NGG transition temperature to the higher temperature in the ST specimen.

According to the above discussion, cold-deformation results in the fine and near-equilibrium precipitation of the AlN particles, which are readily dissolved during the heating process, due to the increase in the solubilities of Al and N in austenite. However, non-equilibrium and coarse AlN precipitates formed in the undeformed specimen, which were resistant to dissolution during the heating process. These should be the reasons for the suppression of the AGG temperature range by the prior cold-deformation.

#### **4. Summary**

In summary, the study of austenite grain growth behavior in a solution-treated low-alloy steel has shown that cold-deformation prior to annealing shifts the abnormal grain growth (AGG) temperature range of austenite toward a lower temperature region. The amount of shift increases with the increase of the reduction ratio. The lowered AGG occurrence temperature is attributed to the effects of cold deformation on the evolution of the AlN precipitates during annealing. TEM observations confirmed that a much lower volume fraction than the equilibrium and coarse AlN precipitates formed in the undeformed steel, which is resistant to dissolution during annealing. However, prior cold-deformation results in the fine and near-equilibrium precipitation of AlN, which is readily dissolved at elevated temperatures. This results in the shift of the AGG temperature range to lower temperatures from the application of prior cold-deformation.

**Author Contributions:** X.Z., K.M. and M.O. conceived and designed the experiments; X.Z. performed the experiments and collected the data; X.Z., and K.M. analyzed the data; X.Z. wrote the paper; K.M. and M.O. discussed with the results and revised the manuscript.

**Funding:** This work was partially supported by "Nanotechnology Platform" Program of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

**Conflicts of Interest:** The authors declare no conflict of interest.

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