# A Statistical Approach to Discovering Process Regime Shifts and Their Determinants
## 3 Methodology
### Statistical Modeling and Testing  
The above procedure requires an appropriate statistical modeling approach and a general testing procedure accommodating varied modeling bases. We discuss both the issues in this section; however, we first refer to the following notations needed in our discussion:  
\(i:\) Time index used with the time series event log data (i.e., \(i=1,2,\ldots,n\) for \(n\) observations) \(p^{\prime}_{i}:\) Time series event log data of evaluated dependent property at time \(i\)  
\(d_{i}:\) A vector of determinant regressors of size \(|D^{\prime}|\times 1\), at the time \(i\)  
\(\alpha_{i}:\) A vector of model coefficients for all \(d_{i}\)  
\(\varepsilon_{i}:\) Random noise at the time \(i\)  
\(m:\) The number of regime shifts in data  
\(M_{m,n}:\) Set of all regime shift points  
\(s:\) Regime index. With \(m\) regime shift points, we have \(m+1\) regimes. In the above notations, \(p^{\prime}_{i}\) is the time series event log for the dependent system property and \(d_{i}\) is the same for the considered determinants. As demonstrated in the initialization phase of the algorithm (Figure 3), we identified the need for an evaluation criterion, which is used to form the base statistical model. Specifically, the base model form is dependent on the nature of the considered properties, determinants and evaluation criteria. To illustrate, we present two scenarios and their corresponding general model forms. The first scenario involves a property \(p_{i}\in\mathbb{R}\), which is assumed to be normally distributed. Figure 3: Procedure to find regime shifts and related determinants. This may be the case of a product's quality characteristics produced by a manufacturing process. The second one is the case in which \(p_{i}\in I\) represents some count data, which thus follow a Poisson or a negative binomial distribution. This case is prevalent in, e.g., traffic accident data involving accident counts. For the two cases, we require separate modeling bases, which in generalized terms are represented, respectively, as:  
\[p_{i}^{\prime}=d_{i}^{T}\alpha_{i}+\varepsilon_{i}\ \ \ \ \ (i=1,2,\ldots,n) \tag{1a}\] \[\log(p_{i}^{\prime})=d_{i}^{T}\alpha_{i}+\varepsilon_{i}\ \ \ \ \ (i=1,2,\ldots,n) \tag{1b}\]  
As demonstrated above, modeling bases for other situations can be formed. Although the above generalized and fully defined models show a relationship structure between the independent and all the dependent determinant variables, the initial requirement of the procedure is the minimal model form, which is used to discover all regime changes in \(p_{i}^{\prime}\). Thus, the above generalized but fully defined model form can be rewritten for its minimal form as:  
\[p_{i}^{\prime}=t_{i}\alpha_{i}+\varepsilon_{i}\ \ \ \ \ (i=1,2,\ldots,n) \tag{2a}\] \[\log(p_{i}^{\prime})=t_{i}\alpha_{i}+\varepsilon_{i}\ \ \ \ \ (i=1,2,\ldots,n) \tag{2b}\]  
Here, we removed variables for all considered determinants in Equation (2a) and Equation (2b) and replaced them with a simple time index term \(t_{i}\). This term in the minimal model captures any trend in the data. Later, respective determinant regressor variables are added when evaluating their roles. **Statistical testing:** For any of the above model structures, the regime shifts' problem is simply to hypothesize any change in the coefficients \(\alpha_{i}\) of the determinants. In other words, if the coefficients remain stable throughout the timeline, there is no regime change. On the other hand, any change in even one of the coefficients means a regime change. Accordingly, we define the null and the alternative hypotheses as follows:  
\[ \begin{array}{c}H_{0}:\ \ \ \ \alpha_{i}=\alpha_{0}\ \ (i=1,\ldots,n)\\ H_{1}:\ \ \ \ \text{at least one}\ \alpha_{i}\
eq\alpha_{0}\ \ (i=1,\ldots,n)\end{array}  \tag{3}\]  
For the general case of \(m\) regime shifts, we have \(m\) combinations of stable coefficients, or, simply, there are \(m+1\) regimes.