# A New Period-Sequential Index Forecasting Algorithm for Time Series Data
## 2 Theoretical Model  
### Period-Sequential Index (PSI) Algorithm  
Finding index-values implicitly carrying structure information, a period-sequential index (PSI) algorithm, is proposed to predict the time series data. The index-values cover the complete period, while the period index and sequential index, as well as small period index and super sequential index, are introduced to describe the dataset structure information in vertical and horizontal dimensions, respectively. In this way, for time series data, the following year's dataset can be predicted using only two consecutive years of reference historical datasets. Figure 1 shows the schematic diagram of the PSI algorithm. \(H_{-2}\), \(H_{-1}\) denote reference historical periods, i.e., the year before last and last year. \(H_{0}\) represents the forecasting period. The period for \(H_{-2}\), \(H_{-1}\), and \(H_{0}\) is uniform, and defined as \(T\) in this paper. At historical time of \(t\), \(Pl(t)\), \(SI(t)\), \(pi(t)\), and \(si(t)\) describe the period index, sequential index, small period index, and super sequential index, respectively. We assume that the forecasting value follows the measurement Equation (1),  
\[F(t_{i})=G(Pl(t_{i}-2T)\,Pl(t_{i}-T)\,SI(t_{i}-2T)\,SI(t_{i}-T)\,pi(t_{i}-2T)\,pi(t_{i}-T)\, si(t_{i}-T)), \tag{1}\]  
where \(F(t_{i})\) is the forecasting value at time of \(t_{i}\) (\(i=1\), \(2\)... \(N\)) during \(H_{0}\) period. \(N\) represents the number of model forecasting samples. \(Pl(t_{i}-2T)\), \(Pl(t_{i}-T)\), \(SI(t_{i}-2T)\), \(SI(t_{i}-T)\), \(pi(t_{i}-2T)\), \(pi(t_{i}-T)\), \(si(t_{i}-2T)\), and \(si(t_{i}-T)\), as the eight variables in Equation (1), are described as follows:  
(1) Period index  
The period index indicates the relationship between the reference historical data and the reference value which can be explained through Equations (2) and (3),  
\[Pl(t_{i}-2T)=\frac{y(t_{i}-2T)}{K_{-2}}, \tag{2}\]  
Figure 1: The schematic diagram of the period-sequential index algorithm (PSI) algorithm. \[PI(t_{i}-T)=\frac{y(t_{i}-T)}{K_{-1}}, \tag{3}\]  
where \(y(t_{i}-2T)\) and \(y(t_{i}-T)\) describe the reference historical data at time of \(t_{i}-2T\) and \(t_{i}-T\), respectively. \(K_{-2}\) and \(K_{-1}\) are reference functions of period index. A standard period average is originally set to be a reference function of period index, and where it is defined as a constant. (2) Sequential Index  
The sequential index indicates the relationship between two adjacent reference historical data with the defined time steps. It is calculated through Equations (4) and (5). \[SI(t_{i}-2T)=\frac{y(t_{i+1}-2T)}{y(t_{i}-2T)} \tag{4}\]  
\[SI(t_{i}-T)=\frac{y(t_{i+1}-T)}{y(t_{i}-T)} \tag{5}\]  
(3) Small Period Index  
The small period index indicates the relationship between the reference historical data and the reference value, which can be explained through Equations (6) and (7),  
\[pi(t_{i}-2T)=\frac{y(t_{i}-2T)}{k_{-2}}, \tag{6}\]  
\[pi(t_{i}-T)=\frac{y(t_{i}-T)}{k_{-2}}, \tag{7}\]  
where \(k_{-1}\) and \(k_{-2}\) are reference functions of a small period index. Here a small period (such as three months, because of seasonal factors) average, is originally set to be a reference function of small period index. (4) Super Sequential Index  
The super sequential index indicates the relationship between two interval reference historical data with the defined steps. It is calculated through Equations (8) and (9).