Integrates the differenced time series and recover the original data (inverse operator of DIFF).
Y is the differenced univariate time series data (a one dimensional array of cells (e.g. rows or columns)).
Order is the time order in the data series (i.e. the first data point's corresponding date (earliest date=1 (default), latest date=0)).
|1||ascending (the first data point corresponds to the earliest date) (default)|
|0||descending (the first data point corresponds to the latest date)|
K is the seasonal difference order of the input time series (e.g. K=0 (no lag), K=1 (1st lag), etc.) If missing, the default value of one is assumed.
D is the number of repeated differencing (e.g. d=0 (none), d=1 (difference once), 2= (difference twice), etc.). If missing, the default value of one is assumed.
XO is the initial (un-differenced) univariate time series data (a one dimensional array of cells (e.g. rows or columns)). If missing, zeros are assumed.
- The input (differenced) time series (i.e. Y) is defined as follow:
- $\left[Y_t\right]$ is the differenced time series.
- $\left[X_t\right]$ is the input time series.
- $L$ is the lag (backward shift or backshift) operator.
- $k$ is the seasonal difference order.
- $d$ is the number of repeated differencing.
- The initial values array is assumed to end at the last non-missing value in the difference array start
- If the difference cell range includes missing values at the beginning, the result array will substute the initial values for missing ones; as we assume the initial values cover up to 1st non-missing value.
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g. #N/A) at either end.
- The integral operator requires an SxD points in the initial time series (i.e. XO). If XO is missing or has fewer points, points with zeros values are appended.
- The time order (i.e. ascending or descending) for the initial (un-differenced) time series (XO) is assumed the same as the differenced time series (Y).
- Similar to the DIFF operator, INTG can be cascaded (i.e. INTG(INTG(INTG...)))), but care must be taken when you specify the initial time series for each level.
- The lag order (i.e. k) must be non-negative and smaller than the time series size.
$$0 \leq K \leq T-1 $$
- D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
- James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896
- Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740
- Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848
- Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568