Integrates the differenced time series and recovers the original data (inverse operator of DIFF).
INTG(Y, Order, K, D, $X_o$)
- is the differenced univariate time series data (a one-dimensional array of cells (e.g. rows or columns)).
- is the time order in the data series (i.e. the first data point's corresponding date (earliest date=1 (default), latest date=0)).
Order Description 1 ascending (the first data point corresponds to the earliest date) (default) 0 descending (the first data point corresponds to the latest date)
- 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.
- 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.
- 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 substitute 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
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