# INTG - Time Series Integration Operator

Integrates the differenced time series and recovers the original data (inverse operator of DIFF).

## Syntax

INTG(Y, Order, K, D, $X_o$)
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)).
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)
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.
$X_o$
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.

## Remarks

1. The input (differenced) time series (i.e. Y) is defined as follow:

$$Y_t=\left(1-L^k\right)^d X_t$$

Where:

Where:
• $\left[Y_t\right]$ 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.
2. The initial values array is assumed to end at the last non-missing value in the difference array start
3. 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.
4. The time series is homogeneous or equally spaced.
5. The time series may include missing values (e.g. #N/A) at either end.
6. 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.
7. 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).
8. 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.
9. The lag order (i.e. k) must be non-negative and smaller than the time series size.

$$0 \leq K \leq T-1$$

## Examples

Example 1:

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A B C D
Date Data DIFF INTG
January 10, 2008 -0.30 #N/A -0.3
January 11, 2008 -1.28 -0.98 -1.28
January 12, 2008 0.24 1.52 0.24
January 13, 2008 1.28 1.04 1.28
January 14, 2008 1.20 -0.08 1.20
January 15, 2008 1.73 0.53 1.73
January 16, 2008 -2.18 -3.91 -2.18
January 17, 2008 -0.23 1.95 -0.23
January 18, 2008 1.10 1.33 1.10
January 19, 2008 -1.09 -2.19 -1.09
January 20, 2008 -0.69 0.40 -0.69
January 21, 2008 -1.69 -1.00 -1.69
January 22, 2008 -1.85 -0.16 -1.85
January 23, 2008 -0.98 0.87 -0.98
January 24, 2008 -0.77 0.21 -0.77
January 25, 2008 -0.30 0.47 -.0.30
January 26, 2008 -1.28 -0.98 -1.28
January 27, 2008 0.24 1.52 0.24
January 28, 2008 1.28 1.04 1.28
January 29, 2008 1.20 -0.08 1.20
January 30, 2008 1.73 0.53 1.73
January 31, 2008 -2.18 -3.91 -2.18
February 1, 2008 -0.23 1.95 -0.23
February 2, 2008 1.10 1.33 1.10
February 3, 2008 -1.09 -2.19 -1.09
February 4, 2008 -0.69 0.40 -0.69
February 5, 2008 -1.69 -1.00 -1.69
February 6, 2008 -1.85 -0.16 -1.85
February 7, 2008 -0.98 0.87 -0.98