# DIFF - Time Series Difference Operator

Returns an array of cells for the differenced time series (i.e., $(1-L^S)^D)$.

## Syntax

DIFF(X, Order, K, D)

X
is the 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 (e.g., K = 0 (no lag), S = 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.

## Remarks

1. The DIFF operator is defined as follows: $$Y_t=\left(1-L^k\right)^d X_t$$ Where:
• $\left[y_t\right]$ is the difference time series.
• $\left[x_t\right]$ is the input time series.
• $L$ is the lag operator.
• $k$ is the seasonality length.
• $d$ is the difference order.
2. The size of the output differenced time series is equal to the input time series, but with the first $s \times d$ observations are set to missing (i.e., #N/A).
3. The seasonal difference order (i.e., $k$) must be non-negative and smaller than the time series size (i.e., $T$). $$0 \leq k \leq T-1$$
4. The input time series is homogenous and equally spaced.
5. The time series may include missing values (e.g., #N/A) at either end.