Returns the moving (rolling/running) average using the previous m data points.

## Syntax

**NxMA**(**X**, Order, **N**, Variant, Return)

**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). **N**- is the number of data points (e.g., days) in a given period.
**Variant**- is the variant/type of the moving average (i.e., 0 = Simple (default), 1 = Cumulative, 2 = Modified, 3 = Weighted). If missing or omitted, a simple moving average (i.e., Variant = 0) is assumed.
Value Description 0 Simple moving average (SMA) (default). 1 Cumulative moving average (CMA). 2 Modified moving average (MMA). 3 Weighted moving average where weights decrease in arithmetic progression. **Return**- is the return type of the function: 0 = last/most recent value (default), 1 = filtered time series (array).
Value Description 0 Return smoothed value of the Last/most-recent observation (default). 1 Return the whole smoothed time series (array).

## Remarks

- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.
- The simple moving average (SMA) is given by: $$\textrm{SMA}_t = \frac{\sum_{i=0}^N x_{t-i}}{N}$$ Where:
- $\textrm{SMA}_t$ is the simple moving average value at time $t$.
- $x_t$ is the value of the time series at time $t$.
- $N$ is the rolling window size (aka. number of points in the moving average).

- The cumulative moving average (CMA) is defined as follows: $$\textrm{CMA}_t = \frac{\sum_{i=1}^t x_i}{t}$$ Where:
- $\textrm{CMA}_t$ is the cumulative moving average value at time $t$.
- $t$ is the given time.
- CMA is basically the average of all of the data points up until the current datum point.

- The Modified moving average (MMA) is given as follows: $$\textrm{MMA}_t = \frac{(N-1)\times \textrm{MMA}_{t-1}+x_t}{N}$$ Where:
- $\textrm{MMA}_t$ is the modified moving average at time $t$.
- $N$ is the rolling window size (aka. number of points in the moving average).
- The MMA is also known as the rolling-moving average (RMA), or the smoothing-moving average (SMMA).
- The MMA is basically a simple exponential smoothing with $\alpha = 1/N$.

- The weighted moving average (WMA) has weights that decrease in arithmetic progression. The WMA can be expressed as follow: $$\textrm{WMA}_t = \frac{N x_t + (N-1)x_{t-1} + (N-2) x_{t-2} + \cdots + 2 x_{t-N+2} + x_{t-N+1}}{N + (N-1) + (N-2) + \cdots + 2 + 1}$$ Where:
- $\textrm{WMA}_t$ is the weighted moving average value at time $t$.
- $x_t$ is the value of the time series at time $t$.
- $N$ is the rolling window size (i.e., the number of data points in the average).
- The denominator is a triangle number equal to ${\displaystyle {\frac {N(N+1)}{2}}}$.

- The NxMA function is available starting with version 1.66 PARSON.

## Files Examples

## Related Links

## References

- R.J. Hyndman, A.B. Koehler, "Another look at measures of forecast accuracy", International Journal of Forecasting, 22 (2006), pp. 679-688.
- 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.
- D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906.

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