# NxMA - Moving (rolling) average using prior data points

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

1. The time series is homogeneous or equally spaced.
2. The time series may include missing values (e.g., #N/A) at either end.
3. 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).
4. 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.
5. 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 $\alpha = 1/N$.
6. 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 ${\frac {N(N+1)}{2}}$.
7. The NxMA function is available starting with version 1.66 PARSON.