# 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)
1 Cumulative moving average (CMA)
2 Modified moving average (MMA)
3 Weighted moving average where weights decrease in arithmetic progression
Return
is 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
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 follow: $$\textrm{CMA}_t = \frac{\sum_{i=1}^t x_i}{t}$$ 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 follow: $$\textrm{MMA}_t = \frac{(N-1)\times \textrm{MMA}_{t-1}+x_t}{N}$$ 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 rolling-moving average (RMA), or 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 arithematic 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. number of data points in the average).
• The denominator is a triangle number equal to ${\displaystyle {\frac {N(N+1)}{2}}.}$
7. The NxMA function is available starting with version 1.66 PARSON.