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 ${\displaystyle {\frac {N(N+1)}{2}}}$.
7. The NxMA function is available starting with version 1.66 PARSON.

Files Examples

Related Links

• Wikipedia - Moving average.

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.