NxEMA - exponentially-weighted moving (rolling/running) average

Returns the exponentially-weighted moving (rolling/running) average using the previous N data points.

 

Syntax

NxEMA(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 smoothing period expressed in number of data points (e.g. days).

Variant is the variant/type of the exponentially-weighted moving average (i.e. 0= Simple (default), 1= Double, 2=Triple, 3=Zero-lagged). If missing or omitted, a simple exponentially-weighted moving average (i.e. Variant = 0) is assumed.

Value Description
0 Plain/simple exponentially-weighted moving average (EMA)
1 Double exponentially-weighted moving average (D-EMA)
2 Triple exponentially-weighted moving average (T-EMA)
3 Zero-lagged exponential-weighted moving average (ZLEMA)

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 formula of exponential moving average definition is expressed in technical analysis terms as follow: $$ \textrm{EMA}_t = \textrm{EMA}_{t-1} + \alpha \times ( x_t - \textrm{EMA}_{t-1}) $$ Where:
    • $\textrm{EMA}_t$ is the exponential moving average at time t.
    • $x_t$ is the value of the time series at time t.
    • $\alpha$ is the smoothing factor (between 0 and 1) represents the degree of weighting decrease. For EMA, the $\alpha$ is exp[ressed as follow: $$ \alpha = \frac{2}{N+2} $$ Where:
      • $N$ is the smoothing period expressed in number of data points (e.g. days)
    • The exponential moving average with a smooth period of N data points is expressed by N-period EMA.
  4. The double exponential moving average (aka. DEMA or D-EMA) is expressed as follow: $$\textrm{D-EMA}_t = 2 \times \textrm{EMA}_t -\textrm{EMA}(\textrm{EMA})_t$$ Where:
    • $\textrm{EMA}(\textrm{EMA})$ is the exponential moving average of the exponential moving average.
  5. The triple exponential moving average (aka. TEMA or T-EMA) is expressed as follow: $$\textrm{T-EMA} = 3\times \textrm{EMA}_t -3\times\textrm{EMA}(\textrm{EMA})_t + \textrm{EMA}(\textrm{EMA}(\textrm{EMA}))_t$$
  6. For zero-lag exponential moving average (ZLEMA),  $$ k = (N-1)/2 $$ $$ Y_t = x_t + (x_t - x_{t-k})$$ $$ \textrm{ZLEMA}_t = \textrm{EMA}_t^N (Y_t) $$ Where:
    • $k$ is the number of periods used to remove cumulative effect of moving average.
    • $Y_t$ is the de-lagged data. Data is de-lagged by removing the data from k-periods ago, thus removing the cumulative effect of the moving average.
  7. In the case the smoothing period ($N$) is an even number, then $k$ and $Y_t$ are calculated as follow: $$ k = \frac{N}{2}$$ $$ Y_t = 2x_t - \left(\frac{x_{t-k}+x_{t-k-1}}{2}\right)$$
  8. The ZLEMA technical indicator was created by John Ehlers and Ric Way.
  9. The NxEMA function is available starting with version 1.66 PARSON.

Files Examples

References

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