QuanTAlib/docs/indicators/averages/ema/calculation.md

EMA - Calculation Method

The EMA calculation utilizes a weighting multiplier, typically denoted as the smoothing factor (alpha). This factor is calculated as:

alpha = \frac{2}{period + 1}

where 'period' represents the chosen period for the EMA.

The general formula for EMA required for arithmetic operations:

EMA_n = (data_{n} \times alpha) + (EMA_{n-1} \times (1 - alpha))

or in optimized form (requires only three arithmetic operations instead of four):

EMA_n = {alpha}\times ({data_{n}} - EMA_{n-1}) + EMA_{n-1}

When calculating the Exponential Moving Average (EMA) and there is not enough data (n < period), several approaches can be considered. Each method has its own pros and cons:

1. Assume all previous values were 0

EMA_0 = 0
EMA_n = alpha \times (data_n - EMA_{n-1}) + EMA_{n-1}

• Will lead to significant underestimation of EMA in early periods

2. Calculate as if all previous values were the same as the first value

EMA_0 = data_0
EMA_n = alpha \times (data_n - EMA_{n-1}) + EMA_{n-1}

• Will overestimate early EMA if initial data point is far from representative

3. Use SMA instead of EMA for the first period

$EMA_n = \left{ \begin{array}{cl} \frac{1}{p}\left( data_{n}-data_{n-p}\right)+SMA_{n-1} & : \ n \leq period \ {alpha}\times ({data_{n}} - EMA_{n-1}) + EMA_{n-1} & : \ n > period \end{array} \right.$

• Creates a discontinuity when switching from SMA to EMA

Conclusion

The choice of method depends on the specific requirements of the application:

• Method 1 is suitable for applications where underestimation in early periods is acceptable.
• Method 2 is beneficial when a smooth transition is crucial and the initial data point is representative.
• Method 3 is appropriate when simplicity is preferred and a clear distinction between SMA and EMA is acceptable.
• Method 4 offers a good balance between adaptability and maintaining the EMA concept, but may require additional explanation to users.