# What is the benefit of using Euler's number e to calculate Exponential Moving Average (EMA)?

I have found three use cases that people use EMA or Exponential smoothing to analyze time-series data in the computer world. The use case 1 & 2 uses very simple $$\alpha$$, and the last one uses a complicated expression to act as $$\alpha$$.

When the sequence of observations begins at time $$t=0$$, the simplest form of exponential smoothing is given by the formulas, which the raw data sequence is often represented by $$x_t$$, and the output of the exponential smoothing algorithm is commonly written as $$s_{t}$$

$$s_{0}=x_{0}\\ s_{t}=\alpha x_{t}+(1-\alpha )s_{t-1}$$

where $$\alpha$$ is the smoothing factor, and $$0<\alpha <1$$

Use case 1 RFC 2988 Computing TCP’s Retransmission Timer

... a TCP sender maintains two state variables, SRTT (smoothed round-trip time) and RTTVAR (round-trip time variation).

... When the first RTT measurement R is made, the host MUST set

SRTT <- R


When a subsequent RTT measurement R’ is made, a host MUST set

RTTVAR <- (1 - beta) * RTTVAR + beta * |SRTT - R’|
SRTT <- (1 - alpha) * SRTT + alpha * R’


The above SHOULD be computed using alpha=1/8 and beta=1/4...

Use case 2 Java Virtual Machine source code

#define DEFAULT_ALPHA_VALUE 0.7

// ...

void AbsSeq::add(double val) {
if (_num == 0) {
// ...
} else {
// otherwise, calculate both
_davg = (1.0 - _alpha) * val + _alpha * _davg;
// ...
}
}


The default value of $$\alpha$$ is 0.7 in this case.

Use case 3 How Linux calculate system load avgerage

static inline unsigned long
calc_load(unsigned long load, unsigned long exp, unsigned long active)
{

newload = load * exp + active * (FIXED_1 - exp);
if (active >= load)

return newload / FIXED_1;
}


The $$\alpha$$(corresponding the 2nd parameter exp) in this situation is complicated, which exp is

$$exp=e^{-\frac{\Delta{t}}{T}}\\$$

where $$e$$ is Euler's number, and $$\Delta{t}$$ is the sampling interval(e.g. 5 second), and $$T$$ means the exponential moving average of system load is in $$T$$ time constant(e.g. 1 minute, 5 minute or 15 minute).

I wonder why Linux prefer to use such a complex expression concerning Euler's number $$e$$ to act as simple, pure decimal $$\alpha(0<\alpha <1)$$.

• I haven't looked deeply into it but there is every reason to believe that the value of alpha in the third model is not to be chosen, that the model is simply subject to natural constraints whereby the value of alpha is what it has to be. Jun 28, 2021 at 10:18