Calculating exponential moving averages

Question

I'm wanting to calculate exponential moving averages of a variable, distance. Is the logic (and math) correct in the below code?

time and lastTime are millisecond-precise timestamps in seconds – the former is the current time, the latter is the time of the last calculations.

lastA etc are the exponential moving averages from the last calculations.

a etc will be left with the calculated exponential moving averages.

var distance = ...;
var a = Math.pow(1.16, -(time-lastTime)),
    b = Math.pow(1.19, -(time-lastTime)),
    c = Math.pow(1.22, -(time-lastTime)),
    d = Math.pow(1.26, -(time-lastTime)),
    e = Math.pow(1.30, -(time-lastTime)),
    f = Math.pow(1.35, -(time-lastTime)),
    g = Math.pow(1.40, -(time-lastTime));
a = a*lastA + (1-a)*distance;
b = b*lastB + (1-b)*distance;
c = c*lastC + (1-c)*distance;
d = d*lastD + (1-d)*distance;
e = e*lastE + (1-e)*distance;
f = f*lastF + (1-f)*distance;
g = g*lastG + (1-g)*distance;

Answer 1

This is border line a bad question, as not enough code is given to properly review it.

The variables a -> g look terrible, I would create an array with the numbers you need:

var dataPoints = [1.16,1.19,1.22,1.26,1.30,1.35,1.40];

Then I would would loop over those points and create an averages object

var averages = {},
    value, x;
for(var i = 0, length = dataPoints.length ; i < length ; i++ ){
  value = dataPoints[i];
  x = Math.pow(value, -(time-lastTime));
  averages[value] = x * lastAverages[value] + (1-x) * distance;
}

I cannot tell whether the math is correct, if it is not correct, then this question does not belong here :)

Answer 2

Either I'm confused by your notation, or you may have implemented something completely different from an exponential moving average, which is traditionally defined as

\$S_{t} = \alpha Y_{t-1} + (1-\alpha) S_{t-1}\$

where

• \$\alpha\$ is the decay rate
• \$Y_{t}\$ is the value at time \$t\$
• \$S_{t}\$ is the exponential moving average at time \$t\$.

How do your variables correspond to those in the definition? Let's just consider one of your letters instead of all seven:

var distance = ...;
var a = Math.pow(1.16, -(time-lastTime));
a = a*lastA + (1-a)*distance;

I'm guessing

• a corresponds to \$\alpha\$, and you adjust the decay per timeslice based on the duration of the timeslice
• lastA corresponds to \$Y_{t-1}\$
• distance corresponds to \$S_{t-1}\$

But then, I'm confused:

1. What's the purpose of the seven letters a … g? To track the results using multiple decay rates? If so, wouldn't the different decay rates result in a different series S_t for each decay rate?
2. Why do all seven cases all share the same distance — isn't it the point to have a different distance series for each case?
3. Why do you assign the final result to a (= the decay rate) rather than to distance or something?