如何在PHP中优化指数移动平均算法？

Question

I'm trying to retrieve the last EMA of a large dataset (15000+ values). It is a very resource-hungry algorithm since each value depends on the previous one. Here is my code :

$k = 2/($range+1);
for ($i; $i<$size_data; ++$i) {
    $lastEMA = $lastEMA + $k * ($data[$i]-$lastEMA);
}

What I already did:

1. Isolate $k so it is not computed 10000+ times
2. Keep only the latest computed EMA, and not keep all of them in an array
3. use for() instead of foreach()
4. the $data[] array doesn't have keys; it's a basic array

This allowed me to reduced execution time from 2000ms to about 500ms for 15000 values!

What didn't work:

1. Use SplFixedArray(), this shaved only ~10ms executing 1,000,000 values
2. Use PHP_Trader extension , this returns an array containing all the EMAs instead of just the latest, and it's slower

Writing and running the same algorithm in C# and running it over 2,000,000 values takes only 13ms! So obviously, using a compiled, lower-level language seems to help ;P

Where should I go from here? The code will ultimately run on Ubuntu, so which language should I choose? Will PHP be able to call and pass such a huge argument to the script?

Answer 1

Clearly implementing with an extension gives you a significant boost. Additionally the calculus can be improved as itself and that gain you can add in whichever language you choose.

It is easy to see that lastEMA can be calculated as follows:

$lastEMA = 0;
$k = 2/($range+1);
for ($i; $i<$size_data; ++$i) {
    $lastEMA = (1-$k) * $lastEMA + $k * $data[$i];
}

This can be rewritten as follows in order to take out of the loop as most as possible:

$lastEMA = 0;
$k = 2/($range+1);
$k1m = 1 - $k;
for ($i; $i<$size_data; ++$i) {
    $lastEMA = $k1m * $lastEMA + $data[$i];
}
$lastEMA = $lastEMA * $k;

To explain the extraction of the "$k" think that in the previous formulation is as if all the original raw data are multiplied by $k so practically you can instead multiply the end result.

Note that, rewritten in this way, you have 2 operations inside the loop instead of 3 (to be precise inside the loop there are also $i increment, $i comparison with $size_data and $lastEMA value assignation) so this way you can expect to achieve an additional speedup in the range between the 16% and 33%.

Further there are other improvements that can be considered at least in some circumstances:

Consider only last values

The first values are multiplied several times by $k1m = 1 - $k so their contribute may be little or even go under the floating point precision (or the acceptable error).

This idea is particularly helpful if you can do the assumption that because if you consider only the last $n values the error that you make is

$err = $EMA_of_discarded_data * (1-$k) ^ $n .

So if order of magnitude is broadly the same we can tell that the relative error done is

$rel_err = $err / $lastEMA = $EMA_of_discarded_data * (1-$k) ^ $n / $lastEMA

that is almost equal to simply (1-$k) ^ $n .

Under the assumption that :

• Let's say that you can accept a relative error $rel_err
  • you can safely consider only the last $n values where (1 - $k)^$n < $rel_err.
  • Means that you can pre-calculate (before the loop) $n = log($rel_err) / log (1-$k) and compute all only considering the last $n values.
  • If the dataset is very big this can give a sensible speedup.
• Consider that for 64 bit floating point numbers you have a relative precision (related to the mantissa) that is 2^-53 (about 1.1e-16 and only 2^-24 = 5.96e-8 for 32 bit floating point numbers) so you cannot obtain better than this relative error
  • so basically you should never have an advantage in calculating more than $n = log(1.1e-16) / log(1-$k) values.
  • to give an example if $range = 2000 then $n = log(1.1e-16) / log(1-2/2001) = 36'746.
    • I think that is interesting to know that extra calculations would go lost inside the roundings ==> it is useless ==> is better not to do.
• now one example for the case where you can accept a relative error larger than floating point precision $rel_err = 1ppm = 1e-6 = 0.00001% = 6 significant decimal digits you have $n = log(1.1e-16) / log(1-2/2001) = 13'815
  • I think is quite a little number compared to your last samples numbers so in that cases the speedup could be evident (I'm assuming that $range = 2000 is meaningful or high for your application but thi I cannot know).
• just other few numbers because I do not know what are your typical figures:
  • $rel_err = 1e-3; $range = 2000 => $n = 6'907
  • $rel_err = 1e-3; $range = 200 => $n = 691
  • $rel_err = 1e-3; $range = 20 => $n = 69
  • $rel_err = 1e-6; $range = 2000 => $n = 13'815
  • $rel_err = 1e-6; $range = 200 => $n = 1'381
  • $rel_err = 1e-6; $range = 20 => $n = 138

If the assumption cannot be taken things are less easy but since the advantage cam be significant it can be meaningful to go on:

• we need to re-consider the full formula: $rel_err = $EMA_of_discarded_data * (1-$k) ^ $n / $lastEMA
• so $n = log($rel_err * $lastEMA / $EMA_of_discarded_data) / log (1-$k) = (log($rel_err) + log($lastEMA / $EMA_of_discarded_data)) / log (1-$k)
• the central point is to calculate $lastEMA / $EMA_of_discarded_data (without actually calculating $lastEMA nor $EMA_of_discarded_data of course)
  • one case is when we know a-priori that for example $EMA_of_discarded_data / $lastEMA < M (for example M = 1000 or M = 1e6)
    • in that case $n < (log($rel_err/M)) / log (1-$k)
  • if you cannot give any M number
    • you have to find a good idea to over-estimate $EMA_of_discarded_data / $lastEMA
    • one quick way could be to take M = max(data) / min(data)

Parallelization

The calculation can be re-written in a form where it is a simple addition of independent terms:

$lastEMA = 0;
$k = 2/($range+1);
$k1m = 1 - $k;
for ($i; $i<$size_data; ++$i) {
    $lastEMA += $k1m ^ ($size_data - 1 - $i) *  $data[$i];
}
$lastEMA = $lastEMA * $k;

So if the implementing language supports parallelization the dataset can be divided in 4 (or 8 or n ...basically the number of CPU cores available) chunks and it can be computed the sum of terms on each chunk in parallel summing up the individual results at the end.

I do not go in detail with this since this reply is already terribly long and I think the concept is already expressed.

Answer 2

Building your own extension definitely improves performance. Here's a good tutorial from the Zend website.

Some performance figures: Hardware: Ubuntu 14.04, PHP 5.5.9, 1-core Intel CPU@3.3Ghz, 128MB RAM (it's a VPS).

• Before (PHP only, 16,000 values) : 500ms
• C Extension, 16,000 values : 0.3ms
• C Extension (100,000 values) : 3.7ms
• C Extension (500,000 values) : 28.0ms

But I'm memory limited at this point, using 70MB. I will fix that and update the numbers accordingly.