Entropy. I still shiver when I hear that word, since I never fully understood that concept. Today marks the first time I was kind of forced to look into it in more detail. And by “in detail”, I mean I found a StackOverflow question that had something to do with a problem I am having (sound familiar?). The problem was is about complexity of time series and one of the suggested methods was Sample Entropy.

#used packages
library(tidyverse)  # for data wrangling
library(pracma) # for Sample Entropy code
library(Rcpp) # integrate C++ in R

# Sample Entropy

Sample entropy is similar to Approximate Entropy and used for assessing the complexity of time-series. The less “complex” the time series is the easier it may be to forecast it.

# Sample Entropy in R

I found two packages that implement sample entropy, pracma and nonlinearTimeSeries. I looked into nonlinearTimeSeries first but the data structure seemed a bit too complex on first glance (for me!). So I decided to go for pracma. When you are ok with the default parameters, then you can simple call sample_entropy().

set.seed(1886)
ts <- rnorm(200)
sample_entropy(ts)
## [1] 2.302585

Simple. Problem is, I need to calculate the sample entropy of 150,000 time series. Can the function handle that in reasonable time?

#calculate sample entropy for 500 time series
set.seed(1886)
A <- matrix(runif(500*200),500,200)
system.time(apply(A,1,function(x)sample_entropy(x)))
##    user  system elapsed
##  40.775   0.004  40.782

This translates to several hours for 150,000 time series, which is kind of not ok. I would prefer it a little faster.

# Sample Entropy with Rcpp

Sample Entropy is actually super easy to implement. So I used my rusty c++ skills and implemented the function myself with the help of Rcpp.

cppFunction(
"double SampleEntropy(NumericVector data, int m, double r, int N, double sd)
{
int Cm = 0, Cm1 = 0;
double err = 0.0, sum = 0.0;

err = sd * r;

for (unsigned int i = 0; i < N - (m + 1) + 1; i++) {
for (unsigned int j = i + 1; j < N - (m + 1) + 1; j++) {
bool eq = true;
//m - length series
for (unsigned int k = 0; k < m; k++) {
if (std::abs(data[i+k] - data[j+k]) > err) {
eq = false;
break;
}
}
if (eq) Cm++;

//m+1 - length series
int k = m;
if (eq && std::abs(data[i+k] - data[j+k]) <= err)
Cm1++;
}
}

if (Cm > 0 && Cm1 > 0)
return std::log((double)Cm / (double)Cm1);
else
return 0.0;

}"
)

The code can also be found on github.

Let’s see if it produces the same output as the pracma version.

set.seed(1886)
ts <- rnorm(200)
sample_entropy(ts)
## [1] 2.302585
SampleEntropy(ts,2L,0.2,length(ts),sd(ts))
## [1] 2.302585

Perfect. Now let’s check if we gained some speed up.

system.time(apply(A,1,function(x)SampleEntropy(x,2L,0.2,length(ts),sd(ts))))
##    user  system elapsed
##   0.084   0.000   0.084

The speed up is actually ridiculous. Remember that the pracma code ran 40 seconds. The Rcpp code not even a tenth of a second. This is definitely good enough for 150,000 time series.