Time Collection Complexity evaluation utilizing Entropy | by Piero Paialunga | Sep, 2023

Right here is a technique to grasp how advanced your time sequence are, in a number of traces of code

Each knowledge scientist is aware of this: step one to the answer of a Machine Studying downside is the exploration of the info.

And it’s not solely about understanding which options might help you in fixing the issue. That’s truly one thing that requires area data, a number of effort, a number of asking round and looking for out. That may be a mandatory step, however for my part, is step quantity two.

Step one is in a roundabout way, form, or type, primarily based on the evaluation of how advanced your knowledge is. Are they asking you to search out effective particulars and sample in one thing that’s form of all the time the identical, or the outputs are utterly completely different from one another? Do they need you to search out the space between 0.0001 and 0.0002 or do they need you to search out the space between 0 and 10?

Let me clarify myself higher.

For instance, I’m a sign processing man. I studied Fourier Remodel, Chirplet Remodel, Wavelet Remodel, Hilbert Remodel, Time Collection Forecasting, Time Collection Clustering, 1D CNN, RNN, and a number of different scary names.

A quite common downside within the Time Collection area goes from an enter (which may certainly be one other time sequence) to a time sequence output. For instance:

• You could have a property of an experimental setup and also you wish to simulate your experiment utilizing Machine Studying: that is truly my PhD Thesis and it’s referred to as surrogate modelling

• You could have the values of the inventory market as much as day 300 and also you wish to predict day 301: that is very well-known and it’s referred to as time sequence forecasting

• You could have a sign that could be very soiled or noisy and also you wish to clear it: that is referred to as encoder-decoder sign denoising, and it is usually very well-known.

And in these issues, the very first thing that I have a look at, surprisingly, is the output (not the enter) time sequence.

Let’s say that I take a random time sequence in my dataset. Is the time sequence a mild and clean mixture of sines and cosines? Is it a polynomial operate? Is it a logarithmic operate? Is it a operate I can’t even title?

And if I take one other random time sequence, how does it change? Is the duty primarily based on taking a look at small modifications from an apparent baseline or is the duty to determine utterly completely different behaviors all throughout the dataset?

In a really single phrase, we try to grasp how advanced our activity is: we’re estimating the complexity of our time sequence. Now the phrase “advanced” can imply one thing completely different for every one among us.

When my spouse exhibits me her anatomy classes I discover them extraordinarily advanced, however for her it’s simply one other Tuesday

The excellent news is that there’s a manner of describing the complexity in a extra scientific and distinctive manner: the idea of Entropy

Let’s outline the entropy ranging from a quite simple instance: a time sequence that may solely have values 1 and 0. I do know that it’s not precisely the kind of Time Collection that we’re used to treating, however you’ll be able to think about it as if each minute you step in your room you flip a coin: if it’s head you may have measured 1, if it’s tail you may have measured 0 (or the other, I don’t have a selected desire of 1 being head, to be sincere…)

Now, if you concentrate on it, one thing is extra “advanced” when it doesn’t actually ring a bell in our mind whenever you don’t perceive it absolutely, or when it does not actually provide you with a considerable amount of info.

I’ll cease teasing you, and I provides you with the equation of this rattling entropy:

Let’s break it down:

• X is the area of our time sequence, in our case X = {0,1}

• p(x) is the likelihood of verifying the worth x that’s in X

Why do we’ve a logarithm in there? What does it imply? Why there’s that minus signal?

Let’s study by instance.

Think about that the likelihood of X being 0 (tail) is 0 and the likelihood of X being 1 (head) is 1. This isn’t even actually a time sequence, as it’s all the time 1. What’s the worth of entropy?

Now, p(x=0)=0, so the primary contribution is 0. p(x=1)=1, however the logarithm of 1 is 0. Which means the second contribution is 0 as effectively, and the entropy is, certainly, 0.

What does it imply that the entropy is 0? That the time sequence shouldn’t be advanced in any respect, and it is smart as a result of it seems to be like this:

There isn’t a “complexity” on this time sequence proper? That’s why it’s entropy is 0.

Let’s make the identical instance if we all know that p(x=0)=p(x=1)=0.5, which means the identical precise likelihood of getting 1 and 0 (head or tail)

That is positively extra advanced, isn’t it? The entropy now turns into:

That’s greater than 0. This worth has no that means per se, however it’s the highest worth that you may have. That signifies that when you change p(x=0) to be completely different than 0.5 the entropy is decrease*.

Now let’s take into consideration our findings.

• When the likelihood is 0, which means there’s no complexity, as we already know every thing: you may have one worth and one worth solely.

• When the likelihood is, let’s say 0.0001, which means the complexity could be very little as a result of it may occur that x=0, however the majority of the time x could be equals to 1

• When the likelihood is 0.5, now the complexity is most as a result of you may have severely no concept of what’s going to occur subsequent: it may be 1 or 0 with the identical likelihood

That is the thought of what’s “advanced” for us. In a easy 1/0 manner, you’ll be able to retrospectively discover the likelihood primarily based on the occurrences, and retrieve the entropy.

In our code, we’ll use Python, and we can even use very fundamental libraries:

Let’s write the code to search out the identical resolution however utilizing the likelihood “retrospectively”, or if you need, utilizing their frequency definition:

The place:

• x isa price within the area: in our case, we solely have 0 and 1, so x is both 0 or 1

• n(x) is the variety of instances that we’ve x in our time sequence

• N is the size of our timeseries

We’re going to discover p(x=0) and p(x=1) after which use equation 1 above…Wonderful, I’ll re-paste it for you:

In Python, you are able to do that with this quite simple code:

Does it work? Let’s take a look at it!

Let’s generate a 100 very long time sequence, with a likelihood=0.5 of getting 0:

Lovely. So we’ve our balanced time sequence. The truth that we’ve set 0.5 as a likelihood doesn’t imply precisely 50 and 50 as you’ll be able to see, so that’s going to offer us some form of error in estimating the likelihood. That’s the unperfect world we reside in

The equation to compute the theoretical entropy is the next:

Let’s see if the theoretical and actual entropy match:

Lovely! They do!

Now let’s change p_0 and see in the event that they maintain matching:

They match with a little or no diploma of error, proper?And the enjoyable half is that if we do that thrice rising the scale of our time sequence, the error can be smaller and smaller:

After dimension = 10k we mainly have 0 variations between actual and predicted entropy

Now, if we nonetheless assume that our time sequence has discrete values (0,1,2,…) we are able to prolong our definition of entropy into far more than solely 2 values of a time sequence.

For instance, let’s decide a three worth case. So our time sequence might be 0,1, or 2.

Let’s create a brand new likelihood vector p_0,p_1 and p_2. To try this we’ll generate 3 random numbers between 0 and 100, and retailer them in a vector, after which divide it by the sum:

We are able to apply the identical equation (and the identical code) as earlier than to search out the actual and predicted entropy.

Let’s prolong the definition of the entropy in the actual entropy definition:

This additionally works for less than the 0/1 case:

And as we are able to see the theoretical and predicted entropy match even for the three worth case:

And to point out you that I’m not dishonest, we are able to see that it really works for quite a lot of instances. If we modify p_vector (and the time sequence) iteratively, we nonetheless see that the actual and predicted entropy match:

On this weblog put up we:

• Mirrored on analyzing the time sequence complexity earlier than making use of any machine studying

• Mirrored on the thought of entropy and dysfunction of a time sequence

• Outlined the mathematical equation of entropy and defined it by instance

• Utilized it in apply for each a 0/1 time sequence and 0,1,2, time sequence, exhibiting how the theoretical definition matches with our computational approximation

Now, the issue (limitation) with this strategy is that typically the time sequence might be too steady for this methodology to work. However don’t panic! There’s a steady entropy definition that fixes the entropy for a Time Collection.

I’ll deal with within the subsequent blogpost!

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