# Correlation between multiple time series

For research, we put some test samples through a physical process for a certain period of time and make measurements. The general structure of the data we collect is as follows:

Experiment A
- Measurement A1
- Measurement A2
- Measurement A3
...


and similarly

Experiment B
- Measurement B1
- Measurement B2
- Measurement B3
...


Note that each measurement is an independent time-series data on a test sample (might be univariate or multivariate). By the nature of the test samples and the process, it is expected that some measurements from different experiments should somehow match or correlate.

I would like to calculate the matching pairs from different experiments (ex. A2 and B4). So far, I was doing this by iterating through measurements and calculating a cross-correlation function. But I want to switch to a Deep Learning/Neural Network approach using PyTorch.

I have some prior experience on time-series data classification using PyTorch. However, I couldn't come up a with a solution to that problem. So here are my questions:

• What kind of a problem is this? I would say it is a classification problem? What are the keywords should I use for further research?
• How should I structure the architecture of my model?

In general, any kind of leads are appreciated. Thanks!

• Just to understand the setting: (a) each measurement is a time series or (b) each measurement is a point on a time series and all measurements of one experient belong to the same time series? If (b) then how do you compute correlation between Measurements as they might just be scalars and correlation between two scalars does not make sense. Jun 23, 2023 at 15:57
• The correct one is (a). Every measurement is an independent time series on a different test sample. Just edited the post to be more clear. Jun 23, 2023 at 17:20

There is not one easy solution to your problem, as it is not well-defined, but there are ways to narrow it down and come closer to a solution. Here are some points that can help you to do so.

##### 1. Something like correlation

To my understanding, you are looking for something like correlation (but more complex) that identifies time series that belong together. This is a vague description. Can you

• describe what makes a match (correlation looks for linear dependencies)
• maybe even mathematically define a similarity or distance measure between ? (In this case, you might not need deep learning and just use the similarity/distance measure)
##### 2. Supervised or unsupervised

Do you have labeled training data, i.e. a set of time series where you know which series match and which don't? If you want to train a neural network, you might need quite an amount of such labeled pairs. In case you do have such labeled data, a supervised approach might work (e.g. classification).

Note: If you use correlation to identifiy matching pairs for the training, then your neural network will just learn to reproduce the correlation results. You need to put manual effort into creating better matches.

If you do not have such labeled data, i.e. you just have the set of time series, you need to look for an unsupervised method.

##### 3. Classification

If you want to turn this into a classification task, you could use two time series as input and have a binary target (matching: yes/no). Note that in this case you need to test all pairs of series to find the matching pairs.

##### 4. Unsupervised methods

There are two classes of unsupversised algorithms that might be of help, here.

• Clustering groups your samples (=time series) in subgroups. You might need a clustering algorithm that creates many small clusters (ideally each cluster is one matching pair). Hierarchical clustering might be able to do so. The downside is, that you need a suitable distance function (see 1. above)
• Embeddings transform your data (e.g. your time series) into a low-dimensional fixed-size vector. Famous algorithms are t-sne and umap. This can be used as preprocessing step for a clustering. A euclidean distance might already work well on the low-dimensional embeddings.