# Similarity Measure of Simulated Time Series vs Observed time Series

In my work I have an observed Time Series and Simulated ones. I want to compare the Light Curves and check for similarityto find out which simulated curve fits best respectivley which parameters simulate the Light Curve the best.

At the moment I do it with the Cross-Correlation function from numpy. But I am not sure if that is the best option, due to the fact that the Light Curve with the highest Cross-Correlation Coefficient not always looks like the best fit/simulation compared to other simulations with a lower CC-Coefficient. Is there a another way to measure for similarity? I read something about the Chi-Square Statistics, but I am not sure how that works and how this could be applied to my problem.

The observation data I use is not evenly binned, so I used the interpolation function of Scipy. Should I also smooth the observation data or would I lose true features of my data. I thought about using the savitzky golay smoothing.

At the moment I am using a brute force method to try out all possible parameters and simulated the corresponding light curve. The problem is this takes a lot of time with 20 parameters. The parameters are more or less dependent on each other. So I cant use a least-square fit method, because there are multiple possible minimas. Is there a simple method that I overlooked. Or is a restricted brute force fit my best option?

In the picture below you'll see one plot with the simulation and the observaton data. Thanks for all suggestions.

in my opinion you can do the following to compare your simulated with your actual data.

Use usual measurements like those you would use in predictions vs actual data:

• RMSE
• MAE (Mean Absolut Error)
• MSE (Mean Squared Error) --> Gives greater weight to larger errors/gaps

Another statistical test that comes to my mind to test "prediction" quality between different models is the Diebold-Mariano test. It is fully implemented in R. As far as I know, not in any Python Library so far.

As a statistical test you could have a look at the distribution (but as you dealing with time-series you may face "memory", meaning that observations t and t+1 face autocorrelation (against most of statistical tests, as they assume independent observations). Autocorrelation seems to be the case for your data series. Therefore you should have a look on the % change of your sample data (from t to t1) and look if both series are drawn from the same distirbution - You can perform the kolmogorov smirnov test.

Hope this helps.

EDIT

Regarding your question concerning smoothing and interpolation. Empiricis (not statistics) is more a trial and error concept. It really depends on your data behavior.

It looks like a signal processing question, where you want to denoise a periodic signal. May I suggest you to look at specific litterature ?

More specifically : https://github.com/unpingco/Python-for-Signal-Processing, the Signal Processing Reading List. You'll learn how to model periodic signals, denoise them and compress them.