Correlation coefficient is a similarity measure in an angular space, being correspondent to the cosine between the two vectors (your time series) that you correlate. This implies that maximally negatively correlated vectors have lowest similarity (r=-1) and two maximally positively correlated vectors have highest similarity (1). If you do not mind orientation you can use r correlation coefficent in absolute numbers so you will have a measure going from 0 to 1, but I do not think this is your case.

Clearly you are in an angular space, this implies you make the hypothesis the radius is normalized to 1 and then the time series (vectors): 1,2,3 and 10,20,30 are identical (r=1). If you do not want to make this assumption you must shift to the Euclidean distance between time series. In general we are more interested in angular similarities (or distances) but this is apt to you and depends on what you are looking for..

Dear Alessandro Giuliani

Thanks for your answer

look, I don't want to calculate the relation between two datasets. for example I prepare two datasets as you can see. The correlation coefficient of them is equal to 1.

But they have far away from each other.

I am looking for an index to calculate for me the differences of these two time series.

For example Nash-Coefficient use for evaluation between observations and simulations datasets, but I am not sure that we can use it or not.

It looks like that the statistical question is: "are the two sets of data drawn from the same distribution or from different distribution functions?". An accepted test for differences between two binned distributions designated ti and ti' is the modified chi-square test with the following statistics in this case:

ch-sq=sum[(ti-ti')^2/(ti+ti')] over all 30 days (or any other number of days).

If the ch-sq value less than the critical one for the degree of freedom 30 in this case and significance level, say, 0.05, then there is no statistical difference between the two distributions (and two time series) and vice versa.

Dear Alexander Kolker

Thanks for your answer

Unfortunately I don't want to check the distribution similarity of them. You can suppose that they have a same distribution function(e.g. Normal).

It seems that I couldn't tell you my purpose. I have an idea to solve this problem:

Can we use this formula to calculate the differences between two datasets?

D= sqrt[ 1/n((ti -t'i)^2) ]

where n is the number of points (n=30 in this problem)

Above formula is Root Mean Square Error (RMSE) but we usually use that for calculate errors of a model outputs and actual values. I am not sure that we allowed to use this formula to this work, so I'm looking for a trusty one.

Thanks a lot

Amirhossein Haghighat

1. Disproving the null hypothesis that two sets are drawn from the same distribution means that the data belong to different distributions, i.e. they are NOT similar in your terminology. If you say that "You can suppose that they have the same distribution function(e.g. Normal)" then it means that there is NO statistical difference between them, i.e. they can be considered similar. No other test is needed.

2. I am not sure that your use of the formulae for D is correct. The RMSE value D must be compared with some value (critical value statistics) in order to tell how "big" it is. This value alone does not tell you anything about how big it is, hence nothing about the difference between the two sets. RMSE formulae above does not include information of the variability of the data sets and degrees of freedom.

You could try instead a paired t-test taking into account all its limitations.

Hi Amirhossein,

I am currently also looking into this topic, but I'm working with kinematic data. In gait analysis literature, there is a metric called Coefficient of Multiple Correlation, which gives the degree of similarity between two curves. It is often interpreted as: the closer CMC is to 1 the more similar the two curves are and the closer to 0 the higher the dissimilarity. It is not the same as the correlation coefficient you mentioned above, it will vary with systematic bias and offset between the curves. The disadvantage of this metric is that it can give you complex numbers (invalid results) for curves with small amplitudes, but some authors suggest interpreting this result as complete dissimilarity.

Here is a reference:

Ferrari, A., Cutti, A. G., & Cappello, A. (2010). A new formulation of the coefficient of multiple correlation to assess the similarity of waveforms measured synchronously by different motion analysis protocols. Gait and Posture, 31(4), 540–542. https://doi.org/10.1016/j.gaitpost.2010.02.009

Hi dear Ana de Oliveira

Thanks for your effective suggestion too much

I will read CMC method and test it in my dataset

Best wishes