# How to represent periodic paired data?

I'm working on paired data, of which absolute values plotted over time, show periodicity with varying peak heights as shown in the following graph: The data represent the absolute distance between two feet, so peaks are maximum steplengths.
Both the blue and red line belong to the same participant that moves in different environments (red: virtual reality; blue: reality).

I need to compare each data to its counterpart and I am wondering which of measures of center, mean, median or the mode, is an appropriate representative value for the varying peak heights of such type of data?
Purpose of comparison: Discovery of difference between step lengths in reality compared to step lengths in virtual reality.

I have thought of z-transforming the data in order to make peak variance more comparable in addition to a measure of center - But I'm not sure if this is a sufficient representation of the data?

• What's your goal for performing this comparison? – caveman Dec 21 '16 at 13:46
• Thanks for the comment, will edit my question to make the purpose of comparison more clear! – Aliakbar Ahmadi Dec 21 '16 at 13:52
• There are several ways to do some analysis. As for periodic data it is sensible to look into fourier transformed data (sometime called "spectral analysis"). I posted some basic stuff here stats.stackexchange.com/questions/241091/… You could use characteristics of curves to compare them in some manner. But as caveman noted, a goal may narrow down the approach. – Drey Dec 21 '16 at 13:58
• @Drey Thanks! In essence, I am only interested in (unfitted/unapproximated) peaks of the step length data. As steps are differently long for each participant and each environment(reality vs. virtual reality), I am looking for a measure that is a most suitable representative for these data, taking in account their variability. – Aliakbar Ahmadi Dec 21 '16 at 14:09
• @AliakbarAhmadi So I guess that your goal is to classify whether two different periodic step data collections belong to a real person or a VR person? – caveman Dec 21 '16 at 14:09

I would suggest to estimate the phase linearly between the two adjacent peaks within one signal, and then plot and analyze in phase space. This fixes your peak positions to 0, but gives you a distribution of how your signal develops per cycle

My suggestion is to analyse each single step of each partecipant, separately, in the two conditions. You can do this using local minimum (=start) and local maximum (=end of the step). In this way you can (i) have a mean step curve for each subject, (ii) exclude outlier steps (I can see a couple of them in your graph).

Then, plot the mean step curve in the two conditions, separately for each partecipant. This would help to let you know if the difference is stable across subjects, and if some of the subjects behave in the opposite way. For example, I can suppose that subjects more 'Novelty Seekers' can have an inclination to move more in virtual reality, while subjects which are more 'Harm Avoiders' will tend to do shorter steps in an unknown (virtual) envirnoment [that is particularly interesting if you have some behavioral scale of your subjects].

Finally, plot the overall mean in the two conditions. You can test the difference between the conditions just using peak value, or dividing the step in more finite phases. That's your decision. Take care of the proper alignment of each subject: maybe the best way to align is using the peak.

If you want to standardize or not each timeline before the analysis, it probably depends on the reliability of the data, but I suggest you to do it (or even try both ways).

Edit: this can also allow a comparison between the standard deviations. In fact, it is possible that the difference between the two environments is not only in the mean peak (or mean anything), but also in the uncertainity of movement of the subject. This will reflect in a different standard deviation. It is an interesting study/analysis, hope this helps.

I think if you show that the difference of the peaks alone are different in a statistically significant way, then the distributions are different because of a systematic difference. But I have no proof for this, I'd like to be corrected by the Gurus please.

I'd suggest to list all the peak values for each step. Probably exclude the outliers. Then represent each line (red, blue) by their average peak distance.

Then measure the absolute difference between those means --- I suggest this absolute difference of the means to be your test statistic.

Then identify the distribution of such test statistic under the null hypothesis in order to calculate the $p$ value. To do this, I suggest you to simulate it using approximate randomization.

• Systematic differences of the samples does not always imply a difference in the distribution (see the so-called Simpson paradox). – smndpln Dec 27 '16 at 11:30
• @smndpln Why not? How is Simpson's paradox related to "whether two samples are from different distributions"? I think Simpson's paradox only tells that unfair observations can draw wrong conclusions (e.g. samples in X is better than samples in Y), but I think it doesn't deny the fact that samples in X are from a different distribution than those of Y --- am I right? Please enlighten. – caveman Jan 1 '17 at 12:00