How to compare two random signals

I have two simulations of ship motion, and measured ship motion. Generally, the motion is random. I've been asked to compare the three sets of data. I was trying the goodness of fit (sum((fi-Fi)^2/Fi) where Fi is count in bins like histogram) and evaluating Fi using Chi Squared cdf to get the probability of being the different. I'm also comparing the mean and standard deviation to see if they are statistically the same. Are there other approaches to compare random data. I'd like to do something with the power spectrum, but what is the statistical comparison between two psd?

The signals are not recorded simultaneously, they represent ship motion in a similar seaway. So I think correlation and coherence don't apply.

Oh, maybe I misunderstood. You are suggesting to do corr on the PSD signals.

Edit>

I have a little trouble understanding some of the comments, so I'll expand on what I am doing. I have three sets of data, two are simulations, and one is measured. The data is ship motion, pitch, roll, heave displacements and rates. The basic question is are the two simulations similar, and use the measured data for reference. So, there are two random signals. The signals differ in duration, and also sample rate.

1) probability of same of mean and standard deviation

2) probability distribution function (histogram) of response with probability of same

4) comparison of autocorrelation of response

5) Comparison between frequency / spectral content of response

6) Quintile-quintile plot of two simulation, include slope and correlation coefficient of linear fit

I'm just starting to write up the results, but it appears that the QQ plot may be the best comparison. The frequency comparison is good to, and can potentially help guide resolution. The autocorrelation is also helpful, but I don't have experience in reading the figure.

[KSSV]

have you tried corelation ?

[John]

Random signals are not correlated, so corr(r1,r2) is near zero for any two random signals. I'm looking for an analysis that shows a high value when the signals have some commonality, like similar probability density function distribution, or similar power spectrum density distribution.

[John BG]

John

is is possible to elaborate a bit?

1.- are you working with ship motion as in ship data: bearing speed locations?

or are you actually simulating and listening to ship movements?

2.- If you mean audio, then the mean and std (var) are the statistical moments of 1st and 2nd order

In Discrete Time Speech Processing J.Deller J.Hansen J.Proakis, in their basics review of stochastic processes, they pack the data in blocks, doesn't matter you only have one feed, and then find the covariance matrix:

3.- pg40 conditional expectation of y given event x.

cov_xy=xcorr_xy-mean_x*mean_y

If the signals have similar pdf then they would have distinctive xcorr(r1,r2),

4.- find out what moments are stationary

5.- pg56 find distance between signals,

6.- zero crossings

7.- Entropy: look for repeating chains

[John]

1) The ship motion is the motion of the deck, pitch, roll, heave displacements and rates.

2) I understand how to compare the mean / std using the distribution of mean / std. But, I need more detailed comparison than just mean / std.

3) I don't think that equation applies in my case, partially because the signals are not coincident, measured at different times, but also because the sampling rates are different.

I need to work on 4. But, can you define stationary, cause I can see that is going to come up with the measured data.

5) I need to get a copy of this book to understand better.

6) While I can find the zero (or mean or low frequency fit) crossings, how do I compare the times or time intervals.

7) Can you explain entropy chain more.

Answer 1

[Florian Bendl]

How about correlation? Should be close to zero for random data.

[John]

Correlation is near zero for random signals. I'm looking for a function that can find similarities between random signals.

Answer 2

[Star Strider]

You may have actually answered your own question with the power spectral density idea. Comparing them by convolving them or convolving the inverse of one with the other in the complex frequency domain (a sort of transfer function calculation) could give you the information you want. That would be my first approach.

If they have different d-c offsets (mean values), that could be enough to give them different probability density functions. Binning each of them with histfit might give you some information.

EDIT — Comparing three sets of histogram data might be more appropriate for an analysis-of-variance than chi-squared.

Answer 3

[Image Analyst]

Can you take 3 signals, 2 that are similar and one that is different from those two, and use pwelch() to plot the spectrum, and attach your screenshot?

Can you take the moments (1st, 2nd, 3rd, 4th, 5th) of the spectrum and see how they change/differ?