Essentially, you have one input function, energy, and an output, damage index. You can try first a simple AR model. If it does not give any significative resutl, try a nonlinear model. A simple approach that can give you a fair idea is the cross-correlation. You can also compute the time-frequency representation, for example using the wavelet transform.

I think we should first isolate the minimum earthquake energy that would not cause any structural damage. The excess energy that would occur may be perhaps random or vary in pulses or spikes. Each quantum of excess energy can be related with a single time variable. Of course then the damage supposed to occur is consequential and would be cumulative. But if failure itself changes with each damage occurrence, then the prediction will be painstaking.

Essentially, you have one input function, energy, and an output, damage index. You can try first a simple AR model. If it does not give any significative resutl, try a nonlinear model. A simple approach that can give you a fair idea is the cross-correlation. You can also compute the time-frequency representation, for example using the wavelet transform.

Mutual information is used for identification of depedencies between non parametric distributions

You should use cross-correlations (performed with FFT to be faster) , scatter plots to check the linear behavior (if it exsists), and CWT (continuos wavelet transform)

Create CSV files of your data & time lag, then if you have access to SPSS port the data. SPSS offers the option of adjusting the time & even super-imposing the cause & affect. Interesting topic, often small quakes do not cause structural damage that is visisble but do introduce the micro-damage that will manifest itself at the next stage. Therefore I would not discount low energy quakes. In Iran (by the name of the University) a lot of construction carried is either old Khesht or frequently reinforced concrete that will become suceptible. Look forward to reading the paper when complete.

The first thing is plot the two quantities and look at them to see what further can be done next. It is possible that the correlations even change with time!

If you have the values of the parameters with tile, just paste them in an excel sheet and plot ont against the other. Then you get an idea of what kind of correlation exists. You can then you what you want to refine, find mathematical correlations (matlab for example.

Chi square test may be used to test association between two variables.

You may employ ANN for this kind of studies. ANN is easy to learn and apply to anything composed of input and output sets, irrespective of size and complexity.

If the variables change (relatively) slowly, then any time series model can be applied, but if they don't change slowly, then you can use, for instance, their derivatives or logarithms or whatever make them stationary..and then apply any regression model..

Hi Arman, I would suggest you to try Dynamic Time Warping. You can find an application to seismic signals on my profile. The classic cross-correlation doesn't work for non linear or time varying signals. As your are looking for a similarity metric I highly recommend to take a look at Dr Eamonn Keogh's work on DTW. Google it! He has pretty amazing tutorials and lectures.

Just to make the correlation matrices in the discrete time points and try to calculate the time fluctuations of the correlation factor.

I think that the first step should be to partition available data into, possibly overlapping, sections ("time windows"). The correlation coefficient is then readily available for each individual time window, computed directly from its definition. Surely, you may not be fully satisfied with the precision of such a result, but requesting more than two significant digits does not make too much sense anyway.

Regression methods

You can use several methods such as: Regression methods, Artificial Neural Networks (ANN), genetic algorithms (GA),...

Note that for the ANN: the number of neurons in the input layer is 1 (time), for the number of those in the output layer are 2 (the earthquake input energy and the Park-Ang structural damage index). For the GA method you must have necessarily the mathematical form of the relationship between input and output. I can helped you, with joy, to determine this relationship.

Interesting question. Did not hear earlier.

Don't know whether the best, but sufficiently robust would be:

1) Detect autocorrelation times of both signals and define sampling frequencies of non autocorrelated signals as inverse of both times

2) Resample both your signals with the smallest of the two sampling frequences, to get two time series, free of auto correlation

3) Estimate correlation coefficient for both time series obtained in this way

4) Make your own decision according to your criterion of correlation significance (threshold value of corrcoef)

the simplest thing is to carry out linear time series models. If you find there is a nonlinearity in the signal then choose appropriate non linear models. Probably artificial neural network may provide good results

Do a time series on one parameter, say x, i.e. test if there is a correlation with x = f(t).

Then consider y, test if there is correlation with y = g(x) which is y = g(f(t)).

Hope that makes sense.

2 Lito Cruz ·

А very exciting method. Does it exist without any specification of function g(x) ? Are you sure that all the parameters x(t) are known? If not specified during setting the problem, the answer "NO" is assumed..And x(t) are not specified !

I suggest you the use of a semivariogram analysis. You can obtain the semivariogram of each serie (f and g) and the cross-semivariogram in the points where both series information were available. The semivariogram analysis provide you very interesting information about variability of the series and periodic information. Another suggestion could be to divide the serie into months, and convert it into a 2D series (months, years, o days,weeks) in order to detect correlation in month or in years.

Arman,

Looking again at your almost surely well correlated variables, I don't see why at all you want to include explicitly the time into your analysis.

I think that it is a typical space-time correlation problem. All the variables vary on time and space, and logically they must be correlated (more energy implies more damages). Also additional variables as lithology or building characteristics could be considered.

I am agree with Jorge Delgado. Cross correlation, and semivariograms could give a first idea of the mutual relationship between signals. Other more complicated models are the correlation of noisy signals, because it is necesary to correlate frecuencies, that will use the same cross correlation and semivariograms over the results of FFT of each signal.

Impulse test, or other methods could be used in the frequency space. Finally, if there are noise in the signal, a smart filter should be used to remove noise, prior to any analisys. for example Hilbert huang transform.

@Vassili,

Finding g(x) in the context of discussion depends on the nature of the data, i.e precisely the use of correlation analysis techniques to find a model for g(x). Also the questioner wants a simple and effective as much as possible hence, my suggestion.

LPC

Maybe you could try adding a random effect parameter for the different time steps using generalized linear mixed moddeling (GLMM) procedures (a generalization of GLM). With GLMM you can add a random effect paramater to your function specification. Instead of a GLM function E(Yi )=µi (specification of the GLM can be found for instance in McCullagh and Nelder 1989 or wikipedia), you get a function E(Yit )=µit*θt where θt is the random effect parameter for each time step t. This way the output of your regression will be the correlation parameters for the predictors and a 'correction paramter' for each time step t.

GLMM is availiable in SPSS thus not so complicated to use.

2 Lito Cruz

I like examples, e.g.

%P and P1 constant Parameters

P=0.2;

P1=1000;

t=1:.5:2000;

x=.5*cos(t)+P1./t+randn(size(t));

y=cos(t+P*sin(t))+2*P1./t+randn(size(t));

C=xcorr(x,y);

plot(C);

shg

Here the crosscorrelation is produced by time dependency of both signals x and y whereas both parameters P and P1 are mutually independent constants

I don't know weather it would be best but give correlation between two parameters.

By using software (like design expert, minitab) it would be easy to correlate the response variable to input parameter(s). The software facilitate to change different functions (such as linear, quadratic, cubic, logarithmic, exponential etc.) and check the best possible out of them by various statistics approximation.

I may help with the generated model, if required.

If one function is related to other Then we can use predictive identification technique..

Arman,

Marek is right.

You have earthquake events and the related damage index. Am I right?

So, you don't need to consider time, just to analyse the possible degree of correlation of the two data vectors.

There is plenty of methods to do this but first, you could use a simple test such a Chi Square.

I would calculate the coherence to see if the correlation is significant at some set of frequencies. Also to investigate, a time-dependent trend removed from the time series may produce a stationary correlation value. cheers, arthur

One way is to measure two-points correlation C(t) (your process must be stationary):

C(t) = ( - )/ ( - )

where denotes an average on time and A and B your two observables.

So you will get a function starting from 1 and going asymptotically to 0. Thus, you can define a relaxation time \tau by taking C(\tau) = exp(-1) for example (because typical relaxation process are exponential). \tau will represent the typical timescale of your correlation.

1) If the correlation is strong, you can link in a graph plot the values ​​of two parameters for the each time point . It is a easy and intuitive method. You can also vary the phase shift between the corresponding values of the parameters.

2) You can try to find the weak relationship of those parameters using the Fourier series. It is assumed that such a representation is physically justified. This method permit to identify the common harmonics and the phase shifts.

(I used this method for similar task for study of the relationship between Volcanic activity and great earthquakes in the Kuril-Kamchatka region).

The suggested approaches are easy and intuitive, but there is difficult to substantiate found relations. They can be used for preliminary analysis.

use the Matlab function: CORRCOEF Correlation coefficients.

CORRCOEF(X,Y), where X and Y are column vectors

I Think that the relation between energie and damage is non linear. if power =Energie/time) is small damage will be small. If power is hight, it will appear a non-linearity in the material (cracking) so the damage at this time will increase strongly.

I thing ther is a non-linear correlation between the dérivate of energy and damage.

You have also to extract the active part of energy (which comes from frequencies near of resonance value or the objet of study), the correlation will be much better ..

If you are studying a non-linear system the time is an important variable because your structure keeps memory of the inputted/dissipated energy; if your are working with linear (or very stiff) systems I don't see the interest in tracking the time history.

In the first case , the time frequency analysis of the signal can give very interesting insights on the damage process. However, pay attention to the frequency range used to evaluate the input energy: it's important to center your analysis around the frequencies of interest for your structure (at least for the first deformation mode), seismic signals have a broad energy content and they could be very energetic in not really interesting (for the structure) frequency ranges.

I'd say that the very first thing to do is to have a look at a scatter plot of the damage index against the eq energy... Then you could just observe the way the damage (index) increases with the energy and decide how you are going to explain and analyze/model it.

You may try time frequency analysis, with Gabor transform (you should program it) o wavelet transform (with matlab). It will give you an easy visu. of the phase difference between both function (see Fichtner et al. Full seismic waveform tomography for upper-mantle structure in

the Australasian region.)

Use Artificial Neural Networks