Atmospheric Sciences 552: Objective Analysis Homework Assignment #6

I have constructed two more data sets for you to analyze. You can get them by anonymous ftp to 'eos.atmos.washington.edu'. Change directory to cd pub/dennis/552_Class/, and get the text files x552x.dat and x552y.dat and put them in your working directory. Now you are ready to start matlab. Once matlab is up you can ingest the data files by typing 'load x552x.dat' and this will give you the data as a matrix called x552x. x552x.dat is 1024x32 and x552y.dat is 1024x48. Again, you can think of the longer dimension as time.

1) Do a standard eof/pc analysis of the two arrays I have given you, much as you did in a previous homework. You can probably guess which modes are significant now just by looking at the eigenvalue spectra. Note what fraction of the variance of the data sets is explained by each eof for later reference.

2) OK, now that you know what basic structures are in both data sets, you can ask whether the two sets are related by using SVD of the covariance matrix. Suppose for brevity, we set x=x552x; and y=x552y;. Construct the covariance matrix by typing xy=x'*y;. Divide by the sample size. Calculate the Frobenius norm of this covariance matrix, which is the sum of the squares of the elements, or the total mean squared covariance. Divide this by the product of the mean squared variance and take the square root to form the Normalized Root Mean Square Covariance=RMSC. Is it large or small?

3) Take the svd this covariance matrix with '[u,s,v]=svd(xy);' where we are using the semicolon to suppress annoying printing. Take a look at the singular value spectrum with 'plot(s)'. The singular values are the time covariance explained by the individual paired modes. Take a look at the important modes with 'plot(u(:,1:3))' and 'plot(v(:,1:3))'. How many significant modes do you think you have? How are the left singular vector and the right singular vector different for the dominant mode(s)? How do these compare with the structures you found in the eof analysis in part 1. Are they the same or different? Calculate the fraction of the covariance explained by the first few modes.

4) You can investigate the temporal evolution of the left and right singular vectors by projecting them on the original data. The original data matrices x and y have 1024 rows and either 32 or 48 columns, respectively. u and v contain the left(x) and right(y) singular vectors. How do you construct the time series of the amplitudes of the singular vectors? You project the singular vectors onto the spatial dimension. Draw a picture to convince yourself that the way to do this is ut=x*u; and vt=y*v;. Plot the time series of the amplitudes of the first few left and right singular vectors (e.g. 'plot(ut(:,1:3))'). See anything interesting? Go back and look at the time series of principle components of the eof's of x and y, and use these results to explain why some eof's show up strongly in SVD analysis and some do not.

5) Next let's suppose the fields input are temperatures and have the dimensions of Kelvins. How could you present the structures of the singular vectors in a way that properly shows the amplitude that they represent? One way to do this is to normalize the time series of the right singular vector amplitude and calculate the covariance in time of this with the left field (and vice versa (see class notes pages 77-78). This is pretty easy to do in Matlab. First get the norm of the 1st right singular vector amplitude time series. "v1=sqrt(vt(:,1)'*vt(:,1)/1024)." which is a scalar. Then construct the normalized time series of the first right singular vector amplitude time series. "vt1n=vt(:,1)/v1" Finally, project this time series onto the original data, "xss1=vt1n'*x/1024." Repeat the procedure to get "yss1=ut1n'*y/1024." Plot the singular structure(with amplitude in units of K) "plot(xss1)". How does the amplitude explained compare with the variance of the data set. How does it compare with the amplitude of the corresponding eofs in the original data fields? Do this for the second and third singular vectors of the MCA analysis. Does this change your opinion about the relative merit of the first three pairs from MCA?

6) Put your results in a little report, and include relevant figures and discussion.
Note you can do these calculations most easily by adapting mca.m, which you can download from the sample software page, if you are using Matlab.

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