The following answer might give a solution of general type. Let H be a separable Hilbert space and {e_n} n in N_1, (where N_1 is contained in N) a finite or infinite  orthonormal family in H.  Assume that by measuring, we know the Fourier coefficients of an unknown function h with respect to the above mentioned family. Complete the initial orthonormal family up to a Hilbert base B in H. Increasing the number of measurements, we increase the number of known Fourier coefficients. The corresponding Fourier series converges to the exact unknown h. Thus h can be well approximated in quadratic mean by known elements. If H is an L^2(M) space, then there exists a sub - sequence of the sequence of partial sums of the Fourier series attached to h, which converges almost everywhere in M to h. See W. Rudin, "Real and complex analysis", third edition.

Hi Hind,

Your data set is not familiar to me since I deal with most of psychological or educational data, but when you analyze nonlinear data with PCA, you want to know how to replicate your nonlinear data (multidimensional) with the single dimensional (first) component solution. If I understand what you try to do correctly, you may try PCA with a "covariance" option and then project your individual datum points (that would be usually rows in your data matrix) onto the first principal component. Then read off the projected values. These projected values are in fact replicated (multidimensional) individual points. 

This projection technique is a well-known BIPLOT approach and the biplot book by Greenacre (2010) would be a great help and I am attaching a reference here. I assume you may purchase it via Amazon.com: Greenacre M. J. (2010). Biplots in Practice. Fundación BBVA, Madrid. In this book, see especially Chapters 5 and 6 for Singular Value Decomposition and PCA biplots. OR if you have a hard time to purchase this book, you may e-mail me and I can send some relevant materials for you privately.

There is a more complicated and technical book by Gower, J., Lubbe, S, & le Roux, N. (2011). Understanding Biplots. West Sussex, UK: John Wiley & Sons, Ltd. doi: 10.1002/9780470973196. However, I would recommend Greenacre's one. I hope my explanations are helpful and wish you the best luck for your research!

Cheers,

Se-Kang

Dear Prof. Al-Mishabi,

                                         I have developed a Dbranes String Methodology for certain types of composite (coupled) engineering solution like nuclear coal mining with stock market engineering and of course information-energy coupled networks providing the quantum mechanical systems classification and systems integration for a safe and sound mining application by using potentially nano metallurged material science to generate Dbranes String and obtain M (Econophysics)  closure along with Sandipan Mallick and Nick Hamburger and provide prototypes for quantum genetic engineering algorithms and design. All these mechanical systems physics require storage reservoirs because of the meta spacextime entropy dynamics as in Alfred Nobel's blasting potential reservoirs. You need some physical structure to blast right, you cannot blast space, one of the indeterminacies in this type of Physics. The application of the Arrow of Time with the increasing entropy and ubiquitous symmetry postulates. So this type of Physics which we have developed, you can take a look at my String Theory of Coal Mining paper on my RG page wwhich is implicitly extended by all three of us, which will give you a rough idea of why String Theory as a technology of analysis is so very important to our day to day physical existence when we move to and fro between various dimensions of time and space. Early morning you are sleeping in your bedroom but at 9.30 you are going up in an elevator and at 2 o'clock you have stepped out into the open streets for a short walk to the deli and at 6pm you are travelling in a tube rail holding on to the overhead handle to control your level of inertia listening to the announcement from the public adress system. What is also very interesting from Economic Sciences is that if you can control for interest charged on using capital you will probably go through more of these quantum econophysical systems and relativity will keep on changing the dimesions of your space of reference with higher "metavelocity". Now imagine what the home minister and his engineers will do if there is a blast somewhere in all this. I am just discussing a mind experiment in systems physics command and control. Earl Chair Prof. Dr. SKM QC EPS Fellow (In) MES MRES MAICTE