What an amazing set of scientific findings you've presented, mindchild! You and your colleagues may well be awarded a Nobel Prize!
For larc, spectral analysis is a technique often used to analyze data in order to determine whether periodicities exist. It can determine a lot more besides, but that's the basic idea.
To give a simple illustration, suppose you wanted to analyze tree ring growth data for a few hundred years to see if you could find any periodicities in growth, which would in turn imply periodicities in climate. You measure the width of growth rings on thousands of trees going back as far as you like, average the results and then plot the average width versus time on an x-y graph. You'll see a squiggly line with humps and dips, but most likely won't see much more than a faint trace of periodicity. Spectral analysis lets you extract the individual periodicities and gives you information about the magnitude of each periodicity in your data. In this case, most likely the strongest periodicity would be at some multiple of the period of the sunspot cycle, i.e., 11 years. Scientists have done this and actually found the strongest period at 22 years.
The term "spectral analysis" comes from the "spectrum" of light, i.e., white light is made up of many frequencies all munged together into the normal ROYGBIV spectrum. When a prism splits light into its various frequencies (colors), it's doing a kind of spectral analysis. More sophisticated equipment can measure the intensity of each color. Light from the sun and stars contains not only a huge number of low-intensity frequencies that make up almost a continuum, but many particularly intense frequencies, or nulls at a frequency, that represent emission or absorption of light at those frequencies by a particular molecule. If you've ever viewed pictures of spectra from the sun or stars, you'll have seen those emission or absorption lines.
Mathematically, any set of data can be analyzed to see the "frequencies" (or equivalently, the periods) inherent in the data. This is done via the technique called "Fourier analysis", which breaks data down into a set of sine waves of various magnitudes, frequencies and phases. In the case of measured data, computer programs exist to do this. The set of extracted sine waves is called the spectrum of the data. If you simply add up all of the extracted sine waves, you get back the shape of the original data curve.
"Least squares" is another set of mathematical techniques that allows one to find the best fit to a predetermined curve of a set of "noisy" data. Once could use this, for example, to find the best fit sine wave to a set of data that look something like a noisy sine wave. I'm not familiar with details of the "least squares spectrum" technique described by our esteemed mindchild, but I imagine it would combine these two techniques to yield a "most probable" spectrum.
Hope this helps.
AlanF