I have bought a book on factor analysis (on PCA more precisely) because it seems to be the algorithm that fits the most to mine my data.
Here is the context: I have a table of "subject x variables", each line is a subject (there are numerous subjects) and each column is a variable (here it is network variables but it shouldn't change anything). All variables are quantitative. My goal is to determine which variables are the most correlate each other and which one are not. I would like to find a typology of my variables (and why not after find clusters among subjects) and have a summary view of all variables giving a good idea if relations are hidden in the amount of data. Do you think PCA is good approach for doing it?
In my book, they often propose two summary graphs to problems: One for subjects and one for variables, and they say, the two graphs are two point of views of the same problem and must be interpreted considering each other. Nevertheless they study a small amount of subjects on examples (for example 10 countries...). In my case I can have thousands of subjects. The number of variables can be up to 50. In this case, I don't see how I could interpret the "subject graph". Is it a problem? Should I only get the "variables graph"? Moreover, in the examples, the subjects have a meaning (countries, socioprofessionnal categories...), in my case, they don't really have meaning (every subject is a set-top-box for example).
I can't comment your answer to give all details to each point you ask, so I make a new answer:
1&2) The book I bought is a french one but if I translate the title in english, it would give "simple and multiple factor analysis", the authors are Brigitte Escofier and Jérôme Pagès, the collection is Dunod. The book deals mainly with 2 methods of what you call "data reduction": Principal Component Analysis (PCA) and Multiple Correspondance Analysis (MCA). The first method is applied to quantitative data and the second one to qualitative data. I agree with the fact that PCA is a data reduction method (one of the main result is a 2 dimensionnal plot that represents the original data set) but it seems to be more than this, for example it gives the correlation circle which seems to be what I am looking for.
3) A correlation analysis is possible but it is applied each time to only two variables and I am looking for a method that could analyze correlation between many variables simultaneousy (and give a reduction view of the data set additionally)
4) Forget about cluster analysis for the time being, It will come in a second time but just to explain, I have read that it was possible to apply hierarchical clustering after a PCA but it's indeed another subject...
Here I am trying to explain more precisely the context of my study, it will certainly help the reader to better understand what I want to do: I have thousands of STB (Set-Top-Box) located in different cities of a country for example: Paris, Lyon and Marseille in France. Each STB gives a set of variables each hour. Each STB per hour is a "subject" and is described by many quantitative and qualitative variables like average bitrate, freezetime on the hour (duration of freezing screen during the hour...), localisation, mean opinion score (quality of the video). I would like to apply a method (that I would repeat "each hour") on the data set to better understand if correlations are hidden. I give you an example: If there is a network problem in Paris which would imply low bitrate and high freeze time, I would like it to be detected by the method. If a certain channel is having a very low mean opinion score (bad quality) I would like it to "appear" thanks to the method (either by a plot or by an indicator). In the book I bought there are hybrid method working with quantitative and qualitative data in the same time (a mix between PCA and MCA) and seems to be what I want to do. I would like that people who have already worked with such problems comfort me if I take the right direction.
5) What I call summary views is the correlation circle and the projection of data on the two first axis that gives a bidimensionnal plot representing at best the data set.
Thanks for the first answer, if you need further information, let me know, I would be delighted to do so.