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I have various object vectors (PC1, PC2, PC3) representing objects ater normalisation and PCA. I also have 'axis' vectors (PC1, PC2, PC3) representing the axis that the objects were originally placed along e.g. hot/cold, like/dislike. What I would like to know is if I do a projection of the object vectors on to the 'axis' vectors does this effectively give a representation of the original (normalised for all datasets) data? Can I then take object-axis projections for many entries to give average and s.d. values?

I add that the PCA and normalisation was done with multiple entries for the same bunch of objects and axis i.e. 20 people x 16 objects x 5 description axis (but description axis considered seperately i.e. vectors all different). Unfortunately it was also done in software which is rather black box in what it does (please dont just tell me to use something else though).

Richard B
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The first question you can easily check yourself: can you reconstruct the original data from obects (scores) and axes (loadings)? Keep in mind that PCA usually centers as a first step.

The knowledge gained from that will probably answer the rest of the questions as well.

Second question: Sure you can give average and standard deviation. The question is: what is their meaning. And this I cannot answer, because I'm not sure about the direction of projection you propose: PCA is a coordinate transformation, so you have original axes and "new" axes... But acutally checking the calculations will probably make things clear for you.

Thirdly, normalization can refer to different things. You need to clarify what you mean.

cbeleites unhappy with SX
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  • thanks @cbeleites for your help. I think that is good advice however I am not entirely sure how to reconstruct the original data, from what i understand i need the transpose of the eigenvectors x the object scores? the program does not give me the eigenvectors and potentially given the size of data not all will be used...or can i get this from the axis? if so how? – Richard B Jan 23 '13 at 13:29
  • @RichardB: Did you read this: http://stats.stackexchange.com/a/34725/4598 ? I'm still not clear what your "axis" is. If it's the loadings or rotation matrix, then it contains the eigenvectors. If you are not sure what your black box software spits out, you can a) calculate the same problem with another software where you know what happens, and/or b) have the black box do a PCA on a small data set where you can calculate the PCA by hand. This should allow you to find out what is going on. – cbeleites unhappy with SX Jan 23 '13 at 15:06
  • Hi @cbeleites I guess the problem is that I dont fully understand how the 'axis' relate to loadings/eigenvectors. If you consider objects plotted against an 'x axis' then after PCA a vector of the 'x axis' in terms of (PC1, PC2, PC3) is given. What is this vector? Is it the loadings (or in terms of the link, part of a matrix of loadings)? Similarly the same objects were originally also plotted against 4 more axis e.g. a 'y axis', after PCA given in terms (PC1, PC2, PC3). This then relates back to my original question if by projecting these vectors are you effectively doing this reconstruction? – Richard B Jan 24 '13 at 09:36
  • Richard, maybe you should give us a picture of your software. As it is, nothing but wild guesses are possible. – cbeleites unhappy with SX Jan 28 '13 at 18:24