# Is it sensible to do PCA on a distance matrix?

I have 10x10 distance matrix where the distance metrics is (1 - overlap coefficient).

I want to represent the observations in this matrix in a low dimensional space to see how observations relate to each other.

My understanding is that one of the go-to methods for this is multidimensional scaling (would you agree with this?). I have done it and the results make sense - sort of.

I have also tried PCA on the distance matrix and plotted the PC1 scores vs PC2 scores. The results seem better then MDS' in terms of agreement with our background knowledge of the data.

Now, I cannot quite wrap my head around whether:

• Is PCA on a distance matrix a sensible thing to do? I mean, PCA doesn't care where the data comes from but it feels odd to use it in this case. In particular:
1. PCA computes the eigenvectors of the covariance matrix of the input. Since my input is a distance matrix, does it make sense to do that?

2. My distances are bound between 0 and 1 - is this a concern?

• Should I be worried that PCA gives "better" results than MDS? Is it a hint that something may be off?

In case anyone wants to have a look, here's the matrix, also as an R object that can be copied & pasted into R

         [,1]   [,2]   [,3]    [,4]    [,5]    [,6]    [,7]  [,8]   [,9] [,10]
[1,] 0.00000 0.1135 0.6255 0.45565 0.46215 0.34426 0.00847 0.937 0.7055 0.629
[2,] 0.11348 0.0000 0.3715 0.25532 0.28646 0.21691 0.01176 0.941 0.5857 0.587
[3,] 0.62550 0.3715 0.0000 0.04264 0.02731 0.07734 0.16880 0.972 0.1226 0.492
[4,] 0.45565 0.2553 0.0426 0.00000 0.00909 0.13687 0.13333 0.955 0.1350 0.522
[5,] 0.46215 0.2865 0.0273 0.00909 0.00000 0.09141 0.14948 0.971 0.0535 0.455
[6,] 0.34426 0.2169 0.0773 0.13687 0.09141 0.00000 0.00763 0.977 0.1164 0.189
[7,] 0.00847 0.0118 0.1688 0.13333 0.14948 0.00763 0.00000 0.951 0.2835 0.175
[8,] 0.93694 0.9412 0.9719 0.95455 0.97093 0.97683 0.95122 0.000 0.9762 0.975
[9,] 0.70547 0.5857 0.1226 0.13498 0.05354 0.11639 0.28350 0.976 0.0000 0.345
[10,] 0.62887 0.5866 0.4924 0.52228 0.45503 0.18887 0.17478 0.975 0.3452 0.000


structure(c(0, 0.113475177304965, 0.625498007968127, 0.455645161290323,
0.462151394422311, 0.344262295081967, 0.00847457627118597, 0.936936936936937,
0.705467372134039, 0.628865979381443, 0.113475177304965, 0, 0.371490280777538,
0.25531914893617, 0.286458333333333, 0.216911764705882, 0.011764705882353,
0.941176470588235, 0.585741811175337, 0.586638830897704, 0.625498007968127,
0.371490280777538, 0, 0.042643923240938, 0.027310924369748, 0.077338129496403,
0.168803418803419, 0.971861471861472, 0.122599704579025, 0.49238578680203,
0.455645161290323, 0.25531914893617, 0.042643923240938, 0, 0.00909090909090904,
0.136871508379888, 0.133333333333333, 0.954545454545454, 0.134980988593156,
0.522281639928699, 0.462151394422311, 0.286458333333333, 0.027310924369748,
0.00909090909090904, 0, 0.091412742382271, 0.149484536082474,
0.970930232558139, 0.053537284894837, 0.455026455026455, 0.344262295081967,
0.216911764705882, 0.077338129496403, 0.136871508379888, 0.091412742382271,
0, 0.00763358778626, 0.976833976833977, 0.116385911179173, 0.188870151770658,
0.00847457627118597, 0.011764705882353, 0.168803418803419, 0.133333333333333,
0.149484536082474, 0.00763358778626, 0, 0.951219512195122, 0.283495145631068,
0.174778761061947, 0.936936936936937, 0.941176470588235, 0.971861471861472,
0.954545454545454, 0.970930232558139, 0.976833976833977, 0.951219512195122,
0, 0.976237623762376, 0.975225225225225, 0.705467372134039, 0.585741811175337,
0.122599704579025, 0.134980988593156, 0.053537284894837, 0.116385911179173,
0.283495145631068, 0.976237623762376, 0, 0.345195729537367, 0.628865979381443,
0.586638830897704, 0.49238578680203, 0.522281639928699, 0.455026455026455,
0.188870151770658, 0.174778761061947, 0.975225225225225, 0.345195729537367,
0), .Dim = c(10L, 10L), .Dimnames = list(NULL, NULL))