Can we estimate value of missing correlation in a given correlation matrix?

This question has no practical importance for me, but anyway: I am interested whether we can predict somehow one missing value in a correlation matrix. Obviously, classical regression modeling is not appropriate here. Is there any other way, how we can say something about missing correlation? In addition, does the answer depend on the type of correlation (Pearson, Spearman, Kendall, MIC)? Here is an example (real data, Pearson correlation).

matrix <- structure(c(1, NA, 0.126, -0.163, 0.203, 0.172, -0.101, 0.068,
-0.125, 0.066, 0.195, NA, 1, 0.019, -0.169, 0.211, 0.265, -0.023,
0.088, 0.016, 0.14, 0.275, 0.126, 0.019, 1, 0.218, 0.094, -0.053,
0.189, 0.179, -0.07, -0.021, 0.172, -0.163, -0.169, 0.218, 1,
-0.066, -0.182, 0.302, 0.193, -0.116, -0.08, -0.151, 0.203, 0.211,
0.094, -0.066, 1, 0.251, 0.064, 0.122, 0.093, 0.214, 0.378, 0.172,
0.265, -0.053, -0.182, 0.251, 1, -0.101, 0.02, 0.192, 0.353,
0.304, -0.101, -0.023, 0.189, 0.302, 0.064, -0.101, 1, 0.652,
0.126, 0.023, -0.016, 0.068, 0.088, 0.179, 0.193, 0.122, 0.02,
0.652, 1, 0.127, 0.01, 0.083, -0.125, 0.016, -0.07, -0.116, 0.093,
0.192, 0.126, 0.127, 1, 0.295, 0.023, 0.066, 0.14, -0.021, -0.08,
0.214, 0.353, 0.023, 0.01, 0.295, 1, 0.299, 0.195, 0.275, 0.172,
-0.151, 0.378, 0.304, -0.016, 0.083, 0.023, 0.299, 1), .Dim = c(11L,
11L), .Dimnames = list(NULL, NULL))

• It is not very clear what you mean by "estimating" here. Given the rest correlations, the unknown correlation may usually vary within some bounds; the bounds being dependent on the rest correlations. Is your question about those bounds? – ttnphns Oct 5 '13 at 9:33
• @ttnphns, yes, this is exactly what I need - whether we can mine any information from rest correlations or we can only say that missing value is within [-1,1]. – Miroslav Sabo Oct 5 '13 at 9:36
• Technically, these bounds is the interval which will keep the whole matrix being positive semidefinite (gramian). This is how it is with Pearson r (and with Spearman, since basically it is the same formula). With other types of correlation the issue seems to be more complex... – ttnphns Oct 5 '13 at 9:47
• And more problems will also arise when many values will be missing. – Miroslav Sabo Oct 5 '13 at 10:05
• I wouldn't put it this way "more problems arise". The less number of entries are present the less constrained is the "task". I we have all correlations missing but one, we are free to compose almost the entire matrix in great many ways. – ttnphns Oct 5 '13 at 10:11

The principle is that you want to impute no special relation between the two variables other than that which is implied by their known relations to the other variables. This implies that you should plug in the value that zeros the partial correlation of the two variables with the $p-2$ other variables held constant. This is the same as maximizing the determinant of the new matrix; the new inverse will have zeros in the corresponding positions. (This works for any symmetric positive definite matrix, not just a Pearson correlation matrix.) In the sample matrix the imputed value is .119695.