Understanding the Correlation[] function in Mathematica

Question

I have this data for four quantities measured in different temperatures:
data = {{729, 656, 2156, 2035}, {761, 677, 2178, 2060}, {952, 856, 
    2651, 2465}, {888, 796, 2311, 2139}, {747, 645, 1807, 1658}, {863,
     737, 1975, 1817}, {844, 734, 1841, 1654}, {705, 602, 1466, 
    1305}, {788, 686, 1491, 1303}, {927, 779, 1732, 1547}, {910, 742, 
    1616, 1429}};
N[Correlation[data], 2]

There are 11 temperature measurements and 4 quantities measured at the same time. The correlation function gives as output:
{{1.0, 0.95, 0.36, 0.30}, {0.95, 1.0, 0.59, 0.54}, {0.36, 0.59, 1.0, 
  1.0}, {0.30, 0.54, 1.0, 1.0}}

As I understand, the output of Correlation[] gives a matrix of the Pearson coefficient for each element in the data. Is that correct? Does it mean that the first and last entries in each data point have a correlation of 0.30?

Chris Degnen · Answer

data = {
   {729, 656, 2156, 2035}, {761, 677, 2178, 2060},
   {952, 856, 2651, 2465}, {888, 796, 2311, 2139},
   {747, 645, 1807, 1658}, {863, 737, 1975, 1817},
   {844, 734, 1841, 1654}, {705, 602, 1466, 1305},
   {788, 686, 1491, 1303}, {927, 779, 1732, 1547},
   {910, 742, 1616, 1429}};
matrix = N[Correlation[data], 4];

rules = (N[Correlation @@
       Transpose[data[[All, #]]], 4] -> #) & /@ Tuples[Range[4], {2}];

{1,1} {1,2} {1,3} {1,4}
{1,2} {1,1} {2,3} {2,4}
{1,3} {2,3} {1,1} {3,4}
{1,4} {2,4} {3,4} {1,1}

This grid shows which columns correspond to the correlations in the matrix.
E.g. Grid[matrix]

1.000   0.9504  0.3553  0.3012
0.9504  1.000   0.5918  0.5411
0.3553  0.5918  1.000   0.9977
0.3012  0.5411  0.9977  1.000

0.3012 corresponds to the correlation of columns 1 & 4
N[Correlation @@ Transpose[data[[All, {1, 4}]]], 4]

0.3012

Understanding the Correlation[] function in Mathematica

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