# Correlation Matrix

Imagery analysis menu, Multi-grids raster analysis.  You must first have converted the satellite image into a series of grids for numerical computations.

A correlation matrix takes each pair of satellite bands (or other related grids) and computes the correlation coefficient between the reflectances in the two bands for each pixel in the image. Because the matrix is symmetrical (the correlation between Bands 1 and 2 does not depend on the order of considering the two bands), often only the upper half is presented. The correlations along the principal diagonal are 1.0, since anything allows perfect prediction of itself.

Correlation Matrix

 Band Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 Band 9 Band 10 Band 11 Band 1 1.0000 0.9804 0.7420 0.6295 -0.0220 0.1765 0.3439 0.5334 -0.2145 -0.2889 Band 2 0.9804 1.0000 0.8318 0.7498 0.0425 0.2912 0.4626 0.4757 -0.0816 -0.1565 Band 3 0.7420 0.8318 1.0000 0.9440 0.4954 0.6991 0.7934 0.3144 0.4001 0.3149 Band 4 0.6295 0.7498 0.9440 1.0000 0.4255 0.7220 0.8325 0.2141 0.4690 0.3973 Band 5 -0.0220 0.0425 0.4954 0.4255 1.0000 0.8132 0.6598 0.0533 0.6981 0.6347 Band 6 0.1765 0.2912 0.6991 0.7220 0.8132 1.0000 0.9565 0.0279 0.7919 0.7300 Band 7 0.3439 0.4626 0.7934 0.8325 0.6598 0.9565 1.0000 0.0689 0.7228 0.6606 Band 9 0.5334 0.4757 0.3144 0.2141 0.0533 0.0279 0.0689 1.0000 -0.3577 -0.4422 Band 10 -0.2145 -0.0816 0.4001 0.4690 0.6981 0.7919 0.7228 -0.3577 1.0000 0.9875 Band 11 -0.2889 -0.1565 0.3149 0.3973 0.6347 0.7300 0.6606 -0.4422 0.9875 1.0000

In this example from a Landsat 8 scene, the best correlation of 0.98 is between bands 1 and 2 (both of which lie in the visible), and bands 10 and 11 in the thermal IR.   Several bands have negative correlations (especially 9 and the two TIR bands), and some have essentially no correlation (bands 1 and 5).  The nature of this matrix depends entirely on what is in the scene; there might be be no negative correlations.

If two bands were perfectly correlated they would be redundant; the recorded reflectance in one channel could be used to predict exactly the value in the other.  Principal Components  analysis can suggest a new set of bands that reduces data volume with little loss of information. When bands are highly correlated, the outliers are unusual and might have the most interest to determine what is different about them.

Correlation Matrix For A Different Scene (Rif of Morocco).  Note there are no negative correlations
 L8 Band 1 L8 Band 2 L8 Band 3 L8 Band 4 L8 Band 5 L8 Band 6 L8 Band 7 L8 Band 9 L8 Band 10 L8 Band 11 L8 Band 1 1.0000 0.9799 0.8637 0.7910 0.6444 0.6496 0.6830 0.2110 0.4909 0.4638 L8 Band 2 0.9799 1.0000 0.9427 0.8835 0.7638 0.7650 0.7925 0.3041 0.6167 0.5966 L8 Band 3 0.8637 0.9427 1.0000 0.9792 0.9143 0.9141 0.9282 0.4366 0.7901 0.7786 L8 Band 4 0.7910 0.8835 0.9792 1.0000 0.9429 0.9632 0.9744 0.4677 0.8420 0.8289 L8 Band 5 0.6444 0.7638 0.9143 0.9429 1.0000 0.9651 0.9519 0.5164 0.8577 0.8458 L8 Band 6 0.6496 0.7650 0.9141 0.9632 0.9651 1.0000 0.9956 0.5146 0.8894 0.8752 L8 Band 7 0.6830 0.7925 0.9282 0.9744 0.9519 0.9956 1.0000 0.5025 0.8844 0.8713 L8 Band 9 0.2110 0.3041 0.4366 0.4677 0.5164 0.5146 0.5025 1.0000 0.5187 0.5235 L8 Band 10 0.4909 0.6167 0.7901 0.8420 0.8577 0.8894 0.8844 0.5187 1.0000 0.9957 L8 Band 11 0.4638 0.5966 0.7786 0.8289 0.8458 0.8752 0.8713 0.5235 0.9957 1.0000

The scattergrams below show only two bands, a concession to human abilities, while the Landsat scene has 11, several of which (coastal, cirrus, and thermal) might be less useful.  For Hyperspectral imagery, there could be hundreds of bands, and the challenge is even greater because the correlation matrix overwhelms the user with its size.

 Scatterplot of two bands from a scene in the Rif of Morocco with low correlation.  r: 0.3041 rē: 0.0925 There is a best fit line, but the predictions from it are not very good. Scatterplot of two TIR bands from a scene in the Rif of Morocco with very high correlation r: 0.9957 rē: 0.9915 The graph shows colors with concentrations of points, and most are in the center of the cloud of points. Scatterplot of two bands from a scene in theAtlas of Morocco with negative correlation r: -0.3386 rē: 0.1146 Scatterplot of two bands from a scene including Rabat, Morocco.  The trend is not linear, and reflects the fact that the scene covers two very distinct regions: Atlantic Ocean with clouds, and the coastal plain. r: -0.8351 rē: 0.6974

Last revision 12/29/2017