How to read/visualise this confusion matrix?

Question

I was offered a confusion matrix, which is called Parametrised 5x5 confusion matrix by authors where I extracted the values and where the diagonal has the correct results (0 in all); the results are bad, which can be seen in the confusion matrix; they do not understand why; in ideal situation, all values should be zero that is no failures should occur; I would like to visualise the matrix somehow to explain how interpret such non-standard confusion matrix

Real Prediction
     A  B  C  D  E
A    0  1  1  1  1 
B    1  0  2  2  2 
C    2  2  0  3  3
D    3  3  3  0  4
E    4  4  4  4  0

Claims

1. it is critical to have cells wrong in column(2-5) - row(2-5) i.e. the lower right square matrix; the authors put equal critical grade on all these cells
2. Column A and row A are ok.

Visualisation attempts

I am thinking if cells of column(2-5) - row(2-5) can visualised individually like in normal confusion matrix, since the deviation from the diagonal indicates the severity.

• Unable to use fourfoldplot for the inspection in R 3.1.1

  # http://stackoverflow.com/questions/23891140/r-how-to-visualize-confusion-matrix-using-the-caret-package
  ctable <- as.table(matrix(c(0, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 0, 3, 3, 3, 3, 3, 0, 4, 4, 4, 4, 4, 0), nrow = 5, byrow = TRUE))
  fourfoldplot(ctable, color = c("#CC6666", "#99CC99"),
           conf.level = 0, margin = 1, main = "Confusion Matrix")
  
  > install.packages("fourfoldplot")
  Installing package into ‘/usr/local/lib/R/site-library’
  (as ‘lib’ is unspecified)
  --- Please select a CRAN mirror for use in this session ---
  
  Warning message:
  package ‘fourfoldplot’ is not available (for R version 3.1.1) 
  

• Attempt in Mathematica

  MatrixPlot[{ {0, 1, 1, 1, 1}, {1, 0, 2, 2, 2}, {2, 2, 0, 3, 3}, {3, 3, 3, 0, 4}, {4, 4, 4, 4, 0}}]
  

  • Fig. 1 Output of the Mathematica code where you

    

Fig. 1 proposes me that the critical points are column1-row5 and column1-(row1-4) whch are brightest colored (orange) while the original claim that the whole lower square matrix column(2-5)-row(2-5) are equally all wrong. To claim that column1-row5 is right does not make sense.

How can you classify/interpet such a confusion matrix?

Answer 1

Per @Firebug's comment, this looks like an error cost matrix:

Real Prediction
     A  B  C  D  E
A    0  1  1  1  1 
B    1  0  2  2  2 
C    2  2  0  3  3
D    3  3  3  0  4
E    4  4  4  4  0

It looks like the classes are ordinal, with A high and E low and the cost for a correct prediction is 0, and the cost for predicting higher than actual is lower than predicting lower than actual. So if you predict a C as a C, you get no penalty, if you predict it as an A or B (higher) you get a 2 penalty, and if you predict id as a D or E (lower), you get a 3 penalty.

The penalties wrap around, so that predicting high on an E is the same as predicting low on a D, and the penalty for predicting high on a D is the same as predicting low on a C.

Sound right? I think this is being mixed up with a confusion matrix, which normally indicates the number of predictions of each type versus the actual classes. The best-case scenario is all numbers on the diagonal (prediction and actual agree) and all zero off-diagonal.