How to understand confusion matrix for 3x3

Question

I used Sklearn logistic regression for multiclass classifier to classify as Male , Female and Infant on abalone data set Below is my sample Logistic regression for multi classifier

    log_reg=LogisticRegression()
    log_model=log_reg.fit(x_train,y_train)
    pred=log_model.predict(x_test)
    confusion_matrix(y_test,pred)

Below is my confusion Matrix

    M   F   I           --- predicted
M   [[  64, 46, 39],
F   [   12, 237,42],
I   [   52, 79, 165]] actual vs Predicted 

Consider a case of 2X2 where I classified patient as HIV positive --1

    1       0       --- predicted
1   [[  1--TP,  0--FN],
0   [   1--FP,  0--TN ]]  Act vs Predicted 

unlike 2 x 2 I am unable to extrapolate it to N X N only I can make out is 64 I predicted as Male and which is Actually male as True Positive My question is how can I identify True Negative , False Positive , false Negative .

Answer 1

Based on the 3x3 confusion matrix in your example (assuming I'm understanding the labels correctly) the columns are the predictions and the rows must therefore be the actual values. The main diagonal (64, 237, 165) gives the correct predictions. That is, the cases where the actual values and the model predictions are the same.

The first row are the actual males. The model predicted 64 of these correctly and incorrectly predicted 46 of the males to be female and 139 of the males to be infants.

Looking at the male column, of the 128 males predicted by the model (sum of column M), 64 were actually males, while 12 were females incorrectly predicted to be males and 52 were infants incorrectly predicted to be males.

Analogous interpretations apply to the other columns and rows.

          Predicted
          M    F    I 
Actual M 64   46  139
       F 12  237   42
       I 52   79  165

If this is a multinomial logistic regression model, then the model output would be predicted probabilities that each observation belongs to a particular class, rather than predicted classes. The links in @StephenKolassa's answer discuss the issue of scoring rules and you may want to consider what scoring rule will result in classifications that minimize a loss function tailored to your specific needs.

Answer 2

True Positive, False Positive and similar counts and rates only make sense if there is a notion of "positive" and "negative" classes in your data. That is, only if you have exactly two classes. You have three classes, not two.

In your case, you can more or less reasonably discuss analogues, like "True Male" numbers: take the number of cases you correctly (!) classify as male and divide by the total number of males in the test sample.

Note that TPR, FPR, accuracy and similar KPIs have major problems if used to evaluate classification models.

Answer 3

TP, TN, FP, FN - in 3x3 matrix could be defined PER CLASS

In the above example: For M class: TP - real M predicted as M (64) TN - real F predicted as F and real I predicted as I (237+165) FP - real F and I predicted as M (12+52) FN - real M predicted as F or I (46+139)

Then you can calculate Precision and Recall metrics (per class).