True Positive Rate and False Positive Rate (TPR, F

问题:

How do you compute the true- and false- positive rates of a multi-class classification problem? Say,

y_true = [1, -1,  0,  0,  1, -1,  1,  0, -1,  0,  1, -1,  1,  0,  0, -1,  0]
y_prediction = [-1, -1,  1,  0,  0,  0,  0, -1,  1, -1,  1,  1,  0,  0,  1,  1, -1]

The confusion matrix is computed by metrics.confusion_matrix(y_true, y_prediction), but that just shifts the problem.

---

EDIT after @seralouk's answer. Here, the class -1 is to be considered as the negatives, while 0 and 1 are variations of positives.

回答1:

Using your data, you can get all the metrics for all the classes at once:

import numpy as np
from sklearn.metrics import confusion_matrix

y_true = [1, -1,  0,  0,  1, -1,  1,  0, -1,  0,  1, -1,  1,  0,  0, -1,  0]
y_prediction = [-1, -1,  1,  0,  0,  0,  0, -1,  1, -1,  1,  1,  0,  0,  1,  1, -1]
cnf_matrix = confusion_matrix(y_true, y_prediction)
print(cnf_matrix)
#[[1 1 3]
# [3 2 2]
# [1 3 1]]

FP = cnf_matrix.sum(axis=0) - np.diag(cnf_matrix)  
FN = cnf_matrix.sum(axis=1) - np.diag(cnf_matrix)
TP = np.diag(cnf_matrix)
TN = cnf_matrix.sum() - (FP + FN + TP)

FP = FP.astype(float)
FN = FN.astype(float)
TP = TP.astype(float)
TN = TN.astype(float)

# Sensitivity, hit rate, recall, or true positive rate
TPR = TP/(TP+FN)
# Specificity or true negative rate
TNR = TN/(TN+FP) 
# Precision or positive predictive value
PPV = TP/(TP+FP)
# Negative predictive value
NPV = TN/(TN+FN)
# Fall out or false positive rate
FPR = FP/(FP+TN)
# False negative rate
FNR = FN/(TP+FN)
# False discovery rate
FDR = FP/(TP+FP)
# Overall accuracy
ACC = (TP+TN)/(TP+FP+FN+TN)

---

For a general case where we have a lot of classes, these metrics are represented graphically in the following image:

回答2:

Since there are several ways to solve this, and none is really generic (see https://stats.stackexchange.com/questions/202336/true-positive-false-negative-true-negative-false-positive-definitions-for-mul?noredirect=1&lq=1 and https://stats.stackexchange.com/questions/51296/how-do-you-calculate-precision-and-recall-for-multiclass-classification-using-co#51301), here is the solution that seems to be used in the paper which I was unclear about:

to count confusion between two foreground pages as false positive

So the solution is to import numpy as np, use y_true and y_prediction as np.array, then:

FP = np.logical_and(y_true != y_prediction, y_prediction != -1).sum()  # 9
FN = np.logical_and(y_true != y_prediction, y_prediction == -1).sum()  # 4
TP = np.logical_and(y_true == y_prediction, y_true != -1).sum()  # 3
TN = np.logical_and(y_true == y_prediction, y_true == -1).sum()  # 1
TPR = 1. * TP / (TP + FN)  # 0.42857142857142855
FPR = 1. * FP / (FP + TN)  # 0.9

回答3:

Another simple way is PyCM (by me), that supports multi-class confusion matrix analysis.

Applied to your Problem :

>>> from pycm import ConfusionMatrix
>>> y_true = [1, -1,  0,  0,  1, -1,  1,  0, -1,  0,  1, -1,  1,  0,  0, -1,  0]
>>> y_prediction = [-1, -1,  1,  0,  0,  0,  0, -1,  1, -1,  1,  1,  0,  0,  1,  1, -1]
>>> cm = ConfusionMatrix(actual_vector=y_true,predict_vector=y_prediction)
>>> print(cm)
Predict          -1       0        1        
Actual
-1               1        1        3        
0                3        2        2        
1                1        3        1        




Overall Statistics : 

95% CI                                                           (0.03365,0.43694)
Bennett_S                                                        -0.14706
Chi-Squared                                                      None
Chi-Squared DF                                                   4
Conditional Entropy                                              None
Cramer_V                                                         None
Cross Entropy                                                    1.57986
Gwet_AC1                                                         -0.1436
Joint Entropy                                                    None
KL Divergence                                                    0.01421
Kappa                                                            -0.15104
Kappa 95% CI                                                     (-0.45456,0.15247)
Kappa No Prevalence                                              -0.52941
Kappa Standard Error                                             0.15485
Kappa Unbiased                                                   -0.15405
Lambda A                                                         0.2
Lambda B                                                         0.27273
Mutual Information                                               None
Overall_ACC                                                      0.23529
Overall_RACC                                                     0.33564
Overall_RACCU                                                    0.33737
PPV_Macro                                                        0.23333
PPV_Micro                                                        0.23529
Phi-Squared                                                      None
Reference Entropy                                                1.56565
Response Entropy                                                 1.57986
Scott_PI                                                         -0.15405
Standard Error                                                   0.10288
Strength_Of_Agreement(Altman)                                    Poor
Strength_Of_Agreement(Cicchetti)                                 Poor
Strength_Of_Agreement(Fleiss)                                    Poor
Strength_Of_Agreement(Landis and Koch)                           Poor
TPR_Macro                                                        0.22857
TPR_Micro                                                        0.23529

Class Statistics :

Classes                                                          -1                      0                       1                       
ACC(Accuracy)                                                    0.52941                 0.47059                 0.47059                 
BM(Informedness or bookmaker informedness)                       -0.13333                -0.11429                -0.21667                
DOR(Diagnostic odds ratio)                                       0.5                     0.6                     0.35                    
ERR(Error rate)                                                  0.47059                 0.52941                 0.52941                 
F0.5(F0.5 score)                                                 0.2                     0.32258                 0.17241                 
F1(F1 score - harmonic mean of precision and sensitivity)        0.2                     0.30769                 0.18182                 
F2(F2 score)                                                     0.2                     0.29412                 0.19231                 
FDR(False discovery rate)                                        0.8                     0.66667                 0.83333                 
FN(False negative/miss/type 2 error)                             4                       5                       4                       
FNR(Miss rate or false negative rate)                            0.8                     0.71429                 0.8                     
FOR(False omission rate)                                         0.33333                 0.45455                 0.36364                 
FP(False positive/type 1 error/false alarm)                      4                       4                       5                       
FPR(Fall-out or false positive rate)                             0.33333                 0.4                     0.41667                 
G(G-measure geometric mean of precision and sensitivity)         0.2                     0.30861                 0.18257                 
LR+(Positive likelihood ratio)                                   0.6                     0.71429                 0.48                    
LR-(Negative likelihood ratio)                                   1.2                     1.19048                 1.37143                 
MCC(Matthews correlation coefficient)                            -0.13333                -0.1177                 -0.20658                
MK(Markedness)                                                   -0.13333                -0.12121                -0.19697                
N(Condition negative)                                            12                      10                      12                      
NPV(Negative predictive value)                                   0.66667                 0.54545                 0.63636                 
P(Condition positive)                                            5                       7                       5                       
POP(Population)                                                  17                      17                      17                      
PPV(Precision or positive predictive value)                      0.2                     0.33333                 0.16667                 
PRE(Prevalence)                                                  0.29412                 0.41176                 0.29412                 
RACC(Random accuracy)                                            0.08651                 0.14533                 0.10381                 
RACCU(Random accuracy unbiased)                                  0.08651                 0.14619                 0.10467                 
TN(True negative/correct rejection)                              8                       6                       7                       
TNR(Specificity or true negative rate)                           0.66667                 0.6                     0.58333                 
TON(Test outcome negative)                                       12                      11                      11                      
TOP(Test outcome positive)                                       5                       6                       6                       
TP(True positive/hit)                                            1                       2                       1                       
TPR(Sensitivity, recall, hit rate, or true positive rate)        0.2                     0.28571                 0.2