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.
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:
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
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
来源：https://stackoverflow.com/questions/50666091/true-positive-rate-and-false-positive-rate-tpr-fpr-for-multi-class-data-in-py