Calculation method of AUC

About AUC Part of the understanding and calculation can refer to ：https://blog.csdn.net/lieyingkub99/article/details/81266664?utm_medium=distribute.pc_relevant.none-task-blog-title-1&spm=1001.2101.3001.4242

But the above calculation AUC The explanation of the second method is not very good , Let's understand here .

Based on Method 1 above ：
（1） Suppose there is $M$ A positive sample ,$N$ Negative samples , Positive and negative samples are combined in pairs , common $M\ast N$ Group .
（2） initialization $P=0$. Two classifiers are used to predict two samples of each group ,
If the classifier predicts The probability that the positive sample is a positive sample is greater than the probability that the negative sample is a positive sample , be $P+1$;
If the classifier predicts The probability that the positive sample is a positive sample is less than the probability that the negative sample is a positive sample , be $P$ unchanged ;
If the classifier predicts The probability that the positive sample is a positive sample is equal to the probability that the negative sample is a positive sample , be $P+0.5$.
（3）$AUC=P/(M\ast N)$.

Deform method 1 , Get formula method 2 ：
（1） Use two classifiers to predict all positive and negative samples , Get the probability that they are positive samples ;
（2） Sort according to the probability from small to large ;
（3） Give each sample an attribute $R$,$R$ Indicates the number of samples whose probability is less than this sample +1.
For example 100 Samples , The probability of a sample is the highest , by 0.99, Of this sample $R=100$.$R$ At the same time, it is equal to the index of the sorted sample +1.
（4） For positive samples with the greatest probability ,$R$ Equal to the sum of the number of positive and negative samples whose probability is less than this sample +1, The number of positive samples is $M-1$, Then the number of negative samples is $R_1-(M-1)-1=R_1-M$.
For positive samples with the second largest probability ,$R$ Equal to the sum of the number of positive and negative samples whose probability is less than this sample +1, The number of positive samples is $M-2$, Then the number of negative samples is $R_2-(M-2)-1=R_2-(M-1)$.
By analogy , consider AUC Is the sum of the number of groups in which the probability of positive samples is greater than the probability of negative samples , Divided by $M\ast N$. Then the probability of positive samples is greater than the sum of the number of groups of the probability of negative samples , be equal to $[R_1-M]+[R_2-(M-1)]+[R_3-(M-2)]+...+[R_m-(1)]$.
According to the summation formula ：$1+2+..+M=M\ast (1+M)/2$,
$AUC=[sum(R_i)+M\ast (1+M)/2]/M\ast N$.
（5） When the probability of positive and negative samples is equal , Of this positive sample $R_i$ Equal to all that have the same probability $R$ The average of .