I have been studying multi-class classification metrics. In the process, I discovered that in the case of the parameter average='micro', accuracy, precision, recall, and the f-1 score are all the same. Through various investigations and examples, I was able to confirm that the values of precision, recall, and f-1 score are the same. However, I don't understand why accuracy is also the same.
In multi-class classification, I heard that TN is usually not considered, and only the TP of each class is taken into account. So, I think the formula goes like this. (The formula might be incorrect). I already know that denominators of precision/recall are consequently same. But when simply checking with the formulas, the denominators of accuracy can not be the same.
I'm sorry that I can't show the images directly :(
I have brought an example from a link
ttl_TP = 15 + 15 + 45 = 75
ttl_FP = 9 + 5 + 11 = 25
ttl_FN = 5 + 15 + 5 = 25.
ttl_samples = 100
micro-averaging precision = 75 / 75 + 25 = 75 / 100.
micro-averaging recall = 75 / 75 + 25 = 75 / 100.
micro-averaging accuracy = 75 / 75 + 25 +25 = 75 / 125. (not the same as above)
Some articles say that \sum_{i=1}^{n} (TP_i + FP_i + FN_i) = \text{ttl_samples} is equal to ttl_samples. This calculation would make sense, as it would make the micro-averaging accuracy = 75 / 100.
I don't understand how \sum_{i=1}^{n} (TP_i + FP_i + FN_i) equals the ttl_samples. Additionally, I am curious to know if the formula I proposed is correct.
I would be very grateful for any help.