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The confusion matrix is a performance measurement method that can be used for binary classification and multi-class classifications.  It is usually shown shown in a table format with the four possible outcomes: True Positive (TP), True Negative (TN), False Positive (FP), and False Negative (FN).  

Confusion Matrix

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Chapter 10: Evaluating Model Performance
Section: A closer look at confusion matrices

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How to read confusion matrix for binary classification

- True Positive (TP)
- True Negative (TN)
- False Positive (FP)
- False Negative (FN)

Components of Confusion Matrix

The multi-class confusion matrix is a straightforward extension of the binary classifier two by two confusion matrix.
For example, for classification of hand-written digits, there are ten classes for each digit, zero through nine. So, the ten-class confusion matrix is a 10x10 matrix with the true digit class indexed by row and the predicted digit class indexed by column. Like the 2x2 case, the correct prediction is by the classifier where the true class matches the predicted class are all along the diagonal and misclassifications are off the diagonal.

In this example, most of the predictions are correct with only a few misclassifications. The most frequent type of mistake here is misclassifying the true digit, 8 as a predicted digit 1 which happened three times.
And the overall accuracy is high, about 97%.

Muti-class Confusion Matrix

Given the following confusion matrix, compute the accuracy.

Classification accuracy and error rate are primary metrics used to evaluate the performance of a classification model. Let TP, FP, TN, and FN denote the number of true positives, false positives, true negatives, and false negatives, respectively.

The classification Accuracy is the proportion of true results (both true positives and true negatives) among the total number of cases examined:
$$\text{Accuracy} = \frac{\text{TP} + \text{TN}}{\text{TP} + \text{TN} + \text{FP} + \text{FN}}$$

The Error Rate is the proportion of misclassified cases (both false positives and false negatives) among the total number of cases examined, which is also equal to 1 minus Accuracy:
$$\text{Error Rate} = \frac{\text{FP} + \text{FN}}{\text{TP} + \text{TN} + \text{FP} + \text{FN}} = 1 - \text{Accuracy}$$

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