Confusion Matrix

A confusion matrix counts how predicted classes differ from true classes. It is the base table behind many classification metrics.

For binary classification:

	Predicted positive	Predicted negative
True positive class	TP	FN
True negative class	FP	TN

The total number of evaluated examples is:

$$n=\mathrm{TP}+\mathrm{FP}+\mathrm{TN}+\mathrm{FN}$$

Derived Quantities

True positive rate, or recall:

$$\mathrm{TPR}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}$$

False positive rate:

$$\mathrm{FPR}=\frac{\mathrm{FP}}{\mathrm{FP}+\mathrm{TN}}$$

Positive predictive value, or precision:

$$\mathrm{PPV}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}$$

True negative rate, or specificity:

$$\mathrm{TNR}=\frac{\mathrm{TN}}{\mathrm{TN}+\mathrm{FP}}$$

False negative rate:

$$\mathrm{FNR}=\frac{\mathrm{FN}}{\mathrm{TP}+\mathrm{FN}}$$

Accuracy:

$$\mathrm{Accuracy}=\frac{\mathrm{TP}+\mathrm{TN}}{n}$$

Balanced accuracy:

$$\mathrm{BalancedAccuracy}=\frac{1}{2}(\mathrm{TPR}+\mathrm{TNR})$$

Balanced accuracy is often more informative than raw accuracy under class imbalance.

Why It Matters

Aggregate metrics hide which error type dominates. In imbalanced problems, a model can have high accuracy while missing most positives or producing too many false positives.

Threshold and Prevalence

The confusion matrix is defined after a decision rule. For a score $s(x)$:

$$\hat{y}=1[s(x)\ge \tau ]$$

Changing $\tau$ changes TP, FP, TN, and FN. A test-set confusion matrix should use a threshold chosen from validation data or a fixed application rule.

Prevalence changes how errors feel operationally. With rare positives, even a low FPR can produce many false positives:

$$\mathrm{FP}\approx (1-\pi )N\mathrm{FPR}$$

where $\pi$ is the positive prevalence and $N$ is the evaluation size.

Multi-Class Form

For $K$ classes, the confusion matrix is:

$$C_{ab}=\#\{i:y_i=a,{\hat{y}}_i=b\}$$

Rows often represent true classes and columns predicted classes. State the convention because transposed matrices invert the reading of common errors.

Checks

• Which class is treated as positive?
• Are classes imbalanced?
• Is the threshold fixed before test evaluation?
• Are false positives and false negatives equally costly?
• Should metrics be reported per class, macro-averaged, or micro-averaged?
• Does the paper show the confusion matrix or only a single derived metric?

Related

• Classification metrics
• Precision and recall
• Threshold selection
• Error analysis
• Confidence interval
• Classification