Micro F1 Score

1. F1 Score Recap (Binary Case)

• F1 score = harmonic mean of precision and recall:

$F1 = \frac{2 \times Precision \times Recall}{Precision + Recall}$

• Balances both precision (low false positives) and recall (low false negatives).

---

2. Multiclass Extension

For K classes, there are two main averaging strategies: macro and micro.

• Macro F1: Compute F1 per class (OvR), then average equally.
• Micro F1: Pool all predictions across classes → compute global TP, FP, FN → then calculate F1.

---

3. Definition of Micro F1

Step 1: Compute global precision and global recall:

$Precision_{micro} = \frac{\sum TP_i}{\sum (TP_i + FP_i)}$

$Recall_{micro} = \frac{\sum TP_i}{\sum (TP_i + FN_i)}$

Step 2: Compute F1 using those:

$F1_{micro} = \frac{2 \times Precision_{micro} \times Recall_{micro}}{Precision_{micro} + Recall_{micro}}$

---

4. Key Property

• In multiclass (single-label) classification, Micro Precision = Micro Recall = Micro F1.
  • Reason: When each sample has exactly one label, TP + FN = total positives, and TP + FP = total predicted positives. This symmetry makes all three equal.

---

5. Example

Suppose we have 3 classes (A, B, C):

• TP (true positives): A=40, B=30, C=10
• FP (false positives): A=10, B=20, C=20
• FN (false negatives): A=10, B=20, C=30

Micro Precision:

$\frac{40+30+10}{(40+10)+(30+20)+(10+20)} = \frac{80}{130} \approx 0.615$

Micro Recall:

$\frac{40+30+10}{(40+10)+(30+20)+(10+30)} = \frac{80}{140} \approx 0.571$

Micro F1:

$\frac{2 \times 0.615 \times 0.571}{0.615 + 0.571} \approx 0.592$

So here: Micro F1 ≈ 0.59.

---

6. When to Use

• Micro F1 is useful when you want to measure overall system performance, giving more weight to majority classes.
• Macro F1 is better if you want fairness across classes, including minority ones.

---

Summary

• Micro F1 = F1 score computed from global TP, FP, FN across all classes.
• In multiclass (single-label), Micro Precision = Micro Recall = Micro F1.
• Good for imbalanced datasets where majority class performance dominates the evaluation.