Implement Micro-F1

Metrics & Evaluation

Easy

Problem

Theory

The F1 score is the harmonic mean of precision and recall:

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

For binary classification:

• Precision = TP / (TP + FP)
• Recall = TP / (TP + FN)

F1 balances the trade-off between precision and recall in a single metric.

The Multi-Class Challenge

With more than two classes, we need to aggregate F1 scores across all classes. There are three main approaches:

1. Micro-averaged F1
2. Macro-averaged F1
3. Weighted F1

Each gives different insights and is appropriate for different scenarios.

Micro-Averaged F1

Micro-averaging computes global precision and recall by summing all true positives, false positives, and false negatives across all classes, then computes F1 from these totals.

\text{Precision}_{\text{micro}} = \frac{\sum_i TP_i}{\sum_i (TP_i + FP_i)}

\text{Recall}_{\text{micro}} = \frac{\sum_i TP_i}{\sum_i (TP_i + FN_i)}

F1_{\text{micro}} = 2 \times \frac{\text{Precision}_{\text{micro}} \times \text{Recall}_{\text{micro}}}{\text{Precision}_{\text{micro}} + \text{Recall}_{\text{micro}}}

Key Insight: Micro F1 Equals Accuracy

For multi-class classification where each sample belongs to exactly one class:

\sum_i TP_i = \text{total correct predictions}

\sum_i (TP_i + FP_i) = \sum_i (TP_i + FN_i) = \text{total samples}

Therefore:

\text{Precision}_{\text{micro}} = \text{Recall}_{\text{micro}} = \text{Accuracy}

And:

F1_{\text{micro}} = \text{Accuracy}

Micro F1 and accuracy are the same for single-label multi-class problems.

Worked Example

3-class problem with 100 samples:

Class A: 50 samples, 45 correct (TP=45, FN=5)

Class B: 30 samples, 24 correct (TP=24, FN=6)

Class C: 20 samples, 16 correct (TP=16, FN=4)

Computing per-class metrics:

For Class A:

• FP = samples predicted A but actually B or C
• Let us say FP_A = 3

For Class B: FP_B = 4

For Class C: FP_C = 8

Micro-averaged precision:

\text{Precision}_{\text{micro}} = \frac{45 + 24 + 16}{(45+3) + (24+4) + (16+8)} = \frac{85}{100} = 0.85

Micro-averaged recall:

\text{Recall}_{\text{micro}} = \frac{45 + 24 + 16}{(45+5) + (24+6) + (16+4)} = \frac{85}{100} = 0.85

Micro F1:

F1_{\text{micro}} = 2 \times \frac{0.85 \times 0.85}{0.85 + 0.85} = 0.85

This equals accuracy: 85 correct out of 100.

Micro vs. Macro F1

Micro F1:

• Aggregates contributions from all classes
• Dominated by majority classes
• Good when you care about overall performance
• Equals accuracy for single-label classification

Macro F1:

• Computes F1 for each class, then averages
• Each class weighted equally regardless of size
• Good when all classes are equally important
• Sensitive to performance on minority classes

When Classes Are Imbalanced

Dataset: 1000 samples

• Class A: 900 samples
• Class B: 80 samples
• Class C: 20 samples

Model predicts everything as Class A:

Micro F1 = 900/1000 = 0.90 (looks good!)

Macro F1:

• F1_A = high (most correct)
• F1_B = 0 (none predicted)
• F1_C = 0 (none predicted)
• Macro F1 = (high + 0 + 0) / 3 = low

Micro F1 hides the failure on minority classes. Macro F1 exposes it.

The Formulas Side by Side

Micro F1:

F1_{\text{micro}} = \frac{2 \times \sum_i TP_i}{2 \times \sum_i TP_i + \sum_i FP_i + \sum_i FN_i}

Macro F1:

F1_{\text{macro}} = \frac{1}{n} \sum_{i=1}^{n} F1_i

Weighted F1:

F1_{\text{weighted}} = \sum_{i=1}^{n} w_i \times F1_i

where w_i is the proportion of samples in class i.

When to Use Micro F1

All samples matter equally: If correctly classifying any sample is equally valuable, regardless of class.

Class imbalance reflects real-world distribution: If the test distribution matches deployment, overall accuracy matters.

Comparing to baselines: Micro F1 (accuracy) is a standard, widely understood metric.

When to Use Macro F1 Instead

All classes matter equally: In medical diagnosis, correctly identifying rare diseases is as important as common ones.

Evaluating fairness: Ensure the model performs well across all groups, not just majority.

Minority class performance is critical: When failing on rare classes is unacceptable.

Multi-Label Classification

For multi-label problems (each sample can have multiple labels):

• Micro F1 aggregates TP, FP, FN across all samples and all labels
• Micro F1 no longer equals accuracy
• The formulas remain the same, but interpretation differs

Example: Sample has labels [A, B], model predicts [A, C]

• TP = 1 (A)
• FP = 1 (C)
• FN = 1 (B missed)

Implementation Notes

Computing micro F1:

1. Build confusion matrix (n_classes x n_classes)
2. Sum diagonal for total TP
3. Sum each column for (TP + FP) per class
4. Sum each row for (TP + FN) per class
5. Apply micro formulas

Efficient computation:

For single-label classification:

• Micro F1 = (correct predictions) / (total predictions)
• Just count matches between predictions and labels

Common Pitfalls

Confusing micro and macro: They can give very different values on imbalanced data. Always specify which you are using.

Ignoring class distribution: High micro F1 can mask poor minority class performance.

Not reporting both: Best practice is to report both micro and macro (or weighted) to give a complete picture.

Examples

Hints

1. sum(actual == predicted for actual, predicted in zip(y_true, y_pred)) counts correct single-label predictions.
2. For single-label multiclass data, every mismatch contributes one false positive and one false negative.

Requirements

• y_true and y_pred contain one integer class label per sample
• Aggregate counts across every class
• Return a Python float rounded to four decimal places

Constraints

• y_true and y_pred have the same nonzero length
• At most 100{,}000 labels are provided
• Do not use an external machine learning library

Starter Code

def f1_micro(y_true: list[int], y_pred: list[int]) -> float:
    """
    Returns the micro-averaged F1 score as a Python float rounded to four decimals.
    """
    # Write code here
    pass

Test Cases