Fβ-Score Measure

An Fβ-Score Measure is a harmonic mean balanced bounded monotonic binary classification measure that computes a parameterized weighted harmonic mean of precision and recall where the parameter β represents the relative importance of recall to precision.

• AKA: Van Rijsbergen's F.
• Context:
  • It can typically compute F-Measure Scores through f-measure parameterized harmonic mean of f-measure precision and f-measure recall with f-measure beta weight parameter.
  • It can typically weight F-Measure Precision Components through f-measure beta-squared factor in f-measure denominator term.
  • It can typically weight F-Measure Recall Components through f-measure unit factor in f-measure denominator term.
  • It can typically behave as a Bounded Monotonic Utility Measure through f-measure normalized range from zero to one.
  • It can typically provide F-Measure Interpretable Scores through f-measure higher-is-better principle.
  • It can typically handle F-Measure Class Imbalances through f-measure precision-recall focus rather than f-measure accuracy bias.
  • It can typically support F-Measure Parameter Tuning through f-measure beta adjustment for f-measure application-specific requirements.
  • It can typically express F-Measure User Preferences through f-measure beta parameter representing f-measure recall-to-precision importance ratio.
  • It can typically achieve F-Measure Maximum Values only when both f-measure precision and f-measure recall are maximized.
  • ...
  • It can often penalize F-Measure Extreme Values through f-measure harmonic mean properties.
  • It can often guide F-Measure Model Selections through f-measure performance comparisons across f-measure candidate models.
  • It can often enable F-Measure Threshold Optimizations through f-measure score maximization over f-measure decision boundarys.
  • It can often facilitate F-Measure Cross-Validations through f-measure fold evaluations in f-measure model assessment.
  • It can often support F-Measure Multi-Class Extensions through f-measure averaging strategies like f-measure macro-averaging and f-measure micro-averaging.
  • It can often be preferred over Accuracy Metrics when f-measure positive classes are rare or f-measure class costs differ.
  • It can often be computed from F-Measure Confusion Matrix Elements: f-measure true positives, f-measure false positives, and f-measure false negatives.
  • ...
  • It can range from being a Precision-Focused F-Measure to being a Recall-Focused F-Measure, depending on its f-measure beta parameter value.
  • It can range from being a Binary F-Measure to being a Multi-Class F-Measure, depending on its f-measure classification scope.
  • It can range from being a Micro-Averaged F-Measure to being a Macro-Averaged F-Measure, depending on its f-measure aggregation strategy.
  • It can range from being a Standard F-Measure to being a Weighted F-Measure, depending on its f-measure class importance.
  • It can range from being a Point F-Measure to being an Interval F-Measure, depending on its f-measure temporal scope.
  • It can range from being a Hard F-Measure to being a Soft F-Measure, depending on its f-measure prediction type.
  • It can range from being a Single-Label F-Measure to being a Multi-Label F-Measure, depending on its f-measure label assignment.
  • ...
  • It can be calculated using F-Measure General Formula: F_β = (1 + β²) × (precision × recall) / (β² × precision + recall).
  • It can be derived from Van Rijsbergen's Effectiveness Measure through f-measure complementary transformation.
  • It can be interpreted as F-Measure Weighted Average biased toward the f-measure lower value between f-measure precision and f-measure recall.
  • It can be optimized directly as F-Measure Loss Functions in f-measure machine learning training.
  • It can be approximated through F-Measure Surrogate Functions for f-measure gradient-based optimization.
  • ...
  • It can evaluate F-Measure Classification Models for f-measure performance assessment.
  • It can optimize F-Measure Decision Thresholds for f-measure operating point selection.
  • It can compare F-Measure Detection Systems for f-measure relative ranking.
  • It can measure F-Measure Information Retrieval Systems for f-measure relevance evaluation.
  • It can assess F-Measure Named Entity Recognition Systems for f-measure extraction quality.
  • It can benchmark F-Measure Machine Translation Systems for f-measure translation accuracy.
  • It can validate F-Measure Medical Diagnostic Systems for f-measure clinical performance.
  • ...
• Example(s):
  • Balanced F-Measures, such as:
    • F1-Score Metric (β=1), which weights f-measure precision and f-measure recall equally for f-measure balanced evaluation.
    • F1-Measure for f-measure equal importance of f-measure false positives and f-measure false negatives.
    • Standard F1 Score used as default in most f-measure classification evaluations.
  • Recall-Emphasized F-Measures, such as:
    • F2-Measure (β=2), which weights f-measure recall twice as much as f-measure precision.
    • F3-Measure (β=3), which strongly emphasizes f-measure recall over f-measure precision.
    • F5-Measure (β=5), which heavily prioritizes f-measure recall for f-measure high-sensitivity applications.
    • F10-Measure (β=10), which almost exclusively focuses on f-measure recall.
  • Precision-Emphasized F-Measures, such as:
    • F0.5-Measure (β=0.5), which weights f-measure precision twice as much as f-measure recall.
    • F0.3-Measure (β=0.3), which strongly emphasizes f-measure precision over f-measure recall.
    • F0.2-Measure (β=0.2), which heavily prioritizes f-measure precision for f-measure high-precision applications.
    • F0.1-Measure (β=0.1), which almost exclusively focuses on f-measure precision.
  • Domain-Specific F-Measures, such as:
    • Information Retrieval F-Measures, such as:
      • Document Retrieval F1-Measure for f-measure document relevance in f-measure search engines.
      • Query-Specific F2-Measure for f-measure recall-oriented search in f-measure legal discovery.
      • Passage Retrieval F0.5-Measure for f-measure precision-oriented extraction.
    • Medical Diagnosis F-Measures, such as:
      • Cancer Screening F2-Measure prioritizing f-measure sensitivity to avoid f-measure missed cases.
      • Confirmation Test F0.5-Measure prioritizing f-measure specificity to avoid f-measure false alarms.
      • Emergency Triage F3-Measure for f-measure critical case detection.
      • Surgical Planning F0.3-Measure for f-measure precise diagnosis.
    • Natural Language Processing F-Measures, such as:
      • Named Entity Recognition F1-Measure for f-measure balanced entity extraction.
      • Machine Translation F0.5-Measure for f-measure precision-oriented translation quality.
      • Sentiment Analysis F1-Measure for f-measure opinion mining.
      • Part-of-Speech Tagging F1-Measure for f-measure grammatical analysis.
    • Computer Vision F-Measures, such as:
      • Object Detection F1-Measure for f-measure bounding box evaluation.
      • Image Segmentation F0.5-Measure for f-measure precise boundary detection.
      • Face Recognition F2-Measure for f-measure identity verification.
    • Security System F-Measures, such as:
      • Intrusion Detection F2-Measure prioritizing f-measure threat detection.
      • Fraud Detection F1-Measure balancing f-measure false alarms and f-measure missed frauds.
      • Spam Filter F0.5-Measure prioritizing f-measure legitimate email preservation.
  • Temporally-Extended F-Measures, such as:
    • Delayed F-Measures with f-measure temporal tolerance window.
    • Time-Window F-Measures for f-measure sliding window evaluation.
    • Event-Detection F-Measures with f-measure temporal alignment.
    • Streaming F-Measures for f-measure online evaluation.
  • Multi-Class F-Measure Extensions, such as:
    • Micro-F1 Measure aggregating f-measure global counts across all classes.
    • Macro-F1 Measure averaging f-measure class-wise scores unweighted.
    • Weighted F1-Measure using f-measure class support weights.
    • Per-Class F-Measures computing separate f-measure scores for each class.
  • Multi-Label F-Measures, such as:
    • Instance-Based F-Measure for f-measure multi-label instance evaluation.
    • Label-Based F-Measure for f-measure multi-label macro-averaging.
    • Hierarchical F-Measure for f-measure taxonomic classification.
  • ...
• Counter-Example(s):
  • Accuracy Metric, which treats all classification errors equally without precision-recall balance.
  • AUC-ROC Metric, which evaluates threshold-independent performance rather than at a specific operating point.
  • Mean Squared Error, which measures regression errors rather than classification performance.
  • Matthews Correlation Coefficient, which uses all confusion matrix elements including true negatives.
  • G-Mean, which uses geometric mean of sensitivity and specificity rather than harmonic mean.
  • Precision Metric, which only considers false positives without false negatives.
  • Recall Metric, which only considers false negatives without false positives.
  • Logarithmic Loss, which evaluates probabilistic predictions rather than hard classifications.
• See: Bounded Monotonic Utility Measure, Classification Performance Measure, Harmonic Mean Function, Precision Metric, Recall Metric, Binary Classification Measure, F1-Score Metric, F2-Measure, F0.5-Measure, Macro-F1 Measure, Micro-F1 Measure, Van Rijsbergen's Effectiveness Measure, Confusion Matrix, True Positive, False Positive, False Negative, Multi-Class Classification Task, Information Retrieval Evaluation.

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References

2011

• (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/F1_score
  • QUOTE:In statistics, the F₁ score (also F-score or F-measure) is a measure of a test's accuracy. It considers both the precision p and the recall r of the test to compute the score: p is the number of correct results divided by the number of all returned results and r is the number of correct results divided by the number of results that should have been returned. The F₁ score can be interpreted as a weighted average of the precision and recall, where an F₁ score reaches its best value at 1 and worst score at 0.

    The traditional F-measure or balanced F-score (F₁ score) is the harmonic mean of precision and recall: :[math]\displaystyle{ F = 2 \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{\mathrm{precision} + \mathrm{recall}} }[/math].

    The general formula for positive real [math]\displaystyle{ β }[/math] is: :[math]\displaystyle{ F_\beta = (1 + \beta^2) \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{(\beta^2 \cdot \mathrm{precision}) + \mathrm{recall}} }[/math].

    The formula in terms of Type I and type II errors: :[math]\displaystyle{ F_\beta = \frac {(1 + \beta^2) \cdot \mathrm{true\ positive} }{((1 + \beta^2) \cdot \mathrm{true\ positive} + \beta^2 \cdot \mathrm{false\ negative} + \mathrm{false\ positive})}\, }[/math].

    Two other commonly used F measures are the [math]\displaystyle{ F_{2} }[/math] measure, which weights recall higher than precision, and the [math]\displaystyle{ F_{0.5} }[/math] measure, which puts more emphasis on precision than recall.

    The F-measure was derived so that [math]\displaystyle{ F_\beta }[/math] "measures the effectiveness of retrieval with respect to a user who attaches [math]\displaystyle{ β }[/math] times as much importance to recall as precision" . It is based on van Rijsbergen's effectiveness measure :[math]\displaystyle{ E = 1 - \left(\frac{\alpha}{P} + \frac{1-\alpha}{R}\right)^{-1} }[/math].

    Their relationship is [math]\displaystyle{ F_\beta = 1 - E }[/math] where [math]\displaystyle{ \alpha=\frac{1}{1 + \beta^2} }[/math].

2009

• (Hu et al., 1999) ⇒ Xiaohua Hu, Xiaodan Zhang, Caimei Lu, E. K. Park, and Xiaohua Zhou. (2009). “Exploiting Wikipedia as External Knowledge for Document Clustering.” In: Proceedings of ACM SIGKDD Conference (KDD-2009). doi:10.1145/1557019.1557066
  • QUOTE:Cluster quality is evaluated by three metrics, purity [14], F-score [10], and normalized mutual information (NMI) [15]. … F-score combines the information of precision and recall which is extensively applied in information retrieval. … All the three metrics range from 0 to 1, and the higher their value, the better the clustering quality is.

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