TPR vs FPR in Machine Learning: What’s the Difference?

📊 TPR vs FPR Correlation Explained (Simple + Mathematical View)

When True Positive Rate (TPR) and False Positive Rate (FPR) are correlated, it means they tend to increase or decrease together as the classification threshold changes.

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📚 Table of Contents

• Basic Definitions
• Mathematical Formulas
• Why TPR and FPR Are Correlated
• ROC Curve Intuition
• Real-Life Example
• Code Example
• CLI Output
• Key Takeaways
• Related Articles

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🧠 Basic Definitions

✔ True Positive Rate (TPR)

Also called Recall:

It measures how many actual positives are correctly identified.

✔ False Positive Rate (FPR)

It measures how many actual negatives are incorrectly predicted as positive.

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📐 Mathematical Formulas

TPR (Recall)

\[ TPR = \frac{TP}{TP + FN} \]

FPR

\[ FPR = \frac{FP}{FP + TN} \]

Explanation:

• TP = True Positives
• FP = False Positives
• TN = True Negatives
• FN = False Negatives

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🔗 Why TPR and FPR Are Correlated

Both metrics depend on the classification threshold.

If we lower the threshold:

• More cases are predicted as positive
• TP increases → TPR increases
• FP also increases → FPR increases

This creates a positive correlation.

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📈 ROC Curve Intuition

The ROC (Receiver Operating Characteristic) curve plots:

• X-axis → FPR
• Y-axis → TPR

As the threshold changes, the model moves along the curve.

\[ ROC = (FPR, TPR) \]

👉 A good model tries to stay in the top-left corner (high TPR, low FPR).

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🔥 Real-Life Example: Spam Detection

Scenario	Effect of Lower Threshold
Spam Email Detection	More spam caught (↑TPR) but more normal emails marked as spam (↑FPR)

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📊 Smoke Alarm Analogy

• High sensitivity → catches real fire (high TPR)
• But also alarms for toast (high FPR)

This shows why both move together.

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💻 Code Example (Python - ROC Calculation)

from sklearn.metrics import roc_curve y_true = [0,0,1,1] y_scores = [0.1,0.4,0.35,0.8] fpr, tpr, thresholds = roc_curve(y_true, y_scores) print("FPR:", fpr) print("TPR:", tpr) print("Thresholds:", thresholds)

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🖥️ CLI Output (Example)

Click to expand output

FPR: [0.  0.  0.5 1. ]
TPR: [0.  0.5 1.  1. ]
Thresholds: [inf 0.8 0.4 0.1]

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💡 Key Takeaways

• TPR and FPR depend on classification threshold
• Lower threshold increases both TPR and FPR
• They are positively correlated in practice
• ROC curve shows this trade-off visually
• Best models maximize TPR while minimizing FPR

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🎯 Final Insight

TPR and FPR are not independent. They are two sides of the same threshold decision. Improving one often impacts the other, and understanding this trade-off is essential for building reliable classification systems.