====== Validation Metrics ======

Validation metrics are used to evaluate the performance of a model on a validation dataset. These metrics help assess how well a model generalizes to unseen data. Below are common validation metrics categorized by problem type.

===== 1. Regression Problems =====
  * **Mean Absolute Error (MAE):**  
    Measures the average absolute difference between predicted and actual values.  
    Formula:  
    **MAE = (1/n) Σ |y_i - ẏ_i|**

  * **Mean Squared Error (MSE):**  
    Averages the squared differences between predicted and actual values.  
    Formula:  
    **MSE = (1/n) Σ (y_i - ẏ_i)²**

  * **Root Mean Squared Error (RMSE):**  
    The square root of MSE, with the same units as the target variable.

  * **R² Score (Coefficient of Determination):**  
    Measures the proportion of variance explained by the model.  
    Formula:  
    **R² = 1 - (Σ(y_i - ẏ_i)² / Σ(y_i - ȳ)²)**

===== 2. Classification Problems =====
  * **Accuracy:**  
    Proportion of correct predictions.  
    Formula:  
    **Accuracy = (Correct Predictions / Total Predictions)**

  * **Precision:**  
    Measures the proportion of true positives among predicted positives.  
    Formula:  
    **Precision = TP / (TP + FP)**

  * **Recall (Sensitivity):**  
    Measures the proportion of true positives identified.  
    Formula:  
    **Recall = TP / (TP + FN)**

  * **F1 Score:**  
    Harmonic mean of precision and recall.  
    Formula:  
    **F1 = 2 × (Precision × Recall) / (Precision + Recall)**

  * **ROC-AUC:**  
    Measures the trade-off between true positive and false positive rates at various thresholds.

  * **Log Loss (Cross-Entropy Loss):**  
    Evaluates the accuracy of predicted probabilities.

===== 3. Clustering Problems =====
  * **Silhouette Score:**  
    Measures how similar an object is to its cluster compared to other clusters.

  * **Adjusted Rand Index (ARI):**  
    Evaluates similarity between true labels and clustering results.

  * **Davies-Bouldin Index:**  
    Assesses compactness and separation of clusters.

  * **Inertia:**  
    Measures how tightly grouped the clusters are.

===== 4. Time Series Problems =====
  * **Mean Absolute Percentage Error (MAPE):**  
    Expresses prediction error as a percentage.  
    Formula:  
    **MAPE = (100/n) Σ |(y_i - ẏ_i) / y_i|**

  * **Symmetric Mean Absolute Percentage Error (sMAPE):**  
    Reduces bias for small values in MAPE.

  * **Mean Squared Logarithmic Error (MSLE):**  
    Penalizes under- and over-predictions logarithmically.

===== 5. Ranking Problems =====
  * **Mean Reciprocal Rank (MRR):**  
    Evaluates ranking quality based on the reciprocal of the rank of the first relevant result.

  * **Normalized Discounted Cumulative Gain (NDCG):**  
    Considers the position of relevant results in a ranked list.

  * **Precision at k (P@k):**  
    Measures precision for the top-k predictions.

===== 6. Multi-Label Problems =====
  * **Hamming Loss:**  
    Proportion of misclassified labels.  
    Formula:  
    **Hamming Loss = (1/nL) ΣΣ I(y_ij ≠ ẏ_ij)**

  * **Subset Accuracy:**  
    Measures the percentage of samples where all labels are correctly predicted.

  * **Macro/Micro Averaged Metrics:**  
    Aggregate metrics across labels (macro) or weight by support (micro).

===== Summary =====
The choice of validation metric depends on the problem type, dataset characteristics, and business goals.