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478 lines
17 KiB
Python
478 lines
17 KiB
Python
import numpy as np
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def binary_cross_entropy(
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y_true: np.ndarray, y_pred: np.ndarray, epsilon: float = 1e-15
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) -> float:
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"""
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Calculate the mean binary cross-entropy (BCE) loss between true labels and predicted
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probabilities.
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BCE loss quantifies dissimilarity between true labels (0 or 1) and predicted
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probabilities. It's widely used in binary classification tasks.
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BCE = -Σ(y_true * ln(y_pred) + (1 - y_true) * ln(1 - y_pred))
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Reference: https://en.wikipedia.org/wiki/Cross_entropy
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Parameters:
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- y_true: True binary labels (0 or 1)
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- y_pred: Predicted probabilities for class 1
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- epsilon: Small constant to avoid numerical instability
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>>> true_labels = np.array([0, 1, 1, 0, 1])
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>>> predicted_probs = np.array([0.2, 0.7, 0.9, 0.3, 0.8])
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>>> binary_cross_entropy(true_labels, predicted_probs)
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0.2529995012327421
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>>> true_labels = np.array([0, 1, 1, 0, 1])
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>>> predicted_probs = np.array([0.3, 0.8, 0.9, 0.2])
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>>> binary_cross_entropy(true_labels, predicted_probs)
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Traceback (most recent call last):
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...
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ValueError: Input arrays must have the same length.
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"""
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if len(y_true) != len(y_pred):
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raise ValueError("Input arrays must have the same length.")
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y_pred = np.clip(y_pred, epsilon, 1 - epsilon) # Clip predictions to avoid log(0)
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bce_loss = -(y_true * np.log(y_pred) + (1 - y_true) * np.log(1 - y_pred))
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return np.mean(bce_loss)
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def binary_focal_cross_entropy(
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y_true: np.ndarray,
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y_pred: np.ndarray,
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gamma: float = 2.0,
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alpha: float = 0.25,
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epsilon: float = 1e-15,
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) -> float:
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"""
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Calculate the mean binary focal cross-entropy (BFCE) loss between true labels
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and predicted probabilities.
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BFCE loss quantifies dissimilarity between true labels (0 or 1) and predicted
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probabilities. It's a variation of binary cross-entropy that addresses class
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imbalance by focusing on hard examples.
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BCFE = -Σ(alpha * (1 - y_pred)**gamma * y_true * log(y_pred)
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+ (1 - alpha) * y_pred**gamma * (1 - y_true) * log(1 - y_pred))
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Reference: [Lin et al., 2018](https://arxiv.org/pdf/1708.02002.pdf)
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Parameters:
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- y_true: True binary labels (0 or 1).
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- y_pred: Predicted probabilities for class 1.
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- gamma: Focusing parameter for modulating the loss (default: 2.0).
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- alpha: Weighting factor for class 1 (default: 0.25).
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- epsilon: Small constant to avoid numerical instability.
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>>> true_labels = np.array([0, 1, 1, 0, 1])
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>>> predicted_probs = np.array([0.2, 0.7, 0.9, 0.3, 0.8])
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>>> binary_focal_cross_entropy(true_labels, predicted_probs)
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0.008257977659239775
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>>> true_labels = np.array([0, 1, 1, 0, 1])
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>>> predicted_probs = np.array([0.3, 0.8, 0.9, 0.2])
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>>> binary_focal_cross_entropy(true_labels, predicted_probs)
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Traceback (most recent call last):
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...
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ValueError: Input arrays must have the same length.
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"""
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if len(y_true) != len(y_pred):
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raise ValueError("Input arrays must have the same length.")
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# Clip predicted probabilities to avoid log(0)
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y_pred = np.clip(y_pred, epsilon, 1 - epsilon)
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bcfe_loss = -(
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alpha * (1 - y_pred) ** gamma * y_true * np.log(y_pred)
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+ (1 - alpha) * y_pred**gamma * (1 - y_true) * np.log(1 - y_pred)
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)
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return np.mean(bcfe_loss)
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def categorical_cross_entropy(
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y_true: np.ndarray, y_pred: np.ndarray, epsilon: float = 1e-15
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) -> float:
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"""
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Calculate categorical cross-entropy (CCE) loss between true class labels and
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predicted class probabilities.
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CCE = -Σ(y_true * ln(y_pred))
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Reference: https://en.wikipedia.org/wiki/Cross_entropy
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Parameters:
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- y_true: True class labels (one-hot encoded)
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- y_pred: Predicted class probabilities
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- epsilon: Small constant to avoid numerical instability
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>>> true_labels = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
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>>> pred_probs = np.array([[0.9, 0.1, 0.0], [0.2, 0.7, 0.1], [0.0, 0.1, 0.9]])
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>>> categorical_cross_entropy(true_labels, pred_probs)
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0.567395975254385
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>>> true_labels = np.array([[1, 0], [0, 1]])
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>>> pred_probs = np.array([[0.9, 0.1, 0.0], [0.2, 0.7, 0.1]])
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>>> categorical_cross_entropy(true_labels, pred_probs)
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Traceback (most recent call last):
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...
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ValueError: Input arrays must have the same shape.
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>>> true_labels = np.array([[2, 0, 1], [1, 0, 0]])
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>>> pred_probs = np.array([[0.9, 0.1, 0.0], [0.2, 0.7, 0.1]])
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>>> categorical_cross_entropy(true_labels, pred_probs)
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Traceback (most recent call last):
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...
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ValueError: y_true must be one-hot encoded.
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>>> true_labels = np.array([[1, 0, 1], [1, 0, 0]])
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>>> pred_probs = np.array([[0.9, 0.1, 0.0], [0.2, 0.7, 0.1]])
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>>> categorical_cross_entropy(true_labels, pred_probs)
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Traceback (most recent call last):
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...
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ValueError: y_true must be one-hot encoded.
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>>> true_labels = np.array([[1, 0, 0], [0, 1, 0]])
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>>> pred_probs = np.array([[0.9, 0.1, 0.1], [0.2, 0.7, 0.1]])
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>>> categorical_cross_entropy(true_labels, pred_probs)
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Traceback (most recent call last):
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...
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ValueError: Predicted probabilities must sum to approximately 1.
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"""
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if y_true.shape != y_pred.shape:
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raise ValueError("Input arrays must have the same shape.")
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if np.any((y_true != 0) & (y_true != 1)) or np.any(y_true.sum(axis=1) != 1):
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raise ValueError("y_true must be one-hot encoded.")
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if not np.all(np.isclose(np.sum(y_pred, axis=1), 1, rtol=epsilon, atol=epsilon)):
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raise ValueError("Predicted probabilities must sum to approximately 1.")
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y_pred = np.clip(y_pred, epsilon, 1) # Clip predictions to avoid log(0)
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return -np.sum(y_true * np.log(y_pred))
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def hinge_loss(y_true: np.ndarray, y_pred: np.ndarray) -> float:
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"""
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Calculate the mean hinge loss for between true labels and predicted probabilities
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for training support vector machines (SVMs).
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Hinge loss = max(0, 1 - true * pred)
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Reference: https://en.wikipedia.org/wiki/Hinge_loss
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Args:
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- y_true: actual values (ground truth) encoded as -1 or 1
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- y_pred: predicted values
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>>> true_labels = np.array([-1, 1, 1, -1, 1])
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>>> pred = np.array([-4, -0.3, 0.7, 5, 10])
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>>> hinge_loss(true_labels, pred)
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1.52
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>>> true_labels = np.array([-1, 1, 1, -1, 1, 1])
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>>> pred = np.array([-4, -0.3, 0.7, 5, 10])
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>>> hinge_loss(true_labels, pred)
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Traceback (most recent call last):
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...
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ValueError: Length of predicted and actual array must be same.
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>>> true_labels = np.array([-1, 1, 10, -1, 1])
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>>> pred = np.array([-4, -0.3, 0.7, 5, 10])
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>>> hinge_loss(true_labels, pred)
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Traceback (most recent call last):
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...
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ValueError: y_true can have values -1 or 1 only.
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"""
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if len(y_true) != len(y_pred):
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raise ValueError("Length of predicted and actual array must be same.")
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if np.any((y_true != -1) & (y_true != 1)):
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raise ValueError("y_true can have values -1 or 1 only.")
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hinge_losses = np.maximum(0, 1.0 - (y_true * y_pred))
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return np.mean(hinge_losses)
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def huber_loss(y_true: np.ndarray, y_pred: np.ndarray, delta: float) -> float:
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"""
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Calculate the mean Huber loss between the given ground truth and predicted values.
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The Huber loss describes the penalty incurred by an estimation procedure, and it
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serves as a measure of accuracy for regression models.
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Huber loss =
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0.5 * (y_true - y_pred)^2 if |y_true - y_pred| <= delta
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delta * |y_true - y_pred| - 0.5 * delta^2 otherwise
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Reference: https://en.wikipedia.org/wiki/Huber_loss
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Parameters:
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- y_true: The true values (ground truth)
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- y_pred: The predicted values
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>>> true_values = np.array([0.9, 10.0, 2.0, 1.0, 5.2])
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>>> predicted_values = np.array([0.8, 2.1, 2.9, 4.2, 5.2])
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>>> np.isclose(huber_loss(true_values, predicted_values, 1.0), 2.102)
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True
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>>> true_labels = np.array([11.0, 21.0, 3.32, 4.0, 5.0])
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>>> predicted_probs = np.array([8.3, 20.8, 2.9, 11.2, 5.0])
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>>> np.isclose(huber_loss(true_labels, predicted_probs, 1.0), 1.80164)
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True
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>>> true_labels = np.array([11.0, 21.0, 3.32, 4.0])
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>>> predicted_probs = np.array([8.3, 20.8, 2.9, 11.2, 5.0])
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>>> huber_loss(true_labels, predicted_probs, 1.0)
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Traceback (most recent call last):
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...
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ValueError: Input arrays must have the same length.
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"""
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if len(y_true) != len(y_pred):
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raise ValueError("Input arrays must have the same length.")
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huber_mse = 0.5 * (y_true - y_pred) ** 2
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huber_mae = delta * (np.abs(y_true - y_pred) - 0.5 * delta)
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return np.where(np.abs(y_true - y_pred) <= delta, huber_mse, huber_mae).mean()
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def mean_squared_error(y_true: np.ndarray, y_pred: np.ndarray) -> float:
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"""
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Calculate the mean squared error (MSE) between ground truth and predicted values.
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MSE measures the squared difference between true values and predicted values, and it
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serves as a measure of accuracy for regression models.
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MSE = (1/n) * Σ(y_true - y_pred)^2
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Reference: https://en.wikipedia.org/wiki/Mean_squared_error
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Parameters:
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- y_true: The true values (ground truth)
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- y_pred: The predicted values
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>>> true_values = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
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>>> predicted_values = np.array([0.8, 2.1, 2.9, 4.2, 5.2])
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>>> np.isclose(mean_squared_error(true_values, predicted_values), 0.028)
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True
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>>> true_labels = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
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>>> predicted_probs = np.array([0.3, 0.8, 0.9, 0.2])
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>>> mean_squared_error(true_labels, predicted_probs)
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Traceback (most recent call last):
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...
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ValueError: Input arrays must have the same length.
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"""
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if len(y_true) != len(y_pred):
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raise ValueError("Input arrays must have the same length.")
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squared_errors = (y_true - y_pred) ** 2
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return np.mean(squared_errors)
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def mean_absolute_error(y_true: np.ndarray, y_pred: np.ndarray) -> float:
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"""
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Calculates the Mean Absolute Error (MAE) between ground truth (observed)
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and predicted values.
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MAE measures the absolute difference between true values and predicted values.
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Equation:
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MAE = (1/n) * Σ(abs(y_true - y_pred))
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Reference: https://en.wikipedia.org/wiki/Mean_absolute_error
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Parameters:
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- y_true: The true values (ground truth)
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- y_pred: The predicted values
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>>> true_values = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
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>>> predicted_values = np.array([0.8, 2.1, 2.9, 4.2, 5.2])
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>>> np.isclose(mean_absolute_error(true_values, predicted_values), 0.16)
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True
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>>> true_values = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
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>>> predicted_values = np.array([0.8, 2.1, 2.9, 4.2, 5.2])
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>>> np.isclose(mean_absolute_error(true_values, predicted_values), 2.16)
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False
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>>> true_labels = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
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>>> predicted_probs = np.array([0.3, 0.8, 0.9, 5.2])
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>>> mean_absolute_error(true_labels, predicted_probs)
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Traceback (most recent call last):
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...
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ValueError: Input arrays must have the same length.
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"""
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if len(y_true) != len(y_pred):
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raise ValueError("Input arrays must have the same length.")
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return np.mean(abs(y_true - y_pred))
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def mean_squared_logarithmic_error(y_true: np.ndarray, y_pred: np.ndarray) -> float:
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"""
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Calculate the mean squared logarithmic error (MSLE) between ground truth and
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predicted values.
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MSLE measures the squared logarithmic difference between true values and predicted
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values for regression models. It's particularly useful for dealing with skewed or
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large-value data, and it's often used when the relative differences between
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predicted and true values are more important than absolute differences.
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MSLE = (1/n) * Σ(log(1 + y_true) - log(1 + y_pred))^2
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Reference: https://insideaiml.com/blog/MeanSquared-Logarithmic-Error-Loss-1035
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Parameters:
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- y_true: The true values (ground truth)
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- y_pred: The predicted values
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>>> true_values = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
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>>> predicted_values = np.array([0.8, 2.1, 2.9, 4.2, 5.2])
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>>> mean_squared_logarithmic_error(true_values, predicted_values)
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0.0030860877925181344
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>>> true_labels = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
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>>> predicted_probs = np.array([0.3, 0.8, 0.9, 0.2])
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>>> mean_squared_logarithmic_error(true_labels, predicted_probs)
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Traceback (most recent call last):
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...
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ValueError: Input arrays must have the same length.
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"""
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if len(y_true) != len(y_pred):
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raise ValueError("Input arrays must have the same length.")
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squared_logarithmic_errors = (np.log1p(y_true) - np.log1p(y_pred)) ** 2
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return np.mean(squared_logarithmic_errors)
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def mean_absolute_percentage_error(
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y_true: np.ndarray, y_pred: np.ndarray, epsilon: float = 1e-15
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) -> float:
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"""
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Calculate the Mean Absolute Percentage Error between y_true and y_pred.
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Mean Absolute Percentage Error calculates the average of the absolute
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percentage differences between the predicted and true values.
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Formula = (Σ|y_true[i]-Y_pred[i]/y_true[i]|)/n
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Source: https://stephenallwright.com/good-mape-score/
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Parameters:
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y_true (np.ndarray): Numpy array containing true/target values.
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y_pred (np.ndarray): Numpy array containing predicted values.
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Returns:
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float: The Mean Absolute Percentage error between y_true and y_pred.
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Examples:
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>>> y_true = np.array([10, 20, 30, 40])
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>>> y_pred = np.array([12, 18, 33, 45])
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>>> mean_absolute_percentage_error(y_true, y_pred)
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0.13125
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>>> y_true = np.array([1, 2, 3, 4])
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>>> y_pred = np.array([2, 3, 4, 5])
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>>> mean_absolute_percentage_error(y_true, y_pred)
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0.5208333333333333
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>>> y_true = np.array([34, 37, 44, 47, 48, 48, 46, 43, 32, 27, 26, 24])
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>>> y_pred = np.array([37, 40, 46, 44, 46, 50, 45, 44, 34, 30, 22, 23])
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>>> mean_absolute_percentage_error(y_true, y_pred)
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0.064671076436071
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"""
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if len(y_true) != len(y_pred):
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raise ValueError("The length of the two arrays should be the same.")
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y_true = np.where(y_true == 0, epsilon, y_true)
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absolute_percentage_diff = np.abs((y_true - y_pred) / y_true)
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return np.mean(absolute_percentage_diff)
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def perplexity_loss(
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y_true: np.ndarray, y_pred: np.ndarray, epsilon: float = 1e-7
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) -> float:
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"""
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Calculate the perplexity for the y_true and y_pred.
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Compute the Perplexity which useful in predicting language model
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accuracy in Natural Language Processing (NLP.)
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Perplexity is measure of how certain the model in its predictions.
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Perplexity Loss = exp(-1/N (Σ ln(p(x)))
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Reference:
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https://en.wikipedia.org/wiki/Perplexity
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Args:
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y_true: Actual label encoded sentences of shape (batch_size, sentence_length)
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y_pred: Predicted sentences of shape (batch_size, sentence_length, vocab_size)
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epsilon: Small floating point number to avoid getting inf for log(0)
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Returns:
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Perplexity loss between y_true and y_pred.
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>>> y_true = np.array([[1, 4], [2, 3]])
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>>> y_pred = np.array(
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... [[[0.28, 0.19, 0.21 , 0.15, 0.15],
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... [0.24, 0.19, 0.09, 0.18, 0.27]],
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... [[0.03, 0.26, 0.21, 0.18, 0.30],
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... [0.28, 0.10, 0.33, 0.15, 0.12]]]
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... )
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>>> perplexity_loss(y_true, y_pred)
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5.0247347775367945
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>>> y_true = np.array([[1, 4], [2, 3]])
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>>> y_pred = np.array(
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... [[[0.28, 0.19, 0.21 , 0.15, 0.15],
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... [0.24, 0.19, 0.09, 0.18, 0.27],
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... [0.30, 0.10, 0.20, 0.15, 0.25]],
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... [[0.03, 0.26, 0.21, 0.18, 0.30],
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... [0.28, 0.10, 0.33, 0.15, 0.12],
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... [0.30, 0.10, 0.20, 0.15, 0.25]],]
|
|
... )
|
|
>>> perplexity_loss(y_true, y_pred)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: Sentence length of y_true and y_pred must be equal.
|
|
>>> y_true = np.array([[1, 4], [2, 11]])
|
|
>>> y_pred = np.array(
|
|
... [[[0.28, 0.19, 0.21 , 0.15, 0.15],
|
|
... [0.24, 0.19, 0.09, 0.18, 0.27]],
|
|
... [[0.03, 0.26, 0.21, 0.18, 0.30],
|
|
... [0.28, 0.10, 0.33, 0.15, 0.12]]]
|
|
... )
|
|
>>> perplexity_loss(y_true, y_pred)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: Label value must not be greater than vocabulary size.
|
|
>>> y_true = np.array([[1, 4]])
|
|
>>> y_pred = np.array(
|
|
... [[[0.28, 0.19, 0.21 , 0.15, 0.15],
|
|
... [0.24, 0.19, 0.09, 0.18, 0.27]],
|
|
... [[0.03, 0.26, 0.21, 0.18, 0.30],
|
|
... [0.28, 0.10, 0.33, 0.15, 0.12]]]
|
|
... )
|
|
>>> perplexity_loss(y_true, y_pred)
|
|
Traceback (most recent call last):
|
|
...
|
|
ValueError: Batch size of y_true and y_pred must be equal.
|
|
"""
|
|
|
|
vocab_size = y_pred.shape[2]
|
|
|
|
if y_true.shape[0] != y_pred.shape[0]:
|
|
raise ValueError("Batch size of y_true and y_pred must be equal.")
|
|
if y_true.shape[1] != y_pred.shape[1]:
|
|
raise ValueError("Sentence length of y_true and y_pred must be equal.")
|
|
if np.max(y_true) > vocab_size:
|
|
raise ValueError("Label value must not be greater than vocabulary size.")
|
|
|
|
# Matrix to select prediction value only for true class
|
|
filter_matrix = np.array(
|
|
[[list(np.eye(vocab_size)[word]) for word in sentence] for sentence in y_true]
|
|
)
|
|
|
|
# Getting the matrix containing prediction for only true class
|
|
true_class_pred = np.sum(y_pred * filter_matrix, axis=2).clip(epsilon, 1)
|
|
|
|
# Calculating perplexity for each sentence
|
|
perp_losses = np.exp(np.negative(np.mean(np.log(true_class_pred), axis=1)))
|
|
|
|
return np.mean(perp_losses)
|
|
|
|
|
|
if __name__ == "__main__":
|
|
import doctest
|
|
|
|
doctest.testmod()
|