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Error-Based Metrics

  • Error-based metrics measure the difference between predicted and actual values to evaluate the accuracy of regression models.

  • Mean Absolute Error (MAE)

    MAE calculates the average of absolute differences between actual and predicted values.

    MAE=1n∑i=1n∣yi−y^i∣MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|MAE=n1​i=1∑n​∣yi​−y^​i​∣

    Where:

    • yiy_iyi​ = Actual value

    • y^i\hat{y}_iy^​i​ = Predicted value

    • nnn = Number of observations

    Characteristics

    • Measures average magnitude of errors

    • Simple to understand

    • Less sensitive to outliers than MSE

    Example

    • Actual = [3, 5, 2]

    • Predicted = [2, 4, 3]

    MAE=∣3−2∣+∣5−4∣+∣2−3∣3=1+1+13=1MAE = \frac{|3-2| + |5-4| + |2-3|}{3} = \frac{1+1+1}{3} = 1MAE=3∣3−2∣+∣5−4∣+∣2−3∣​=31+1+1​=1

    Mean Squared Error (MSE)

    MSE calculates the average of squared differences between actual and predicted values.

    MSE=1n∑i=1n(yi−y^i)2MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2MSE=n1​i=1∑n​(yi​−y^​i​)2

    Characteristics

    • Penalizes larger errors more heavily than MAE

    • Sensitive to outliers

    • Often used in optimization (gradient descent)

    Example

    • Actual = [3, 5, 2]

    • Predicted = [2, 4, 3]

    MSE=(3−2)2+(5−4)2+(2−3)23=1+1+13=1MSE = \frac{(3-2)^2 + (5-4)^2 + (2-3)^2}{3} = \frac{1+1+1}{3} = 1MSE=3(3−2)2+(5−4)2+(2−3)2​=31+1+1​=1


    Root Mean Squared Error (RMSE)

    RMSE is the square root of MSE, giving error in the same units as target variable.

    RMSE=MSE=1n∑i=1n(yi−y^i)2RMSE = \sqrt{MSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}RMSE=MSE​=n1​i=1∑n​(yi​−y^​i​)2​

    Characteristics

    • Gives intuitive interpretation in original units

    • Penalizes large errors more than MAE

    Example

    • MSE = 1

    RMSE=1=1RMSE = \sqrt{1} = 1RMSE=1​=1

    Example: MAE, MSE, RMSE

MAE, MSE, and RMSE Calculation in Python for Model Evaluation

This Python example demonstrates how to evaluate a machine learning model using common regression error metrics: Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE). The code compares actual and predicted values, calculates each metric using scikit-learn and NumPy, and prints the results to measure the prediction accuracy of the model. 

# Step 1: Import Libraries
from sklearn.metrics import mean_absolute_error, mean_squared_error
import numpy as np

# Step 2: Actual vs Predicted
y_true = np.array([3, 5, 2])
y_pred = np.array([2, 4, 3])

# Step 3: Calculate MAE
mae = mean_absolute_error(y_true, y_pred)
print("Mean Absolute Error (MAE):", mae)

# Step 4: Calculate MSE
mse = mean_squared_error(y_true, y_pred)
print("Mean Squared Error (MSE):", mse)

# Step 5: Calculate RMSE
rmse = np.sqrt(mse)
print("Root Mean Squared Error (RMSE):", rmse)

  • Output:


    Mean Absolute Error (MAE): 1.0

    Mean Squared Error (MSE): 1.0

    Root Mean Squared Error (RMSE): 1.0

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