Prediction Error Magnitude Calculator

In the realm of predictive modeling and regression analysis, understanding the magnitude of prediction errors is essential for evaluating model performance. The Root Mean Square Deviation (RMSD), also known as the Root Mean Square Error, is a fundamental statistical measure that quantifies the average magnitude of discrepancies between predicted values and observed ground truth values.

Unlike simple error averaging, RMSD applies quadratic weighting to deviations, making it particularly sensitive to large errors. This property makes it invaluable when large prediction mistakes are especially undesirable, such as in financial forecasting, weather prediction, or safety-critical systems.

Mathematical Foundation:

The RMSD is computed by taking the square root of the arithmetic mean of squared residuals (differences between predictions and actual observations):

$$RMSD = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i^{actual} - y_i^{predicted})^2}$$

Where:

• n represents the total count of observations
• $y_i^{actual}$ denotes the ground truth value for observation i
• $y_i^{predicted}$ represents the model's prediction for observation i

Key Properties:

• The RMSD is always non-negative (RMSD ≥ 0)
• A perfect model yields an RMSD of exactly 0
• The RMSD has the same units as the original data, making it directly interpretable
• Squaring amplifies larger deviations, penalizing outlier predictions more heavily

Your Task: Implement a Python function that computes the Root Mean Square Deviation between two arrays representing actual observations and predicted values. The function should:

• Compute the RMSD and return the result rounded to three decimal places
• Handle edge cases appropriately including mismatched dimensions and empty inputs