SE - Squared Error

Squared Error (SE) is an element-wise metric that measures the square of the difference between an actual value and a predicted value for each individual data point.

Unlike Mean Squared Error (MSE), which computes the average over an entire dataset, SE computes an array of squared errors for a pair of lists, tuples, or NumPy arrays.

\[\text{SE}_i = (y_i - \hat{y}_i)^2\]

Description

Advantages:

  • Mathematical tractability: The squaring operation makes the function strictly convex and differentiable everywhere. This is highly advantageous for calculating gradients in optimization algorithms.

  • Aggressive outlier penalization: By squaring the error, SE exponentially magnifies larger deviations. This makes it extremely useful when you want to isolate and identify individual predictions that failed massively.

Disadvantages:

  • Lack of interpretability: The error is expressed in squared units of the target variable, making it completely unintuitive for clinical or business communication.

  • Non-aggregated: SE is an element-wise array of scores. To evaluate the overall predictive performance across an entire model, it must be aggregated into MSE, RMSE, or Sum of Squared Errors (SSE).

Properties

  • Best possible score: 0.0 (Smaller value is better).

  • Range: [0, +inf)

Example Usage

from numpy import array
from permetrics.regression import RegressionMetric

## 1. For 1-D array (Single-output)
y_true = array([3, -0.5, 2, 7])
y_pred = array([2.5, 0.0, 2, 8])

evaluator = RegressionMetric(y_true, y_pred)
# Calculate Squared Error for each element
print("SE: ", evaluator.SE())

## 2. For > 1-D array (Multi-output)
y_true = array([[0.5, 1], [-1, 1], [7, -6]])
y_pred = array([[0, 2], [-1, 2], [8, -5]])

evaluator = RegressionMetric(y_true, y_pred)
# Calculate Squared Error for multi-dimensional array
print("SE (Multi-output): ", evaluator.SE())