SLE - Squared Log Error

Squared Log Error (SLE) is an element-wise metric that measures the squared difference between the natural logarithm of the predicted values and the natural logarithm of the actual (ground truth) values.

By applying a logarithmic transformation before calculating the squared error, SLE inherently penalizes underestimates more heavily than overestimates and focuses on the relative (percentage) difference rather than the absolute scale.

\[\text{SLE}_i = (\log(y_i + 1) - \log(\hat{y}_i + 1))^2\]

Description

Advantages:

  • Outlier dampening: The logarithmic scale compresses massive values, making SLE highly resilient to extreme outliers. It is the ideal metric when targets have an exponential growth trend or right-skewed distribution (e.g., population growth, housing prices).

  • Relative deviation: It measures the ratio of the true to the predicted value. An error between predicting 10 for a true value of 20 is heavily penalized, while an error between 1010 and 1020 is treated as practically negligible.

Disadvantages:

  • Domain restriction (Crucial): Because the mathematical formulation uses \(\log(x + 1)\), the input values for both actual and predicted data must be strictly greater than -1. If your dataset contains negative values (e.g., temperatures below zero), SLE will return undefined bounds (NaN/Inf) and crash your evaluation pipeline.

  • Non-aggregated: SLE computes an array of element-wise errors. To evaluate the entire model, it must be aggregated into Mean Squared Logarithmic Error (MSLE).

Properties

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

  • Range: [0, +inf)

Example Usage

Note: Ensure all target and predicted values are > -1 to avoid math domain errors.

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 Log Error for each element
print("SLE: ", evaluator.SLE())

## 2. For > 1-D array (Multi-output)
y_true = array([[0.5, 1], [1.5, 1], [7, 6]])
y_pred = array([[0, 2], [1.0, 2], [8, 5]])

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