NNSE - Normalized Nash-Sutcliffe Efficiency

The Normalized Nash-Sutcliffe Efficiency (NNSE) [13] is a bounded transformation of the standard Nash-Sutcliffe Efficiency (NSE) metric, widely used in hydrology and water resources engineering.

While standard NSE ranges from negative infinity to 1, NNSE maps this infinite domain into a strictly bounded range between 0 and 1. This transformation allows for safe multi-model or multi-site averaging without the risk of a single catastrophically poor prediction (with an NSE approaching \(-\infty\)) heavily skewing the aggregate results.

\[\text{NNSE}(y, \hat{y}) = \frac{1}{2 - \text{NSE}(y, \hat{y})}\]
Note:

  • When NSE = 1, NNSE = 1.

  • When NSE = 0, NNSE = 0.5.

  • As NSE approaches \(-\infty\), NNSE approaches 0.

Description

Advantages:

  • Stable Aggregation (Crucial Feature): Because the range is strictly bounded, you can safely calculate the mean or median NNSE across dozens of different catchment areas or evaluation periods.

  • Bounded Optimization: Highly useful as an objective/loss function in machine learning algorithms that require bounded metrics (where infinite values would cause gradient explosions).

Disadvantages:

  • Resolution Compression: The normalization heavily compresses negative NSE values. For instance, a slightly bad model (NSE = -2) yields an NNSE of 0.25, while a completely catastrophic model (NSE = -98) yields an NNSE of 0.01. It becomes difficult to differentiate the severity of “bad” models just by looking at their NNSE scores.

  • Loss of Intuition: The intuitive baseline of NSE = 0.0 (where the model is as good as the mean) is shifted to NNSE = 0.5, which can occasionally confuse stakeholders used to standard R2 or NSE scales.

Properties

  • Best possible score: 1.0 (Indicates perfect agreement between observed and simulated data).

  • Baseline score: 0.5 (Model is only as accurate as predicting the observed mean).

  • Range: (0.0, 1.0] (Mathematically, it approaches 0 but never strictly reaches it unless the error is genuinely infinite).

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 Normalized Nash-Sutcliffe Efficiency
print("NNSE: ", evaluator.NNSE())

## 2. For > 1-D array (Multi-output)
y_true = array([[0.5, 1], [-1, 1], [7, -6], [1, 2], [2.1, 2.2], [3.4, 5.5]])
y_pred = array([[0, 2], [-1, 2], [8, -5], [1.1, 1.9], [2.0, 2.3], [3.0, 4.2]])

evaluator = RegressionMetric(y_true, y_pred)
# Return an array of scores for each column
print("NNSE (Multi-output): ", evaluator.NNSE(multi_output="raw_values"))