KGE - Kling-Gupta Efficiency

\[\text{KGE}(y, \hat{y}) = 1 - \sqrt{ (r(y, \hat{y}) - 1)^2 + (\beta(y, \hat{y}) - 1)^2 + (\gamma(y, \hat{y}) - 1)^2 }\]

where: r = correlation coefficient, CV = coefficient of variation, \(\mu\) = mean, \(\sigma\) = standard deviation.

\[ \begin{align}\begin{aligned}\beta = \text{bias ratio} = \frac{\mu_{\hat{y}} }{\mu_{y}}\\ \gamma = \text{variability ratio} = \frac{ CV_{\hat{y}} } {CV_y} = \frac{ \sigma _{\hat{y}} / \mu _{\hat{y}} }{ \sigma _y / \mu _y}\end{aligned}\end{align} \]

• Best possible score is 1, bigger value is better. Range = (-inf, 1]

• Link: https://rstudio-pubs-static.s3.amazonaws.com/433152_56d00c1e29724829bad5fc4fd8c8ebff.html

Latex equation code:

\text{KGE}(y, \hat{y}) = 1 - \sqrt{ (r(y, \hat{y}) - 1)^2 + (\beta(y, \hat{y}) - 1)^2 + (\gamma(y, \hat{y}) - 1)^2 }
where:
        r = correlation coefficient
        \beta = \text{bias ratio} = \frac{\mu_{\hat{y}} }{\mu_{y}}
        \gamma = \text{variability ratio} = \frac{ CV_{\hat{y}} } {CV_y} = \frac{ \sigma _{\hat{y}} / \mu _{\hat{y}} }{ \sigma _y / \mu _y}
and:
        CV = coefficient of variation
        \mu = mean
        \sigma = standard deviation

Example to use KGE metric:

from numpy import array
from permetrics.regression import RegressionMetric

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

evaluator = RegressionMetric(y_true, y_pred, decimal=5)
print(evaluator.kling_gupta_efficiency())

## For > 1-D array
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, decimal=5)
print(evaluator.KGE(multi_output="raw_values"))