Objective functions

Objective functions available in Kalix for hydrological model calibration. All objective functions are configured for minimisation during optimisation.

---

NSE - Nash-Sutcliffe Efficiency

Description

The Nash-Sutcliffe Efficiency (NSE) is one of the most widely used statistics for assessing the goodness-of-fit of hydrological models. It compares the performance of the model against a simple baseline (the mean of observed values).

Formula

NSE=1−∑i=1n(Qo[i]−Qo)2∑i=1n(Qo[i]−Qm[i])2

Where: - Qo[i] = observed value at timestep i - Qm[i] = modeled/simulated value at timestep i - \(\overline{Q\_o}\) = mean of observed values - n = number of timesteps

Optimisation

• Original range: (−∞, 1] where 1 = perfect fit

• Minimisation: NSE is negated so objective = −NSE

• Optimal value: -1.0 (corresponds to NSE = 1.0)

When to Use

• General-purpose calibration metric

• Emphasizes high flows (squared errors give more weight to large deviations)

• NSE = 1: perfect fit

• NSE = 0: model performs as well as using the mean

• NSE < 0: model performs worse than using the mean

---

LNSE - Log Nash-Sutcliffe Efficiency

Description

Log-transformed version of NSE that applies logarithmic transformation to flows before calculating the Nash-Sutcliffe metric. This gives more weight to low flows compared to standard NSE.

Formula

LNSE=1−∑i=1n(ln(Qo[i]+ϵ)−ln(Qo+ϵ))2∑i=1n(ln(Qo[i]+ϵ)−ln(Qm[i]+ϵ))2

Where ϵ = 0.01 is a small constant added to avoid ln (0)

Optimisation

• Original range: (−∞, 1] where 1 = perfect fit

• Minimization: LNSE is negated so objective = −LNSE

• Optimal value: -1.0 (corresponds to LNSE = 1.0)

When to Use

• When accurate simulation of low flows is important

• Baseflow-dominated catchments

• Water quality or ecological flow studies

• Reduces influence of extreme high flows

---

RMSE - Root Mean Square Error

Description

RMSE measures the average magnitude of errors between observed and simulated values. It gives more weight to larger errors due to squaring.

Formula

RMSE=n1∑i=1n(Qo[i]−Qm[i])2

Optimization

• Range: [0, ∞) where 0 = perfect fit

• Minimization: Already a minimization metric (lower is better)

• Optimal value: 0.0

When to Use

• When you want errors in the same units as your data

• Penalizes large errors more heavily than MAE

• Standard error metric across many fields

• More sensitive to outliers than MAE

---

MAE - Mean Absolute Error

Description

MAE measures the average absolute difference between observed and simulated values. Unlike RMSE, it treats all errors proportionally.

Formula

MAE=n1∑i=1n∣Qo[i]−Qm[i]∣

Optimization

• Range: [0, ∞) where 0 = perfect fit

• Minimization: Already a minimization metric (lower is better)

• Optimal value: 0.0

When to Use

• When outliers should not dominate the error metric

• More robust to extreme values than RMSE

• Easier to interpret (average error magnitude)

• When all errors should be weighted equally

---

KGE - Kling-Gupta Efficiency

Description

The Kling-Gupta Efficiency provides a decomposed analysis of model performance by separately evaluating correlation, variability bias, and mean bias. It was developed to address limitations in NSE.

Formula

KGE=1−(r−1)2+(α−1)2+(β−1)2

Where: - r = Pearson correlation coefficient between observed and simulated - α=σoσm = ratio of simulated to observed standard deviations - β=μoμm = ratio of simulated to observed means

Optimization

• Original range: (−∞, 1] where 1 = perfect fit

• Minimization: KGE is negated so objective = −KGE

• Optimal value: -1.0 (corresponds to KGE = 1.0)

When to Use

• Provides balanced assessment of three model aspects:
• Correlation (timing and dynamics)
• Variability (capturing the spread)

• Bias (volume errors)

• Preferred over NSE in many recent hydrological studies

• Less sensitive to systematic bias than NSE

• Good for diagnosing specific model deficiencies

---

PBIAS - Percent Bias

Description

Percent Bias measures the average tendency of simulated values to be larger or smaller than observed values. Positive values indicate model overestimation, negative values indicate underestimation.

Formula

PBIAS=100×∑i=1nQo[i]∑i=1n(Qm[i]−Qo[i])

Optimization

• Range: (−∞, ∞) where 0 = no bias

• Minimization: Absolute value is taken, so objective = |PBIAS|

• Optimal value: 0.0

When to Use

• Simple measure of volume balance

• Easy to interpret (percentage over/under prediction)

• Often used in conjunction with other metrics

• Important: PBIAS alone doesn’t assess timing or variability

---

SDEB - Sorted Data Error with Bias

Description

SDEB is a specialized objective function that combines temporal error (SD), distributional error (SE), and volume bias (B). It uses square-root transformation to balance emphasis between high and low flows and compares both chronological sequences and flow duration curves.

Formula

SDEB = (0.1 × SD + 0.9 × SE) × B

Where:

Temporal Error (SD): SD=∑i=1n(Qo[i]−Qm[i])2

Distributional Error (SE): SE=∑i=1n(Ro[i]−Rm[i])2

Where Ro and Rm are observed and simulated values sorted in ascending order (exceedance).

Bias Penalty (B): B=1+∑i=1nQo[i]∣∑i=1nQo[i]−∑i=1nQm[i]∣

Optimization

• Range: [0, ∞) where 0 = perfect fit

• Minimization: Already a minimization metric (lower is better)

• Optimal value: 0.0

When to Use

• Balanced emphasis on flow distribution (90%) and timing (10%)

• Square-root transformation reduces dominance of high flows

• Better for matching flow duration curves

• Includes volume balance through bias penalty

• Good for applications where flow distribution matters (e.g., hydropower, ecology)

Special Features

• Uses lazy caching for efficiency

• Penalizes volume errors through multiplicative bias term

---

PEARS_R - Pearson’s Correlation Coefficient

Description

Pearson’s R measures the linear correlation between observed and simulated values. It assesses how well the model captures the dynamics and timing of the observed data, independent of bias or scale.

Formula r=∑i=1n(Qo[i]−Qo)2×∑i=1n(Qm[i]−Qm)2∑i=1n(Qo[i]−Qo)(Qm[i]−Qm)

Where: - \(\overline{Q\_o}\) = mean of observed values - \(\overline{Q\_m}\) = mean of simulated values

Optimization

• Original range: [−1, 1] where 1 = perfect positive correlation

• Minimization: R is negated so objective = −r

• Optimal value: -1.0 (corresponds to R = 1.0)

When to Use

• Focus on timing and dynamics rather than volume

• Insensitive to systematic bias (can have high R but large bias)

• Insensitive to scale differences

• Best used with other metrics (e.g., PBIAS for volume balance)

• Good for pattern matching

• R > 0.9: very high correlation

• R > 0.7: high correlation

• R > 0.5: moderate correlation

Limitations

• Does not penalize bias or variability differences

• Can give high scores even with systematic over/under prediction

• Should be combined with bias metrics for comprehensive assessment

---

Summary Table

Objective	Config Name	Optimal	Emphasizes
NSE	NSE	-1.0	High flows
LNSE	LNSE	-1.0	Low flows
RMSE	RMSE	0.0	Large errors
MAE	MAE	0.0	All errors equally
KGE	KGE	-1.0	Correlation + variability + bias
PBIAS	PBIAS	0.0	Volume balance only
SDEB	SDEB	0.0	Flow distribution + timing
PEARS_R	PEARS_R	-1.0	Timing/dynamics only

---

Configuration Example

To use an objective function in your calibration INI file:

objective_function = KGE

Valid options: NSE, LNSE, RMSE, MAE, KGE, PBIAS, SDEB, PEARS_R

---

Recommendations

For General Hydrological Calibration

• First choice: SDEB - provides balanced assessment

• Alternative: KGE - provides balanced assessment

• Alternative: NSE - widely used

For Low-Flow Focused Applications

• First choice: LNSE - emphasizes baseflow

• Alternative: SDEB - balanced high/low flows

For Volume Balance

• Use PBIAS in combination with other metrics

For Timing Matching

• First choice: PEARS_R - focuses on timing

For Error Measurement

• MAE - less sensitive to outliers

• RMSE - more sensitive to large errors