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Bivariate Statistics SRG
Introduction
Many of the bivariate statistics available in Source are intended for the purpose of calibrating hydrological models, particularly for evaluating the fit between observed and modelled streamflow. The discussion below assumes that they are being used in this context. However, most of the statistics are generic and can be applied to evaluate the relationships other time series, or ordered variable, types.
The main types of bivariate statistics available in Source are:
Source also offers a range of composite statistics that combine the NSE with other metrics. These are discussed in the section on Composite Bivariate Statistics Involving the NSE, are listed below:
Each of the bivariate statistics is described below. For further information, the reader is referred to Vaze et al. (2011, Section 6), who discuss the use and interpretation of most of the bivariate statistics available in Source.
Treatment of Missing Data
It is common for hydrological time series to contain missing values and to have differing start and end dates. Generally, Source calculates bivariate statistics using only data from those time steps for which there are complete data pairs. The Bivariate Statistics tool in the Results Manager allows the user to calculate some statistics using all data (as opposed to just overlapping data). However, the results should be interpreted with caution.
Nash-Sutcliffe Efficiency (NSE)
Definition
The NSE is a normalised metric that measures the relative magnitude of the model error variance compared to the measured data variance (Nash and Sutcliffe, 1970). It is defined as:
where:
Qobs,i is the observed flow for time step i
Qmod,i is the modelled flow for time step i
N is the number of time steps
The time step size is arbitrary.
Interpretation
The NSE can range between -∞ and 1, where:
NSE = 1 corresponds to a perfect match between modelled and observed data
NSE = 0 indicates that the model predictions are as accurate as the mean of the observed data
NSE < 0 indicates that the mean of the observed data is a better predictor than the model
The NSE is sensitive to the timing of flow events and to extreme values. It is often applied on a daily time step (or shorter) to evaluate the model's ability to represent the timing of flow peaks and recession rates. Applying it on a longer time step, such as monthly, can be used to evaluate the fit to the pattern of flows without considering individual runoff events. The NSE is not suitable for evaluating a model's fit to low flows as the statistic will tend to be dominated by errors in the high flows.
The NSE is not very sensitive to systematic model over- or under-prediction, especially during low flow periods.
Discussion
An alternative, but equivalent, formulation of the NSE is:
Equation 2 |  |
This formulation obviates the necessity to calculate the average of the observed flows before evaluating the denominator in the traditional version.
NSE of Log Data (NSE Log)
Definition
NSE Log is the standard NSE metric (Equation 1) applied to the logarithm of flow data:
where:
c is a positive constant equal to the maximum of 1 ML and the 10th percentile of the observed flow
other terms are as defined in Equation 1
As with the standard NSE, the time step size is arbitrary. The NSE Log cannot be applied to time series with negative values, as the logarithm of a number less than or equal to zero is undefined.
Interpretation
Using the logarithm of flows has the effect of reducing the sensitivity of the metric to high flows and increasing the sensitivity to low and mid-range flows. For this reason, NSE Log is often used for model calibration when low-flow performance is important. The use of the constant c de-emphasises very small flows, which tend to be unreliable, and avoids numerical problems with attempting to calculate the logarithm of zero flows.
The NSE Log can range between -∞ and 1 and the interpretation is the same as for the NSE, but applied to log data.
Relative Bias
Definition
The relative bias measures the magnitude of the model errors relative to the magnitude of the observations. It has the form: