R-factor
rfactor.RdR-factor represents the average width of the given uncertainty bounds divided by the standard deviation of the observations.
Ideally, i.e., with a combination of model structure and parameter values that perfectly represents the catchment under study, and in absence of measurement errors and other additional sources of uncertainty, all the simulated values should be in a perfect match with the observations, leading to a P-factor equal to 1, and an R-factor equal to zero. However, in real-world applications we aim at encompassing as much observations as possible within the given uncertainty bounds (P-factor close to 1) while keeping the width of the uncertainty bounds as small as possible (R-factor close to 0), in order to avoid obtaining a good bracketing of observations at expense of uncertainty bounds too wide to be informative for the decision-making process.
Usage
rfactor(x, ...)
# Default S3 method
rfactor(x, lband, uband, na.rm=TRUE, ...)
# S3 method for class 'data.frame'
rfactor(x, lband, uband, na.rm=TRUE, ...)
# S3 method for class 'matrix'
rfactor(x, lband, uband, na.rm=TRUE, ...)Arguments
- x
ts or zoo object with the observed values.
- lband
numeric, ts or zoo object with the values of the lower uncertainty bound
- uband
numeric, ts or zoo object with the values of the upper uncertainty bound
- na.rm
logical value indicating whether 'NA' values should be stripped before the computation proceeds.
- ...
further arguments passed to or from other methods.
Details
The R-factor quantifies the average width of the prediction uncertainty band relative to the variability of the observed data. It is a measure of the magnitude of predictive uncertainty associated with a model simulation.
Mathematically, the R-factor is defined as:
$$ R\text{-factor} = \frac{\overline{d_x}}{\sigma_x} $$
where \(\sigma_x\) is the standard deviation of the observed variable \(x\), and \(\overline{d_x}\) is the average thickness of the uncertainty band, computed as:
$$ \overline{d_x} = \frac{1}{N} \sum_{i=1}^{N} \left( uband_i - lband_i \right) $$
where \(N\) is the total number of observations, and \(lband_i\) and \(uband_i\) are the lower and upper uncertainty bounds, respectively, at time step \(i\).
The R-factor ranges from 0 to infinity, with an optimal value of 0 indicating perfect agreement between simulated and observed values (i.e., zero prediction uncertainty). In practical applications, the R-factor represents the width of the uncertainty interval and should be as small as possible. Values close to or smaller than 1 are commonly considered indicative of an acceptable level of predictive uncertainty, although acceptable thresholds depend on the quality of observations and the modelling context.
Because a larger fraction of observations can often be bracketed by widening the uncertainty bounds, the R-factor is typically interpreted jointly with the P-factor. A balance between a high P-factor (good coverage) and a low R-factor (narrow uncertainty bounds) is therefore sought during model calibration and uncertainty analysis.
Value
Average width of the given uncertainty bounds, given by lband and uband, divided by the standard deviation of the observations x
If sim and obs are matrixes, the returned value is a vector, with the R-factor between each column of sim and obs.
References
Abbaspour, K.C.; Faramarzi, M.; Ghasemi, S.S.; Yang, H. (2009), Assessing the impact of climate change on water resources in Iran, Water Resources Research, 45(10), W10,434, doi:10.1029/2008WR007615.
Abbaspour, K.C., Yang, J. ; Maximov, I.; Siber, R.; Bogner, K.; Mieleitner, J. ; Zobrist, J.; Srinivasan, R. (2007), Modelling hydrology and water quality in the pre-alpine/alpine Thur watershed using SWAT, Journal of Hydrology, 333(2-4), 413-430, doi:10.1016/j.jhydrol.2006.09.014.
Schuol, J.; Abbaspour, K.C.; Srinivasan, R.; Yang, H. (2008b), Estimation of freshwater availability in the West African sub-continent using the SWAT hydrologic model, Journal of Hydrology, 352(1-2), 30, doi:10.1016/j.jhydrol.2007.12.025
Abbaspour, K.C. (2007), User manual for SWAT-CUP, SWAT calibration and uncertainty analysis programs, 93pp, Eawag: Swiss Fed. Inst. of Aquat. Sci. and Technol. Dubendorf, Switzerland.
Examples
x <- 1:10
lband <- x - 0.1
uband <- x + 0.1
rfactor(x, lband, uband)
#> [1] 0.06605783
lband <- x - rnorm(10)
uband <- x + rnorm(10)
rfactor(x, lband, uband)
#> [1] 0.04194472
#############
# Loading daily streamflows of the Ega River (Spain), from 1961 to 1970
data(EgaEnEstellaQts)
obs <- EgaEnEstellaQts
# Selecting only the daily values belonging to the year 1961
obs <- window(obs, end=as.Date("1961-12-31"))
# Generating the lower and upper uncertainty bounds, centred at the observations
lband <- obs - 5
uband <- obs + 5
rfactor(obs, lband, uband)
#> [1] 0.3312039
# Randomly generating the lower and upper uncertainty bounds
uband <- obs + rnorm(length(obs))
lband <- obs - rnorm(length(obs))
rfactor(obs, lband, uband)
#> [1] -0.0006138383