Unbiased Root Mean Square Error

unbiased Root Mean Square Error (ubRMSE) between sim and obs, in the same units of sim and obs, with treatment of missing values.

ubRMSE was introduced by Entekhabi et al. (2010) to improve the evaluation of the temporal dynamic of volumentric soil moisture, by removing from the traditional RMSE the mean bias error caused by the mistmatch between the spatial representativeness of in situ soil moisture and the corresponding gridded values.

A smaller value indicates better model performance.

Usage

ubRMSE(sim, obs, ...)

# Default S3 method
ubRMSE(sim, obs, na.rm=TRUE, fun=NULL, ...,
              epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
              epsilon.value=NA)

# S3 method for class 'data.frame'
ubRMSE(sim, obs, na.rm=TRUE, fun=NULL, ...,
              epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
              epsilon.value=NA)

# S3 method for class 'matrix'
ubRMSE(sim, obs, na.rm=TRUE, fun=NULL, ...,
              epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
              epsilon.value=NA)

# S3 method for class 'zoo'
ubRMSE(sim, obs, na.rm=TRUE, fun=NULL, ...,
              epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
              epsilon.value=NA)

Arguments

sim

numeric, zoo, matrix or data.frame with simulated values

obs

numeric, zoo, matrix or data.frame with observed values

na.rm

a logical value indicating whether 'NA' should be stripped before the computation proceeds.
When an 'NA' value is found at the i-th position in obs OR sim, the i-th value of obs AND sim are removed before the computation.

fun

function to be applied to sim and obs in order to obtain transformed values thereof before computing the Root Mean Square Error.

The first argument MUST BE a numeric vector with any name (e.g., x), and additional arguments are passed using ....

...

arguments passed to fun, in addition to the mandatory first numeric vector.

epsilon.type

argument used to define a numeric value to be added to both sim and obs before applying FUN.

It is was designed to allow the use of logarithm and other similar functions that do not work with zero values.

Valid values of epsilon.type are:

1) "none": sim and obs are used by fun without the addition of any numeric value. This is the default option.

2) "Pushpalatha2012": one hundredth (1/100) of the mean observed values is added to both sim and obs before applying fun, as described in Pushpalatha et al. (2012).

3) "otherFactor": the numeric value defined in the epsilon.value argument is used to multiply the the mean observed values, instead of the one hundredth (1/100) described in Pushpalatha et al. (2012). The resulting value is then added to both sim and obs, before applying fun.

4) "otherValue": the numeric value defined in the epsilon.value argument is directly added to both sim and obs, before applying fun.

epsilon.value

numeric value to be added to both sim and obs when epsilon.type="otherValue".

Details

The traditional root mean square error (RMSE) is severely compromised if there are biases in either the mean or the amplitude of fluctuations of the simulated values. If it can be estimated reliably, the mean-bias (BIAS) can easily be removed from RMSE, leading to the unbiased RMSE:

$$ ubRMSE = \sqrt{ RMSE^2 - BIAS^2 } $$

Value

Unbiased Root mean square error (ubRMSE) between sim and obs.

If sim and obs are matrixes or data.frames, the returned value is a vector, with the ubRMSE between each column of sim and obs.

References

Entekhabi, D.; Reichle, R.H.; Koster, R.D.; Crow, W.T. (2010). Performance metrics for soil moisture retrievals and application requirements. Journal of Hydrometeorology, 11(3), 832-840. doi: 10.1175/2010JHM1223.1.

Ling, X.; Huang, Y.; Guo, W.; Wang, Y.; Chen, C.; Qiu, B.; Ge, J.; Qin, K.; Xue, Y.; Peng, J. (2021). Comprehensive evaluation of satellite-based and reanalysis soil moisture products using in situ observations over China. Hydrology and Earth System Sciences, 25(7), 4209-4229. doi:10.5194/hess-25-4209-2021.

Author

Mauricio Zambrano Bigiarini <mzb.devel@gmail.com>

Note

obs and sim has to have the same length/dimension

The missing values in obs and sim are removed before the computation proceeds, and only those positions with non-missing values in obs and sim are considered in the computation

Examples

##################
# Example 1: basic ideal case
obs <- 1:10
sim <- 1:10
ubRMSE(sim, obs)
#> [1] 0

obs <- 1:10
sim <- 2:11
ubRMSE(sim, obs)
#> [1] 0

##################
# Example 2: 
# Loading daily streamflows of the Ega River (Spain), from 1961 to 1970
data(EgaEnEstellaQts)
obs <- EgaEnEstellaQts

# Generating a simulated daily time series, initially equal to the observed series
sim <- obs 

# Computing the 'ubRMSE' for the "best" (unattainable) case
ubRMSE(sim=sim, obs=obs)
#> [1] 0

##################
# Example 3: ubRMSE for simulated values equal to observations plus random noise 
#            on the first half of the observed values. 
#            This random noise has more relative importance for ow flows than 
#            for medium and high flows.
  
# Randomly changing the first 1826 elements of 'sim', by using a normal distribution 
# with mean 10 and standard deviation equal to 1 (default of 'rnorm').
sim[1:1826] <- obs[1:1826] + rnorm(1826, mean=10)
ggof(sim, obs)


ubRMSE(sim=sim, obs=obs)
#> [1] 5.04438

##################
# Example 4: ubRMSE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying (natural) 
#            logarithm to 'sim' and 'obs' during computations.

ubRMSE(sim=sim, obs=obs, fun=log)
#> [1] 0.5525138

# Verifying the previous value:
lsim <- log(sim)
lobs <- log(obs)
ubRMSE(sim=lsim, obs=lobs)
#> [1] 0.5525138

##################
# Example 5: ubRMSE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying (natural) 
#            logarithm to 'sim' and 'obs' and adding the Pushpalatha2012 constant
#            during computations

ubRMSE(sim=sim, obs=obs, fun=log, epsilon.type="Pushpalatha2012")
#> [1] 0.5375517

# Verifying the previous value, with the epsilon value following Pushpalatha2012
eps  <- mean(obs, na.rm=TRUE)/100
lsim <- log(sim+eps)
lobs <- log(obs+eps)
ubRMSE(sim=lsim, obs=lobs)
#> [1] 0.5375517

##################
# Example 6: ubRMSE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying (natural) 
#            logarithm to 'sim' and 'obs' and adding a user-defined constant
#            during computations

eps <- 0.01
ubRMSE(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)
#> [1] 0.5515315

# Verifying the previous value:
lsim <- log(sim+eps)
lobs <- log(obs+eps)
ubRMSE(sim=lsim, obs=lobs)
#> [1] 0.5515315

##################
# Example 7: ubRMSE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying (natural) 
#            logarithm to 'sim' and 'obs' and using a user-defined factor
#            to multiply the mean of the observed values to obtain the constant
#            to be added to 'sim' and 'obs' during computations

fact <- 1/50
ubRMSE(sim=sim, obs=obs, fun=log, epsilon.type="otherFactor", epsilon.value=fact)
#> [1] 0.5236909

# Verifying the previous value:
eps  <- fact*mean(obs, na.rm=TRUE)
lsim <- log(sim+eps)
lobs <- log(obs+eps)
ubRMSE(sim=lsim, obs=lobs)
#> [1] 0.5236909

##################
# Example 8: ubRMSE for simulated values equal to observations plus random noise 
#            on the first half of the observed values and applying a 
#            user-defined function to 'sim' and 'obs' during computations

fun1 <- function(x) {sqrt(x+1)}

ubRMSE(sim=sim, obs=obs, fun=fun1)
#> [1] 0.7082036

# Verifying the previous value, with the epsilon value following Pushpalatha2012
sim1 <- sqrt(sim+1)
obs1 <- sqrt(obs+1)
ubRMSE(sim=sim1, obs=obs1)
#> [1] 0.7082036