Root Mean Square Error

Root Mean Square Error (RMSE) between sim and obs, in the same units of sim and obs, with treatment of missing values.
RMSE gives the standard deviation of the model prediction error. A smaller value indicates better model performance.

Usage

rmse(sim, obs, ...)

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

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

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

# S3 method for class 'zoo'
rmse(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

-) when epsilon.type="otherValue" it represents the numeric value to be added to both sim and obs before applying fun.
-) when epsilon.type="otherFactor" it represents the numeric factor used to multiply the mean of the 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.

Details

$$ rmse = \sqrt{ \frac{1}{N} \sum_{i=1}^N { \left( S_i - O_i \right)^2 } } $$

Value

Root mean square error (rmse) between sim and obs.

If sim and obs are matrixes, the returned value is a vector, with the RMSE between each column of sim and obs.

References

https://en.wikipedia.org/wiki/Root_mean_square_deviation

Willmott, C.J.; Matsuura, K. (2005). Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance, Climate Research, 30, 79-82, doi:10.3354/cr030079.

Chai, T.; Draxler, R.R. (2014). Root mean square error (RMSE) or mean absolute error (MAE)? - Arguments against avoiding RMSE in the literature, Geoscientific Model Development, 7, 1247-1250. doi:10.5194/gmd-7-1247-2014.

Hodson, T.O. (2022). Root-mean-square error (RMSE) or mean absolute error (MAE): when to use them or not, Geoscientific Model Development, 15, 5481-5487, doi:10.5194/gmd-15-5481-2022.

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
rmse(sim, obs)
#> [1] 0

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

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

##################
# Example 3: rmse 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)


rmse(sim=sim, obs=obs)
#> [1] 7.157178

##################
# Example 4: rmse 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.

rmse(sim=sim, obs=obs, fun=log)
#> [1] 0.6975895

# Verifying the previous value:
lsim <- log(sim)
lobs <- log(obs)
rmse(sim=lsim, obs=lobs)
#> [1] 0.6975895

##################
# Example 5: rmse 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

rmse(sim=sim, obs=obs, fun=log, epsilon.type="Pushpalatha2012")
#> [1] 0.6804015

# 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)
rmse(sim=lsim, obs=lobs)
#> [1] 0.6804015

##################
# Example 6: rmse 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
rmse(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)
#> [1] 0.6964637

# Verifying the previous value:
lsim <- log(sim+eps)
lobs <- log(obs+eps)
rmse(sim=lsim, obs=lobs)
#> [1] 0.6964637

##################
# Example 7: rmse 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
rmse(sim=sim, obs=obs, fun=log, epsilon.type="otherFactor", epsilon.value=fact)
#> [1] 0.6644011

# Verifying the previous value:
eps  <- fact*mean(obs, na.rm=TRUE)
lsim <- log(sim+eps)
lobs <- log(obs+eps)
rmse(sim=lsim, obs=lobs)
#> [1] 0.6644011

##################
# Example 8: rmse 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)}

rmse(sim=sim, obs=obs, fun=fun1)
#> [1] 0.9654451

# Verifying the previous value, with the epsilon value following Pushpalatha2012
sim1 <- sqrt(sim+1)
obs1 <- sqrt(obs+1)
rmse(sim=sim1, obs=obs1)
#> [1] 0.9654451