dr: Refined Index of Agreement

In hydroGOF: Goodness-of-Fit Functions for Comparison of Simulated and Observed Hydrological Time Series

Refined Index of Agreement

Description

Refined Index of Agreement (dr) between sim and obs, with treatment of missing values.

Usage

dr(sim, obs, ...)

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

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

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

## S3 method for class 'zoo'
dr(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 Nash-Sutcliffe efficiency. 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 nummeric value. 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

c = 2

A = \sum_{i=1}^N {\left| S_i - O_i \right|}

B = c \sum_{i=1}^N {\left| O_i - \bar{O} \right|}

dr = 1 - \frac{A} { B } ; A \leq B

dr = 1 - \frac{B} { A } ; A > B

The Refined Index of Agreement (dr, Willmott et al., 2012) is a reformulation of the orginal Willmott's index of agreement developed in the 1980s (Willmott, 1981; Willmott, 1984; Willmott et al., 1985)

The Refined Index of Agreement (dr) is dimensionless, and it varies between -1 to 1 (in contrast to the original d, which varies in [0, 1]).

The Refined Index of Agreement (dr) is monotonically related with the modified Nash-Sutcliffe (E1) desribed in Legates and McCabe (1999).

In general, dr is more rationally related to model accuracy than are other existing indices (Willmott et al., 2012; Willmott et al., 2015). It also is quite flexible, making it applicable to a wide range of model-performance problems (Willmott et al., 2012)

Value

Refined Index of Agreement (dr) between sim and obs.

If sim and obs are matrixes or data.frames, the returned value is a vector, with the Refined Index of Agreement (dr) between each column of sim and obs.

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

Author(s)

Mauricio Zambrano Bigiarini <[e-mail]>

References

Willmott, C.J.; Robeson, S.M.; Matsuura, K. (2012). A refined index of model performance. International Journal of climatology, 32(13), pp.2088-2094. doi:10.1002/joc.2419.

Willmott, C.J.; Robeson, S.M.; Matsuura, K.; Ficklin, D.L. (2015). Assessment of three dimensionless measures of model performance. Environmental Modelling & Software, 73, pp.167-174. doi:10.1016/j.envsoft.2015.08.012

Willmott, C.J. (1981). On the validation of models. Physical Geography, 2, 184–194. doi:10.1080/02723646.1981.10642213.

Willmott, C.J. (1984). On the evaluation of model performance in physical geography. Spatial Statistics and Models, G. L. Gaile and C. J. Willmott, eds., 443-460. doi:10.1007/978-94-017-3048-8_23.

Willmott, C.J.; Ackleson, S.G. Davis, R.E.; Feddema, J.J.; Klink, K.M.; Legates, D.R.; O'Donnell, J.; Rowe, C.M. (1985), Statistics for the Evaluation and Comparison of Models, J. Geophys. Res., 90(C5), 8995-9005. doi:10.1029/JC090iC05p08995.

See Also

d, md, rd, gof, ggof

Examples

##################
# Example 1: basic ideal case
obs <- 1:10
sim <- 1:10
dr(sim, obs)

obs <- 1:10
sim <- 2:11
dr(sim, obs)

##################
# 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 'dr' for the "best" (unattainable) case
dr(sim=sim, obs=obs)

##################
# Example 3: dr 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)

dr(sim=sim, obs=obs)

##################
# Example 4: dr 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.

dr(sim=sim, obs=obs, fun=log)

# Verifying the previous value:
lsim <- log(sim)
lobs <- log(obs)
dr(sim=lsim, obs=lobs)

##################
# Example 5: dr 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

dr(sim=sim, obs=obs, fun=log, epsilon.type="Pushpalatha2012")

# 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)
dr(sim=lsim, obs=lobs)

##################
# Example 6: dr 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
dr(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)

# Verifying the previous value:
lsim <- log(sim+eps)
lobs <- log(obs+eps)
dr(sim=lsim, obs=lobs)

##################
# Example 7: dr 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
dr(sim=sim, obs=obs, fun=log, epsilon.type="otherFactor", epsilon.value=fact)

# Verifying the previous value:
eps  <- fact*mean(obs, na.rm=TRUE)
lsim <- log(sim+eps)
lobs <- log(obs+eps)
dr(sim=lsim, obs=lobs)

##################
# Example 8: dr 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)}

dr(sim=sim, obs=obs, fun=fun1)

# Verifying the previous value, with the epsilon value following Pushpalatha2012
sim1 <- sqrt(sim+1)
obs1 <- sqrt(obs+1)
dr(sim=sim1, obs=obs1)

hydroGOF documentation built on Nov. 4, 2024, 5:08 p.m.