md {hydroGOF}R Documentation

Modified index of agreement


This function computes the modified Index of Agreement between sim and obs, with treatment of missing values.
If 'x' is a matrix or a data frame, a vector of the modified index of agreement among the columns is returned.


md(sim, obs, ...)

## Default S3 method:
md(sim, obs, j=1, na.rm=TRUE, ...)

## S3 method for class 'data.frame'
md(sim, obs, j=1, na.rm=TRUE, ...)

## S3 method for class 'matrix'
md(sim, obs, j=1, na.rm=TRUE, ...)

## S3 method for class 'zoo'
md(sim, obs, j=1, na.rm=TRUE, ...)



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


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


numeric, with the exponent to be used in the computation of the modified index of agreement. The default value is j=1.


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.


further arguments passed to or from other methods.


md = [ 1 - ( sum( (abs(obs - sim))^j ) ] / sum( ( abs(sim - mean(obs)) + abs(obs - mean(obs)) )^j )

The Index of Agreement (d) developed by Willmott (1981) as a standardized measure of the degree of model prediction error and varies between 0 and 1.
A value of 1 indicates a perfect match, and 0 indicates no agreement at all (Willmott, 1981).

The index of agreement can detect additive and proportional differences in the observed and simulated means and variances; however, it is overly sensitive to extreme values due to the squared differences (Legates and McCabe, 1999).


Modified index of agreement between sim and obs.

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


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


Mauricio Zambrano Bigiarini <>


Krause, P., Boyle, D. P., and Base, F.: Comparison of different efficiency criteria for hydrological model assessment, Adv. Geosci., 5, 89-97, 2005

Willmott, C. J. 1981. On the validation of models. Physical Geography, 2, 184–194

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

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

Legates, D. R., and G. J. McCabe Jr. (1999), Evaluating the Use of "Goodness-of-Fit" Measures in Hydrologic and Hydroclimatic Model Validation, Water Resour. Res., 35(1), 233–241

See Also

d, rd, gof, ggof


obs <- 1:10
sim <- 1:10
md(sim, obs)

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

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

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

# Computing the modified index of agreement for the "best" (unattainable) case
md(sim=sim, obs=obs)

# Randomly changing the first 2000 elements of 'sim', by using a normal distribution 
# with mean 10 and standard deviation equal to 1 (default of 'rnorm').
sim[1:2000] <- obs[1:2000] + rnorm(2000, mean=10)

# Computing the new 'd1'
md(sim=sim, obs=obs)

[Package hydroGOF version 0.3-10 Index]