R2: Coefficient of determination

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

Coefficient of determination

Description

coefficient of determination between sim and obs, with treatment of missing values.

Usage

R2(sim, obs, ...)

## Default S3 method:
R2(sim, obs, fun=NULL, ..., 
            epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
            epsilon.value=NA)

## S3 method for class 'matrix'
R2(sim, obs, na.rm=TRUE, fun=NULL, ..., 
            epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
            epsilon.value=NA)

## S3 method for class 'data.frame'
R2(sim, obs, na.rm=TRUE, fun=NULL, ..., 
            epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"), 
            epsilon.value=NA)

## S3 method for class 'zoo'
R2(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 this goodness-of-fit index. 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 coefficient of determination (R2) is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).

It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model.

The coefficient of determination is a statistical measure of how well the regression predictions approximate the real data points. An R2 of 1 indicates that the regression predictions perfectly fit the data.

Values of R2 outside the range 0 to 1 occur when the model fits the data worse than the worst possible least-squares predictor (equivalent to a horizontal hyperplane at a height equal to the mean of the observed data). This occurs when a wrong model was chosen, or nonsensical constraints were applied by mistake.

Value

Coefficient of determination between sim and obs.

If sim and obs are matrixes, the returned value is a vector, with the coefficient of determination 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

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

Box, G.E. (1966). Use and abuse of regression. Technometrics, 8(4), 625-629. doi:10.1080/00401706.1966.10490407.

Hahn, G.J. (1973). The coefficient of determination exposed. Chemtech, 3(10), 609-612. Aailable online at: https://www2.hawaii.edu/~cbaajwe/Ph.D.Seminar/Hahn1973.pdf.

Barrett, J.P. (1974). The coefficient of determination-some limitations. The American Statistician, 28(1), 19-20. doi:10.1080/00031305.1974.10479056.

See Also

cor

Examples

##################
# Example 1: basic ideal case
obs <- 1:10
sim <- 1:10
R2(sim, obs)

obs <- 1:10
sim <- 2:11
R2(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 'R2' for the "best" (unattainable) case
R2(sim=sim, obs=obs)

##################
# Example 3: R2 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)

R2(sim=sim, obs=obs)

##################
# Example 4: R2 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.

R2(sim=sim, obs=obs, fun=log)

# Verifying the previous value:
lsim <- log(sim)
lobs <- log(obs)
R2(sim=lsim, obs=lobs)

##################
# Example 5: R2 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

R2(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)
R2(sim=lsim, obs=lobs)

##################
# Example 6: R2 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
R2(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)

# Verifying the previous value:
lsim <- log(sim+eps)
lobs <- log(obs+eps)
R2(sim=lsim, obs=lobs)

##################
# Example 7: R2 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
R2(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)
R2(sim=lsim, obs=lobs)

##################
# Example 8: R2 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)}

R2(sim=sim, obs=obs, fun=fun1)

# Verifying the previous value, with the epsilon value following Pushpalatha2012
sim1 <- sqrt(sim+1)
obs1 <- sqrt(obs+1)
R2(sim=sim1, obs=obs1)

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