br2 | R Documentation |
Coefficient of determination (r2) multiplied by the slope of the regression line between sim
and obs
, with treatment of missing values.
br2(sim, obs, ...)
## Default S3 method:
br2(sim, obs, na.rm=TRUE, use.abs=FALSE, fun=NULL, ...,
epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"),
epsilon.value=NA)
## S3 method for class 'data.frame'
br2(sim, obs, na.rm=TRUE, use.abs=FALSE, fun=NULL, ...,
epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"),
epsilon.value=NA)
## S3 method for class 'matrix'
br2(sim, obs, na.rm=TRUE, use.abs=FALSE, fun=NULL, ...,
epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"),
epsilon.value=NA)
## S3 method for class 'zoo'
br2(sim, obs, na.rm=TRUE, use.abs=FALSE, fun=NULL, ...,
epsilon.type=c("none", "Pushpalatha2012", "otherFactor", "otherValue"),
epsilon.value=NA)
sim |
numeric, zoo, matrix or data.frame with simulated values |
obs |
numeric, zoo, matrix or data.frame with observed values |
na.rm |
logical value indicating whether 'NA' should be stripped before the computation proceeds. |
use.abs |
logical value indicating whether the condition to select the formula used to compute |
fun |
function to be applied to The first argument MUST BE a numeric vector with any name (e.g., |
... |
arguments passed to |
epsilon.type |
argument used to define a numeric value to be added to both It is was designed to allow the use of logarithm and other similar functions that do not work with zero values. Valid values of 1) "none": 2) "Pushpalatha2012": one hundredth (1/100) of the mean observed values is added to both 3) "otherFactor": the numeric value defined in the 4) "otherValue": the numeric value defined in the |
epsilon.value |
-) when |
br2 = |b| R2 , b <= 1 ; br2 = \frac{R2}{|b|}, b > 1
A model that systematically over or under-predicts all the time will still result in "good" R2
(close to 1), even if all predictions were wrong (Krause et al., 2005).
The br2
coefficient allows accounting for the discrepancy in the magnitude of two signals (depicted by 'b') as well as their dynamics (depicted by R2
)
br2 between sim
and obs
.
If sim
and obs
are matrixes, the returned value is a vector, with the br2
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
The slope b
is computed as the coefficient of the linear regression between sim
and obs
, forcing the intercept be equal to zero.
Mauricio Zambrano Bigiarini <mzb.devel@gmail.com>
Krause, P.; Boyle, D.P.; Base, F. (2005). Comparison of different efficiency criteria for hydrological model assessment, Advances in Geosciences, 5, 89-97. doi:10.5194/adgeo-5-89-2005.
Krstic, G.; Krstic, N.S.; Zambrano-Bigiarini, M. (2016). The br2-weighting Method for Estimating the Effects of Air Pollution on Population Health. Journal of Modern Applied Statistical Methods, 15(2), 42. doi:10.22237/jmasm/1478004000
R2
, rPearson
, rSpearman
, cor
, lm
, gof
, ggof
##################
# Example 1:
# Looking at the difference between r2 and br2 for a case with systematic
# over-prediction of observed values
obs <- 1:10
sim1 <- 2*obs + 5
sim2 <- 2*obs + 25
# The coefficient of determination is equal to 1 even if there is no one single
# simulated value equal to its corresponding observed counterpart
r2 <- (cor(sim1, obs, method="pearson"))^2 # r2=1
# 'br2' effectively penalises the systematic over-estimation
br2(sim1, obs) # br2 = 0.3684211
br2(sim2, obs) # br2 = 0.1794872
ggof(sim1, obs)
ggof(sim2, obs)
# Computing 'br2' without forcing the intercept be equal to zero
br2.2 <- r2/2 # br2 = 0.5
##################
# 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 'br2' for the "best" (unattainable) case
br2(sim=sim, obs=obs)
##################
# Example 3: br2 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)
br2(sim=sim, obs=obs)
##################
# Example 4: br2 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.
br2(sim=sim, obs=obs, fun=log)
# Verifying the previous value:
lsim <- log(sim)
lobs <- log(obs)
br2(sim=lsim, obs=lobs)
##################
# Example 5: br2 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
br2(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)
br2(sim=lsim, obs=lobs)
##################
# Example 6: br2 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
br2(sim=sim, obs=obs, fun=log, epsilon.type="otherValue", epsilon.value=eps)
# Verifying the previous value:
lsim <- log(sim+eps)
lobs <- log(obs+eps)
br2(sim=lsim, obs=lobs)
##################
# Example 7: br2 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
br2(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)
br2(sim=lsim, obs=lobs)
##################
# Example 8: br2 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)}
br2(sim=sim, obs=obs, fun=fun1)
# Verifying the previous value, with the epsilon value following Pushpalatha2012
sim1 <- sqrt(sim+1)
obs1 <- sqrt(obs+1)
br2(sim=sim1, obs=obs1)
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