hydroGOF (version 0.6-0)

rPearson: Pearson correlation coefficient

Description

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

Usage

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

Value

Pearson correlation coefficient between sim and obs.

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

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".

Author

Mauricio Zambrano Bigiarini <[e-mail]>

Details

It is a wrapper to the cor function.

The Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data.

It is the ratio between the covariance of two variables and the product of their standard deviations; thus, it is essentially a normalized measurement of the covariance, such that the result always has a value between -1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationships or correlations.

The correlation coefficient ranges from -1 to 1. An absolute value of exactly 1 implies that a linear equation describes the relationship between sim and obs perfectly, with all data points lying on a line. The correlation sign is determined by the regression slope: a value of +1 implies that all data points lie on a line for which sim increases as obs increases, and vice versa for -1. A value of 0 implies that there is no linear dependency between the variables.

References

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

Pearson, K. (1920). Notes on the history of correlation. Biometrika, 13(1), 25-45. doi:10.2307/2331722.

Schober, P.; Boer, C.; Schwarte, L.A. (2018). Correlation coefficients: appropriate use and interpretation. Anesthesia and Analgesia, 126(5), 1763-1768. doi:10.1213/ANE.0000000000002864.

See Also

cor

Examples

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

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

##################
# Example 3: rPearson 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)

rPearson(sim=sim, obs=obs)

##################
# Example 4: rPearson 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.

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

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

##################
# Example 5: rPearson 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

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

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

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

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

##################
# Example 8: rPearson 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)}

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

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