Srho {tseriesEntropy}

Entropy Measure Of Serial And Cross Dependence

Description

Entropy based measure of serial and cross dependence for integer or categorical data. Implements a normalized version of the Hellinger/Matusita distance. As shown in the references the metric measure is a proper distance.

Usage

Srho(x, y, lag.max, stationary = TRUE, plot = TRUE, version = c("FORTRAN","R"),
 nor = FALSE)

Arguments

x, y

integer or factor time series objects or vectors. (y is missing in the univariate case).

lag.max

maximum lag at which to calculate Srho; default is round(N/4) where N is the number of observations.

stationary

logical. If TRUE assumes stationarity and computes marginal probabilities by using N observations. If FALSE uses N-k observations where k is the lag.

plot

logical. If TRUE (the default) Srho is plotted.

version

either "FORTRAN" or "R". FORTRAN version is the default and is preferred over the pure R version which is considerably slower but is included in case of portability issues.

nor

logical. If TRUE normalizes Srho with respect to its attainable maximum. Defaults to FALSE.

Details

Univariate version: serial entropy

Srho(x, lag.max,
stationary = TRUE, plot = TRUE, version = c("FORTRAN","R"), nor = FALSE)

Bivariate version: cross entropy

Srho(x, y, lag.max,
stationary = TRUE, plot = TRUE, version = c("FORTRAN","R"), nor = FALSE)

This implementation of the measure is normalized to take values in [0, 1]. Normalization is performed with respect to the maximum attainable value computed analytically. This makes the results of Srho comparable among different series.

Value

An object of S4 class "Srho", which is a list with the following elements:

.Data

vector of lag.max elements containing Srho computed at each lag.

lags

integer vector that contains the lags at which Srho is computed.

stationary

Object of class "logical": TRUE if the stationary version is computed.

data.type

Object of class "character": contains the data type.

notes

Object of class "character": additional notes.

Warning

Unlike ccf the lag k value returned by Srho(x,y) estimates Srho between x[t] and y[t+k]. The result is returned invisibly if plot is TRUE.

Author(s)

Simone Giannerini<[e-mail]>

References

Granger C. W. J., Maasoumi E., Racine J., (2004) A dependence metric for possibly nonlinear processes. Journal of Time Series Analysis, 25(5), 649–669.

Giannerini S., Maasoumi E., Bee Dagum E., (2015), Entropy testing for nonlinear serial dependence in time series, Biometrika, 102(3), 661–675 doi:10.1093/biomet/asv007.

Maasoumi E., (1993) A compendium to information theory in economics and econometrics. Econometric Reviews, 12(2), 137–181.

See Also

See Also Srho.test. The function Srho.ts implements the same measure for numeric data.

Examples

## UNIVARIATE VERSION
x <- as.integer(rbinom(n=20,size=4,prob=0.5))
Srho(x,lag.max=4)

## BIVARIATE VERSION
y <- as.integer(rbinom(n=20,size=4,prob=0.5))
Srho(x,y,lag.max=4)

## EXAMPLE  1: the effect of normalization
## computes the maximum attainable value by correlating x with itself

set.seed(12)
K    <- 5           # number of categories
smax <- 1-1/sqrt(K) # theoretical maximum under the uniform distribution
x    <- as.integer(sample(1:K,size=1e3,replace=TRUE)) # generates the sequence
S    <- Srho(x,x,lag.max=2,nor=FALSE,plot=FALSE)

plot(S,lwd=2,col=4)
abline(h=smax,col=2,lty=2)
text(x=-1,y=0.54,labels=paste("theoretical maximum = ",round(smax,4),sep=""),col=2)
text(x=-1,y=0.45,labels=paste("estimated maximum = ",round(S[3],4),sep=""),col=4)