pracma (version 1.9.9)

hurstexp: Hurst Exponent

Description

Calculates the Hurst exponent using R/S analysis.

Usage

hurstexp(x, d = 50, display = TRUE)

Arguments

x

a time series.

d

smallest box size; default 50.

display

logical; shall the results be printed to the console?

Value

hurstexp(x) returns a list with the following components:

• Hs - simplified R over S approach
• Hrs - corrected R over S Hurst exponent
• He - empirical Hurst exponent
• Hal - corrected empirical Hurst exponent
• Ht - theoretical Hurst exponent

Details

hurstexp(x) calculates the Hurst exponent of a time series x using R/S analysis, after Hurst, with slightly different approaches, or corrects it with small sample bias, see for example Weron.

These approaches are a corrected R/S method, an empirical and corrected empirical method, and a try at a theoretical Hurst exponent. It should be mentioned that the results are sometimes very different, so providing error estimates will be highly questionable.

Optimal sample sizes are automatically computed with a length that possesses the most divisors among series shorter than x by no more than 1 percent.

References

H.E. Hurst (1951) Long-term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers 116, 770-808.

R. Weron (2002) Estimating long range dependence: finite sample properties and confidence intervals, Physica A 312, 285-299.

See Also

fractal::hurstSpec, RoverS, hurstBlock and fArma::LrdModelling

Examples

##  Computing the Hurst exponent
data(brown72)
x72 <- brown72                          #  H = 0.72
xgn <- rnorm(1024)                      #  H = 0.50
xlm <- numeric(1024); xlm[1] <- 0.1     #  H = 0.43
for (i in 2:1024) xlm[i] <- 4 * xlm[i-1] * (1 - xlm[i-1])

hurstexp(brown72, d = 128)           # 0.72
# Simple R/S Hurst estimation:         0.6590931 
# Corrected R over S Hurst exponent:   0.7384611 
# Empirical Hurst exponent:            0.7068613 
# Corrected empirical Hurst exponent:  0.6838251 
# Theoretical Hurst exponent:          0.5294909

hurstexp(xgn)                        # 0.50
# Simple R/S Hurst estimation:         0.5518143 
# Corrected R over S Hurst exponent:   0.5982146 
# Empirical Hurst exponent:            0.6104621 
# Corrected empirical Hurst exponent:  0.5690305 
# Theoretical Hurst exponent:          0.5368124 

hurstexp(xlm)                        # 0.43
# Simple R/S Hurst estimation:         0.4825898 
# Corrected R over S Hurst exponent:   0.5067766 
# Empirical Hurst exponent:            0.4869625 
# Corrected empirical Hurst exponent:  0.4485892 
# Theoretical Hurst exponent:          0.5368124 


##  Compare with other implementations
## Not run: 
# library(fractal)
# 
# x <- x72
# hurstSpec(x)                    # 0.776   # 0.720
# RoverS(x)                       # 0.717
# hurstBlock(x, method="aggAbs")  # 0.648
# hurstBlock(x, method="aggVar")  # 0.613
# hurstBlock(x, method="diffvar") # 0.714
# hurstBlock(x, method="higuchi") # 1.001
# 
# x <- xgn
# hurstSpec(x)                    # 0.538   # 0.500
# RoverS(x)                       # 0.663
# hurstBlock(x, method="aggAbs")  # 0.463
# hurstBlock(x, method="aggVar")  # 0.430
# hurstBlock(x, method="diffvar") # 0.471
# hurstBlock(x, method="higuchi") # 0.574
# 
# x <- xlm
# hurstSpec(x)                    # 0.478   # 0.430
# RoverS(x)                       # 0.622
# hurstBlock(x, method="aggAbs")  # 0.316
# hurstBlock(x, method="aggVar")  # 0.279
# hurstBlock(x, method="diffvar") # 0.547
# hurstBlock(x, method="higuchi") # 0.998
# ## End(Not run)