fnmSim Simulates fractional Brownian motion,
- mvn from the numerical approximation of the stochastic integral,
- chol from the Choleski's decomposition of the covariance matrix,
- lev using the method of Levinson,
- circ using the method of Wood and Chan,
- wave using the wavelet synthesis,
fgnSim Simulates fractional Gaussian noise,
- beran using the method of Beran,
- durbin using the method Durbin and Levinson,
- paxson using the method of Paxson,
farimaSim simulates FARIMA time series processes. }
Functions to estimate the Hurst exponent:
aggvarFit Aggregated variance method,
diffvarFit Differenced aggregated variance method,
absvalFit aggregated absolute value (moment) method,
higuchiFit Higuchi's or fractal dimension method,
pengFit Peng's or variance of residuals method,
rsFit R/S Rescaled Range Statistic method,
perFit periodogram method,
boxperFit boxed (modified) periodogram method,
whittleFit Whittle estimator,
hurstSlider Interactive Display of Hurst Estimates. }
Function for the wavelet estimator:
waveletFit wavelet estimator. }fbmSim(n = 100, H = 0.7, method = c("mvn", "chol", "lev", "circ", "wave"),
waveJ = 7, doplot = TRUE, fgn = FALSE)
fgnSim(n = 1000, H = 0.7, method = c("beran", "durbin", "paxson"))
farimaSim(n = 1000, model = list(ar = c(0.5, -0.5), d = 0.3, ma = 0.1),
method = c("freq", "time"), ...)
aggvarFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5),
doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
diffvarFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5),
doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
absvalFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5), moment = 1,
doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
higuchiFit(x, levels = 50, minnpts = 2, cut.off = 10^c(0.7, 2.5),
doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
pengFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5),
method = c("mean", "median"),
doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
rsFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5),
doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
perFit(x, cut.off = 0.1, method = c("per", "cumper"),
doplot = FALSE, title = NULL, description = NULL)
boxperFit(x, nbox = 100, cut.off = 0.10,
doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
whittleFit(x, order = c(1, 1), subseries = 1, method = c("fgn", "farma"),
trace = FALSE, spec = FALSE, title = NULL, description = NULL)
hurstSlider(x = fgnSim())
waveletFit(x, length = NULL, order = 2, octave = c(2, 8),
doplot = FALSE, title = NULL, description = NULL)
## S3 method for class 'fHURST':
show(object)FALSE, the functions returns a FBM
series otherwise a FGN series.NULL, the previous power will be used."mvn",
"chol", "lev", "circ", or "wave".
[fgnSim] -ar, ma and d.
ar is a numeric vector giving the AR coefficients,
d is an integer value giving the degree of differencing,
and mafHurst.c(2, 8). If the
upper value is too large, it will be replaced by the maximum
allowed value.2.timeSeries,
or any other object which can be transofrmed into a numeric
vector by the function as.vector.fgnSim and farimaSim return a numeric vector of length
n, the FGN or FARIMA series.
*Fit returns an S4 object of class fHURST with the
following slots:H. Optional values may be the value of the fitted
slope beta, or information from the fit.waveletFitfgnSim simulates a series of fractional
Gaussian noise, FGN. FGN provides a parsimonious model for
stationary increments of a self-similar process parameterised
by the Hurst exponent H and variance. Fractional Gaussian noise
with H < 0.5 demonstrates negatively autocorrelated or
anti-persistent behaviour, and FGN with H > 0.5 demonstrates
1/f , long memory or persistent behaviour, and the special
case. The case H = 0.5 corresponds to the classical Gaussian
white noise. One can select from three different
methods. The first generator named "beran" uses
the fast Fourier transform to generate the series based on
SPLUS code written originally by J. Beran [1994]. The second
generator named "durbin" produces a FGN series by
using the Durbin-Levinson coefficients. The algorithm was
reimplemented in pure S based on the C source code written by
V. Teverovsky [199x]. The third generator named
"paxson" was proposed by V. Paxson [199x], this
approaximate method is a very fast and requires low storage.
However, the algorithm reveals some weakness in the method
which was discussed by D.A. Rolls [2001].
Fractional ARIMA Processes:
The function farimaSim is a generator for fractional
ARIMA time series processes. A Gaussian FARIMA(0,d,0) series
can be created, where d is related to the the Hurst
exponent H through d=H-0.5. This is a particular
case of the more general Gaussian FARIMA(p,d,q) process which
follows the same asymptotic relations for their autocovariance
and the spectral density as do the Gaussian FARIMA(0,d,0)
processes. Two different generators are implement in S. The
first named "freq" works in the frequence domain and
generates the series from the fast Fourier transform based on
SPLUS code written originally by J. Beran [1994]. The second
method creates the series in the time domain, therefore named
"time". The algorithm was reimplemented in pure S based
on the Fortran source code from the R's fracdiff package
originally written by C. Fraley [1991]. Details for the algorithm
are given in J Haslett and A.E. Raftery [1989].
Functions to Estimate the Hurst Exponent:
These are 9 functions as described by M.S. Taqqu, V. Teverovsky,
and W. Willinger [1995] to estimate the self similarity parameter
and/or the intensity of long-range dependence in a time series.
Aggregated Variance Method:
The function aggvarFit computes the Hurst exponent from
the variance of an aggregated FGN or FARIMA time series process.
The original time series is divided into blocks of size m.
Then the sample variance within each block is computed. The slope
beta=2*H-2 from the least square fit of the logarithm of
the sample variances versus the logarithm of the block sizes
provides an estimate for the Hurst exponent H.
Differenced Aggregated Variance Method:
To distinguish jumps and slowly decaying trends which are two
types of non-stationary, from long-range dependence, the function
diffvarFit differences the sample variances of successive
blocks. The slope beta=2*H-2 from the least square fit of
the logarithm of the differenced sample variances versus the
logarithm of the block sizes provides an estimate for the Hurst
exponent H.
Aggregated Absolute Value/Moment Method:
The function absvalFit computes the Hurst exponent from
the moments moment=M of absolute values of an aggregated
FGN or FARIMA time series process. The first moment M=1
coincides with the absolute value method, and the second moment
M=2 with the aggregated variance method. Again, the slope
beta=M*(H-1) of the regression line of the logarithm of
the statistic versus the logarithm of the block sizes provides
an estimate for the Hurst exponent H.
Higuchi or Fractal Dimension Method:
The function higuchiFit implements a technique which is
very similar to the absolute value method. Instead of blocks a
sliding window is used to compute the aggregated series. The
function involves the calculation the calculation of the length
of a path and, in principle, finding its fractal Dimension D.
The slope D=2-H from the least square fit of the logarithm
of the expected path lengths versus the logarithm of the block
(window) sizes provides an estimate for the Hurst exponent H.
Peng or Variance of Residuals Method:
The function pengFit uses the method described by peng.
In Peng's variance of residuals method the series is also divided
into blocks of size m. Within each block the cumulated
sums are computed up to t and a least-squares line
a+b*t is fitted to the cumulated sums. Then the sample
variance of the residuals is computed which is proportional to
m^(2*H). The "mean" or "median" are
computed over the blocks. The slope beta=2*H from the
least square provides an estimate for the Hurst exponent H.
The R/S Method:
The function rsFit implements the algorithm named
rescaled range analysis which is dicussed for example
in detail by B. Mandelbrot and Wallis [199x], B. Mandelbrot [199x]
and B. Mandelbrot and M.S. Taqqu [199x].
The Periodogram Method:
The function perFit estimates the Hurst exponent from the
periodogram. In the finite variance case, the periodogram is an
estimator of the spectral density of the time series. A series
with long range dependence will show a spectral density with a
lower law behavior in the frequency. Thus, we expect that a
log-log plot of the periodogram versus frequency will display
a straight line, and the slopw can be computed as 1-2H.
In practice one uses only the lowest 10% of the frequencies,
since the power law behavior holds only for frequencies close to
zero. Varying this cut off may provide additional information.
Plotting H versus the cut off, one should select that
cut off where the curve flattens out to estimate H.
This approach can be selected by the argument method="per".
Alternatively we can select method="cumper". In this case,
instead of using the periodgram itself, the cmulative periodgram
will be investigated. The slope of the double logarithmic fit
is given by 2-2H. More details can be found in the work
of J. Geweke and S. Porter-Hudak [1983] and in Taqqu [?].
The Boxed or Modified Periodogram Method:
The function boxperFit is a modification of the periodogram
method. The algorithm devides the frequency axis into logarithmically
equally spaced boxes and averages the periodogram values corresponding
to the frequencies inside the box.
The Whittle Estimator:
The function whittleFit performs also a periodogram analysis.
The algorithm is based on the minimization of a likelihood function
defined in the frequency domain. For FGN and FARIMA(0,d,0) processes
the parameter H or d is the unknown parameter which
minimizes the function. This approach also allows to compute confidence
intervals. Unlike the previous eight estimators the Whittle estimator
is not a graphical method, it just returns the values of H
or d together with their confidence intervals. The function
allows also to investigate FARIMA(p,d,q) models, then the parameter
set to be optimized is enlarged by the AR and MA coefficients. It
is worth to remark, that the empirical series is required to be a
Gaussian process and that the underlying form must be specified.
The original functions were written by V. Teverovsky and W. Willinger
for SPLUS calling internal functions written in C. The software can
be found on M. Taqqu's home page:
http://math.bu.edu/people/murad/
In addition the Whittle estimator uses SPlus functions written
by J. Beran. They can be found in the appendix of his book or on
the StatLib server:
http://lib.stat.cmu.edu/S/
Note, all nine R functions and internal utility functions are
reimplemented entirely in S.
Functions to perform a Wavelet Analysis:
The function waveletFit computes the Discrete Wavelet
Transform, averages the squares of the coefficients of the transform,
and then performs a linear regression on the logarithm of the
average, versus the log of the scale parameter of the transform.
The result should be directly proportional to H providing
an estimate for the Hurst exponent.Paxson V. (1995); Fast Approximation of Self-Similar Network Traffic, Technical report, LBL-36750/UC-405, Berkeley, and Computer Communcation Review27, p.5--18, 1997. Rolls D.A. (2001); Improved Fast Approximate Synthesis of Fractional Gaussian Noise, Thesis, Department of Mathematics and Statistics, Queen's University at Kingston, Kingston, Ontario, Canada, 5 pages. Taqqu M., et al. Hurst Exponent, Several Preprints.
## fgnSim -
par(mfrow = c(3, 1), cex = 0.75)
# Beran's Method:
plot(fgnSim(n = 200, H = 0.75), type = "l",
ylim = c(-3, 3), xlab = "time", ylab = "x(t)", main = "Beran")
# Durbin's Method:
plot(fgnSim(n = 200, H = 0.75, method = "durbin"), type = "l",
ylim = c(-3, 3), xlab = "time", ylab = "x(t)", main = "Durbin")
# Paxson's Method:
plot(fgnSim(n = 200, H = 0.75, method = "paxson"), type = "l",
ylim = c(-3, 3), xlab = "time", ylab = "x(t)", main = "Paxson")Run the code above in your browser using DataLab