LongRangeDependence {fSeries} | R Documentation

## Long Range Dependence Modelling

### Description

A collection and description of functions to investigate the long range dependence or long memory behavior of an univariate time series process. Included are functions to simulate fractional Gaussian noise and fractional ARMA processes, functions to model the true autocorrelations and the spectrum of these processes, and functions to compute the Hurst exponent by several different methods.

The Functions and methods are:

Functions to simulate long memory time series processes:

`fnmSim` | Simulates fractio nal 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 model the true autocorrelation function and spectrum:

`fgnTrueacf` | Returns true FGN covariances,
`fgnTruefft` | returns true FGN fast Fourier transform,
`farimaTrueacf` | returns true FARIMA covariances,
`farimaTruefft` | returns true FARIMA fast Fourier transform.

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.

Function for the wavelet estimator:

`waveletFit` | wavelet estimator.

### Usage

    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"), ...)

    fgnTrueacf(n, H)
    fgnTruefft(n, H)
    farimaTrueacf(n, H)
    farimaTruefft(n, H)

    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)

    waveletFit(x, length = NULL, order = 2, octave = c(2, 8),
        doplot = FALSE, title = NULL, description = NULL)

    show.fHURST(object)

### Arguments

`cut.off` | [*Fit] - a numeric vector with the lower and upper cut-off points for the estimation. They should be chosen to define a linear range. The default values are c(0.7, 2.5), i.e. 10^0.7 and 10^2.5, respectively.
`description` | [*Fit] - a character string which allows for a brief description.
`doplot` | [*Fit] - a logical flag, by default FALSE. Should a plot be displayed?
`fgn` | [fbmSim] - a logical flag, if `FALSE`, the functions returns a FBM series otherwise a FGN series.
`H` | [fgnSim] - the Hurst exponent, a numeric value between 0.5 and 1, by default 0.7.
`length` | [waveletFit] - the length of data to be used, must be power of 2. If set to `NULL`, the previous power will be used.
`levels` | [*Fit] - the number of aggregation levels or number of blocks from which the variances or moments are computed.
`method` | [fbmSim] - the method how to generate the FBM time series sequence, one of the following five character strings: `"mvn"`, `"chol"`, `"lev"`, `"circ"`, or `"wave"`. [fgnSim] - the method how to generate the FGN time series sequence, one of the following three character strings: `"beran"`, `"durbin"`, or `"paxson"`. [farimaSim] - the method how to generate the time series sequence, one of the following tow character strings: `"freq"`, or `"time"`. [pengFit] - a string naming the method how to do the averaging, either calculating the `"mean"` or the `"median"`. [perFit] - a string naming the method how to fit the data, either using the peridogram itself `"per"`, or using the cumulated periodogram `"cumper"`. [whittleFit] - a string naming the underlying time series process to be estimated, either `"fgn"` for FGN processes, or `"farima"`for FARIMA models.
`minnpts` | [*Fit] - the minimum number of points or blocksize to be used to estimate the variance or moments at any aggregation level.
`model` | a list with model parameters `ar`, `ma` and `d`. `ar` is a numeric vector giving the AR coefficients, `d` is an integer value giving the degree of differencing, and `ma` is a numeric vector giving the MA coefficients. Thus the order of the time series process is FARMA(p, d, q) with `p=length(ar)` and `q=length(ma)`. `d` is a fractional value for FARMA models. By default an FARMA(2, d, 1) model with coefficients `ar=c(0.5, -0.5)`, `ma=0.1`, and `d=0.3` will be generated.
`moment` | [absvalHurst] - an integer value, by default 1 which denotes absolute values. For values larger than one this argument determines what absolute moment should be calculated.
`n` | [fgnSim][farimaSim] - number of data points to be simulated, a numeric value, by default 1000.
`nbox` | [boxperFit] - is the number of boxes to divide the data into. A numeric value, by default 100.
`object` | an object of class `fHurst`.
`octave` | [waveletFit] - beginning and ending octave for estimation. An integer vector with two elements. By default `c(2, 8)`. If the upper value is too large, it will be replaced by the maximum allowed value.
`order` | [waveletFit] - the order of the wavelet. An integer value, by default `2`.
`spec` | [whittleFit] - Should the periodogram be returned? A logical flag, by default FALSE.
`subseries` | [whittleFit] - allows optionally to subdivide the series into subseries. A numeric value, by default 1.
`title` | a character string which allows for a project title.
`trace` | a logical value, by defaul FALSE. Should the estimation process be traced?
`waveJ` | [fbmSim] - an integer parameter for the simulation of FBM using the wavelet method.
`x` | [*Fit] - the numeric vector of data, an object of class `timeSeries`, or any other object which can be transofrmed into a numeric vector by the function `as.vector`.
`...` | arguments to be passed.

### Details

I: Functions to Simulate Long Memory Processes:

Fractional Gaussian Noise:
The function `fgnSim` 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].

II: Functions to Model True Correlations and Spectrum:

The functions `fgnTrueacf` and `farimaTrueacf` return the true covariances of an FGN and Gaussian FARIMA(0,d,0) time series process. The functions `fgnTruefft` and `farimaTruefft` return the true fast Fourier transform of an FGN and Gaussian FARIMA(0,d,0) time series process. The R functions are implemented from SPlus code written by J. Beran [1994].

III: 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.

IV: 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.

V: Statistical Tests:

The ...

### Value

`fgnSim` and `farimaSim` return a numeric vector of length `n`, the FGN or FARIMA series.
`fgnTrueacf(n, H)`, `fgnTruefft(n, H)`, `farimaTrueacf(n, H)`, and `farimaTruefft(n, H)` return the true covariance and the true spectrum of the FGN or FARIMA time series process.
`*Fit` returns an S4 object of class `fHURST` with the following slots:

`@call` | the function call.
`@method` | a character string with the selected method string.
`@hurst` | a list with at least one element, the Hurst exponent named `H`. Optional values may be the value of the fitted slope `beta`, or information from the fit.
`@parameters` | a list with a varying number of elements describing the input parameters from the argument list.
`@data` | a list holding the input data.
`@fit` | a list object with all information of the fit.
`@plot` | a list object which holds information to create a plot of the fit.
`@title` | a character string with the name of the test.
`@description` | a character string with a brief description of the test.

`waveletFit`

### Author(s)

V. Paxson, code as listed in the Appendix of his paper 1995,
J. Beran, ported by Maechler, code as listed in the Appendix of his Book,
M.S. Taqqu et al. for the S-Plus and C code concerned with the Hurst exponent,
C. Fraley for the FARIMA simulation code,
Guy Nason for the functions from the R package 'wavethresh',
Diethelm Wuertz for the Rmetrics R-port.

### References

Beran J. (1992); Statistics for Long-Memory Processes, Chapman and Hall, New York, 1994.

Haslett J., Raftery A.E. (1989); Space-Time Modelling with Long-Memory Dependence: Assessing Ireland's Wind Power Resource, Applied Statistics 38, pp. 1–50.

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.

### Examples

    ## SOURCE("fSeries.33A-LongRangeDependence")

    ## 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")

[Package fSeries version 221.10065 Index]