lmModel {fractal} | R Documentation

## Constructor function for objects of class "lmModel"

### Description

Packs the parameters defining a specfied stochastic fractal time series model into a list an returns the result.

### Usage

    lmModel(model, variance.=1.0, delta=0.45,
        alpha=-0.9, HG=0.95, HB=0.95,
        innovations.var=NULL, Cs=NULL,
        bterms=10, dterms=10, M=100)

### Arguments

`model` | a character string defining the model type. Choices are `"ppl"` Pure power law (PPL) process. A process {X(t)} is a PPL process if its SDF is given by S(X,f) = Cs * |f|^alpha where Cs > 0. The innovations variance for this process is given by Cs * exp(-alpha*(log(2)+1)) (this is the variance of the best linear predictor of the process given its infinite past). `"fdp"` Fractionally differenced (FD) process. A process {X(t)} is a FD process if its SDF is given by S(X,f) = sigma^2 / (2 * sin(pi*f) )^(2 * delta) for |f| <= 1/2 where sigma^2 is the innovations variance, and delta is the FD parameter. Thus, an FD model is completely defined by the innovations variance and FD parameter. `"fgn"` Fractional Gaussian noise (FGN) process. An FGN process {X(t)} is a stationary Gaussian process if its ACVF is given by s(X,k) = var{X}/2 *(|k + 1|^(2*Hg) - 2*|k|^(2*Hg) + |k + 1|^(2*Hg)) where var{X} > 0 is the variance of the process, while Hg is the so-called Hurst coefficient. The coefficient Hg is sometimes called the self-similarity parameter for a FGN process and is usually designated in the literature as simply H. `"dfbm"` Discrete Fractional Brownian Motion. i.e., regularly-spaced samples from a FBM process that is defined over the entire real axis.
`Cs` | pure power law constant. If supplied, this argument is used to compute `variance` and `innovations.var`. If not supplied and `innovations.var` is supplied, then `Cs` and `variance` are determined from the `innovations.var`. Default: `NULL`.
`HB` | the Hurst coefficient for a DFBM process. Default: `0.95`.
`HG` | the Hurst coefficient for an FGN process. Default: `0.95`.
`M` | sets the number of terms used in the Euler-Maclaurin summation for calculating the SDF of an FGN process and DFBM process. The default value should be adequate for all values of the Hurst coefficient. Default: `100`.
`alpha` | power law exponent for a PPL model. Default: `-0.9`.
`bterms` | an integer used to control the number of primary terms cumulatively summed in computing an ACVS for a PPL process. Default: `10`.
`delta` | the FD parameter. Default: `0.45`.
`dterms` | an integer used to control the number of secondary terms cumulatively summed in computing an ACVS for a PPL process. Default: `10`.
`innovations.var` | innovations variance for an FD or PPL model. If supplied, this argument is used to compute `variance` and, for a PPL model, `Cs`. If not supplied and `Cs` is supplied for a PPL model, then `Cs` determines `innovations.var`. If not supplied and `Cs` is also not supplied for a PPL model or if not supplied for an FD model, then `variance` determines `innovations.var`. Default: `NULL`.
`variance.` | the process variance with a default of unity. If `cs` or `innovations.var` is specified, this parameter is set in agreement with those. If the process is nonstationary but has stationary differences, i.e., incrementally stationary, then the process variance is taken to be the variance of the stationary process that is formed by appropriately differencing the nonstationary process.

### Value

an object of class `lmModel` containing a list of model parameters.

### References

D. Percival and A. Walden (2000), Wavelet Methods for Time Series Analysis, Cambridge University Press, Chapter 7.

J. Beran (1994), Statistics for Long-Memory Processes, Chapman and Hall, Chapter 2.

D. Percival and A. Walden (1993), Spectral Analysis for Physical Applications, Cambridge University Press, 1993, Chapter 9.

### See Also

`lmACF`, `lmSDF`, `lmSimulate`, `lmConvert`, `lmConfidence`, `FDWhittle`.

### Examples

    lmModel("ppl", alpha=-2.0)
    lmModel("fdp", delta=0.45, innov=1.3)
    lmModel("fgn", HG=0.98)
    lmModel("dfbm", HB=0.35)

[Package fractal version 1.0-1 Index]