G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05ZTF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G05ZTF produces realisations of a fractional Brownian motion, using the circulant embedding method. The square roots of the extended covariance matrix (or embedding matrix) need to be input, and can be calculated using G05ZMF or G05ZNF.

## 2  Specification

 SUBROUTINE G05ZTF ( NS, S, M, XMAX, H, LAM, RHO, STATE, Z, XX, IFAIL)
 INTEGER NS, S, M, STATE(*), IFAIL REAL (KIND=nag_wp) XMAX, H, LAM(M), RHO, Z(NS+1,S), XX(NS+1)

## 3  Description

The routines G05ZMF or G05ZNF and G05ZTF are used to simulate a fractional Brownian motion process with Hurst parameter $H$ over an interval $\left[0,{x}_{\mathrm{max}}\right]$, using a set of equally spaced gridpoints. Fractional Brownian motion itself cannot be simulated directly using this method, since it is not a stationary Gaussian random field; however its increments can be simulated like a stationary Gaussian random field. The circulant embedding method is described in the documentation for G05ZMF or G05ZNF.
G05ZTF takes the square roots of the eigenvalues of the embedding matrix as returned by G05ZMF or G05ZNF, and its size $M$, as input and outputs $S$ realisations of the fractional Brownian motion in $Z$.
One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05ZTF.

## 4  References

Dietrich C R and Newsam G N (1997) Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix SIAM J. Sci. Comput. 18 1088–1107
Schlather M (1999) Introduction to positive definite functions and to unconditional simulation of random fields Technical Report ST 99–10 Lancaster University
Wood A T A and Chan G (1994) Simulation of stationary Gaussian processes in ${\left[0,1\right]}^{d}$ Journal of Computational and Graphical Statistics 3(4) 409–432

## 5  Parameters

1:     NS – INTEGERInput
On entry: the number of sample points (grid points) to be generated in realisations of the increments of the fractional Brownian motion. This must be the same value as supplied to G05ZMF or G05ZNF when calculating the eigenvalues of the embedding matrix.
Constraint: ${\mathbf{NS}}\ge 1$.
2:     S – INTEGERInput
On entry: the number of realisations of the fractional Brownian motion to simulate.
Constraint: ${\mathbf{S}}\ge 1$.
3:     M – INTEGERInput
On entry: the size of the embedding matrix, as returned by G05ZMF or G05ZNF.
Constraint: ${\mathbf{M}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{NS}}-1\right)\right)$.
4:     XMAX – REAL (KIND=nag_wp)Input
On entry: the upper bound for the interval over which the fractional Brownian motion is to be simulated, as returned by G05ZMF or G05ZNF.
Constraint: ${\mathbf{XMAX}}>0.0$.
5:     H – REAL (KIND=nag_wp)Input
On entry: the Hurst parameter for the fractional Brownian motion. This must be the same value as supplied to G05ZMF or G05ZNF when calculating the eigenvalues of the embedding matrix.
Constraint: $0.0<{\mathbf{H}}<1.0$.
6:     LAM(M) – REAL (KIND=nag_wp) arrayInput
On entry: contains the square roots of the eigenvalues of the embedding matrix, as returned by G05ZNF.
Constraint: ${\mathbf{LAM}}\left(i\right)=0$, $i=1,2,\dots ,{\mathbf{M}}$.
7:     RHO – REAL (KIND=nag_wp)Input
On entry: indicates the scaling of the covariance matrix, as returned by G05ZMF or G05ZNF.
Constraint: $0.0<{\mathbf{RHO}}\le 1.0$.
8:     STATE($*$) – INTEGER arrayCommunication Array
Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
9:     Z(${\mathbf{NS}}+1$,S) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the realisations of the fractional Brownian motion. Each column of Z contains one realisation of the fractional Brownian motion, with ${\mathbf{Z}}\left(i,j\right)$, for $j=1,2,\dots ,{\mathbf{S}}$, corresponding to the gridpoint ${\mathbf{XX}}\left(i\right)$.
10:   XX(${\mathbf{NS}}+1$) – REAL (KIND=nag_wp) arrayOutput
On exit: the gridpoints at which values of the fractional Brownian motion are output. The first gridpoint is always zero, and the subsequent NS gridpoints represent the equispaced steps towards the last gridpoint, XMAX. Note that in G05ZMF and G05ZNF, the returned NS sample points are the mid-points of the grid returned in XX here.
11:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{NS}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{NS}}\ge 1$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{S}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{S}}\ge 1$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$, and ${\mathbf{NS}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{M}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{NS}}-1\right)\right)$.
${\mathbf{IFAIL}}=4$
On entry, ${\mathbf{XMAX}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{XMAX}}>0.0$.
${\mathbf{IFAIL}}=5$
On entry, ${\mathbf{H}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0<{\mathbf{H}}<1.0$.
${\mathbf{IFAIL}}=6$
On entry, at least one element of LAM was negative.
Constraint: all elements of LAM must be non-negative.
${\mathbf{IFAIL}}=7$
On entry, ${\mathbf{RHO}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0<{\mathbf{RHO}}\le 1.0$.
${\mathbf{IFAIL}}=8$
On entry, STATE vector has been corrupted or not initialized.

Not applicable.

None.

## 9  Example

This example calls G05ZTF to generate $5$ realisations of a fractional Brownian motion over $10$ steps from $x=0.0$ to $x=2.0$ using eigenvalues generated by G05ZNF with ${\mathbf{ICOV1}}=14$.

### 9.1  Program Text

Program Text (g05ztfe.f90)

### 9.2  Program Data

Program Data (g05ztfe.d)

### 9.3  Program Results

Program Results (g05ztfe.r)