1 Introduction
Rough phenomena arise in texture simulations for image processing or medical imaging, natural scenes simulations (clouds, mountains) and geophysical morphology modeling, financial mathematics, ethernet traffic, etc. Some are time-indexed, some others, like texture or natural scene simulations, should be indexed by subsets of the Euclidean spaces \(\mathbb{R}^2\) or \(\mathbb{R}^3\). Recent data (as the Cosmic Microwave Background or solar data) are even indexed by a manifold.
The fractional Brownian motion (fBm), introduced by (Kolmogorov 1940) (and developed by Mandelbrot and Van Ness (1968)) is nowadays widely used to model this roughness. Fractional Brownian motions have been extended in many directions: higher dimensions with fields, anisotropy, multifractionality, etc. This paper is devoted to a simulation method for conditional Gaussian fields. This could improve, in the future, natural scene simulations by fixing for instance the valleys.
The simulation of fractional Gaussian processes is not difficult in dimension one (see a review of Coeurjolly (2000)). Let us recall the numerical complexity of some classical methods: the Cholesky method has a complexity of \(O(N^3)\) where \(N\) is the size of the simulated sample path. For specific stationary processes (on a regular grid) the Levinson’s algorithm has a complexity of \(O(N^2 \log N)\) and the Wood and Chan algorithm (see Wood and Chan (1994)) a complexity of \(O(N \log N)\).
In higher dimensions, the Wood and Chan method has been extended to stationary increments fields with the Stein’s method (Stein 2002) ; the fractional Brownian field can therefore be simulated on a regular grid of the plane. For general Gaussian fields on a general discrete grid, the Cholesky method is costly and exact simulations are no longer tractable. Approximate methods have been intensively developed (midpoint, Peitgen and Saupe (1988); turning bands, Yin (1996); truncated wavelet decomposition) but for specific fields. On manifolds, simulation procedures based on truncated series of eigenfunctions of the Laplace-Beltrami operator are discussed in (Gelbaum and Titus 2014).
Our approach, presented in Brouste et al. (2007, 2010), is based on a 2-steps method
with an exact simulation step plus a refined fast step, that is an
improvement of the midpoint method. It has been implemented in the
FieldSim package
(Brouste and Lambert-Lacroix. 2015). The fieldsim simulation method can be applied to general
Gaussian processes on general simulation grids (regular and non regular)
on Euclidean spaces and even on some manifolds (see Figure 1).
It is worth mentioning that another package,
RandomFields
(Schlather et al. 2016), allows the simulation of a large class of random fields
such as Gaussian random fields, Poisson fields, binary fields,
chi-square fields, \(t\) fields and max-stable fields (see Schlather et al. (2015)). In
RandomFields, conditional random fields (which are the purpose of the
present paper) are given for a wide range of spatial and spatio-temporal
Gaussian random fields. Some of the default models of the FieldSim
package cannot be simulated with the help of default models of the
RandomFields package. Nevertheless, it is still possible to simulate
them with the RMuser() and RFsimulate() commands of the
RandomFields package. It may be noted that the FieldSim package does
not allow for the simulation of more than the RandomFields package.
FieldSim package is an alternative in which the underlying methods of
simulation are generic.
We propose here to adapt the FieldSim package
to conditional simulations. Definitions and
notation will be introduced in the following
section with the “process” class, the setProcess procedure and the
fieldsim procedure. The fieldsim procedure adapted to conditional
Gaussian fields is described in the next section.
Simulations with the package FieldSim are presented in the last
section.
2 Notation and preliminaries
Fractional Gaussian fields
Let \(d\) be a positive integer and \(X(\cdot)=\left\{X(M),M\in \mathbb{R} ^d\right\}\) be a real valued non stationary field with zero mean and second order moments. It is worth emphasizing that we consider in this paper the metric space \(\mathbb{R}^d\) with the Euclidean norm but the method can be generalized to a smooth and complete Riemannian manifold equipped with its geodesic distance (Brouste et al. 2010).
The covariance function \(R(\cdot,\cdot)\) is defined by: \[R(M_1,M_2)= \mbox{cov} \left( X(M_1), X(M_2)\right),\quad M_1,\; M_2\in\mathbb{R} ^d.\] This function is nonnegative definite (n.n.d.). Conversely, for any n.n.d. function \(R(\cdot,\cdot)\), there exists an unique centered Gaussian field of second order structure given by \(R(\cdot,\cdot).\)
Different classical fractional Gaussian fields have been simulated to illustrate the FieldSim package in (Brouste et al. 2007, 2010). In the sequel, \(M\) and \(M'\) are two points of \(\mathbb{R}^d\) and \(\|\cdot\|\) is the usual norm on \(\mathbb{R}^d,\) \(d=1,\) \(2\). We can cite:
The standard fractional Brownian fields are defined through their covariance function (e.g., Samorodnitsky and Taqqu (1994)): \[R(M,M')=\frac 1 2 \left( \|M\|^{2H}+\|M'\|^{2H}-\|M-M'\|^{2H}\right),\] where the Hurst parameter \(H\) is real in \((0,1)\).
The standard multifractional Brownian fields are defined through their covariance function (see Peltier and Levy-Véhel (1996; Benassi et al. 1997)):
\[R(M,M')=\alpha(M,M') \left( \|M\|^{\tilde{H}(M,M')}+\|M'\|^{\tilde{H}(M,M')} -\|M-M'\|^{\tilde{H}(M,M')}\right),\] where \[\begin{aligned} \tilde{H}(M,M') &=H(M)+H(M'),\\ \alpha(M,M') &= \frac{C\left( \frac{H(M)+H(M')}{2} \right)^2}{2C\left(H(M) \right)C\left( H(M')\right)},\\ C(h) &= \left(\frac{\pi^{\frac{d+1}{2}}\Gamma\left(h+\frac{1}{2}\right)}{h\sin \left(\pi h \right)\Gamma\left( 2h\right)\Gamma\left(h+\frac{d}{2} \right) } \right)^{\frac{1}{2}}, \end{aligned}\] and the Hurst parameter is a continuous function \(H:\mathbb{R}^d \longrightarrow (0,1),\) where \(\Gamma\) is the usual Gamma function.
The standard fractional Brownian sheets are defined through their covariance function (see Kamont (1996)): \[R(M,M')=\frac{1}{2^d} \prod_{i=1}^d \left\{|M_{i}|^{2H_i}+|M'_{i}|^{2H_i}-|M_{i}-M'_{i}|^{2H_i}\right\},\] where \(\left(H_1,\ldots,H_d\right)\) stands for the multivariate Hurst index in \(\mathbb{R}^d\), \(0<H_i< 1.\)
The anisotropic fractional Brownian fields are defined through their covariance function (see Bonami and Estrade (2003)): \[R(M,M') = v_H (M) + v_H(M') - v_H (M- M'),\] where the variogram \[v_H(x)= 2^{2H-1}\gamma(H) C_{H,\vartheta_1,\vartheta_2} (x) \| x \|^{2H},\] with \(H \in(0,1)\), \(\gamma(H)\) depends explicitly on \(H\) and \(C_{H,\vartheta_1,\vartheta_2}(.)\) implies incomplete Beta functions and two constants \(-\frac{\pi}{2} \leq \vartheta_1 < \vartheta_2 \leq \frac{\pi}{2}\).
The FieldSim package
In the new version 3.2 of the package FieldSim, new features have been
added. The most important add is the “process” class and the
setProcess function.
An object of class “process” has different slots:
The
nameof the process. Several names are reserved for classical fractional Gaussian processes: see Table 1 for details."cond"is used for all kind of conditional simulations (see further).The slot
valuesstores the values of the process on the simulation (and visualization) grid.An object of class “manifold” which is the Riemannian manifold on which the process is lying; an object of the class “manifold” has four slots:
namewhich is the name of the manifold we consider. The name"line","plane","sphere"and"hyperboloid"are taken for the eponymous manifolds.atlaswhich is the union of discretized domains that cover the manifold (must be a matrix where the number of rows is the dimension of the space where the manifold lives).distancewhich is the distance considered on the manifold.originwhich is the origin considered on the manifold (must be a point on the manifold).
The setter
setManifoldpermits the user to create an object of class “manifold” with all its slots. This class is already described in Brouste et al. (2010).The slot
covfwhich contains the covariance function of the Gaussian process.The slot
parameterwhich contains all the parameters associated to the covariance function of the process. Here are the classical parameters associated to the classical process.
All the examples presented can be defined with the setProcess command
(see Table 1). With the following command, the user can set
a fBm with Hurst parameter 0.7 on a regular grid of the interval \([0,1]\)
(of size 256).
R> linefBm <- setProcess("fBm-line", 0.7)
R> str(linefBm)Formal class 'process' [package "FieldSim"] with 7 slots
..@ name : chr "fBm"
..@ values : num 0
..@ manifold :Formal class 'manifold' [package "FieldSim"] with 4 slots
.. .. ..@ name : chr "line"
.. .. ..@ atlas : num [1, 1:256] 0 0.00392 0.00784 0.01176 0.01569 ...
.. .. ..@ distance:function (xi, xj)
.. .. ..@ origin : num [1, 1] 0
..@ covf :function (xi, xj)
..@ parameter: num 0.7
..@ values2 : num 0
..@ manifold2:Formal class 'manifold' [package "FieldSim"] with 4 slots
.. .. ..@ name : chr "line"
.. .. ..@ atlas : num [1, 1:256] 0 0.00392 0.00784 0.01176 0.01569 ...
.. .. ..@ distance:function (xi, xj)
.. .. ..@ origin : num [1, 1] 0It is worth mentioning that the slot values is empty since there is no
simulation done. Then as usual, the user can use the fieldsim function
in order to simulate the Gaussian process associated to covf on the
manifold grid defined in manifold.
R> fieldsim(linefBm)In the fieldsim function, we can add the quantity Ne, the number of
points of the grid to be simulated in the exact step, and nbNeighbor,
the number of neighbors used in the refined step. By default, Ne is
equal to the size of the grid given in atlas. The slot values are
now set with the simulated values. There exist different visualization
procedures to draw the results, for instance:
R> plot(linefbm, "default")We recall that the discretization grids can be modified with the
setAtlas command. Depending on the manifold, there are several types
of grids: "regular", "random" and "visualization". For instance,
R> setAtlas(linefBm, "regular", 1000)
R> fieldsim(linefBm)
R> plot(linefBm, "default")3 The fieldsim procedure for conditional Gaussian fields
In order to build conditional fractional Gaussian fields, we consider a conditioning set \({\cal N}= \left\{N_1, \ldots,\right.\) \(\left. N_k \right\},\) \(N_i\in \mathbb{R} ^d,\) \(i=1,\ldots,k,\) and the conditioning values \(\mathbf x=\left(x_1, \ldots, x_k\right)^T \in \mathbb{R}^k\). Then we will say that \(\widetilde{X}(\cdot)=\{\widetilde{X}(M),M\in \mathbb{R} ^d\}\) is the conditional Gaussian field associated to the field \(X(\cdot)\) (of covariance function \(R\)) and to the conditioning pair \(\left({\cal N},\mathbf x\right)\) if the finite dimensional laws of \(\widetilde{X}(\cdot)\) is the same as the finite dimensional laws of \(X(\cdot)\) given the event \(\{(X(N_1), \ldots, X(N_k))^T=:\mathbf X_{\cal N} =\mathbf x\}\). We denote by \(\widetilde{m}(\cdot)\) (resp. \(\widetilde{R}(\cdot,\cdot)\)) the mean (resp. covariance) function of the process \(\widetilde{X}(\cdot)\). The following lemma allows us to determine \(\widetilde{m}(\cdot)\) and \(\widetilde{R}(\cdot,\cdot)\) according to \({R}(\cdot,\cdot)\) (sketch of proof is given in Piterbag (1996 A.1)).
Lemma 1. *Let us consider the centered Gaussian vector \(\left(Y_1,Y_2,\mathbf Z^T\right)^T\in \mathbb{R}\times\mathbb{R}\times \mathbb{R}^k\) with the covariance matrix \[\Sigma^2= \begin{pmatrix} {\mathbb{E}} (Y_1^2) & {\mathbb{E}} (Y_1 Y_2) & {\mathbb{E}} (Y_1 \mathbf Z^T) \\ {\mathbb{E}} (Y_1 Y_2) & {\mathbb{E}} (Y_2^2) & {\mathbb{E}} (Y_2 \mathbf Z^T) \\ {\mathbb{E}} (\mathbf Z Y_1) & {\mathbb{E}} (\mathbf Z Y_2)& {\mathbb{E}} (\mathbf Z \mathbf Z^T) \end{pmatrix}.\] Suppose that \({\mathbb{E}} (\mathbf Z \mathbf Z^T)\) is invertible. Then the conditional law of \((Y_1,Y_2)^T\) given the event \(\{\mathbf Z=\mathbf z \in \mathbb{R}^k\}\) is Gaussian with mean
\[\widetilde{m} = \begin{pmatrix} {\mathbb{E}} (Y_1 \mathbf Z^T) \\ {\mathbb{E}} (Y_2 \mathbf Z^T) \end{pmatrix} \{{\mathbb{E}} (\mathbf Z \mathbf Z^T)\}^{-1} \mathbf z, \tag{1} \] and covariance matrix \[\label{eq:V} \widetilde{\Sigma}^2= \begin{pmatrix} {\mathbb{E}} (Y_1^2) & {\mathbb{E}} (Y_1Y_2) \\ {\mathbb{E}} (Y_1Y_2) & {\mathbb{E}} (Y_2^2) \end{pmatrix} - \begin{pmatrix} {\mathbb{E}} (Y_1 \mathbf Z^T) \\ {\mathbb{E}} (Y_2 \mathbf Z^T) \end{pmatrix} \{{\mathbb{E}} (\mathbf Z \mathbf Z^T)\}^{-1} \begin{pmatrix} {\mathbb{E}} (\mathbf Z Y_1) & {\mathbb{E}} (\mathbf Z Y_2) \end{pmatrix}. \tag{2} \]
In the Gaussian field context, Lemma 1 allows us to write down an explicit expression of the mean function and the autocovariance function of the conditional Gaussian field associated to \(R(\cdot,\cdot)\) and to \(({\cal N},\mathbf x)\). Let us put \(Y_1=X(M_1)\) and \(Y_2=X(M_2)\) the values of the field \(X(\cdot)\) at points \(M_1 \in \mathbb{R}^d\) and \(M_2 \in \mathbb{R}^d\) respectively, and \(\mathbf Z=\mathbf X_{\cal N} \in \mathbb{R}^k\). Therefore, all quantities in (1) and (2) can be expressed in terms of the autocovariance function \(R\). Precisely, \[{\mathbb{E}} (Y_i Y_j) = R(M_i, M_j), \quad (i,j) \in \{1,2\}^2,\] and \[{\mathbb{E}} (Y_i\mathbf Z_\ell) = R(M_i, N_\ell), \quad i \in \{1,2\}, \quad \ell =1,\ldots,k.\] Consequently, the mean function of the conditional Gaussian field is given by
\[\label{eq:mean} \widetilde{m}(M)= {\mathbb{E}} (X(M) \mathbf X_{\cal N}^T)\{{\mathbb{E}} (\mathbf X_{\cal N} \mathbf X_{\cal N}^T)\}^{-1} \mathbf x,\quad M\in\mathbb{R} ^d. \tag{3} \]
Then the autocovariance function of a conditional Gaussian field (using the \((1, 2)\)-coordinate of Equation (2)) is given by \[\label{eq:var} \widetilde{R}(M_1,M_2) = R(M_1,M_2) - {\mathbb{E}} (X(M_1) \mathbf X_{\cal N}^T)\{{\mathbb{E}} (\mathbf X_{\cal N} \mathbf X_{\cal N}^T)\}^{-1} {\mathbb{E}} ( \mathbf X_{\cal N}X(M_2) ). \tag{4} \] For instance, for \(k=1\), we get \[\widetilde{m}(M)= \frac{R(M,N_1)}{R(N_1,N_1)}x_1,\] and \[\widetilde{R}(M_1,M_2) = R(M_1,M_2) - \frac{ R(M_1,N_1) R(M_2,N_1)} {R(N_1,N_1)}.\]
Let us recall that the goal of this paper is to give a procedure that yields discretization of the sample path of the conditional Gaussian field over a space discretization \(\{{\cal S}_e, {\cal S}_r\}\) of \({\mathbb{R}^d}\) associated to the n.n.d. autocovariance function \(R\) and the conditioning set and values \(({\cal N},\mathbf x)\) . In the sequel, we denote by \(\widetilde{X}(\cdot)\) this sample path. Since the mean function (3) is known, we can consider the centered field \(\overline{X}(\cdot)= \widetilde{X}(\cdot) - \widetilde{m}(\cdot)\). The fieldsim procedure for conditional Gaussian fields proceeds as follows.
Exact simulation step.
Given a space discretization \({\cal S}_e\), a sample of a centered Gaussian vector \((\overline{X}(M))_{M\in {\cal S}_e}\) with covariance matrix \(\widetilde{\mathbf{R}}\) given by \(\{\widetilde{\mathbf{R}}\}_{i,j}=\widetilde{R}(M_i,M_j),\) \(M_i,M_j\in {\cal S}_e,\) is simulated. Here \(\widetilde{R}\) is defined by (4). This simulation is obtained by an algorithm based on Cholesky decomposition of the matrix \(\widetilde{\mathbf{R}}\).
Refined simulation step.
Let \({\cal S}_r\) be the remaining space discretization. For each new point \(M\in {\cal S}_r\) at which we want to simulate the field, \(\overline{X}(M)\) is generated by using only a set of neighbors instead of all the simulated components (as in the accurate simulation step). Precisely, let \({\cal O}_M\) be a neighbors set of \(M\) (for the Euclidean distance) and \({\cal X}_{{\cal O}_M}\) be the space generated by the variables \(X(M'),\) \(M'\in {\cal O}_M\). Let us remark that the neighbors set is defined with all the already simulated variables (in the accurate and refined simulation step). Let \(X_{{\cal X}_{{\cal O}_M}}(M)\) be the best linear combination of variables of \({\cal X}_{{\cal O}_M}\) approximating \(\overline{X}(M)\) in the sense that the variance of the innovation \[\varepsilon_{{\cal X}_{N_M}}(M)=\overline{X}(M)-X_{{\cal X}_{{\cal O}_M}}(M),\] is minimal. The new variable \(\overline{X}(M)\) is obtained by \[X_{{\cal X}_{{\cal O}_M}}(M)+\sqrt{Var(\varepsilon_{{\cal X}_{{\cal O}_M}}(M))}U,\] where \(U\) is a centered and reduced Gaussian variable independent of the already simulated components. Note that the variable \(X_{{\cal X}_{{\cal O}_M}}(M)\) and the variance \(Var(\varepsilon_{{\cal X}_{{\cal O}_M}}(M))\) are completely determined by the covariance structure of the sequence \(\overline{X}(M')\), \(M' \in {\cal O}_M\cup \{M\}\).
Adding the mean.
Finally, we compute \(\widetilde{X}(M) = \overline{X}(M) + \widetilde{m}(M)\) for all \(M\in \{{\cal S}_e, {\cal S}_r\}\).
For storage and computing time, the accurate simulation step must concern only a small number of variables whereas the second step can relate to a larger number of variables. That leads to an effective and fast method to simulate any Gaussian field.
It is worth mentioning that the setProcess command will check if
\(\{{\mathbb{E}} (\mathbf X_{\cal N} \mathbf X_{\cal N}^T)\}^{-1}\)
exists for common conditional simulations.
4 Some examples of conditional fractional Gaussian fields
We focus, in this paper, on the conditional Gaussian fields associated to the previously mentioned fields but every other classical Gaussian field can be also simulated: standard bifractional Brownian motion, space-time deformed fractional Brownian motion, etc. (see Brouste et al. (2007)). We also consider conditional simulations associated to fractional Gaussian fields on manifolds (hyperboloid and sphere) (see Brouste et al. (2010) for the covariance function definition).
The procedure fieldsim is extended to the conditional Gaussian
fields. We can find the setProcess reference
short-card in Table 1.
On the line
The fractional Gaussian processes on the line are fast to simulate.
Conditional simulations associated to fractional Brownian motion (fBm) and multifractional Brownian motion (mBm) and to the conditioning set \({\cal N}=\{ {1}/{2},{3}/{4},1 \}\) and conditioning values \(\mathbf x=\{1, 1/2, 0 \}\) are illustrated on Figure 2. Here the Hurst exponent is \(H=0.7\) for the fBm and \(H(t)=0.3+0.6t,\) \(t\in [0,1]\) for the mBm. The processes are simulated on a regular grid of 256 points of \([0,1]\) with only an exact simulation step (\({\cal S}_r= \emptyset\)).
They can be obtained with the fieldsim procedure. For instance, the
mBm in Figure 2 is obtained with:
R> funcH <- function(x) 0.3 + x * 0.6
R> cond.mBm <- setProcess("cond-mBm-line",
+ list(Gamma = matrix(c(1/2, 1, 3/4, 0.5, 1, 0), 2, 3), par = funcH))
R> fieldsim(cond.mBm)
R> plot(cond.mBm)In the simulation below, the points of the set \({\cal N}\) belong to the visualization grid. When this is not the case, the plot could show a failure for the conditioning in the region of high variability. To avoid this, it is possible to add the points of the set \({\cal N}\) to the visualization grid. For instance, in the previous example, to add the point \(1/6\) to the visualization grid, we can use the following lines of code:
R> atlas.cond.mBm <- sort(c(cond.mBm@manifold@atlas[1, ], 1/6))
R> cond.mBm@manifold@atlas <- matrix(atlas.mBm, nrow = 1)Another solution is to use finer grids which contain the points of the set \({\cal N}\).
On the plane
Conditional simulations associated to a fractional Brownian field (for \(H=0.9\)) and multifractional Brownian field (for \(H(\mathbf t)=0.3+0.6t_1\)) are illustrated in Figure 3. Conditional simulations associated to anisotropic fields (fractional Brownian sheet with \(H_1=0.9,\) \(H_2=0.3\), anisotropic fractional Brownian field with \(H=0.7\), \(\vartheta_1=\frac{\pi}{6}\) and \(\vartheta_2=\frac{\pi}{3}\)) are presented in Figure 4. For all the fields, we consider the following conditioning set \[{\cal N}=\left\{\left(1,\frac k{2^6+1}\right),\;\left(\frac k{2^6+1},1\right),\; k=0,\ldots, 2^6+1 \right\},\] and conditioning values \(\mathbf x=\mathbf 0\).
