gencorr: Variogram Models

Description

The function gencorr computes intrinsic or stationary isotropic generalized correlations (= negative semi-variances computed with sill (variance) parameter equal to 1) for a set of basic variogram models formerly made available by the function RFfctn of the now archived R package RandomFields.

Usage

gencorr(x, variogram.model, param)

Value

A numeric vector with generalized correlations (= negative semi-variances computed with variance parameter param["variance"] = 1).

Arguments

x: a numeric vector with scaled lag distances, i.e. lag distances divided by the range parameter param["scale"].
variogram.model: a character keyword defining the variogram model. Currently, the following models are implemented:
"RMaskey", "RMbessel", "RMcauchy", "RMcircular", "RMcubic", "RMdagum",
"RMdampedcos", "RMdewijsian", "RMexp" (default), "RMfbm", "RMgauss",
"RMgencauchy", "RMgenfbm", "RMgengneiting", "RMgneiting", "RMlgd",
"RMmatern", "RMpenta", "RMqexp", "RMspheric", "RMstable", "RMwave",
"RMwhittle", see Details.
param: a named numeric vector with values of the additional parameters of the variogram models such as the smoothness parameter of the Whittle-Matérn model, see param.names for the names of these parameters. Note that some variogram models (e.g. "RMcircular") do not have any additional parameters.

Author

Andreas Papritz papritz@retired.ethz.ch

Details

The name and parametrization of the variogram models originate from the function RFfctn of RandomFields. The equations and further information are taken (with minor modifications) from the help pages of the respective functions of the archived R package RandomFields, version 3.3.14 (Schlather et al., 2022). Note that the variance (sill, param["variance"]) and the range parameters (param["scale"]) are assumed to be equal to 1 in the following formulae, and $x$ is the lag distance. The variogram functions are stationary and are valid for any number of dimensions if not mentioned otherwise.

The following intrinsic or stationary isotropic variogram functions $\gamma(x)$ are implemented in gencorr:

RMaskey $$ \gamma(x)= 1 - (1-x)^\alpha 1_{[0,1]}(x) $$ $1_{[0,1]}(x)$ is the indicator function equal to 1 for $x \in [0,1]$ and 0 otherwise. This variogram function is valid for dimension $d$ if $\alpha \ge (d+1)/2$. For $\alpha=1$ we get the well-known triangle (or tent) model, which is only valid on the real line.
RMbessel $$ \gamma(x) = 1 - 2^\nu \Gamma(\nu+1) x^{-\nu} J_\nu(x) $$ where $\nu \ge \frac{d-2}2$, $\Gamma$ denotes the gamma function and $J_\nu$ is a Bessel function of first kind. This models a hole effect (see Chilès and Delfiner, 1999, p. 92). An important case is $\nu=-0.5$ which gives the variogram function $$\gamma(x)= 1 - \cos(x)$$ and which is only valid for $d=1$ (this equals RMdampedcos for $\lambda = 0$). A second important case is $\nu=0.5$ with variogram function $$ \gamma(x) = \left(1 - \frac{\sin(x)}{x}\right) 1_{x>0} $$ which is valid for $d \le 3$. This coincides with RMwave.
RMcauchy $$\gamma(x) = 1 - (1 + x^2)^{-\gamma}$$

where $\gamma > 0$. The parameter $\gamma$ determines the asymptotic power law. The smaller $\gamma$, the longer the long-range dependence. The generalized Cauchy Family (RMgencauchy) includes this family for the choice $\alpha = 2$ and $\beta = 2 \gamma$.
RMcircular $$ \gamma(x) = 1 - \left(1 -\frac{2}{\pi} \left(x \sqrt{1-x^2} + \arcsin(x)\right)\right) 1_{[0,1]}(x) $$ This variogram function is valid only for dimensions $d \le 2$.
RMcubic $$ \gamma(x) = 1 - (1-7 x^2 + 8.75 x^3 - 3.5 x^5 + 0.75 x^7) 1_{[0,1]}(x) $$ The model is only valid for dimensions $d \le 3$. It is a 2 times differentiable variogram function with compact support (see Chilès and Delfiner, 1999, p. 84).
RMdagum $$ \gamma(x) = (1+x^{-\beta})^{-\gamma / \beta} $$ The parameters $\beta$ and $\gamma$ can be varied in the intervals $(0,1]$ and $(0,1)$, respectively. Like the generalized Cauchy model (RMgencauchy) the Dagum family can be used to model separately fractal dimension and Hurst effect (see Berg et al., 2008).
RMdampedcos $$ \gamma(x) = 1 - \exp(-\lambda x) \cos(x) $$ The model is valid for any dimension $d$. However, depending on the dimension of the random field the following bound $ \lambda \ge 1/{\tan(\pi/(2d))} $ has to be respected. This variogram function models a hole effect (see Chilès and Delfiner, 1999, p. 92). For $\lambda = 0$ we obtain $$\gamma(x)= 1 - \cos(x)$$ which is only valid for $d=1$ and corresponds to RMbessel for $\nu=-0.5$.
RMdewijsian $$ \gamma(x) = \log(1 + x^{\alpha}) $$ where $\alpha \in (0,2]$. This is an intrinsic variogram function. Originally, the logarithmic model $\gamma(x) = \log(x)$ was named after de Wijs and reflects a principle of similarity (see Chilès and Delfiner, 1999, p. 90). But note that $\gamma(x) = \log(x)$ is not a valid variogram function.
RMexp $$\gamma(x) = 1 - e^{-x}$$ This model is a special case of the Whittle model (RMwhittle) if $\nu=0.5$ and of the stable family (RMstable) if $\nu = 1$. Moreover, it is the continuous-time analogue of the first order auto-regressive time series covariance structure.
RMfbm $$\gamma(x) = x^\alpha$$ where $\alpha \in (0,2)$. This is an intrinsically stationary variogram function. For $\alpha=1$ we get a variogram function corresponding to a standard Brownian Motion. For $\alpha \in (0,2)$ the quantity $H = \frac{\alpha}{2}$ is called Hurst index and determines the fractal dimension $D = d + 1 - H$ of the corresponding Gaussian sample paths where $d$ is the dimension of the random field (see Chilès and Delfiner, 1999, p. 89).
RMgauss $$\gamma(x) = 1 - e^{-x^2}$$

The Gaussian model has an infinitely differentiable variogram function. This smoothness is artificial. Furthermore, this often leads to singular matrices and therefore numerically instable procedures (see Stein, 1999, p. 29). The Gaussian model is included in the stable class (RMstable) for the choice $\alpha = 2$.
RMgencauchy $$\gamma(x) = 1 - (1 + x^\alpha)^{-\beta/\alpha}$$ where $\alpha \in (0,2]$ and $\beta > 0$. This model has a smoothness parameter $\alpha$ and a parameter $\beta$ which determines the asymptotic power law. More precisely, this model admits simulating random fields where fractal dimension D of the Gaussian sample path and Hurst coefficient H can be chosen independently (compare also with RMlgd): Here, we have $D = d + 1 - \alpha/2, \alpha \in (0,2]$ and $ H = 1 - \beta/2, \beta > 0$. The smaller $\beta$, the longer the long-range dependence. The variogram function is very regular near the origin, because its Taylor expansion only contains even terms and reaches its sill slowly. Note that the Cauchy Family (RMcauchy) is included in this family for the choice $\alpha = 2$ and $\beta = 2 \gamma$.
RMgenfbm $$ \gamma(x) = (1 + x^{\alpha})^{\delta/\alpha} - 1 $$ where $\alpha \in (0,2)$ and $\delta \in (0,1)$. This is an intrinsic variogram function.
RMgengneiting This is a family of stationary models whose elements are specified by the two parameters $\kappa$ and $\mu$ with $\kappa$ being a non-negative integer and $\mu \ge \frac{d}{2}$ with $d$ denoting the dimension of the random field (the models can be used for any dimension). Let $\beta = \mu + 2\kappa +1/2$.

For $\kappa = 0$ the model equals the Askey model (RMaskey) and is therefore not implemented.

For $\kappa = 1$ the model is given by

$$ \gamma(x) = 1 - \left(1+\beta x \right)(1-x)^{\beta} 1_{[0,1]}(x), \qquad \beta = \mu +2.5, $$

If $\kappa = 2$ $$ \gamma(x) = 1 - \left(1 + \beta x + \frac{\beta^{2} - 1}{3} x^{2} \right)(1-x)^{\beta} 1_{[0,1]}(x), \qquad \beta = \mu+4.5, $$

and for $\kappa = 3$ $$ \gamma(x) = 1 - \left( 1 + \beta x + \frac{(2\beta^{2}-3)}{5} x^{2}+ \frac{(\beta^2 - 4)\beta}{15} x^{3} \right)(1-x)^\beta 1_{[0,1]}(x), \beta = \mu+6.5, $$

A special case of this family is RMgneiting (with $s = 1$ there) for the choice $\kappa = 3, \mu = 3/2$.
RMgneiting

$$ \gamma(x) = 1 - (1 + 8 s x + 25 s^2 x^2 + 32 s^3 x^3)(1-s x)^8 $$

if $0 \le x \le \frac{1}{s}$ and $$\gamma(x)= 1$$ otherwise. Here, $s=0.301187465825$. This variogram function is valid only for dimensions less than or equal to 3. It is 6 times differentiable and has compact support. This model is an alternative to RMgauss as its graph is hardly distinguishable from the graph of the Gaussian model, but possesses neither the mathematical nor the numerical disadvantages of the Gaussian model. It is a special case of RMgengneiting for the choice $\kappa=3, \mu=1.5$.
RMlgd $$ \gamma(x) = \frac{\beta}{\alpha + \beta} x^{\alpha} 1_{[0,1]}(x) + (1 - \frac{\alpha}{\alpha + \beta} x^{-\beta}) 1_{x>1}(x) $$ where $\beta >0$ and $0 < \alpha \le (3-d)/2$, with $d$ denoting the dimension of the random field. The model is only valid for dimension $d=1,2$. This model admits simulating random fields where fractal dimension $D$ of the Gaussian sample and Hurst coefficient $H$ can be chosen independently (compare also RMgencauchy): Here, the random field has fractal dimension $D = d+1 - \alpha/2$ and Hurst coefficient $H = 1-\beta/2$ for $0< \beta \le 1$.
RMmatern $$ \gamma(x) = 1 - \frac{2^{1-\nu}}{\Gamma(\nu)} (\sqrt{2\nu}x)^\nu K_\nu(\sqrt{2\nu}x) $$ where $\nu > 0$ and $K_\nu$ is the modified Bessel function of second kind. This is one of 3 possible parametrizations (Whittle, Matérn, Handcock-Wallis) of the Whittle-Matérn model. The Whittle-Matérn model is the model of choice if the smoothness of a random field is to be parametrized: the sample paths of a Gaussian random field with this covariance structure are $m$ times differentiable if and only if $\nu > m$ (see Gneiting and Guttorp, 2010, p. 24). Furthermore, the fractal dimension $D$ of the Gaussian sample paths is determined by $\nu$: We have $D = d + 1 - \nu, \nu \in (0,1)$ and $D = d$ for $\nu > 1$ where $d$ is the dimension of the random field (see Stein, 1999, p. 32). If $\nu=0.5$ the Matérn model equals RMexp. For $\nu$ tending to $\infty$ a rescaled Gaussian model RMgauss appears as limit for the Handcock-Wallis parametrization.
RMpenta $$ \gamma(x) = 1 - \left(1 - \frac{22}{3}x^{2} + 33 x^{4} - \frac{77}{2} x^{5} + \frac{33}{2} x^{7} - \frac{11}{2} x^{9} + \frac{5}{6}x^{11}\right) 1_{[0,1]}(x) $$ The model is only valid for dimensions $d \le 3$. It has a 4 times differentiable variogram function with compact support (cf. Chilès and Delfiner, 1999, p. 84).
RMqexp $$\gamma(x)= 1 - \frac{2 e^{-x} - \alpha e^{-2x}}{ 2 - \alpha }$$ where $\alpha \in [0,1]$.
RMspheric $$ \gamma(x) = 1 - \left(1 - \frac{3}{2} x + \frac{1}{2} x^3\right) 1_{[0,1]}(x) $$ This variogram model is valid only for dimensions less than or equal to 3 and has compact support.
RMstable $$\gamma(x) = 1 - e^{-x^\alpha}$$ where $\alpha \in (0,2]$. The parameter $\alpha$ determines the fractal dimension $D$ of the Gaussian sample paths: $D = d + 1 - \alpha/2$ where $d$ is the dimension of the random field. For $\alpha < 2$ the Gaussian sample paths are not differentiable (see Gelfand et al., 2010, p. 25). The stable family includes the exponential model (RMexp) for $\alpha = 1$ and the Gaussian model ( RMgauss) for $\alpha = 2$.
RMwave

$$ \gamma(x) = \left(1 - \frac{\sin(x)}{x}\right) 1_{x>0} $$

The model is only valid for dimensions $d \le 3$. It is a special case of RMbessel for $\nu = 0.5$. This variogram models a hole effect (see Chilès and Delfiner, 1999, p. 92).
RMwhittle

$$ \gamma(x)=1 - \frac{2^{1- \nu}}{\Gamma(\nu)} x^{\nu}K_{\nu}(x) $$

where $\nu > 0$ and $K_\nu$ is the modified Bessel function of second kind. This is one of 3 possible parametrizations (Whittle, Matérn, Handcock-Wallis) of the Whittle-Matérn model, for further details, see information for entry RMmatern above.

References

Berg, C., Mateau, J., Porcu, E. (2008) The dagum family of isotropic correlation functions, Bernoulli, 14, 1134--1149, tools:::Rd_expr_doi("10.3150/08-BEJ139").

Chilès, J.-P., Delfiner, P. (1999) Geostatistics Modeling Spatial Uncertainty, Wiley, New York, tools:::Rd_expr_doi("10.1002/9780470316993").

Gneiting, T. (2002) Compactly supported correlation functions. Journal of Multivariate Analysis, 83, 493--508, tools:::Rd_expr_doi("10.1006/jmva.2001.2056").

Gneiting, T., Schlather, M. (2004) Stochastic models which separate fractal dimension and Hurst effect. SIAM review 46, 269--282, tools:::Rd_expr_doi("10.1137/S0036144501394387").

Gneiting, T., Guttorp, P. (2010) Continuous Parameter Stochastic Process Theory, In Gelfand, A. E., Diggle, P. J., Fuentes, M., Guttrop, P. (Eds.) Handbook of Spatial Statistics, CRC Press, Boca Raton, p. 17--28, tools:::Rd_expr_doi("10.1201/9781420072884").

Schlather M., Malinowski A., Oesting M., Boecker D., Strokorb K., Engelke S., Martini J., Ballani F., Moreva O., Auel J., Menck P.J., Gross S., Ober U., Ribeiro P., Ripley B.D., Singleton R., Pfaff B., R Core Team (2022). RandomFields: Simulation and Analysis of Random Fields. R package version 3.3.14, https://cran.r-project.org/src/contrib/Archive/RandomFields/.

Stein, M. L. (1999) Interpolation of Spatial Data: Some Theory for Kriging, Springer, New York, tools:::Rd_expr_doi("10.1007/978-1-4612-1494-6").

Examples

Run this code

## scaled lag distances
x <- seq(0, 3, length.out = 100)

## generalized correlations stable model
y <- gencorr(x, variogram.model = "RMstable", param = c(alpha = 1.5))
plot(x, y)

## generalized correlations circular model
y <- gencorr(x, variogram.model = "RMcircular")
plot(x, y)

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