# Create isotropic covariance model Function to generate isotropic covariance models or to add an isotropic covariance model to an existing isotropic model. ```r covmodel(modelname, mev, nugget,variance, scale, parameter, add.covmodel) ## S3 method for class 'covmodel' print(x, ...) ``` ## Arguments - `modelname`: a character scalar with the name of an isotropic covariance model, see **Details** for a list of implemented models. A call of `covmodel()` without any function argument displays a table with all available models and their parameters, see **Examples**. - `mev`: a numeric scalar, variance of the measurement error. - `nugget`: a numeric scalar, variance of microstructure white noise process with range smaller than the minimal distance between any pair of support data. - `variance`: a numeric scalar, partial sill of the covariance model. - `scale`: a numeric scalar, scale ("range") parameter of the covariance model. - `parameter`: a numeric vector of further covariance parameters, missing for some model like `nugget`, `spherical` or `gauss`, etc, see **Details**. If a model has several extra parameters, say `a`, `b`, ... then they must be given as `c(a, b, ...)`. - `add.covmodel`: an object of the class `covmodel` that is added to the covariance model defined by `modelname` (see examples) - `x`: a covariance model generated by `covmodel` - ``: further printing arguments ## Details The name and parametrisation of the covariance models originate from the function `CovarianceFct` of the archived package `RandomFields`, version 2.0.71. The following isotropic covariance functions are implemented (equations taken from help page of function `CovarianceFct` of archived package `RandomFields`, version 2.0.71, note that the variance and range parameters are equal to 1 in the following formulae and $h$ is the lag distance.): * `bessel` $$ C(h)=2^a \Gamma(a+1)h^{-a} J_a(h) $$ For a 2-dimensional region, the parameter $a$ must be greater than or equal to 0. * `cauchy` $$ C(h)=\left(1+h^2\right)^{-a}C(h)=(1+h^2)^(-a) $$ The parameter $a$ must be positive. * `cauchytbm` $$ C(h)= (1+(1-b/3)h^a)(1+h^a)^{(-b/a-1)} $$ The parameter $a$ must be in (0,2] and $b$ positive. The model is valid for 3 dimensions. It allows for simulating random fields where fractal dimension and Hurst coefficient can be chosen independently. * `circular` $$ C(h)=\left(1-\frac 2\pi\left(h \sqrt{1-h^2} +\arcsin(h)\right)\right)1_{[0,1]}(h)C(h)=1-2/pi*(h sqrt(1-h^2)+asin(h)) if0\<=h\<=1, 0 otherwise $$ This isotropic covariance function is valid only for dimensions less than or equal to 2. * `constant` $$ C(h)=1 $$ * `cubic` $$ C(h)=(1- 7h^2+8.75h^3-3.5h^5+0.75 h^7)1_{[0,1]}(h)C(h)=1- 7 h^2 + 8.75 h^3 - 3.5 h^5 + 0.75 h^7 if 0\<=h\<=1,0 otherwise $$ This model is valid only for dimensions less than or equal to 3. It is a 2 times differentiable covariance functions with compact support. * `dampedcosine` (hole effect model) $$ C(h)= e^{-a h} \cos(h) $$ This model is valid for 2 dimensions iff $a \ge 1$. * `exponential` $$ C(h)=e^{-h} $$ This model is a special case of the `whittle` model (for $a=0.5$) and the `stable` model (for $a = 1$). * `gauss` $$ C(h)=e^{-h^2}C(h)=exp(-h^2) $$ This model is a special case of the `stable` model (for $a=2$). See `gneiting` for an alternative model that does not have the disadvantages of the Gaussian model. * `gencauchy` (generalised `cauchy`) $$ C(h)= \left(1+h^a\right)^{(-b/a)}C(h)= (1+h^a)^{(-b/a)} $$ The parameter $a$ must be in (0,2] and $b$ positive. This model allows for random fields where fractal dimension and Hurst coefficient can be chosen independently. * `gengneiting` (generalised `gneiting`) If $a=1$ and let $\beta = b+1$ then $$ C(h)=\left(1+\beta h\right) (1-h)^{\beta}1_{[0,1]}(h)C(h)=(1+\beta h) (1-h)^{\beta}1_{[0,1]}(h) $$ If $a=2$ and let $\beta = b+2$ then $$ C(h)=\left(1+\beta h+\left(\beta ^2-1\right)h^2/3\right)(1-h)^{\beta} 1_{[0,1]}(h)C(h)=(1+\beta h+(\beta ^2-1)h^2/3)(1-h)^{\beta} 1_{[0,1]}(h) $$ If $a=3$ and let $\beta = b+3$ then $$ C(h)=\left(1+\beta h+\left(2\beta ^2-3\right)\frac{h^2}{5}+\left(\beta ^2-4\right)\beta \frac{h^3}{15}\right)(1-h)^{\beta}1_{[0,1]}(h)C(h)=(1+\beta h+(2\beta ^2-3)h^2/5+(\beta ^2-4)\beta h^3/15)(1-h)^{\beta}1_{[0,1]}(h) $$ The parameter $a$ is a positive integer; here only the cases $a=1, 2, 3$ are implemented. For two dimensional regions the parameter $b$ must greater than or equal to $(2 + 2a +1)/2$. * `gneiting` $$ C(h)=\left(1 + 8 sh + 25 (sh)^2 + 32(sh)^3\right)(1-sh)^8 1_{[0,1]}(sh)C(h)=(1 + 8 s h + 25 s^2 h^2 + 32s^3 h^3)*(1-s h)^8 if 0 \<= h \<= 1/s, 0 otherwise $$ where $s=0.301187465825$. This covariance function is valid only for dimensions less than or equal to 3. It is a 6 times differentiable covariance functions with compact support. It is an alternative to the `gaussian` model since its graph is visually hardly distinguishable from the graph of the Gaussian model, but possesses neither the mathematical and nor the numerical disadvantages of the Gaussian model. * `hyperbolic` $$ C(h)= c^{-b}(K_{b}(a c))^{-1}( c^2 + h^2 )^{b/2}K_{b}(a [ c^2 + h^2 ]^{1/2} ) $$ The parameters are such that $c\ge0$, $a\>0$ and $b\>0, $ or $c\>0$, $a\>0$ and $b\>0, $ or $c\>0$, $a\ge0$, and $b\<0$. Note that this class is over-parametrised; always one of the three parameters $a$, $c$, and scale can be eliminated in the formula. * `lgd1` (local-global distinguisher) $$ C(h)=1 - \frac{b}{a+b} h^{a}, h \le 1 \qquad \hbox{and} \qquad\frac{a}{a+b} h^{-b}, h \> 1C(h)=1 - b/(a+b) h^{a}, h \le 1 anda/(a+b) h^{-b}, h \> 1 $$ Here $b\>0$ and $a$ msut be in $(0,0.5]$. The random field has for 2-dimensional regions fractal dimension $3 - a/2$ and Hurst coefficient $1 -b/2$ for $b \in (0,1]$ * `matern` $$ C(h)= 2^{1-a} \Gamma(a)^{-1}(\sqrt{2 a} h)^a K_a(\sqrt{2 a}h) $$ The parameter $a$ must be positive. This is the model of choice if the smoothness of a random field is to be parametrised: if $a \> m$ then the graph is $m$ times differentiable. * `nugget` $$ C(h)=1_{[0]}(h)C(h)=1_[0](h) $$ * `penta` $$ C(h)= \left(1 - \frac{22}3 h^2 +33 h^4 -\frac{77}2 h^5 + \frac{33}2h^7 -\frac{11}2 h^9 + \frac 56 h^{11}\right)1_{[0,1]}(h)C(h)=1 - 22/3 h^2 +33 h^4- 77/2 h^5 + 33/2 h^7 - 11/2 h^9 + 5/6 h^11 if 0\<=h\<=1,0 otherwise $$ valid only for dimensions less than or equal to 3. This is a 4 times differentiable covariance functions with compact support. * `power` $$ C(h)= (1-h)^a 1_{[0,1]}(h) $$ This covariance function is valid for 2 dimensions iff $a \ge 1.5$. For $a=1$ we get the well-known triangle (or tent) model, which is valid on the real line, only. * `qexponential` $$ C(h)= ( 2 e^{-h} - a e^{-2x} ) / ( 2 - a ) $$ The parameter $a$ must be in $[0,1]$. * `spherical` $$ C(h)=\left(1- 1.5 h+0.5 h^3\right) 1_{[0,1]}(h)C(h)=1 - 1.5 h + 0.5 h^3 if 0\<=h\<=1, 0 otherwise $$ This covariance function is valid only for dimensions less than or equal to 3. * `stable` $$ C(h)=\exp\left(-h^a\right) $$ The parameter $a$ must be in $(0,2]$. See `exponential` and `gaussian` for special cases. * `wave` $$ C(h)=\frac{\sin h}{h}, \quad h\>0 \qquad \hbox{and } \qquad C(0)=1C(h)=sin(h)/h if h\>0 and C(0)=1 $$ This isotropic covariance function is valid only for dimensions less than or equal to 3. It is a special case of the `bessel` model (for $a=0.5$). * `whittle` $$ C(h) = 2^{1-a} \Gamma(a)^{-1} h^aK_a(h) $$ The parameter $a$ must be positive. This is the model of choice if the smoothness of a random field is to be parametrised: if $a \> m$ then the graph is $m$ times differentiable. The default values of the arguments `mev`, `nugget`, `variance` and `scale` are eq 0. ## Returns an object of the class `covmodel` that defines a covariance model. ## Author(s) Christoph Hofer, christoph.hofer@alumni.ethz.ch ## Examples ```r # table with all available covariance models and their # parameters covmodel() # exponential model without a measurement error and without a nugget, # partial sill = 10, scale parameter = 15 covmodel(modelname = "exponential", variance = 10, scale = 15) # exponential model with a measurement error ( mev = 0.5) and a # nugget (nugget = 2.1), exponential partial sill (variance = 10) # and scale parameter = 15 covmodel(modelname = "exponential", mev = 0.5, nugget = 2.1, variance = 10, scale = 15) ```

covmodel function