# Fit a spatio-temporal sample variogram to a sample variogram

### Description

Fits a spatio-temporal variogram of a given type to spatio-temporal sample variogram.

### Usage

1 2 | ```
fit.StVariogram(object, model, ..., method = "L-BFGS-B",
lower, upper, fit.method = 6, stAni=NA, wles)
``` |

### Arguments

`object` |
The spatio-temporal sample variogram. Typically output from |

`model` |
The desired spatio-temporal model defined through |

`...` |
further arguments passed to |

`lower` |
Lower limits used by optim. If missing, the smallest well defined values are used (mostly near 0). |

`upper` |
Upper limits used by optim. If missing, the largest well defined values are used (mostly |

`method` |
fit method, pass to |

`fit.method` |
an integer between 0 and 13 determine the fitting routine (i.e. weighting of the squared residuals in the LSE). Values 0 to 6 correspond with the pure spatial version (see |

`stAni` |
The spatio-temporal anisotropy that is used in the weighting. Might be missing if the desired spatio-temporal variogram model does contain a spatio-temporal anisotropy parameter (this might cause bad convergence behaviour). The default is |

`wles` |
Should be missing; only for backwards compatibility, |

### Details

The following list summarizes the meaning of the `fit.method`

argument which is essential a weighting of the squared residuals in the least-squares estimation. Please note, that weights based on the models gamma value might fail to converge properly due to the dependence of weights on the variogram estimate:

`fit.method = 0`

no fitting, however the MSE between the provided variogram model and sample variogram surface is calculated.

`fit.method = 1`

Number of pairs in the spatio-temporal bin:

*N[j]*`fit.method = 2`

Number of pairs in the spatio-temporal bin divided by the square of the current variogram model's value:

*N[j]/gamma(h[j],u[j])^2*`fit.method = 3`

Same as

`fit.method = 1`

for compatibility with`fit.variogram`

but as well evaluated in R.`fit.method = 4`

Same as

`fit.method = 2`

for compatibility with`fit.variogram`

but as well evaluated in R.`fit.method = 5`

Reserved for REML for compatibility with

`fit.variogram`

, not yet implemented.`fit.method = 6`

No weights.

`fit.method = 7`

Number of pairs in the spatio-temporal bin divided by the square of the bin's metric distance. If

`stAni`

is not specified, the model's parameter is used to calculate the metric distance across space and time:*N[j]/(h[j]^2+ stAni^2*u[j]^2)*`fit.method = 8`

Number of pairs in the spatio-temporal bin divided by the square of the bin's spatial distance.

*N[j]/h[j]^2*. Note that the 0 distances are replaced by the smallest non-zero distances to avoid division by zero.`fit.method = 9`

Number of pairs in the spatio-temporal bin divided by the square of the bin's temporal distance.

*N[j]/u[j]^2*. Note that the 0 distances are replaced by the smallest non-zero distances to avoid division by zero.`fit.method = 10`

Reciprocal of the square of the current variogram model's value:

*1/gamma(h[j],u[j])^2*`fit.method = 11`

Reciprocal of the square of the bin's metric distance. If

`stAni`

is not specified, the model's parameter is used to calculate the metric distance across space and time:*1/(h[j]^2+ stAni^2*u[j]^2)*`fit.method = 12`

Reciprocal of the square of the bin's spatial distance.

*1/h[j]^2*. Note that the 0 distances are replaced by the smallest non-zero distances to avoid division by zero.`fit.method = 13`

Reciprocal of the square of the bin's temporal distance.

*1/u[j]^2*. Note that the 0 distances are replaced by the smallest non-zero distances to avoid division by zero.

See also Table 4.2 in the gstat manual for the original spatial version.

### Value

Returns a spatio-temporal variogram model, as S3 class StVariogramModel. It carries the temporal and spatial unit as attributes `"temporal unit"`

and `"spatial unit"`

in order to allow `krigeST`

to adjust for different units. The units are obtained from the provided empirical variogram. Further attributes are the optim output `"optim.output"`

and the always not weighted mean squared error `"MSE"`

.

### Author(s)

Benedikt Graeler

### See Also

`fit.variogram`

for the pure spatial case. `extractParNames`

helps to understand the parameter structure of spatio-temporal variogram models.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
# separable model: spatial and temporal sill will be ignored
# and kept constant at 1-nugget respectively. A joint sill is used.
## Not run:
separableModel <- vgmST("separable",
method = "Nelder-Mead", # no lower & upper needed
space=vgm(0.9,"Exp", 123, 0.1),
time =vgm(0.9,"Exp", 2.9, 0.1),
sill=100)
data(vv)
separableModel <- fit.StVariogram(vv, separableModel,
method="L-BFGS-B",
lower=c(10,0,0.01,0,1),
upper=c(500,1,20,1,200))
plot(vv, separableModel)
## End(Not run) # dontrun
``` |