Builds the *trend* matrix in accordance to a specification
of the mean provided by the user.

`trend.spatial(trend, geodata, add.to.trend)`

trend

specifies the mean part of the model.
See `DETAILS`

below.

geodata

optional. An object of the class `geodata`

as described in
`as.geodata`

.

add.to.trend

optional. Specifies aditional terms to the mean part of the model. See details below.

An object of the class `trend.spatial`

which is an \(n \times p\) *trend*
matrix, where \(n\)
is the number of spatial
locations and \(p\) is the number of mean parameters in the model.

The implicity model assumes that there is an underlying process
with mean \(\mu(x)\), where \(x = (x_1, x_2)\) denotes the coordinates
of a spatial location.
The argument `trend`

defines the form of the mean and the
following options are allowed:

`"cte"`

the mean is assumed to be constant over the region, in which case \(\mu(x)= \mu\). This is the default option.`"1st"`

the mean is assumed to be a first order polynomial on the coordinates: $$\mu(x)= \beta_0 + \beta_1 x_1 + \beta_2 x_2$$`"2nd"`

the mean is assumed to be a second order polynomial on the coordinates: $$\mu(x)= \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 (x_1)^2 + \beta_4 (x_2)^2 + \beta_5 x_1 * x_2$$`~ model`

a model specification. See`formula`

for further details on how to specify a model in R using formulas. Notice that the model term before the`~`

is not necessary. Typically used to include covariates (external trend) in the model.

Denote by \(x_1\) and \(x_2\) the spatial coordinates. The following specifications are equivalent:

`trend = "1st"`

and`trend = ~ x1 + x2`

`trend = "2nd"`

and`trend = ~ x1 + x2 + I(x1^2) + I(x2^2) + I(x1*x2)`

**Search path for covariates**
Typically, functions in the package geoR which calls
`trend.spatial`

will have the arguments `geodata`

,
`coords`

and `data`

.

When the trend is specifed as `trend = ~ model`

the terms included in the model will be searched for in the following
path sequence (modified in version 1.7-6, no longer attach objects):

in the object

`geodata`

(coerced to data-frame)in the users/session Global environment

in the session search path

The argument `add.to.trend`

adds terms to what is specified in
the argument `trend`

. This seems redundant but allow
specifications of the type: `trend="2nd", add.trend=~other.covariates`

.

Further information on the package geoR can be found at: http://www.leg.ufpr.br/geoR.

The section `DETAILS`

in the documentation for
`likfit`

for more about the underlying model.

# NOT RUN { # a first order polynomial trend trend.spatial("1st", sic.100)[1:5,] # a second order polynomial trend trend.spatial("2nd", sic.100)[1:5,] # a trend with a covariate trend.spatial(~altitude, sic.100)[1:5,] # a first degree trend plus a covariate trend.spatial(~coords+altitude, sic.100)[1:5,] # with produces the same as trend.spatial("1st", sic.100, add=~altitude)[1:5,] # and yet another exemple trend.spatial("2nd", sic.100, add=~altitude)[1:5,] # }