# corRatio

##### Rational Quadratic Correlation Structure

This function is a constructor for the `corRatio`

class,
representing a rational quadratic spatial correlation structure. Letting
\(d\) denote the range and \(n\) denote the nugget
effect, the correlation between two observations a distance
\(r\) apart is \(1/(1+(r/d)^2)\) when no nugget effect
is present and \((1-n)/(1+(r/d)^2)\) when a
nugget effect is assumed. Objects created using this constructor need
to be later initialized using the appropriate `Initialize`

method.

- Keywords
- models

##### Usage

`corRatio(value, form, nugget, metric, fixed)`

##### Arguments

- value
an optional vector with the parameter values in constrained form. If

`nugget`

is`FALSE`

,`value`

can have only one element, corresponding to the "range" of the rational quadratic correlation structure, which must be greater than zero. If`nugget`

is`TRUE`

, meaning that a nugget effect is present,`value`

can contain one or two elements, the first being the "range" and the second the "nugget effect" (one minus the correlation between two observations taken arbitrarily close together); the first must be greater than zero and the second must be between zero and one. Defaults to`numeric(0)`

, which results in a range of 90% of the minimum distance and a nugget effect of 0.1 being assigned to the parameters when`object`

is initialized.- form
a one sided formula of the form

`~ S1+...+Sp`

, or`~ S1+...+Sp | g`

, specifying spatial covariates`S1`

through`Sp`

and, optionally, a grouping factor`g`

. When a grouping factor is present in`form`

, the correlation structure is assumed to apply only to observations within the same grouping level; observations with different grouping levels are assumed to be uncorrelated. Defaults to`~ 1`

, which corresponds to using the order of the observations in the data as a covariate, and no groups.- nugget
an optional logical value indicating whether a nugget effect is present. Defaults to

`FALSE`

.- metric
an optional character string specifying the distance metric to be used. The currently available options are

`"euclidean"`

for the root sum-of-squares of distances;`"maximum"`

for the maximum difference; and`"manhattan"`

for the sum of the absolute differences. Partial matching of arguments is used, so only the first three characters need to be provided. Defaults to`"euclidean"`

.- fixed
an optional logical value indicating whether the coefficients should be allowed to vary in the optimization, or kept fixed at their initial value. Defaults to

`FALSE`

, in which case the coefficients are allowed to vary.

##### Value

an object of class `corRatio`

, also inheriting from class
`corSpatial`

, representing a rational quadratic spatial correlation
structure.

##### References

Cressie, N.A.C. (1993), "Statistics for Spatial Data", J. Wiley & Sons.

Venables, W.N. and Ripley, B.D. (2002) "Modern Applied Statistics with S", 4th Edition, Springer-Verlag.

Littel, Milliken, Stroup, and Wolfinger (1996) "SAS Systems for Mixed Models", SAS Institute.

Pinheiro, J.C., and Bates, D.M. (2000) "Mixed-Effects Models in S and S-PLUS", Springer.

##### See Also

##### Examples

```
# NOT RUN {
sp1 <- corRatio(form = ~ x + y + z)
# example lme(..., corRatio ...)
# Pinheiro and Bates, pp. 222-249
fm1BW.lme <- lme(weight ~ Time * Diet, BodyWeight,
random = ~ Time)
# p. 223
fm2BW.lme <- update(fm1BW.lme, weights = varPower())
# p 246
fm3BW.lme <- update(fm2BW.lme,
correlation = corExp(form = ~ Time))
# p. 249
fm5BW.lme <- update(fm3BW.lme, correlation =
corRatio(form = ~ Time))
# example gls(..., corRatio ...)
# Pinheiro and Bates, pp. 261, 263
fm1Wheat2 <- gls(yield ~ variety - 1, Wheat2)
# p. 263
fm3Wheat2 <- update(fm1Wheat2, corr =
corRatio(c(12.5, 0.2),
form = ~ latitude + longitude,
nugget = TRUE))
# }
```

*Documentation reproduced from package nlme, version 3.1-145, License: GPL (>= 2) | file LICENCE*