Matrix of a corStruct Object

This method function extracts the correlation matrix (or its transpose inverse square-root factor), or list of correlation matrices (or their transpose inverse square-root factors) corresponding to covariate and object. Letting \(\Sigma\) denote a correlation matrix, a square-root factor of \(\Sigma\) is any square matrix \(L\) such that \(\Sigma = L'L\). When corr = FALSE, this method extracts \(L^{-t}\).

# S3 method for class 'corStruct'
corMatrix(object, covariate, corr, ...)

Arguments

object

an object inheriting from class "corStruct" representing a correlation structure.

covariate

an optional covariate vector (matrix), or list of covariate vectors (matrices), at which values the correlation matrix, or list of correlation matrices, are to be evaluated. Defaults to getCovariate(object).

corr

a logical value. If TRUE the function returns the correlation matrix, or list of correlation matrices, represented by object. If FALSE the function returns a transpose inverse square-root of the correlation matrix, or a list of transpose inverse square-root factors of the correlation matrices.

...

some methods for this generic require additional arguments. None are used in this method.

Value

If covariate is a vector (matrix), the returned value will be an array with the corresponding correlation matrix (or its transpose inverse square-root factor). If the covariate is a list of vectors (matrices), the returned value will be a list with the correlation matrices (or their transpose inverse square-root factors) corresponding to each component of covariate.

References

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

Author

José Pinheiro and Douglas Bates bates@stat.wisc.edu

See also

corFactor.corStruct, Initialize.corStruct

Examples

cs1 <- corAR1(0.3)
corMatrix(cs1, covariate = 1:4)
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.30 0.09 0.027
#> [2,] 0.300 1.00 0.30 0.090
#> [3,] 0.090 0.30 1.00 0.300
#> [4,] 0.027 0.09 0.30 1.000
corMatrix(cs1, covariate = 1:4, corr = FALSE)
#>            [,1]       [,2]       [,3]     [,4]
#> [1,]  1.0000000  0.0000000  0.0000000 0.000000
#> [2,] -0.3144855  1.0482848  0.0000000 0.000000
#> [3,]  0.0000000 -0.3144855  1.0482848 0.000000
#> [4,]  0.0000000  0.0000000 -0.3144855 1.048285
#> attr(,"logDet")
#> [1] 0.141466

# Pinheiro and Bates, p. 225
cs1CompSymm <- corCompSymm(value = 0.3, form = ~ 1 | Subject)
cs1CompSymm <- Initialize(cs1CompSymm, data = Orthodont)
corMatrix(cs1CompSymm)
#> $M01
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M02
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M03
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M04
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M05
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M06
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M07
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M08
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M09
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M10
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M11
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M12
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M13
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M14
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M15
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $M16
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F01
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F02
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F03
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F04
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F05
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F06
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F07
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F08
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F09
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F10
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 
#> $F11
#>      [,1] [,2] [,3] [,4]
#> [1,]  1.0  0.3  0.3  0.3
#> [2,]  0.3  1.0  0.3  0.3
#> [3,]  0.3  0.3  1.0  0.3
#> [4,]  0.3  0.3  0.3  1.0
#> 

# Pinheiro and Bates, p. 226
cs1Symm <- corSymm(value = c(0.2, 0.1, -0.1, 0, 0.2, 0),
                   form = ~ 1 | Subject)
cs1Symm <- Initialize(cs1Symm, data = Orthodont)
corMatrix(cs1Symm)
#> $M01
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M02
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M03
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M04
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M05
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M06
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M07
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M08
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M09
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M10
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M11
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M12
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M13
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M14
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M15
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $M16
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F01
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F02
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F03
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F04
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F05
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F06
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F07
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F08
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F09
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F10
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 
#> $F11
#>      [,1]         [,2]          [,3]          [,4]
#> [1,]  1.0 2.000000e-01  1.000000e-01 -1.000000e-01
#> [2,]  0.2 1.000000e+00  9.020562e-17  2.000000e-01
#> [3,]  0.1 9.020562e-17  1.000000e+00 -1.040834e-16
#> [4,] -0.1 2.000000e-01 -1.040834e-16  1.000000e+00
#> 

# Pinheiro and Bates, p. 236 
cs1AR1 <- corAR1(0.8, form = ~ 1 | Subject)
cs1AR1 <- Initialize(cs1AR1, data = Orthodont)
corMatrix(cs1AR1)
#> $M01
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M02
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M03
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M04
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M05
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M06
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M07
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M08
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M09
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M10
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M11
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M12
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M13
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M14
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M15
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $M16
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F01
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F02
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F03
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F04
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F05
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F06
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F07
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F08
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F09
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F10
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 
#> $F11
#>       [,1] [,2] [,3]  [,4]
#> [1,] 1.000 0.80 0.64 0.512
#> [2,] 0.800 1.00 0.80 0.640
#> [3,] 0.640 0.80 1.00 0.800
#> [4,] 0.512 0.64 0.80 1.000
#> 

# Pinheiro and Bates, p. 237 
cs1ARMA <- corARMA(0.4, form = ~ 1 | Subject, q = 1)
cs1ARMA <- Initialize(cs1ARMA, data = Orthodont)
corMatrix(cs1ARMA)
#> $M01
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M02
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M03
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M04
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M05
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M06
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M07
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M08
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M09
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M10
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M11
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M12
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M13
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M14
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M15
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $M16
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F01
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F02
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F03
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F04
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F05
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F06
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F07
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F08
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F09
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F10
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 
#> $F11
#>           [,1]      [,2]      [,3]      [,4]
#> [1,] 1.0000000 0.3448276 0.0000000 0.0000000
#> [2,] 0.3448276 1.0000000 0.3448276 0.0000000
#> [3,] 0.0000000 0.3448276 1.0000000 0.3448276
#> [4,] 0.0000000 0.0000000 0.3448276 1.0000000
#> 

# Pinheiro and Bates, p. 238 
spatDat <- data.frame(x = (0:4)/4, y = (0:4)/4)
cs1Exp <- corExp(1, form = ~ x + y)
cs1Exp <- Initialize(cs1Exp, spatDat)
corMatrix(cs1Exp)
#>           [,1]      [,2]      [,3]      [,4]      [,5]
#> [1,] 1.0000000 0.7021885 0.4930687 0.3462272 0.2431167
#> [2,] 0.7021885 1.0000000 0.7021885 0.4930687 0.3462272
#> [3,] 0.4930687 0.7021885 1.0000000 0.7021885 0.4930687
#> [4,] 0.3462272 0.4930687 0.7021885 1.0000000 0.7021885
#> [5,] 0.2431167 0.3462272 0.4930687 0.7021885 1.0000000