ARMA(p,q) Correlation Structure

This function is a constructor for the corARMA class, representing an autocorrelation-moving average correlation structure of order (p, q). Objects created using this constructor must later be initialized using the appropriate Initialize method.

corARMA(value, form, p, q, fixed)

Arguments

value

a vector with the values of the autoregressive and moving average parameters, which must have length p + q and all elements between -1 and 1. Defaults to a vector of zeros, corresponding to uncorrelated observations.

form

a one sided formula of the form ~ t, or ~ t | g, specifying a time covariate t and, optionally, a grouping factor g. A covariate for this correlation structure must be integer valued. 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.

p, q

non-negative integers specifying respectively the autoregressive order and the moving average order of the ARMA structure. Both default to 0, but at least one should be > 0.

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 corARMA, representing an autocorrelation-moving average correlation structure.

References

Box, G.E.P., Jenkins, G.M., and Reinsel G.C. (1994) "Time Series Analysis: Forecasting and Control", 3rd Edition, Holden-Day.

Pinheiro, J.C., and Bates, D.M. (2000) "Mixed-Effects Models in S and S-PLUS", Springer, esp. pp. 236, 397.

Author

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

See also

corAR1, corClasses Initialize.corStruct, summary.corStruct

Examples

## ARMA(1,2) structure, with observation order as a covariate and
## Mare as grouping factor
cs1 <- corARMA(c(0.2, 0.3, -0.1), form = ~ 1 | Mare, p = 1, q = 2)

# 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
#> 

cs2ARMA <- corARMA(c(0.8, 0.4), form = ~ 1 | Subject, p=1, q=1)
cs2ARMA <- Initialize(cs2ARMA, data = Orthodont)
corMatrix(cs2ARMA)
#> $M01
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M02
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M03
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M04
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M05
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M06
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M07
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M08
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M09
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M10
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M11
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M12
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M13
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M14
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M15
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $M16
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F01
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F02
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F03
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F04
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F05
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F06
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F07
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F08
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F09
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F10
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 
#> $F11
#>        [,1]  [,2]  [,3]   [,4]
#> [1,] 1.0000 0.880 0.704 0.5632
#> [2,] 0.8800 1.000 0.880 0.7040
#> [3,] 0.7040 0.880 1.000 0.8800
#> [4,] 0.5632 0.704 0.880 1.0000
#> 

# Pinheiro and Bates use in nlme:  
# from p. 240 needed on p. 396
fm1Ovar.lme <- lme(follicles ~ sin(2*pi*Time) + cos(2*pi*Time),
                   data = Ovary, random = pdDiag(~sin(2*pi*Time)))
#> Error in solve.default(do.call("cbind", Xcols), apply(shifted, 2, fun,     ...)): Lapack routine dgesv: system is exactly singular: U[9,9] = 0
fm5Ovar.lme <- update(fm1Ovar.lme,
                correlation = corARMA(p = 1, q = 1))
#> Error: object 'fm1Ovar.lme' not found
# p. 396
fm1Ovar.nlme <- nlme(follicles~
     A+B*sin(2*pi*w*Time)+C*cos(2*pi*w*Time),
   data=Ovary, fixed=A+B+C+w~1,
   random=pdDiag(A+B+w~1),
   start=c(fixef(fm5Ovar.lme), 1) )
#> Error: object 'fm5Ovar.lme' not found
# p. 397
fm3Ovar.nlme <- update(fm1Ovar.nlme,
         correlation=corARMA(p=0, q=2) )
#> Error: object 'fm1Ovar.nlme' not found