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lotri

The goal of lotri is to easily specify block-diagonal matrices with (lo)wer (tri)angular matrices. Its as if you have won the (badly spelled) lotri (or lottery).

This was made to allow people (like me) to specify lower triangular matrices similar to the domain specific language implemented in nlmixr. Originally I had it included in RxODE, but thought it may have more general applicability, so I separated it into a new package.

Installation

You can install the released version of lotri from CRAN with:

install.packages("lotri")

And the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("nlmixrdevelopment/lotri")

Example

This is a basic example for an easier way to specify matrices in R. For instance to fully specify a simple 2x2 matrix, in R you specify:

mat <- matrix(c(1, 0.5, 0.5, 1),nrow=2,ncol=2,dimnames=list(c("a", "b"), c("a", "b")))

With lotri, you simply specify:

library(lotri)
library(microbenchmark)
library(ggplot2)


mat <- lotri(a+b ~ c(1,
                     0.5, 1))
print(mat)
#>     a   b
#> a 1.0 0.5
#> b 0.5 1.0

I find it more legible and easier to specify, especially if you have a more complex matrix. For instance with the more complex matrix:

mat <- lotri({
    a+b ~ c(1,
            0.5, 1)
    c ~ 1
    d +e ~ c(1,
             0.5, 1)
})
print(mat)
#>     a   b c   d   e
#> a 1.0 0.5 0 0.0 0.0
#> b 0.5 1.0 0 0.0 0.0
#> c 0.0 0.0 1 0.0 0.0
#> d 0.0 0.0 0 1.0 0.5
#> e 0.0 0.0 0 0.5 1.0

To fully specify this in base R you would need to use:

mat <- matrix(c(1, 0.5, 0, 0, 0,
                0.5, 1, 0, 0, 0,
                0, 0, 1, 0, 0,
                0, 0, 0, 1, 0.5,
                0, 0, 0, 0.5, 1),
              nrow=5, ncol=5,
              dimnames= list(c("a", "b", "c", "d", "e"),
                             c("a", "b", "c", "d", "e")))
print(mat)
#>     a   b c   d   e
#> a 1.0 0.5 0 0.0 0.0
#> b 0.5 1.0 0 0.0 0.0
#> c 0.0 0.0 1 0.0 0.0
#> d 0.0 0.0 0 1.0 0.5
#> e 0.0 0.0 0 0.5 1.0

Of course with the excellent Matrix package this is a bit easier:

library(Matrix)
mat <- matrix(c(1, 0.5, 0.5, 1),
              nrow=2,
              ncol=2,
              dimnames=list(c("a", "b"), c("a", "b")))

mat <- bdiag(list(mat, matrix(1), mat))

## Convert back to standard matrix
mat <- as.matrix(mat)
##
dimnames(mat) <- list(c("a", "b", "c", "d", "e"),
                      c("a", "b", "c", "d", "e"))
print(mat)
#>     a   b c   d   e
#> a 1.0 0.5 0 0.0 0.0
#> b 0.5 1.0 0 0.0 0.0
#> c 0.0 0.0 1 0.0 0.0
#> d 0.0 0.0 0 1.0 0.5
#> e 0.0 0.0 0 0.5 1.0

Regardless, I think lotri is a bit easier to use.

Creating lists of matricies with attached properties

lotri also allows lists of matrices to be created by conditioning on an id with the | syntax.

For example:

mat <- lotri({
    a+b ~ c(1,
            0.5, 1) | id
    c ~ 1 | occ
    d + e ~ c(1,
              0.5, 1) | id(lower=3, upper=2, omegaIsChol=FALSE)
})

print(mat)
#> $id
#>     d   e
#> d 1.0 0.5
#> e 0.5 1.0
#> 
#> $occ
#>   c
#> c 1
#> 
#> Properties: lower, upper, omegaIsChol

print(mat$lower)
#> $id
#> d e 
#> 3 3 
#> 
#> $occ
#>    c 
#> -Inf
print(mat$upper)
#> $id
#> d e 
#> 2 2 
#> 
#> $occ
#>   c 
#> Inf
print(mat$omegaIsChol)
#> $id
#> [1] FALSE

This gives a list of matrix(es) conditioned on the variable after the |. It also can add properties to each list that can be accessible after the list of matrices is returned, as shown in the above example. To do this, you simply have to enclose the properties after the conditional variable. That is et1 ~ id(lower=3).

Combining symmetric (named) matrices

Now there is even a faster way to do a similar banded matrix concatenation with lotriMat

testList <- list(lotri({et2 + et3 + et4 ~ c(40,
                            0.1, 20,
                            0.1, 0.1, 30)}),
                     lotri(et5 ~ 6),
                     lotri(et1+et6 ~c(0.1, 0.01, 1)),
                     matrix(c(1L, 0L, 0L, 1L), 2, 2,
                            dimnames=list(c("et7", "et8"),
                                          c("et7", "et8"))))

matf <- function(.mats){
  .omega <- as.matrix(Matrix::bdiag(.mats))
  .d <- unlist(lapply(seq_along(.mats),
                      function(x) {
                        dimnames(.mats[[x]])[2]
                      }))
  dimnames(.omega) <- list(.d, .d)
  return(.omega)
}

print(matf(testList))
#>      et2  et3  et4 et5  et1  et6 et7 et8
#> et2 40.0  0.1  0.1   0 0.00 0.00   0   0
#> et3  0.1 20.0  0.1   0 0.00 0.00   0   0
#> et4  0.1  0.1 30.0   0 0.00 0.00   0   0
#> et5  0.0  0.0  0.0   6 0.00 0.00   0   0
#> et1  0.0  0.0  0.0   0 0.10 0.01   0   0
#> et6  0.0  0.0  0.0   0 0.01 1.00   0   0
#> et7  0.0  0.0  0.0   0 0.00 0.00   1   0
#> et8  0.0  0.0  0.0   0 0.00 0.00   0   1

print(lotriMat(testList))
#>      et2  et3  et4 et5  et1  et6 et7 et8
#> et2 40.0  0.1  0.1   0 0.00 0.00   0   0
#> et3  0.1 20.0  0.1   0 0.00 0.00   0   0
#> et4  0.1  0.1 30.0   0 0.00 0.00   0   0
#> et5  0.0  0.0  0.0   6 0.00 0.00   0   0
#> et1  0.0  0.0  0.0   0 0.10 0.01   0   0
#> et6  0.0  0.0  0.0   0 0.01 1.00   0   0
#> et7  0.0  0.0  0.0   0 0.00 0.00   1   0
#> et8  0.0  0.0  0.0   0 0.00 0.00   0   1

mb <- microbenchmark(matf(testList),lotriMat(testList))

print(mb)
#> Unit: microseconds
#>                expr     min      lq      mean  median       uq      max neval
#>      matf(testList) 463.599 491.021 598.57232 521.231 585.1135 5710.686   100
#>  lotriMat(testList)   1.297   1.646   2.56874   2.732   3.0105    7.734   100

autoplot(mb)
#> Coordinate system already present. Adding new coordinate system, which will replace the existing one.

plot of chunk unnamed-chunk-8

You may also combine named and unnamed matrices, but the resulting matrix will be unnamed, and still be faster than Matrix:

testList <- list(lotri({et2 + et3 + et4 ~ c(40,
                            0.1, 20,
                            0.1, 0.1, 30)}),
                     lotri(et5 ~ 6),
                     lotri(et1+et6 ~c(0.1, 0.01, 1)),
                     matrix(c(1L, 0L, 0L, 1L), 2, 2))

matf <- function(.mats){
  .omega <- as.matrix(Matrix::bdiag(.mats))
  return(.omega)
}

print(matf(testList))
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 40.0  0.1  0.1    0 0.00 0.00    0    0
#> [2,]  0.1 20.0  0.1    0 0.00 0.00    0    0
#> [3,]  0.1  0.1 30.0    0 0.00 0.00    0    0
#> [4,]  0.0  0.0  0.0    6 0.00 0.00    0    0
#> [5,]  0.0  0.0  0.0    0 0.10 0.01    0    0
#> [6,]  0.0  0.0  0.0    0 0.01 1.00    0    0
#> [7,]  0.0  0.0  0.0    0 0.00 0.00    1    0
#> [8,]  0.0  0.0  0.0    0 0.00 0.00    0    1

print(lotriMat(testList))
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 40.0  0.1  0.1    0 0.00 0.00    0    0
#> [2,]  0.1 20.0  0.1    0 0.00 0.00    0    0
#> [3,]  0.1  0.1 30.0    0 0.00 0.00    0    0
#> [4,]  0.0  0.0  0.0    6 0.00 0.00    0    0
#> [5,]  0.0  0.0  0.0    0 0.10 0.01    0    0
#> [6,]  0.0  0.0  0.0    0 0.01 1.00    0    0
#> [7,]  0.0  0.0  0.0    0 0.00 0.00    1    0
#> [8,]  0.0  0.0  0.0    0 0.00 0.00    0    1

mb <- microbenchmark(matf(testList),lotriMat(testList))

print(mb)
#> Unit: nanoseconds
#>                expr    min       lq      mean   median       uq     max neval
#>      matf(testList) 448951 454581.5 535816.43 463802.5 528361.0 2228159   100
#>  lotriMat(testList)    989   1281.5   2690.28   2396.0   2731.5   26770   100

autoplot(mb)
#> Coordinate system already present. Adding new coordinate system, which will replace the existing one.

plot of chunk unnamed-chunk-9

New features

A new feature is the ability to condition on variables by |. This will be useful when simulating nested random effects using the upcoming RxODE