---
title: "Lotri Motivation"
author: "Matthew Fidler"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Lotri Motivation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```
# Motivation 
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. 

For me, specifying the matrices in this way is easier than
specifying them using R's default matrix.  For instance to fully
specify a simple 2x2 matrix, in R you specify:

```{r}
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:

```{r}
library(lotri)
library(microbenchmark)
library(ggplot2)

mat <- lotri(a+b ~ c(1,
                     0.5, 1))
```

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

```{r}
mat <- lotri({
    a+b ~ c(1,
            0.5, 1)
    c ~ 1
    d +e ~ c(1,
             0.5, 1)
})
```

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

```{r}
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")))
```

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

```{r}
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"))
```

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

# Creating lists of matrices with attached properties

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

For example:


```{r}
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)

print(mat$lower)
print(mat$upper)
print(mat$omegaIsChol)
```

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`

```{r}
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))

print(lotriMat(testList))


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

print(mb)
autoplot(mb)
```
