2-24-Improving_performance.Rmd

```{r, include=FALSE}
source("common.R")
```

# Improving performance

## Checking for existing solutions

1. __[Q]{.Q}__: What are faster alternatives to `lm`? Which are specifically designed to work with larger datasets?

   __[A]{.solved}__: In the [Cran Task View](https://cran.rstudio.com/web/views/) for [High Performance Computing](https://cran.rstudio.com/web/views/HighPerformanceComputing.html) we can find for example the `speedglm` package and its `speedlm()` function. For small datasets, there may be only minor performance gains:

    ```{r}
    stopifnot(all.equal(
      coef(speedglm::speedlm(Sepal.Length ~ Sepal.Width + Species, data = iris)),
      coef(lm(Sepal.Length ~ Sepal.Width + Species, data = iris))))

    microbenchmark::microbenchmark(
      speedglm::speedlm(Sepal.Length ~ Sepal.Width + Species, data = iris),
      lm(Sepal.Length ~ Sepal.Width + Species, data = iris)
    )
    ```
    
   However on bigger datasets it can make a difference:
    
    ```{r,collapse = TRUE}
    eps <- rnorm(100000)
    x1 <- rnorm(100000, 5, 3)
    x2 <- rep(c("a", "b"), 50000)
    y <- 7 * x1 + (x2 == "a") + eps
    td <- data.frame(y = y, x1 = x1, x2 = x2, eps = eps)
    
    stopifnot(all.equal(
      coef(speedglm::speedlm(y ~ x1 + x2, data = td)),
      coef(lm(y ~ x1 + x2, data = td))))
    
    microbenchmark::microbenchmark(
      speedglm::speedlm(y ~ x1 + x2, data = td),
      lm(y ~ x1 + x2, data = td)
    )
    ```
    
   For further speed improvements, you might consider switching your linear algebra libraries as stated in `?speedglm::speedlm`
    
   > The functions of class 'speedlm' may speed up the fitting of LMs to large data sets. High performances can be obtained especially if R is linked against an optimized BLAS, such as ATLAS.
    
   Note that there are many other opportunities mentioned in the task view, also some that make it possible to handle data which is not in memory. 
    
   When it comes to pure speed a quick search on [r fastest lm](http://stackoverflow.com/questions/25416413/is-there-a-faster-lm-function) provides a stackoverflow thread where someone already solved this problem for us...

2. __[Q]{.Q}__: What package implements a version of `match()` that's faster for repeated lookups? How much faster is it?
    
   __[A]{.solved}__: Again google gives a good recommendation for the search term [r faster match](http://stackoverflow.com/questions/32934933/faster-in-operator):
    
    ```{r}
    set.seed(1)
    table <- 1L:100000L
    x <- sample(table, 10000, replace = TRUE)
    
    stopifnot(all.equal(match(x, table), fastmatch::fmatch(x, table)))
    
    microbenchmark::microbenchmark(
      match(x, table),
      fastmatch::fmatch(x, table)
    )
    ```
    
   On my laptop `fastmatch::fmatch()` is around 25 times as fast as `match()`.

3. __[Q]{.Q}__: List four functions (not just those in base R) that convert a string into a date time object. What are their strengths and weaknesses?
    
   __[A]{.started}__: At least these functions will do the trick: `as.POSIXct()`, `as.POSIXlt()`, `strftime()`, `strptime()`, `lubridate::ymd_hms()`. There might also be some in the timeseries packages `xts` or `zoo` and in `anytime`. An update on this will follow.

<!-- add fasttime -->

4. __[Q]{.Q}__: Which packages provide the ability to compute a rolling mean?
    
   __[A]{.solved}__: Again google [r rolling mean](http://stackoverflow.com/questions/743812/calculating-moving-average-in-r)
provides us with enough information and guides our attention on solutions in the following packages:
    * `zoo`
    ```{r}
    zoo::rollmean(1:10, 2, na.pad = TRUE, align = "left")
    zoo::rollapply(1:10, 2, mean, fill = NA, align = "left")
    ```
    * `TTR`
    ```{r}
    TTR::SMA(1:10, 2)
    ```
    * `RcppRoll`
    ```{r}
    RcppRoll::roll_mean(1:10, n = 2, fill = NA, align = "left")
    ```
    * `caTools`
    ```{r}
    caTools::runmean(1:10, k = 2, endrule = "NA", align = "left")
    ```

   Note that an exhaustive example on how to create a rolling mean function is provided in the [textbook](http://adv-r.had.co.nz/Functionals.html).

5. __[Q]{.Q}__: What are the alternatives to `optim()`?
    
   __[A]{.solved}__: Depending on the use case a lot of different options might be considered. For a general overview we would suggest the corresponding taskview on [Optimization](https://cran.r-project.org/web/views/Optimization.html).

## Doing as little as possible

1. __[Q]{.Q}__: What's the difference between `rowSums()` and `.rowSums()`?
    
   __[A]{.solved}__: `.rowSums()` is defined as
    
    ```{r}
    .rowSums
    ```
    
   this means, that the internal `.rowSums()` function is called via `.Internal()`.
    
   > .Internal performs a call to an internal code which is built in to the R
    interpreter.
    
   The internal `.rowSums()` is a complete different function than the "normal" `rowSums()` function.
    
   Of course (since they have the same name) in this case these functions are heavily related with each other: If we look into the source code of `rowSums()`, we see that it is a wrapper around the internal `.rowSums()`. Just some input checkings, conversions and the special cases (complex numbers) are added:
    
    ```{r}
    rowSums
    ```

2. __[Q]{.Q}__: Make a faster version of `chisq.test()` that only computes the chi-square test statistic when the input is two numeric vectors with no missing values. You can try simplifying `chisq.test()` or by coding from the [mathematical definition](http://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test).
    
   __[A]{.solved}__: Since `chisq.test()` has a relatively long source code, we try a new implementation from scratch:
    
    ```{r}
    chisq.test2 <- function(x, y){
      
      # Input
      if (!is.numeric(x)) {
        stop("x must be numeric")}
      if (!is.numeric(y)) {
        stop("y must be numeric")}
      if (length(x) != length(y)) {
        stop("x and y must have the same length")}
      if (length(x) <= 1) {
        stop("length of x must be greater one")}
      if (any(c(x, y) < 0)) {
        stop("all entries of x and y must be greater or equal zero")}
      if (sum(complete.cases(x, y)) != length(x)) {
        stop("there must be no missing values in x and y")}
      if (any(is.null(c(x, y)))) {
        stop("entries of x and y must not be NULL")}
      
      # Help variables
      m <- rbind(x, y)
      margin1 <- rowSums(m)
      margin2 <- colSums(m)
      n <- sum(m)
      me <- tcrossprod(margin1, margin2) / n
      
      # Output
      x_stat = sum((m - me)^2 / me)
      dof <- (length(margin1) - 1) * (length(margin2) - 1)
      p <- pchisq(x_stat, df = dof, lower.tail = FALSE)
      
      return(list(x_stat = x_stat, df = dof, `p-value` = p))
    }
    ```
    
   We check if our new implementation returns the same results
    
    ```{r}
    a <- 21:25
    b <- c(21, 23, 25, 27, 29)
    m_test <- cbind(a, b)
    
    chisq.test(m_test)
    chisq.test2(a, b)
    ```
    
   Finally we benchmark this implementation against a compiled version of itself and the original `stats::chisq.test()`:
    
    ```{r}
    chisq.test2c <- compiler::cmpfun(chisq.test2)
    
    microbenchmark::microbenchmark(
      chisq.test(m_test),
      chisq.test2(a, b),
      chisq.test2c(a, b)
    )
    ```

3. __[Q]{.Q}__: Can you make a faster version of `table()` for the case of an input of two integer vectors with no missing values? Can you use it to speed up your chi-square test?
    
   __[A]{.solved}__: When analysing the source code of `table`, we can omit everything unnecessary and extract the main building blocks:
   
    ```{r}
    table2 <- function(a, b){
      u_a <- unique(a)
      u_b <- unique(b)
        
      dims <- c(length(u_a), length(u_b))
      pd <- prod(dims)
      dn <- list(a = as.character(sort(u_a)), 
             b = as.character(sort(u_b)))
      
      bin <- (as.integer(as.factor(a))-1) + length(u_a) * (as.integer(as.factor(b)) - 1) + 1
      
      y <- array(tabulate(bin, pd), dim = dims, dimnames = dn)
      class(y) <- "table"
      
      y
    }
    
    a <- sample(100, 10000, TRUE)
    b <- sample(100, 10000, TRUE)
    
    bench::mark(table(a, b),
                table2(a, b),
                relative = TRUE)
    ```
   
   Unfortunately, the performance gain is very small.

## Vectorise

1. __[Q]{.Q}__: The density functions, e.g., `dnorm()`, have a common interface. Which arguments are vectorised over? What does `rnorm(10, mean = 10:1)` do?
    
   __[A]{.solved}__: We can see the interface of these functions via `?dnorm`:
    
    ```{r, eval = FALSE}
    dnorm(x, mean = 0, sd = 1, log = FALSE)
    pnorm(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
    qnorm(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
    rnorm(n, mean = 0, sd = 1).
    ```
    
   They are vectorised over their numeric arguments, which is always the first argument (`x`, `q`, `p`, `n`), `mean` and `sd`. Note that it's dangerous to supply a vector to `n` in the `rnorm()` function, since the behaviour will change, when `n` has length 1 (like in the second part of this question).
    
   `rnorm(10, mean = 10:1)` generates ten random numbers from different normal distributions. The normal distributions differ in their means. The first has mean 10, the second has mean 9, the third mean 8 etc.

2. __[Q]{.Q}__: Compare the speed of `apply(x, 1, sum)` with `rowSums(x)` for varying sizes of `x`.
    
   __[A]{.solved}__: We compare regarding different sizes for square matrices:
    
    ```{r long calculation, eval=FALSE}
    library(microbenchmark)
    dimensions <- c(1e0, 1e1, 1e2, 1e3, 0.5e4, 1e4)
    matrices <- lapply(dimensions,
                       function(x) tcrossprod(rnorm(x), rnorm(x)))
    
    bench_rs <- lapply(matrices,
                       function(x) fivenum(microbenchmark(rowSums(x),
                                                          unit = "ns")$time))
    
    bench_rs <- data.frame(time = unlist(bench_rs), 
                           call = "rowSums", stringsAsFactors = FALSE)
    
    bench_apply <- lapply(matrices,
                          function(x) fivenum(microbenchmark(apply(x, 1, sum),
                                                             unit = "ns")$time))
    bench_apply <- data.frame(time = unlist(bench_apply),
                              call = "apply", stringsAsFactors = FALSE)
    
    df <- rbind(bench_rs, bench_apply)
    
    df$dimension <- rep(dimensions, each = 5)
    df$aggr <- rep(c("min", "lq", "median", "uq", "max"),
                   times = length(dimensions))
    df$aggr_size <- rep(c(1, 2, 3, 2, 1), times = length(dimensions))
    df$group <- paste(as.character(df$call), as.character(df$aggr), sep = " ")
    
    library(ggplot2)
    
    ggplot(df, aes(x = dimension, y = time, colour = call, group = group)) +
      geom_point() + 
      geom_line(aes(linetype = factor(aggr_size, levels = c("3", "2", "1"))),
            show.legend = FALSE) 
    ```
    
   The graph is a good indicator to notice, that `apply()` is not "vectorised for performance".
  
3. __[Q]{.Q}__: How can you use `crossprod()` to compute a weighted sum? How much faster is it than the naive `sum(x * w)`?
    
   __[A]{.solved}__: We can just give the vectors to `crossprod()` which converts them to row- and column-vectors and then multiplies these. The result is the dot product which is also a weighted sum.
    
    ```{r}
    a <- rnorm(10)
    b <- rnorm(10)
    sum(a * b) - crossprod(a, b)[1]
    ```
    
   A benchmark of both alternatives for different vector lengths indicates, that the `crossprod()` variant is about 2.5 times faster than `sum()`:
    
    ```{r, eval=FALSE}
    dimensions <- c(1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 0.5e7, 1e7)
    xvector <- lapply(dimensions, rnorm)
    weights <- lapply(dimensions, rnorm)
    
    bench_sum <- Map(function(x, y) fivenum(microbenchmark(sum(x * y))$time),
                     xvector, weights)
    bench_sum <- data.frame(time = unlist(bench_sum),
                            call = "sum",
                            stringsAsFactors = FALSE)
    bench_cp <- Map(function(x, y) fivenum(microbenchmark(crossprod(x, y)[1])$time),
                    xvector, weights)
    bench_cp <- data.frame(time = unlist(bench_cp),
                           call = "crossproduct",
                           stringsAsFactors = FALSE)
    
    df <- rbind(bench_sum, bench_cp)
    
    df$dimension <- rep(dimensions, each = 5)
    df$aggr <- rep(c("min", "lq", "median", "uq", "max"), times = length(dimensions))
    df$aggr_size <- rep(c(1, 2, 3, 2, 1), times = length(dimensions))
    df$group <- paste(as.character(df$call), as.character(df$aggr), sep = " ")
    
    ggplot(df, aes(x = dimension, y = time, colour = call, group = group)) +
      geom_point() + 
      geom_line(aes(linetype = factor(aggr_size, levels = c("3", "2", "1"))),
                show.legend = FALSE) +
      scale_y_continuous(expand = c(0, 0))
    ```
    
## Old Exercises

1. __[Q]{.Q}__: How many different ways can you compute a 1d density estimate in R?
    
    __[A]{.solved}__: According to [Deng and Wickham (2011)](https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=0ahUKEwim0sTI9JLRAhVScFAKHdLuBOIQFggxMAI&url=http%3A%2F%2Fvita.had.co.nz%2Fpapers%2Fdensity-estimation.pdf&usg=AFQjCNFBdOT4DvSTXtGoawIFkgf6JlXV9Q&sig2=APnTeIL55Zj_0oATw15mBA&cad=rja) density estimation is implemented in over 20 R packages.


<!-- ## Memory profiling with lineprof -->

1. __[Q]{.Q}__: When the input is a list, we can make a more efficient `as.data.frame()` by using special knowledge. A data frame is a list with class `data.frame` and `row.names` attribute. `row.names` is either a character vector or vector of sequential integers, stored in a special format created by `.set_row_names()`. This leads to an alternative `as.data.frame()`:

    ```{r}
    to_df <- function(x) {
      class(x) <- "data.frame"
      attr(x, "row.names") <- .set_row_names(length(x[[1]]))
      x
    }
    ```

   What impact does this function have on `read_delim()`?  What are the downsides of this function?

1. __[Q]{.Q}__: Line profile the following function with `torture = TRUE`. What is surprising? Read the source code of `rm()` to figure out what's going on.

    ```{r}
    f <- function(n = 1e5) {
      x <- rep(1, n)
      rm(x)
    }
    ```

<!-- ## Do as little as possible -->

1. __[Q]{.Q}__: How do the results change if you compare `mean()` and `mean.default()` on 10,000 observations, rather than on 100?
    
   __[A]{.solved}__: We start with 100 observations as shown in the textbook:
    
    ```{r}
    x <- runif(1e2)
    microbenchmark::microbenchmark(
      mean(x),
      mean.default(x)
    )
    ```
    
   In case of 10,000 observations we can observe that using `mean.default()` preserves only a small advantage over the use of `mean()`:
    
    ```{r,collapse = TRUE}
    x <- runif(1e4)
    microbenchmark::microbenchmark(
      mean(x),
      mean.default(x),
      unit = "ns"
    )
    ```
    
   When using even more observations - like in the next lines - it seems that `mean.default()` doesn't preserve anymore any advantage at all:
    
    ```{r}
    x <- runif(1e6)
    microbenchmark::microbenchmark(
      mean(x),
      mean.default(x),
      unit = "ns"
    )
    ```

2. __[Q]{.Q}__: The following code provides an alternative implementation of `rowSums()`. Why is it faster for this input?

    ```{r}
    rowSums2 <- function(df) {
      out <- df[[1L]]
      if (ncol(df) == 1) return(out)
      
      for (i in 2:ncol(df)) {
        out <- out + df[[i]]
      }
      out
    }
    
    df <- as.data.frame(
      replicate(1e3, sample(100, 1e4, replace = TRUE))
    )
    system.time(rowSums(df))
    system.time(rowSums2(df))
    ```
    
   __[A]{.solved}__:
   
3. __[Q]{.Q}__: Imagine you want to compute the bootstrap distribution of a sample correlation using `cor_df()` and the data in the example below. Given that you want to run this many times, how can you make this code faster? (Hint: the  function has three components that you can speed up.)

    ```{r, eval = FALSE}
    n <- 1e6
    df <- data.frame(a = rnorm(n), b = rnorm(n))

    cor_df <- function(df, n) {
      i <- sample(seq(n), n, replace = FALSE)
      cor(df[i, , drop = FALSE])[2, 1]
      # note also that in the last line the textbook says q[] instead of df[]. Since
      # this is probably just a typo, we changed this to df[].
    }
    ```

   Is there a way to vectorise this procedure?
    
   __[A]{.solved}__: The three components (mentioned in the questions hint) are:

   1. sampling of indices
   2. subsetting the data frame/conversion to matrix (or vector input)
   3. the `cor()` function itself.
    
   Since a run of lineprof like shown in the textbook suggests that `as.matrix()` within the `cor()` function is the biggest bottleneck, we start with that:
    
    ```{r}
    n <- 1e6
    df <- data.frame(a = rnorm(n), b = rnorm(n))
    ```

   Remember the outgoing function:
    
    ```{r}
    cor_df <- function() {
      i <- sample(seq(n), n, replace = FALSE)
      cor(df[i, , drop = FALSE])[2, 1]
    }
    ```
   
   First we want to optimise the second line (without attention to the `cor()` function itself). Therefore we exclude the first line from our optimisation approaches and define `i` within the global environment:
    
    ```{r}
    i <- sample(seq(n), n)
    ```
    
   Then we define our approaches, check that their behaviour is correct and do the first benchmark:
    
    ```{r}
    # old version
    cor_v1 <- function() {
      cor(df[i, , drop = FALSE])[2, 1]
    }
    
    # cbind instead of internal as.matrix
    cor_v2 <- function() {
      m <- cbind(df$a[i], df$b[i])
      cor(m)[2, 1]
    }

    # cbind + vector subsetting of the output matrix
    cor_v3 <- function() {
      m <- cbind(df$a[i], df$b[i])
      cor(m)[2]
    }

    # use vector input within the cor function, so that no conversion is needed
    cor_v4 <- function() {
      cor(df$a[i], df$b[i])
    }
    
    # check if all return the same result (if you don't get the same result on your
    # machine, you might wanna check if it is due to the precision and istead
    # run cor_v1(), cor_v2() etc.)
    cor_list <- list(cor_v1, cor_v2, cor_v3, cor_v4)
    ulapply <- function(X, FUN, ...) unlist(lapply(X, FUN, ...))
    ulapply(cor_list, function(x) identical(x(), cor_v1()))
    
    # benchmark
    set.seed(1)
    microbenchmark::microbenchmark(
      cor_v1(),
      cor_v2(),
      cor_v3(),
      cor_v4()
    )
    ```
    
   According to the resulting medians, lower and upper quartiles of our benchmark all three new versions seem to provide more or less the same speed benefit (note that the maximum and mean can vary a lot for these approaches). Since the second version is most similar to the code we started, we implement this line into a second version of `cor_df()` (if this sounds too arbitrary, note that in the final solution we will come back to the vector input version anyway) and do a benchmark to get the overall speedup:
    
    ```{r}
    cor_df2 <- function() {
      i <- sample(seq(n), n)
      m <- cbind(df$a[i], df$b[i])
      cor(m)[2, 1]
    }
    
    microbenchmark::microbenchmark(
      cor_df(),
      cor_df2()
    )
    ```
    
   Now we can focus on a speedup for the random generation of indices. (Note that a run of linepfrof suggests to optimize `cbind()`. However, after rewriting `cor()` to a version that only works with vector input, this step will be unnecessary anyway). We could try differnt approaches for the sequence generation within `sample()` (like `seq(n)`, `seq.int(n)`, `seq_len(n)`, `a:n`) and a direct call of `sample.int()`. In the following, we will see, that `sample.int()` is always faster (since we don't include the generation of the sequence into our benchmark). When we look into `sample.int()` we see that it calls two different internal sample versions depending on the input. Since in our usecase always the second version will be called, we also provide this version in our benchmark: 
    
    ```{r}
    seq_n <- seq(n)
    
    microbenchmark::microbenchmark(
      sample(seq_n, n),
      sample.int(n, n),
      .Internal(sample(n, n, replace = FALSE, prob = NULL))
    )
    ```
    
   The `sample.int()` versions give clearly the biggest improvement. Since the internal version doesn't provide any clear improvement, but restricts the general scope of our function, we choose to implement `sample.int()` in a third version of `cor_df()` and benchmark our actual achievements:
    
    ```{r}
    cor_df3 <- function() {
      i <- sample.int(n, n)
      m <- cbind(df$a[i], df$b[i])
      cor(m)[2, 1]
    }

    microbenchmark::microbenchmark(
      cor_df(),
      cor_df2(),
      cor_df3()
    )
    ```
    
   As a last step, we try to speedup the calculation of the pearson correlation coefficient. Since quite a lot of functionality is build into the `stats::cor()` function this seems like a reasonable approach. We try this by working with another `cor()` function from the `WGCNA` package and an own implementation which should give a small improvement, because we use `sum(x) / length(x)` instead of `mean(x)` for internal calculations:
    
    ```{r}
    #WGCNA version (matrix and vector). Note that I don't use a local setup which uses
    #the full potential of this function. For furter information see ?WGCNA::cor
    cor_df4m <- function() {
      i <- sample.int(n, n)
      m <- cbind(df$a[i], df$b[i])
      WGCNA::cor(m)[2]
    }
    
    cor_df4v <- function() {
      i <- sample.int(n, n)
      WGCNA::cor(df$a[i], df$b[i], quick = 1)[1]
    }
    
    #New implementation of underlying cor function
    #A definition can be found for example here
    #http://www.socscistatistics.com/tests/pearson/
    cor2 <- function(x, y){
      xm <- sum(x) / length(x)
      ym <- sum(y) / length(y)
      x_xm <- x - xm
      y_ym <- y - ym
      numerator <- sum((x_xm) * (y_ym))
      denominator <- sqrt(sum(x_xm^2)) * sqrt(sum(y_ym^2))
      return(numerator / denominator)
    }
    
    cor2 <- compiler::cmpfun(cor2)
    
    cor_df5 <- function() {
      i <- sample.int(n, n)
      cor2(df$a[i], df$b[i])
    }
    ```
    
   In our final benchmark, we also include compiled verions of all our attempts:
    ```{r, eval = FALSE}
    cor_df_c <- compiler::cmpfun(cor_df)
    cor_df2_c <- compiler::cmpfun(cor_df2)
    cor_df3_c <- compiler::cmpfun(cor_df3)
    cor_df4m_c <- compiler::cmpfun(cor_df4m)
    cor_df4v_c <- compiler::cmpfun(cor_df4v)
    cor_df5_c <- compiler::cmpfun(cor_df5)

    microbenchmark::microbenchmark(
      cor_df(),
      cor_df2(),
      cor_df3(),
      cor_df4m(),
      cor_df4v(),
      cor_df5(),
      cor_df_c(),
      cor_df2_c(),
      cor_df3_c(),
      cor_df4m_c(),
      cor_df4v_c(),
      cor_df5_c()
    )
    #> 
    #> Unit: milliseconds
    #>          expr      min       lq      mean    median        uq        max
    #>      cor_df() 7.885838 9.146994 12.739322  9.844561 10.447188   83.08044
    #>     cor_df2() 1.951941 3.178821  6.101036  3.339742  3.485085   36.25404
    #>     cor_df3() 1.288831 1.943510  4.353773  2.018105  2.145119   66.74022
    #>    cor_df4m() 1.534061 2.156848  6.344015  2.243540  2.370921  234.17307
    #>    cor_df4v() 1.639997 2.271949  5.755720  2.349477  2.453214  196.49897
    #>     cor_df5() 1.281133 1.879911  6.263500  1.932696  2.064292  237.83135
    #>    cor_df_c() 7.600654 9.337239 13.613901 10.009330 10.965688   41.17146
    #>   cor_df2_c() 2.009124 2.878242  5.645224  3.298138  3.469871   34.29037
    #>   cor_df3_c() 1.312291 1.926281  4.426418  1.986948  2.094532   87.97220
    #>  cor_df4m_c() 1.548357 2.179025  3.306796  2.242991  2.355709   23.87195
    #>  cor_df4v_c() 1.669689 2.255087 27.148456  2.341962  2.490419 2263.33230
    #>   cor_df5_c() 1.284065 1.888892  3.884798  1.953774  2.078955   25.81583
    #>  neval cld
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    #>    100   a
    ```
    
   Our final solution benefits most from the switch from data frames to vectors. Working with `sample.int` gives only little improvement. Reimplementing and compiling a new correlation function adds only minimal speedup.
    
   To trust our final result we include a last check for similar return values:
    
    ```{r, eval = FALSE}
    set.seed(1)
    cor_df()
    #> [1] -0.001441277
    set.seed(1)
    cor_df5_c()
    #> [1] -0.001441277
    ```
    
   Vectorisation of this problem seems rather difficult, since attempts of using matrix calculus, always depend on building and handling big matrices in the first place.
    
   We can for example rewrite our correlation function to work with matrices and build a new (vectorised) version of `cor_df()` on top of that
    
    ```{r}
    cor2m <- function(x, y){
      n_row <- nrow(x)
      xm <- colSums(x) / n_row
      ym <- colSums(y) / n_row
      x_xm <- t(t(x) - xm)
      y_ym <- t(t(y) - ym)
      numerator <- colSums((x_xm) * (y_ym))
      denominator <- sqrt(colSums(x_xm^2)) * sqrt(colSums(y_ym^2))
      return(numerator / denominator)
    }
    
    cor_df_v <- function(i){
      indices <- replicate(i, sample.int(n, n), simplify = "array")
      x <- matrix(df$a[indices], ncol = i)
      y <- matrix(df$b[indices], ncol = i)
      cor2m(x, y)
    }
    cor_df_v <- compiler::cmpfun(cor_df_v)
    ```
    
   However this still doesn't provide any improvement over the use of `lapply()`:
    
    ```{r, eval = FALSE}
    ulapply2 <- function(X, FUN, ...) unlist(lapply(X, FUN, ...), use.names = FALSE)
    
    microbenchmark::microbenchmark(
      cor_df5_c(),
      ulapply2(1:100, function(x) cor_df5_c()),
      cor_df_v(100)
    )
    #> Unit: milliseconds
    #>                                      expr        min         lq      mean
    #>                               cor_df5_c()   1.098219   1.148988   2.06762
    #>  ulapply2(1:100, function(x) cor_df5_c()) 228.250897 233.052303 256.56849
    #>                             cor_df_v(100) 263.158918 272.988449 312.43802
    #>      median         uq      max neval cld
    #>    1.864515   2.030201  13.2325   100 a  
    #>  238.665277 247.184900 410.0768   100  b 
    #>  278.648158 291.164035 581.2834   100   c
    ```
    
   Further improvements can be achieved using parallelisation (for example via `parallel::parLapply()`)