# Learning R: A gentle introduction to higher-order functions

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Have you ever thought about why the definition of a function in R is different from many other programming languages? The part that causes the biggest difficulties (especially for beginners of R) is that you state the name of the function at the beginning and use the assignment operator – as if functions were like any other data type, like vectors, matrices or data frames…

Congratulations! You just encountered one of the big ideas of functional programming: functions *are* indeed like any other data type, they are not special – or in programming lingo, functions are first-class members. Now, you might ask: So what? Well, there are many ramifications, for example that you could use functions on other functions by using one function as an argument for another function. Sounds complicated?

In mathematics most of you will be familiar with taking the derivative of a function. When you think about it you could say that you put one function into the derivative function (or operator) and get out another function!

In R there are many applications as well, let us go through a simple example step by step.

Let’s say I want to apply the mean function on the first four columns of the `iris`

dataset. I could do the following:

mean(iris[ , 1]) ## [1] 5.843333 mean(iris[ , 2]) ## [1] 3.057333 mean(iris[ , 3]) ## [1] 3.758 mean(iris[ , 4]) ## [1] 1.199333

Quite tedious and not very elegant. Of course, we can use a for loop for that:

for (x in iris[1:4]) { print(mean(x)) } ## [1] 5.843333 ## [1] 3.057333 ## [1] 3.758 ## [1] 1.199333

This works fine but there is an even more intuitive approach. Just look at the original task: “apply the mean function on the first four columns of the iris dataset” – so let us do just that:

apply(iris[1:4], 2, mean) ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## 5.843333 3.057333 3.758000 1.199333

Wow, this is very concise and works perfectly (the `2`

just stands for “go through the data column wise”, `1`

would be for “row wise”). `apply`

is called a “higher-order function” and we could use it with all kinds of other functions:

apply(iris[1:4], 2, sd) ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## 0.8280661 0.4358663 1.7652982 0.7622377 apply(iris[1:4], 2, min) ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## 4.3 2.0 1.0 0.1 apply(iris[1:4], 2, max) ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## 7.9 4.4 6.9 2.5

You can also use user-defined functions:

midrange <- function(x) (min(x) + max(x)) / 2 apply(iris[1:4], 2, midrange) ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## 6.10 3.20 3.95 1.30

We can even use new functions that are defined “on the fly” (or in functional programming lingo “anonymous functions”):

apply(iris[1:4], 2, function(x) (min(x) + max(x)) / 2) ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## 6.10 3.20 3.95 1.30

Let us now switch to another inbuilt data set, the `mtcars`

dataset with 11 different variables of 32 cars (if you want to find out more, please consult the documentation):

head(mtcars) ## mpg cyl disp hp drat wt qsec vs am gear carb ## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4 ## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4 ## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1 ## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1 ## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2 ## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1

To see the power of higher-order functions let us create a (numeric) matrix with minimum, first quartile, median, mean, third quartile and maximum for all 11 columns of the `mtcars`

dataset with just one command!

apply(mtcars, 2, summary) ## mpg cyl disp hp drat wt qsec vs am gear carb ## Min. 10.40000 4.0000 71.1000 52.0000 2.760000 1.51300 14.50000 0.0000 0.00000 3.0000 1.0000 ## 1st Qu. 15.42500 4.0000 120.8250 96.5000 3.080000 2.58125 16.89250 0.0000 0.00000 3.0000 2.0000 ## Median 19.20000 6.0000 196.3000 123.0000 3.695000 3.32500 17.71000 0.0000 0.00000 4.0000 2.0000 ## Mean 20.09062 6.1875 230.7219 146.6875 3.596563 3.21725 17.84875 0.4375 0.40625 3.6875 2.8125 ## 3rd Qu. 22.80000 8.0000 326.0000 180.0000 3.920000 3.61000 18.90000 1.0000 1.00000 4.0000 4.0000 ## Max. 33.90000 8.0000 472.0000 335.0000 4.930000 5.42400 22.90000 1.0000 1.00000 5.0000 8.0000

Wow, that was easy and the result is quite impressive, is it not!

Or if you want to perform a linear regression for all ten variables separately against mpg and want to get a table with all coefficients – there you go:

sapply(mtcars, function(x) round(coef(lm(mpg ~ x, data = mtcars)), 3)) ## mpg cyl disp hp drat wt qsec vs am gear carb ## (Intercept) 0 37.885 29.600 30.099 -7.525 37.285 -5.114 16.617 17.147 5.623 25.872 ## x 1 -2.876 -0.041 -0.068 7.678 -5.344 1.412 7.940 7.245 3.923 -2.056

Here we used another higher-order function, `sapply`

, together with an anonymous function. `sapply`

goes through all the columns of a data frame (i.e. elements of a list) and tries to simplify the result (here your get back a nice matrix).

Often, you might not even have realised when you were using higher-order functions! I can tell you that it is quite a hassle in many programming languages to program a simple function plotter, i.e. a function which plots another function. In R it has already been done for you: you just use the higher-order function curve and give it the function you want to plot as an argument:

curve(sin(x) + cos(1/2 * x), -10, 10)

I want to give you one last example of another very helpful higher-order function (which not too many people know or use): `by`

. It comes in very handy when you want to apply a function on different attributes split by a factor. So let’s say you want to get a summary of all the attributes of `iris`

split by (!) species – here it comes:

by(iris[1:4], iris$Species, summary) ## iris$Species: setosa ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## Min. :4.300 Min. :2.300 Min. :1.000 Min. :0.100 ## 1st Qu.:4.800 1st Qu.:3.200 1st Qu.:1.400 1st Qu.:0.200 ## Median :5.000 Median :3.400 Median :1.500 Median :0.200 ## Mean :5.006 Mean :3.428 Mean :1.462 Mean :0.246 ## 3rd Qu.:5.200 3rd Qu.:3.675 3rd Qu.:1.575 3rd Qu.:0.300 ## Max. :5.800 Max. :4.400 Max. :1.900 Max. :0.600 ## -------------------------------------------------------------- ## iris$Species: versicolor ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## Min. :4.900 Min. :2.000 Min. :3.00 Min. :1.000 ## 1st Qu.:5.600 1st Qu.:2.525 1st Qu.:4.00 1st Qu.:1.200 ## Median :5.900 Median :2.800 Median :4.35 Median :1.300 ## Mean :5.936 Mean :2.770 Mean :4.26 Mean :1.326 ## 3rd Qu.:6.300 3rd Qu.:3.000 3rd Qu.:4.60 3rd Qu.:1.500 ## Max. :7.000 Max. :3.400 Max. :5.10 Max. :1.800 ## -------------------------------------------------------------- ## iris$Species: virginica ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## Min. :4.900 Min. :2.200 Min. :4.500 Min. :1.400 ## 1st Qu.:6.225 1st Qu.:2.800 1st Qu.:5.100 1st Qu.:1.800 ## Median :6.500 Median :3.000 Median :5.550 Median :2.000 ## Mean :6.588 Mean :2.974 Mean :5.552 Mean :2.026 ## 3rd Qu.:6.900 3rd Qu.:3.175 3rd Qu.:5.875 3rd Qu.:2.300 ## Max. :7.900 Max. :3.800 Max. :6.900 Max. :2.500

This was just a very shy look at this huge topic. There are very powerful higher-order functions in R, like `lapppy`

, `aggregate`

, `replicate`

(very handy for numerical simulations) and many more. A good overview can be found in the answers of this question: stackoverflow (my answer there is on the rather illusive switch function: switch).

For some reason people tend to confuse higher-order functions with recursive functions but that is the topic of another post, so stay tuned…

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