Grouping functions (tapply, by, aggregate) and the *apply family

Whenever I want to do something "map"py in R, I usually try to use a function in the apply family.

However, I've never quite understood the differences between them -- how {sapply, lapply, etc.} apply the function to the input/grouped input, what the output will look like, or even what the input can be -- so I often just go through them all until I get what I want.

Can someone explain how to use which one when?

My current (probably incorrect/incomplete) understanding is...

sapply(vec, f): input is a vector. output is a vector/matrix, where element i is f(vec[i]), giving you a matrix if f has a multi-element output
lapply(vec, f): same as sapply, but output is a list?
apply(matrix, 1/2, f): input is a matrix. output is a vector, where element i is f(row/col i of the matrix)
tapply(vector, grouping, f): output is a matrix/array, where an element in the matrix/array is the value of f at a grouping g of the vector, and g gets pushed to the row/col names
by(dataframe, grouping, f): let g be a grouping. apply f to each column of the group/dataframe. pretty print the grouping and the value of f at each column.
aggregate(matrix, grouping, f): similar to by, but instead of pretty printing the output, aggregate sticks everything into a dataframe.

Side question: I still haven't learned plyr or reshape -- would plyr or reshape replace all of these entirely?

R has many *apply functions which are ably described in the help files (e.g. ?apply). There are enough of them, though, that beginning useRs may have difficulty deciding which one is appropriate for their situation or even remembering them all. They may have a general sense that "I should be using an *apply function here", but it can be tough to keep them all straight at first.

Despite the fact (noted in other answers) that much of the functionality of the *apply family is covered by the extremely popular plyr package, the base functions remain useful and worth knowing.

This answer is intended to act as a sort of signpost for new useRs to help direct them to the correct *apply function for their particular problem. Note, this is not intended to simply regurgitate or replace the R documentation! The hope is that this answer helps you to decide which *apply function suits your situation and then it is up to you to research it further. With one exception, performance differences will not be addressed.

apply - When you want to apply a function to the rows or columns of a matrix (and higher-dimensional analogues); not generally advisable for data frames as it will coerce to a matrix first.

 # Two dimensional matrix
 M <- matrix(seq(1,16), 4, 4)

 # apply min to rows
 apply(M, 1, min)
 [1] 1 2 3 4

 # apply max to columns
 apply(M, 2, max)
 [1]  4  8 12 16

 # 3 dimensional array
 M <- array( seq(32), dim = c(4,4,2))

 # Apply sum across each M[*, , ] - i.e Sum across 2nd and 3rd dimension
 apply(M, 1, sum)
 # Result is one-dimensional
 [1] 120 128 136 144

 # Apply sum across each M[*, *, ] - i.e Sum across 3rd dimension
 apply(M, c(1,2), sum)
 # Result is two-dimensional
      [,1] [,2] [,3] [,4]
 [1,]   18   26   34   42
 [2,]   20   28   36   44
 [3,]   22   30   38   46
 [4,]   24   32   40   48

If you want row/column means or sums for a 2D matrix, be sure to investigate the highly optimized, lightning-quick colMeans, rowMeans, colSums, rowSums.

lapply - When you want to apply a function to each element of a list in turn and get a list back.

This is the workhorse of many of the other *apply functions. Peel back their code and you will often find lapply underneath.
```
 x <- list(a = 1, b = 1:3, c = 10:100) 
 lapply(x, FUN = length) 
 $a 
 [1] 1
 $b 
 [1] 3
 $c 
 [1] 91
 lapply(x, FUN = sum) 
 $a 
 [1] 1
 $b 
 [1] 6
 $c 
 [1] 5005
```
sapply - When you want to apply a function to each element of a list in turn, but you want a vector back, rather than a list.

If you find yourself typing unlist(lapply(...)), stop and consider sapply.
```
 x <- list(a = 1, b = 1:3, c = 10:100)
 # Compare with above; a named vector, not a list 
 sapply(x, FUN = length)  
 a  b  c   
 1  3 91

 sapply(x, FUN = sum)   
 a    b    c    
 1    6 5005 
```
In more advanced uses of sapply it will attempt to coerce the result to a multi-dimensional array, if appropriate. For example, if our function returns vectors of the same length, sapply will use them as columns of a matrix:
```
 sapply(1:5,function(x) rnorm(3,x))
```
If our function returns a 2 dimensional matrix, sapply will do essentially the same thing, treating each returned matrix as a single long vector:
```
 sapply(1:5,function(x) matrix(x,2,2))
```
Unless we specify simplify = "array", in which case it will use the individual matrices to build a multi-dimensional array:
```
 sapply(1:5,function(x) matrix(x,2,2), simplify = "array")
```
Each of these behaviors is of course contingent on our function returning vectors or matrices of the same length or dimension.

vapply - When you want to use sapply but perhaps need to squeeze some more speed out of your code or want more type safety.

For vapply, you basically give R an example of what sort of thing your function will return, which can save some time coercing returned values to fit in a single atomic vector.

 x <- list(a = 1, b = 1:3, c = 10:100)
 #Note that since the advantage here is mainly speed, this
 # example is only for illustration. We're telling R that
 # everything returned by length() should be an integer of 
 # length 1. 
 vapply(x, FUN = length, FUN.VALUE = 0L) 
 a  b  c  
 1  3 91

mapply - For when you have several data structures (e.g. vectors, lists) and you want to apply a function to the 1st elements of each, and then the 2nd elements of each, etc., coercing the result to a vector/array as in sapply.

This is multivariate in the sense that your function must accept multiple arguments.
```
 #Sums the 1st elements, the 2nd elements, etc. 
 mapply(sum, 1:5, 1:5, 1:5) 
 [1]  3  6  9 12 15
 #To do rep(1,4), rep(2,3), etc.
 mapply(rep, 1:4, 4:1)   
 [[1]]
 [1] 1 1 1 1

 [[2]]
 [1] 2 2 2

 [[3]]
 [1] 3 3

 [[4]]
 [1] 4
```

Map - A wrapper to mapply with SIMPLIFY = FALSE, so it is guaranteed to return a list.

 Map(sum, 1:5, 1:5, 1:5)
 [[1]]
 [1] 3

 [[2]]
 [1] 6

 [[3]]
 [1] 9

 [[4]]
 [1] 12

 [[5]]
 [1] 15

rapply - For when you want to apply a function to each element of a nested list structure, recursively.

To give you some idea of how uncommon rapply is, I forgot about it when first posting this answer! Obviously, I'm sure many people use it, but YMMV. rapply is best illustrated with a user-defined function to apply:

 # Append ! to string, otherwise increment
 myFun <- function(x){
     if(is.character(x)){
       return(paste(x,"!",sep=""))
     }
     else{
       return(x + 1)
     }
 }

 #A nested list structure
 l <- list(a = list(a1 = "Boo", b1 = 2, c1 = "Eeek"), 
           b = 3, c = "Yikes", 
           d = list(a2 = 1, b2 = list(a3 = "Hey", b3 = 5)))


 # Result is named vector, coerced to character          
 rapply(l, myFun)

 # Result is a nested list like l, with values altered
 rapply(l, myFun, how="replace")

tapply - For when you want to apply a function to subsets of a vector and the subsets are defined by some other vector, usually a factor.

The black sheep of the *apply family, of sorts. The help file's use of the phrase "ragged array" can be a bit confusing, but it is actually quite simple.

A vector:
```
 x <- 1:20
```
A factor (of the same length!) defining groups:
```
 y <- factor(rep(letters[1:5], each = 4))
```
Add up the values in x within each subgroup defined by y:
```
 tapply(x, y, sum)  
  a  b  c  d  e  
 10 26 42 58 74 
```
More complex examples can be handled where the subgroups are defined by the unique combinations of a list of several factors. tapply is similar in spirit to the split-apply-combine functions that are common in R (aggregate, by, ave, ddply, etc.) Hence its black sheep status.

On the side note, here is how the various plyr functions correspond to the base *apply functions (from the intro to plyr document from the plyr webpage http://had.co.nz/plyr/)

Base function   Input   Output   plyr function 
---------------------------------------
aggregate        d       d       ddply + colwise 
apply            a       a/l     aaply / alply 
by               d       l       dlply 
lapply           l       l       llply  
mapply           a       a/l     maply / mlply 
replicate        r       a/l     raply / rlply 
sapply           l       a       laply

One of the goals of plyr is to provide consistent naming conventions for each of the functions, encoding the input and output data types in the function name. It also provides consistency in output, in that output from dlply() is easily passable to ldply() to produce useful output, etc.

Conceptually, learning plyr is no more difficult than understanding the base *apply functions.

plyr and reshape functions have replaced almost all of these functions in my every day use. But, also from the Intro to Plyr document:

Related functions tapply and sweep have no corresponding function in plyr, and remain useful. merge is useful for combining summaries with the original data.

Grouping functions (tapply, by, aggregate) and the *apply family

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