Grouping functions tapply by aggregate and the apply family

Question

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

User · Answer

First start with Joran s excellent answer -- doubtful anything can better that   Then the following mnemonics may help to remember the distinctions between each   Whilst some are obvious  others may be less so --- for these you ll find justification in Joran s discussions   Mnemonics   lapply is a list apply which acts on a list or vector and returns a list  sapply is a simple lapply  function defaults to returning a vector or matrix when possible  vapply is a verified apply  allows the return object type to be prespecified  rapply is a recursive apply for nested lists  i e  lists within lists tapply is a tagged apply where the tags identify the subsets apply  is  generic  applies a function to a matrix s rows or columns  or  more generally  to dimensions of an array    Building the Right Background  If using the apply family still feels a bit alien to you  then it might be that you re missing a key point of view   These two articles can help   They provide the necessary background to motivate the functional programming techniques that are being provided by the apply family of functions     Users of Lisp will recognise the paradigm immediately   If you re not familiar with Lisp  once you get your head around FP  you ll have gained a powerful point of view for use in R -- and apply will make a lot more sense    Advanced R  Functional Programming  by Hadley Wickham Simple Functional Programming in R  by Michael Barton

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Since I realized that  the very excellent  answers of this post lack of by and aggregate explanations  Here is my contribution   BY  The by function  as stated in the documentation can be though  as a  wrapper  for tapply  The power of by arises when we want to compute a task that tapply can t handle  One example is this code   ct  lt - tapply iris Sepal Width   iris Species   summary   cb  lt - by iris Sepal Width   iris Species   summary     cb iris Species  setosa    Min  1st Qu   Median    Mean 3rd Qu     Max     2 300   3 200   3 400   3 428   3 675   4 400  --------------------------------------------------------------  iris Species  versicolor    Min  1st Qu   Median    Mean 3rd Qu     Max     2 000   2 525   2 800   2 770   3 000   3 400  --------------------------------------------------------------  iris Species  virginica    Min  1st Qu   Median    Mean 3rd Qu     Max     2 200   2 800   3 000   2 974   3 175   3 800    ct  setosa    Min  1st Qu   Median    Mean 3rd Qu     Max     2 300   3 200   3 400   3 428   3 675   4 400    versicolor    Min  1st Qu   Median    Mean 3rd Qu     Max     2 000   2 525   2 800   2 770   3 000   3 400    virginica    Min  1st Qu   Median    Mean 3rd Qu     Max     2 200   2 800   3 000   2 974   3 175   3 800    If we print these two objects  ct and cb  we  essentially  have the same results and the only differences are in how they are shown and the different class attributes  respectively by for cb and array for ct   As I ve said  the power of by arises when we can t use tapply  the following code is one example    tapply iris  iris Species  summary   Error in tapply iris  iris Species  summary       arguments must have same length   R says that arguments must have the same lengths  say  we want to calculate the summary of all variable in iris along the factor Species   but R just can t do that because it does not know how to handle   With the by function R dispatch a specific method for data frame class and then let the summary function works even if the length of the first argument  and the type too  are different   bywork  lt - by iris  iris Species  summary    bywork iris Species  setosa   Sepal Length    Sepal Width     Petal Length    Petal Width          Species    Min     4 300   Min     2 300   Min     1 000   Min     0 100   setosa     50    1st Qu  4 800   1st Qu  3 200   1st Qu  1 400   1st Qu  0 200   versicolor  0    Median  5 000   Median  3 400   Median  1 500   Median  0 200   virginica   0    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          Species    Min     4 900   Min     2 000   Min     3 00   Min     1 000   setosa      0    1st Qu  5 600   1st Qu  2 525   1st Qu  4 00   1st Qu  1 200   versicolor 50    Median  5 900   Median  2 800   Median  4 35   Median  1 300   virginica   0    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          Species    Min     4 900   Min     2 200   Min     4 500   Min     1 400   setosa      0    1st Qu  6 225   1st Qu  2 800   1st Qu  5 100   1st Qu  1 800   versicolor  0    Median  6 500   Median  3 000   Median  5 550   Median  2 000   virginica  50    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        it works indeed and the result is very surprising  It is an object of class by that along Species  say  for each of them  computes the summary of each variable   Note that if the first argument is a data frame  the dispatched function must have a method for that class of objects  For example is we use this code with the mean function we will have this code that has no sense at all    by iris  iris Species  mean  iris Species  setosa  1  NA -------------------------------------------  iris Species  versicolor  1  NA -------------------------------------------  iris Species  virginica  1  NA Warning messages  1  In mean default data x    drop   FALSE            argument is not numeric or logical  returning NA 2  In mean default data x    drop   FALSE            argument is not numeric or logical  returning NA 3  In mean default data x    drop   FALSE            argument is not numeric or logical  returning NA   AGGREGATE  aggregate can be seen as another a different way of use tapply if we use it in such a way   at  lt - tapply iris Sepal Length   iris Species   mean  ag  lt - aggregate iris Sepal Length   list iris Species   mean    at     setosa versicolor  virginica       5 006      5 936      6 588   ag      Group 1     x 1     setosa 5 006 2 versicolor 5 936 3  virginica 6 588   The two immediate differences are that the second argument of aggregate must be a list while tapply can  not mandatory  be a list and that the output of aggregate is a data frame while the one of tapply is an array   The power of aggregate is that it can handle easily subsets of the data with subset argument and that it has methods for ts objects and formula as well   These elements make aggregate easier to work with that tapply in some situations  Here are some examples  available in documentation    ag  lt - aggregate len      data   ToothGrowth  mean    ag   supp dose   len 1   OJ  0 5 13 23 2   VC  0 5  7 98 3   OJ  1 0 22 70 4   VC  1 0 16 77 5   OJ  2 0 26 06 6   VC  2 0 26 14   We can achieve the same with tapply but the syntax is slightly harder and the output  in some circumstances  less readable   att  lt - tapply ToothGrowth len  list ToothGrowth dose  ToothGrowth supp   mean    att        OJ    VC 0 5 13 23  7 98 1   22 70 16 77 2   26 06 26 14   There are other times when we can t use by or tapply and we have to use aggregate    ag1  lt - aggregate cbind Ozone  Temp    Month  data   airquality  mean    ag1   Month    Ozone     Temp 1     5 23 61538 66 73077 2     6 29 44444 78 22222 3     7 59 11538 83 88462 4     8 59 96154 83 96154 5     9 31 44828 76 89655   We cannot obtain the previous result with tapply in one call but we have to calculate the mean along Month for each elements and then combine them  also note that we have to call the na rm   TRUE  because the formula methods of the aggregate function has by default the na action   na omit    ta1  lt - tapply airquality Ozone  airquality Month  mean  na rm   TRUE  ta2  lt - tapply airquality Temp  airquality Month  mean  na rm   TRUE    cbind ta1  ta2         ta1      ta2 5 23 61538 65 54839 6 29 44444 79 10000 7 59 11538 83 90323 8 59 96154 83 96774 9 31 44828 76 90000   while with by we just can t achieve that in fact the following function call returns an error  but most likely it is related to the supplied function  mean    by airquality c  Ozone    Temp     airquality Month  mean  na rm   TRUE    Other times the results are the same and the differences are just in the class  and then how it is shown printed and not only  -- example  how to subset it  object   byagg  lt - by airquality c  Ozone    Temp     airquality Month  summary  aggagg  lt - aggregate cbind Ozone  Temp    Month  data   airquality  summary    The previous code achieve the same goal and results  at some points what tool to use is just a matter of personal tastes and needs  the previous two objects have very different needs in terms of subsetting

User · Answer

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  lt - 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  lt - 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  lt - 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  lt - 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   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  lt - 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  lt - function x       if is character x          return paste x     sep                else        return x   1            A nested list structure l  lt - 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  lt - 1 20   A factor  of the same length   defining groups   y  lt - 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

User · Answer

Despite all the great answers here  there are 2 more base functions that deserve to be mentioned  the useful outer function and the obscure eapply function  outer  outer is a very useful function hidden as a more mundane one  If you read the help for outer its description says   The outer product of the arrays X and Y is the array A with dimension   c dim X   dim Y   where element A c arrayindex x  arrayindex y        FUN X arrayindex x   Y arrayindex y           which makes it seem like this is only useful for linear algebra type things   However  it can be used much like mapply to apply a function to two vectors of inputs  The difference is that mapply will apply the function to the first two elements and then the second two etc  whereas outer will apply the function to every combination of one element from the first vector and one from the second  For example    A lt -c 1 3 5 7 9   B lt -c 0 3 6 9 12   mapply FUN pmax  A  B    gt  mapply FUN pmax  A  B   1   1  3  6  9 12  outer A B  pmax     gt  outer A B  pmax          1    2    3    4    5    1      1    3    6    9   12   2      3    3    6    9   12   3      5    5    6    9   12   4      7    7    7    9   12   5      9    9    9    9   12   I have personally used this when I have a vector of values and a vector of conditions and wish to see which values meet which conditions   eapply  eapply is like lapply except that rather than applying a function to every element in a list  it applies a function to every element in an environment  For example if you want to find a list of user defined functions in the global environment   A lt -c 1 3 5 7 9  B lt -c 0 3 6 9 12  C lt -list x 1  y 2  D lt -function x  x 1    gt  eapply  GlobalEnv  is function   A  1  FALSE   B  1  FALSE   C  1  FALSE   D  1  TRUE    Frankly I don t use this very much but if you are building a lot of packages or create a lot of environments it may come in handy

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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

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It is maybe worth mentioning ave  ave is tapply s friendly cousin  It returns results in a form that you can plug straight back into your data frame   dfr  lt - data frame a 1 20  f rep LETTERS 1 5   each 4   means  lt - tapply dfr a  dfr f  mean      A    B    C    D    E     2 5  6 5 10 5 14 5 18 5      great  but putting it back in the data frame is another line   dfr m  lt - means dfr f   dfr m2  lt - ave dfr a  dfr f  FUN mean    NB argument name FUN is needed  dfr      a f    m   m2      1 A  2 5  2 5      2 A  2 5  2 5      3 A  2 5  2 5      4 A  2 5  2 5      5 B  6 5  6 5      6 B  6 5  6 5      7 B  6 5  6 5            There is nothing in the base package that works like ave for whole data frames  as by is like tapply for data frames   But you can fudge it   dfr foo  lt - ave 1 nrow dfr   dfr f  FUN function x        x  lt - dfr x       sum x m x m2     dfr        a f    m   m2    foo    1   1 A  2 5  2 5    25    2   2 A  2 5  2 5    25    3   3 A  2 5  2 5    25

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From slide 21 of http   www slideshare net hadley plyr-one-data-analytic-strategy      Hopefully it s clear that apply corresponds to  Hadley s aaply and aggregate corresponds to  Hadley s ddply etc  Slide 20 of the same slideshare will clarify if you don t get it from this image     on the left is input  on the top is output

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In the collapse package recently released on CRAN  I have attempted to compress most of the common apply functionality into just 2 functions     dapply  Data-Apply  applies functions to rows or  default  columns of matrices and data frames and  default  returns an object of the same type and with the same attributes  unless the result of each computation is atomic and drop   TRUE   The performance is comparable to lapply for data frame columns  and about 2x faster than apply for matrix rows or columns  Parallelism is available via mclapply  only for MAC      Syntax    dapply X  FUN       MARGIN   2  parallel   FALSE  mc cores   1L          return   c  same    matrix    data frame    drop   TRUE    Examples     Apply to columns  dapply mtcars  log  dapply mtcars  sum  dapply mtcars  quantile    Apply to rows  dapply mtcars  sum  MARGIN   1  dapply mtcars  quantile  MARGIN   1    Return as matrix  dapply mtcars  quantile  return    matrix   dapply mtcars  quantile  MARGIN   1  return    matrix     Same for matrices        BY is a S3 generic for split-apply-combine computing with vector  matrix and data frame method  It is significantly faster than tapply  by and aggregate  an also faster than plyr  on large data dplyr is faster though      Syntax   BY X  g  FUN       use g names   TRUE  sort   TRUE     expand wide   FALSE  parallel   FALSE  mc cores   1L     return   c  same    matrix    data frame    list      Examples     Vectors  BY iris Sepal Length  iris Species  sum  BY iris Sepal Length  iris Species  quantile  BY iris Sepal Length  iris Species  quantile  expand wide   TRUE    This returns a matrix    Data frames BY iris -5   iris Species  sum  BY iris -5   iris Species  quantile  BY iris -5   iris Species  quantile  expand wide   TRUE    This returns a wider data frame BY iris -5   iris Species  quantile  return    matrix     This returns a matrix   Same for matrices       Lists of grouping variables can also be supplied to g    Talking about performance  A main goal of collapse is to foster high-performance programming in R and to move beyond split-apply-combine alltogether  For this purpose the package has a full set of C   based fast generic functions  fmean  fmedian  fmode  fsum  fprod  fsd  fvar  fmin  fmax  ffirst  flast  fNobs  fNdistinct  fscale  fbetween  fwithin  fHDbetween  fHDwithin  flag  fdiff and fgrowth  They perform grouped computations in a single pass through the data  i e  no splitting and recombining    Syntax   fFUN x  g   NULL   w   NULL   TRA   NULL   na rm   TRUE   use g names   TRUE  drop   TRUE    Examples   v  lt - iris Sepal Length f  lt - iris Species    Vectors fmean v                mean fmean v  f             grouped mean fsd v  f               grouped standard deviation fsd v  f  TRA          grouped scaling fscale v  f            grouped standardizing  scaling and centering  fwithin v  f           grouped demeaning  w  lt - abs rnorm nrow iris    fmean v  w   w         Weighted mean fmean v  f  w          Weighted grouped mean fsd v  f  w            Weighted grouped standard-deviation fsd v  f  w            Weighted grouped scaling fscale v  f  w         Weighted grouped standardizing fwithin v  f  w        Weighted grouped demeaning    Same using data frames    fmean iris -5   f                   grouped mean fscale iris -5   f                  grouped standardizing fwithin iris -5   f                 grouped demeaning    Same with matrices       In the package vignettes I provide benchmarks  Programming with the fast functions is significantly faster than programming with dplyr or data table  especially on smaller data  but also on large data

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There are lots of great answers which discuss differences in the use cases for each function  None of the answer discuss the differences in performance  That is reasonable cause various functions expects various input and produces various output  yet most of them have a general common objective to evaluate by series groups  My answer is going to focus on performance  Due to above the input creation from the vectors is included in the timing  also the apply function is not measured     I have tested two different functions sum and length at once  Volume tested is 50M on input and 50K on output  I have also included two currently popular packages which were not widely used at the time when question was asked  data table and dplyr  Both are definitely worth to look if you are aiming for good performance         library dplyr  library data table  set seed 123  n   5e7 k   5e5 x   runif n  grp   sample k  n  TRUE   timing   list      sapply timing   sapply      system time       lt   split x  grp      r sapply   sapply lt  function x  list sum x   length x    simplify   FALSE        lapply timing   lapply      system time       lt   split x  grp      r lapply   lapply lt  function x  list sum x   length x          tapply timing   tapply      system time      r tapply  lt - tapply x  list grp   function x  list sum x   length x         by timing   by      system time      r by  lt - by x  list grp   function x  list sum x   length x    simplify   FALSE       aggregate timing   aggregate      system time      r aggregate  lt - aggregate x  list grp   function x  list sum x   length x    simplify   FALSE       dplyr timing   dplyr      system time       df   data frame x  grp      r dplyr   summarise group by df  grp   sum x   n          data table timing   data table      system time       dt   setnames setDT list x  grp    c  x   grp        r data table   dt     sum x    N   grp        all output size match to group count sapply list sapply r sapply  lapply r lapply  tapply r tapply  by r by  aggregate r aggregate  dplyr r dplyr  data table r data table           function x   if is data frame x   nrow else length  x   k       sapply     lapply     tapply         by  aggregate      dplyr data table         TRUE       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE        print timings as data table sapply timing         elapsed    keep rownames   TRUE                    fun   V1  elapsed   V2                    order -elapsed              fun elapsed  1   aggregate 109 139  2          by  25 738  3       dplyr  18 978  4      tapply  17 006  5      lapply  11 524  6      sapply  11 326  7  data table   2 686

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I recently discovered the rather useful sweep function and add it here for the sake of completeness  sweep The basic idea is to sweep through an array row- or column-wise and return a modified array  An example will make this clear  source  datacamp   Let s say you have a matrix and want to standardize it column-wise  dataPoints  lt - matrix 4 15  nrow   4     Find means per column with  apply    dataPoints means  lt - apply dataPoints  2  mean     Find standard deviation with  apply    dataPoints sdev  lt - apply dataPoints  2  sd     Center the points  dataPoints Trans1  lt - sweep dataPoints  2  dataPoints means  quot - quot      Return the result dataPoints Trans1           1    2    3      1   -1 5 -1 5 -1 5     2   -0 5 -0 5 -0 5     3    0 5  0 5  0 5     4    1 5  1 5  1 5    Normalize dataPoints Trans2  lt - sweep dataPoints Trans1  2  dataPoints sdev   quot   quot      Return the result dataPoints Trans2                 1          2          3      1   -1 1618950 -1 1618950 -1 1618950     2   -0 3872983 -0 3872983 -0 3872983     3    0 3872983  0 3872983  0 3872983     4    1 1618950  1 1618950  1 1618950  NB  for this simple example the same result can of course be achieved more easily by apply dataPoints  2  scale

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

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