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# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - # # Basic Commands and Statistics with R - 1.2 - permutations # path: ~/ownCloud/STA_Statistics/Basic_Statistics/ # file_name: statistic_basics3.R # files_used: #install.packages( 'gtools' ) library( gtools ) #install.packages( "combinat" ) library( combinat ) #install.packages( "magrittr" ) library( "magrittr" ) # - - - - - - - - - - - - - - - - # x <- 6 factorial( x ) # [1] 720 for( i in 1:x ) print( factorial( i ) ) # [1] 1 # [1] 2 # [1] 6 # [1] 24 # [1] 120 # [1] 720 # - - - - - - - - - - - - - - - - # x <- c( "a", "b", "c" ) combinat::permn( x ) # [[1]] # [1] "a" "b" "c" # [[2]] # [1] "a" "c" "b" # [[3]] # [1] "c" "a" "b" # [[4]] # [1] "c" "b" "a" # [[5]] # [1] "b" "c" "a" # [[6]] # [1] "b" "a" "c" # - - - - - - - - - - - - - - - - # string conversion using combinat:: x <- "ABCDEFG" x <- strsplit( x, "" )[[1]] y <- combinat::permn( x ) y x <- LETTERS[1:5] y <- combinat::permn( x ) paste( y[[1]], collapse = "" ) for( i in 1:length( y ) ) print( paste( y[[i]], collapse = "" ) ) # - - - - - - - - - - - - - - - - # using gtools:: for permutations gtools::permutations( 5, 5, x ) # [,1] [,2] [,3] [,4] [,5] # [1,] "A" "C" "I" "M" "P" # [2,] "A" "C" "I" "P" "M" # [3,] "A" "C" "M" "I" "P" # [4,] "A" "C" "M" "P" "I" # [5,] "A" "C" "P" "I" "M" # ... gtools::permutations( 5, 5, x, repeats.allowed = TRUE ) # [,1] [,2] [,3] [,4] [,5] # [1,] "A" "A" "A" "A" "A" # [2,] "A" "A" "A" "A" "C" # [3,] "A" "A" "A" "A" "I" # [4,] "A" "A" "A" "A" "M" # [5,] "A" "A" "A" "A" "P" # [6,] "A" "A" "A" "C" "A" # [7,] "A" "A" "A" "C" "C" # [8,] "A" "A" "A" "C" "I" # [9,] "A" "A" "A" "C" "M" # [10,] "A" "A" "A" "C" "P" # ... # - - - - - - - - - - - - - - - - # using gtools:: for combinations gtools::combinations( 4, 2, x ) gtools::combinations( 5, 2, x ) gtools::combinations( 5, 3, x ) gtools::combinations( 4, 2, x, repeats.allowed = TRUE ) gtools::combinations( 2, 5, x, repeats.allowed = TRUE ) # Martin Stoppacher # # office@martinstoppacher.com # # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - # ################################################################################# |
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# matrix algebra m1 <- matrix( c( 1, 2, 3, 4, 5, 6, 7, 8, 9 ) , 3, 3 ) m1 m2 <- matrix( rep( 2, 9 ), 3, 3 ) m2 # [,1] [,2] [,3] # [1,] 1 4 7 # [2,] 2 5 8 # [3,] 3 6 9 m1 * m1 # element multipliction # [,1] [,2] [,3] # [1,] 1 16 49 # [2,] 4 25 64 # [3,] 9 36 81 m1 * m2 # [,1] [,2] [,3] # [1,] 2 8 14 # [2,] 4 10 16 # [3,] 6 12 18 # Matrix Multiplication m1 %*% m1 # [,1] [,2] [,3] # [1,] 30 66 102 # [2,] 36 81 126 # [3,] 42 96 150 # Transpose t( m1 ) # [,1] [,2] [,3] # [1,] 1 2 3 # [2,] 4 5 6 # [3,] 7 8 9 # Outer Product m1[ 1 , ] m1[ , 1 ] m1[ 1 , ] %o% m1[ , 1 ] m1 %o% m1 # , , 1, 1 # # [,1] [,2] [,3] # [1,] 1 4 7 # [2,] 2 5 8 # [3,] 3 6 9 # # , , 2, 1 # # [,1] [,2] [,3] # [1,] 2 8 14 # [2,] 4 10 16 # [3,] 6 12 18 # # , , 3, 1 # # [,1] [,2] [,3] # [1,] 3 12 21 # [2,] 6 15 24 # [3,] 9 18 27 # # , , 1, 2 # # [,1] [,2] [,3] # [1,] 4 16 28 # [2,] 8 20 32 # [3,] 12 24 36 # # , , 2, 2 # # [,1] [,2] [,3] # [1,] 5 20 35 # [2,] 10 25 40 # [3,] 15 30 45 # # , , 3, 2 # # [,1] [,2] [,3] # [1,] 6 24 42 # [2,] 12 30 48 # [3,] 18 36 54 # # , , 1, 3 # # [,1] [,2] [,3] # [1,] 7 28 49 # [2,] 14 35 56 # [3,] 21 42 63 # # , , 2, 3 # # [,1] [,2] [,3] # [1,] 8 32 56 # [2,] 16 40 64 # [3,] 24 48 72 # # , , 3, 3 # # [,1] [,2] [,3] # [1,] 9 36 63 # [2,] 18 45 72 # [3,] 27 54 81 # kroenecker product m1 %x% m1 # [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] # [1,] 1 4 7 4 16 28 7 28 49 # [2,] 2 5 8 8 20 32 14 35 56 # [3,] 3 6 9 12 24 36 21 42 63 # [4,] 2 8 14 5 20 35 8 32 56 # [5,] 4 10 16 10 25 40 16 40 64 # [6,] 6 12 18 15 30 45 24 48 72 # [7,] 3 12 21 6 24 42 9 36 63 # [8,] 6 15 24 12 30 48 18 45 72 # [9,] 9 18 27 18 36 54 27 54 81 m1 %x% m2 # [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] # [1,] 2 2 2 8 8 8 14 14 14 # [2,] 2 2 2 8 8 8 14 14 14 # [3,] 2 2 2 8 8 8 14 14 14 # [4,] 4 4 4 10 10 10 16 16 16 # [5,] 4 4 4 10 10 10 16 16 16 # [6,] 4 4 4 10 10 10 16 16 16 # [7,] 6 6 6 12 12 12 18 18 18 # [8,] 6 6 6 12 12 12 18 18 18 # [9,] 6 6 6 12 12 12 18 18 18 # cross Product m1[ , 1 ] m1[ , 2 ] crossprod( m1, m2[ , 1 ] ) # [,1] # [1,] 32 # [2,] 77 # [3,] 122 crossprod( m1 ) # [,1] [,2] [,3] # [1,] 14 32 50 # [2,] 32 77 122 # [3,] 50 122 194 crossprod( m1, m2 ) # [,1] [,2] [,3] # [1,] 12 12 12 # [2,] 30 30 30 # [3,] 48 48 48 # Diagonal Matrix with elements of x in the diagonal diag( nrow = 3 ) # [,1] [,2] [,3] # [1,] 1 0 0 # [2,] 0 1 0 # [3,] 0 0 1 diag( 3, nrow = 3 ) # [,1] [,2] [,3] # [1,] 3 0 0 # [2,] 0 3 0 # [3,] 0 0 3 # returns a vector of the principal diagonal elements m1 diag( m1 ) # [,1] [,2] [,3] # [1,] 1 0 0 # [2,] 0 2 0 # [3,] 0 0 3 # k * k identity matrix for m1[ , 1 ] diag( m1[ , 1 ] ) # [,1] [,2] [,3] # [1,] 1 0 0 # [2,] 0 2 0 # [3,] 0 0 # simple combinations cbind( m1, m2 ) rbind( m1, m2 ) # means and sums rowMeans( m1 ) rowSums( m1 ) colMeans( m1 ) colSums( m1 ) # eigenvalues, eigenvectors eigen( m1 ) e <- eigen( m1 ) e$values e$vectors # eigen() decomposition # $values # [1] 1.611684e+01 -1.116844e+00 -5.700691e-16 # # $vectors # [,1] [,2] [,3] # [1,] -0.4645473 -0.8829060 0.4082483 # [2,] -0.5707955 -0.2395204 -0.8164966 # [3,] -0.6770438 0.4038651 0.4082483 # single value decomposition svd( m1 ) # $d singular values of m1 # [1] 1.684810e+01 1.068370e+00 5.543107e-16 # # $u left singular vectors of m1 # [,1] [,2] [,3] # [1,] -0.4796712 0.77669099 0.4082483 # [2,] -0.5723678 0.07568647 -0.8164966 # [3,] -0.6650644 -0.62531805 0.4082483 # # $v right singular vectors of m1 # [,1] [,2] [,3] # [1,] -0.2148372 -0.8872307 0.4082483 # [2,] -0.5205874 -0.2496440 -0.8164966 # [3,] -0.8263375 0.3879428 0.4082483 # QR decomposition qr( m1 ) # $qr # [,1] [,2] [,3] # [1,] -3.7416574 -8.552360 -1.336306e+01 # [2,] 0.5345225 1.963961 3.927922e+00 # [3,] 0.8017837 0.988693 1.776357e-15 # # $rank # [1] 2 # # $qraux # [1] 1.267261e+00 1.149954e+00 1.776357e-15 # # $pivot # [1] 1 2 3 # # attr(,"class") # [1] "qr" # inverse solve( m1 ) # Error in solve.default(m1) : # Lapack routine dgesv: system is exactly singular: U[3,3] = 0 m3 <- matrix( c( 2, 6, 1, 4 ) , 2, 2 ) m3 # [,1] [,2] # [1,] 2 1 # [2,] 6 4 sm3 <- solve( m3 ) # [,1] [,2] # [1,] 2 -0.5 # [2,] -3 1.0 m3 %*% sm3 # [,1] [,2] # [1,] 1 0 # [2,] 0 1 |