Vectors, Matrices, Arrays, Lists, and Data FramesVector – a collection of ordered homogeneous elements.We can think of matrices, arrays, lists and data frames as deviations from a vector. Thedeviaitions are related to the two characteristicsorder and homogeneity.Matrix - a vector with two-dimensional shape information.> xx = matrix(1:6, nrow=3, ncol =2)> xx[,1] [,2][1,] 1 4[2,] 2 5[3,] 3 6> class(xx) [1] "matrix"> is.vector(xx) [1] FALSE> is.matrix(xx) [1] TRUE> length(xx) [1] 6> dim(xx) [1] 3 2– Typeset by FoilTEX – 1Also, matrices can have row and column names, which can be determined and/or assignedbyrownames and colnames. Other functions nrow, ncol, dimnames.– Typeset by FoilTEX – 2> yy = array(1:12, c(2,3,2))> yy, , 1[,1] [,2] [,3][1,] 1 3 5[2,] 2 4 6, , 2[,1] [,2] [,3][1,] 7 9 11[2,] 8 10 12> length(yy) [1] 12> dim(yy) [1] 2 3 2> is.matrix(yy) [1] FALSE> is.array(yy) [1] TRUE– Typeset by FoilTEX – 3ListsA vector with possible heterogeneous elements. The elements of a list can be numericvectors, character vectors, matrices, arrays, and lists.myList = list(a = 1:10, b = ”def”, c(TRUE, FALSE, TRUE))$a[1] 1 2 3 4 5 6 7 8 9 10$b[1] "def"[[3]][1] TRUE FALSE TRUE• length(myList) – there are 3 elements in the list• class(myList) – the class is a “list”• names(myList) – are “a”, “b” and the empty character “”• myList[1:2] – returns a list with two elements• myList[1] – returns a list with one element. What is length(myList[1]) ?• myList[[1]] – returns a vector with ten elements, the numbers 1, 2, ..., 10 What islength(myList[[1]]) ?– Typeset by FoilTEX – 4Data FramesA list with possible heterogeneous vector elements of the same length. The elements of adata frame can be numeric vectors, factor vectors, and logical vectors, but they must all be ofthe same length.> intelDate Transistors Microns Clock speed Data MIPS8080 1974 6000 6.00 2.0 MHz 8 0.648088 1979 29000 3.00 5.0 MHz 16 0.3380286 1982 134000 1.50 6.0 MHz 16 1.0080386 1985 275000 1.50 16.0 MHz 32 5.0080486 1989 1200000 1.00 25.0 MHz 32 20.00Pentium 1993 3100000 0.80 60.0 MHz 32 100.00PentiumII 1997 7500000 0.35 233.0 MHz 32 300.00PentiumIII 1999 9500000 0.25 450.0 MHz 32 510.00Pentium4 2000 42000000 0.18 1.5 GHz 32 1700.00Pentium4x 2004 125000000 0.09 3.6 GHz 32 7000.00– Typeset by FoilTEX – 5• names(intel) – returns the element names of the list, which are the names of each of thevectors: ”Date”, ”Transistors”, ”Microns” etc.• class(intel) – a ”data.frame”• dim(intel) – as a rectangular list, the data frame supports some matrix features: 10 7• length(intel) – the length is the number of elements in the list, NOT the combined numberof elements in the vectors, i.e. it is ?• class of intel[”Date”] versus intel[[”Date”]] – recall the [] returns an object of the sametype, i.e. a list but[[ ]] returns the element in the list.• What is the class of the speed element in intel?> intel[["speed"]][1] MHz MHz MHz MHz MHz MHz MHz MHz GHz GHzLevels: GHz MHz– Typeset by FoilTEX – 6Subsetting a Data FrameUsing the fact that a data frame is a list which also support some matrix features, fill in thetable specifying theclass (data.frame or integer) and the length and dim of the subset of thedata frame. Note that some responses will be NULL.Subset class length dimintelintel[1]intel[[1]]intel[,1]intel[“Date”]intel[, “Date”]intel$Date– Typeset by FoilTEX – 7Computations involving Vectors and Lists• Write code using vectorized function callse.g. nchar(y), x[ ] = 0, z + w• Use the apply mechanism–lapply and sapply for lists–apply for matrices and arrays–tapply for ragged arrays as vectors• With these functions we can avoid looping, and write code that is meaningful in astatistical setting, e.g. if we have a list of rainfall data where each element representsthe measurements taken at a different weather station, when we think about studying theaverage rainfall at each station we don’t think in terms of loops.– Typeset by FoilTEX – 8Applyapply(xx, 1, sum) for the matrix xx, the sum function is applied across the columns so thatthe row dimension (i.e. dim 1) is preserved.> xx[,1] [,2] [,3][1,] 1 3 5[2,] 2 4 6> apply(xx, 1, sum)[1] 9 12– Typeset by FoilTEX – 9apply(aa, c(1,2), sum) for the array aa, the sum function is applied across the pages so thatthe row and column dimensions (i.e. dim 1 and 2) are preserved.> aa, , 1[,1] [,2] [,3][1,] 1 3 5[2,] 2 4 6, , 2[,1] [,2] [,3][1,] 7 9 11[2,] 8 10 12> apply(aa,c(1,2),sum)[,1] [,2] [,3][1,] 8 12 16[2,] 10 14 18apply(aa, 2, sum) apply(aa, c(2, 3), sum) apply(aa, c(3, 2), sum)– Typeset by FoilTEX – 10> apply(aa,c(2),sum)[1] 18 26 34> apply(aa,c(2,3),sum)[,1] [,2][1,] 3 15[2,] 7 19[3,] 11 23> class(apply(aa,c(2,3),sum))[1] "matrix"> apply(aa,c(3,2),sum)[,1] [,2] [,3][1,] 3 7 11[2,] 15 19 23– Typeset by FoilTEX – 11Applying functions to list elementsThe lapply and sapply both apply a specified function to each element of a list. The formerreturns a list object and the latter a vector when possible.> ll[[1]][1] 1 2 3 4 5[[2]][1] 2 2 2[[3]][1] 0.0546 0.6851 0.8388 -0.1199 0.7995 -0.2518[7] -0.0585 -0.1581 0.6912 0.3957> lapply(ll, sum)– Typeset by FoilTEX – 12[[1]][1] 15[[2]][1] 6[[3]][1] 2.87678> sapply(ll, sum)[1] 15.00000 6.00000 2.87678– Typeset by FoilTEX – 13tapplyThis function is useful to apply a function to subsets of a vector.> x[1] 1 2 3 4 5 6 7 8 9 10> v[1] 1 1 1 0 0 0 1 1 1 0> tapply(x, v, mean)0 16.25 5.00> tapply(x, v, median)0 15.5 5.0– Typeset by FoilTEX –