Statistical Programming with R

Transcription

1 Statistical Programming with R Lecture 3: Matrices Bisher M. Iqelan biqelan@iugaza.edu.ps Department of Mathematics, Faculty of Science, The Islamic University of Gaza , Semester 1

2 Matrices A matrix is a rectangular arrangement of values which are all of the same basic type. For example, ( ) arranges the numbers from a vector 1:8 in two rows and four columns. This is a 2 4 matrix of numbers. In R, the matrix is created column by column from the input vector. This is known as column-major order storage. We can also create a matrix row by row. Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

3 Creating Matrices Let us create a matrix from a vector. > mat = matrix(c(1,2,3,4,5,6),ncol=2) > mat [,1] [,2] [1,] 1 4 [2,] 2 5 [3,] 3 6 An optional argument byrow=true can be used to arrange the vector of values by row. > mat = matrix(c(1,2,3,4,5,6),byrow=true, ncol=2) > mat [,1] [,2] [1,] 1 2 [2,] 3 4 [3,] 5 6 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

4 Matrix Elements and Recycling As we said, the rst argument to the matrix() function contains the vector which is to form the matrix. If there are not enough elements in the vector to create the matrix, the recycling rule is applied to obtain more. A 3 2 matrix lled with 1s can be obtained as follows. > Ones.mat = matrix(1,nrow=3, ncol=2) > Ones.mat [,1] [,2] [1,] 1 1 [2,] 1 1 [3,] 1 1 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

5 Optional Dimension Specications Often R can work out the number of rows in a matrix given the number of columns and the elements, or the the number of columns in a matrix given the number of rows and the elements. In such cases it is not necessary to specify both the number of rows and the number of columns. > Grow.mat = matrix(1:6,nrow=3) > Grow.mat [,1] [,2] [1,] 1 4 [2,] 2 5 [3,] 3 6 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

6 Optional Dimension Specications Or simply by given the number of columns and the vector. For example, > Gcol.mat = matrix(1:6,ncol=3) > Gcol.mat [,1] [,2] [,3] [1,] [2,] Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

7 Determining Matrix Dimensions The number of rows and columns of a matrix can be obtained with the functions nrow and ncol. Or obtained together with the function dim which produces a vector containing the number of rows as the rst element and the number of columns as its second. > Gcol.mat [,1] [,2] [,3] [1,] [2,] > nrow(gcol.mat) [1] 2 > ncol(gcol.mat) [1] 3 > dim(gcol.mat) [1] 2 3 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

8 Create Matrix by changing dimensions We can also create the matrices by changing its dimension. For example > mat = matrix(c(1,2,3,4,5,6),byrow=true, ncol=2) > mat [,1] [,2] [1,] 1 2 [2,] 3 4 [3,] 5 6 > dim(mat)= c(2,3) > mat [,1] [,2] [,3] [1,] [2,] Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

9 > mat2 = matrix(7:12,3,2) > mat2 [,1] [,2] [1,] 7 10 [2,] 8 11 [3,] 9 12 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31 Creating Matrices from Rows and Columns We can also create matrices from other matrices and vectors using the rbind (row bind) and cbind (column bind) commands. That is, by gluing together rows with rbind and gluing together columns with cbind. For example, let mat1 and mat2 be two matrices such that > mat1 = matrix(1:6,byrow=true, ncol=2) > mat1 [,1] [,2] [1,] 1 2 [2,] 3 4 [3,] 5 6

10 rbind and cbind Continued... > cbind(mat1,mat2) [,1] [,2] [,3] [,4] [1,] [2,] [3,] > rbind(mat1,mat2) [,1] [,2] [1,] 1 2 [2,] 3 4 [3,] 5 6 [4,] 7 10 [5,] 8 11 [6,] 9 12 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

11 rbind and cbind Continued... >mat3 = rbind(mat1,c(77,77)) >mat3 [,1] [,2] [1,] 1 2 [2,] 3 4 [3,] 5 6 [4,] >mat4 = cbind(1:3, 4:6) >mat4 [,1] [,2] [1,] 1 4 [2,] 2 5 [3,] 3 6 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

12 Binding Rows and Columns; Recycling The arguments to rbind and cbind are not required to be the same length length. When they are not, a matrix is created which is big enough to accommodate the largest argument, and the others have the recycling rule applied to them to supply additional arguments. Apparent mismatches produce warnings. > rbind(1:2, 1:3) [,1] [,2] [,3] [1,] [2,] Warning message: In rbind(1:2, 1:3) : number of columns of result is not a multiple of vector length (arg 1) Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

13 Matrices and Naming It is possible to attach row and column names to matrices. Row names can be attached with rownames, column names with colnames, and both can be attached simultaneously with dimnames. > mat1 = matrix(1:6,byrow=true, ncol=2) > mat1 [,1] [,2] [1,] 1 2 [2,] 3 4 [3,] 5 6 >dimnames(mat1)=list(c("a", "B", "C"),c("First", "Second")) > mat1 First Second A 1 2 B 3 4 C 5 6 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

14 Extracting Names Names can also be extracted with dimnames, rownames and colnames. > dimnames(mat1); rownames(mat1); colnames(mat1) [[1]] [1] "A" "B" "C" [[2]] [1] "First" "Second" [1] "A" "B" "C" [1] "First" "Second" Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

15 Extracting Matrix Elements The ij-th element of a matrix mat can be extracted with the expression mat[i, j]. > M <- matrix(1:6, 2, 3) > M[1, 2] [1] 3 > M[2, 1] [1] 2 Indices can also be missing. > M[1, ] [1] > M[, 2] [1] 3 4 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

16 Extracting Matrix Elements: Example This means, for example, that we could sum all the elements in a matrix mat with the following code. > s = 0 > for(i in 1:nrow(mat)) for(j in 1:ncol(mat)) s = s + mat[i, j] Or more eciently we can use > s = sum(mat) Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

17 Extracting Matrix Elements as a matrix form By default, when a single element of a matrix is retrieved, it is returned as a vector of length 1 rather than a 1 1 matrix. This behavior can be turned o by setting drop = FALSE. > M <- matrix(1:6, 2, 3) > M[1, 2] [1] 3 > M[1, 2, drop = FALSE] [,1] [1,] 3 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

18 Subsetting a Matrix Similarly, subsetting a single column or a single row will give you a vector, not a matrix (by default). > M <- matrix(1:6, 2, 3) > M[1,] [1] > M[1,, drop = FALSE] [,1] [,2] [,3] [1,] Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

19 Matrix Subsets Expressions of the form mat[i, j] can also be used to extract more general subsets of the elements of a matrix by specifying vector subscripts. > mat5 = matrix(1:12, nrow = 3, ncol = 4) > mat5 [,1] [,2] [,3] [,4] [1,] [2,] [3,] > mat5[1:2, c(2, 4)] [,1] [,2] [1,] 4 10 [2,] 5 11 Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

20 Assigning to Matrix Subsets It is also possible to assign to subsets of matrices. > mat5[1:2, c(2, 4)]= 21:24 > mat5 [,1] [,2] [,3] [,4] [1,] [2,] [3,] > mat5[2:1, c(2, 4)]= 21:24 > mat5 [,1] [,2] [,3] [,4] [1,] [2,] [3,] Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

21 More Examples on Indexation As with vectors it is useful to be able to extract sub-components of matrices. In this case, we may wish to pick out individual elements, rows or columns. As before, the [ ] notation is used to subscript. The following examples should make things clear: > (A <- matrix(1:12, 3, 4, byrow = TRUE)) [,1] [,2] [,3] [,4] [1,] [2,] [3,] Try the following, what you expect > A[3,] # Extracting the third row > A[,3] # Extracting the third column > A[3,3] # the third row and the third column > A[-1,] # the matrix except the first row > A[,-2] # the matrix except the second column > A[-1,2] # mix subscripting > A[-1,2:3] # use of negative indices Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

22 Treating Matrices as Vectors As we previously explained, Matrices can be treated as vectors. >(mat= matrix(1:6,nr=2, nc=3,byrow=t)) [,1] [,2] [,3] [1,] [2,] > mat[6:8] [1] 6 NA NA The functions row and col return matrices indicating the row and column of each element. This can be used to extract or change submatrices. > mat[row(mat) < col(mat)] = NA > mat [,1] [,2] [,3] [1,] 1 NA NA [2,] 4 5 NA Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

23 Example: Create your Own Matrix Tri-diagonal matrix Create a 4 4 tri-diagonal matrix with diagonal values being 7 and the o-diagonal elements being 3. > T.diag = matrix(0, nrow = 4, ncol = 4) > T.diag[row(T.diag) == col(t.diag)] = 7 > T.diag[abs(row(T.diag) - col(t.diag)) == 1] = 3 > T.diag Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

24 Example: Create your Own Matrix Tri-diagonal matrix Create a 4 4 tri-diagonal matrix with diagonal values being 7 and the o-diagonal elements being 3. > T.diag = matrix(0, nrow = 4, ncol = 4) > T.diag[row(T.diag) == col(t.diag)] = 7 > T.diag[abs(row(T.diag) - col(t.diag)) == 1] = 3 > T.diag [,1] [,2] [,3] [,4] [1,] [2,] [3,] [4,] Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

25 Some Examples of Matrices > diag(4) Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

26 Some Examples of Matrices > diag(4) [,1] [,2] [,3] [,4] [1,] [2,] [3,] [4,] > diag(c(4,5,6,7)) # or similarly diag(4:7) Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

27 Some Examples of Matrices > diag(4) [,1] [,2] [,3] [,4] [1,] [2,] [3,] [4,] > diag(c(4,5,6,7)) # or similarly diag(4:7) [,1] [,2] [,3] [,4] [1,] [2,] [3,] [4,] > diag(diag(c(4,5,6,7))) Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

28 Some Examples of Matrices > diag(4) [,1] [,2] [,3] [,4] [1,] [2,] [3,] [4,] > diag(c(4,5,6,7)) # or similarly diag(4:7) [,1] [,2] [,3] [,4] [1,] [2,] [3,] [4,] > diag(diag(c(4,5,6,7))) [1] Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

29 Array Arrays generalize matrices by extending the function dim to more than two dimensions. For example, if the rows and columns of a matrix are the length and the width of a rectangular arrangement of values of equal-sized cube, then length, width and height represent the dimensions of three way array. There is no limit to the number of dimensions of an array. Here is an example explaining a 3-dimensional array: > arr1 <- array( c(2:9,12:19,112:119), dim=c(2,4,3)) > # 2 rows, 4 columns, 3 layers or levels Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

30 Array Continued... > arr1,, 1 [,1] [,2] [,3] [,4] [1,] [2,] ,, 2 [,1] [,2] [,3] [,4] [1,] [2,] ,, 3 [,1] [,2] [,3] [,4] [1,] [2,] Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

31 Array Continued... The rst dimension (the rows) is incremented rst. This is the same as placing the values column by column. The second dimension (column) is incremented second. The third dimension is incremented by lling a matrix for each level of the third dimension. > length(arr1) [1] 24 > mode(arr1) [1]"numeric" > class(arr1) [1]"array" > dim(arr1) [1] > dimnames(arr1) NULL Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

32 Named arrays > dimnames(arr1) = list(c("a", "B"), NULL, c("x", "Y", "Z")) > arr1,, X [,1] [,2] [,3] [,4] A B ,, Y [,1] [,2] [,3] [,4] A B ,, Z [,1] [,2] [,3] [,4] A B Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

33 Extension arrays Try to understand arrays with more and more dimensions. For example, > arr2 <- array(0, dim = c(7, 4, 6, 8)) will give you a 4-dimensional array. > arr3 <- array(0, dim = c(3, 4, 5, 3, 7)) will give you a 5-dimensional array. Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

34 Factors Factors are how R handles qualitative or categorical data. Factors can be unordered or ordered. One can think of a factor as an integer vector where each integer has a label. Some examples include gender with values male and female, marital status with values single, married, separated and divorced. pain level where the values are none, mild, medium, severe, eye color where the values are brown, black, blue, green, etc.., Using factors with labels is better than using integers because factors are self-describing; having a variable that has values "Male" and "Female" is better than a variable that has values 1 and 2. Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

35 Factors: Example > gndr <- c("female","female","male","female","male") > gndr <- factor(gndr) > gndr [1] female female male female male Levels: female male # stores gndr as 3 1s and 2 2s and associates # 1=female, 2=male internally (alphabetically) >table(gndr) gndr female male 3 2 >unclass(gndr) [1] attr(,"levels") [1] "female" "male" Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

36 Factors: Cont.. The order of the levels can be set using the levels argument to factor(). > gndr <- factor(c("female", "female", "male", "female", + "male"), + levels = c("male", "female")) > gndr [1] female female male female male Levels: male female Bisher M. Iqelan (IUG) R Programming: Matrices 1 st Semester / 31

37 End of lecture 3. Thank you.!!!