--------------------------------------------------- # # # Chapter 1: Compact Introduction # # to the R-Software # # # --------------------------------------------------- R-homepage: https://cran.r-project.org/ The symbol # is the comment-symbol in R, all code after the #-symbol is ignored. # There are 4 basic data types in R: vectors, # matrices, data frames and lists. Numbers are # vectors of lengths 1: x = 7.7 y = 8.8 x+y # There are basically 3 commands to generate vectors # in R: c(), seq() and rep() v1 = c(2.22,-4.56,29) # "c" for "concatenate" v1 v2 = seq(from=10,to=30,by=2) v2 v3 = seq(from=0,to=2*pi,length=100) v3 plot(sin(v3)) # R always calculates element-wise

plot(sin(v3),type="l") points(cos(v3)) # add points to an existing plot lines(cos(v3),col="red") # same as points, just of line-type

?plot # help-pages v4 = rep(2,10) v4 v5 = rep(c(77,33),4) v5 # for vectors mit step size 1 or -1: v6 = 1:10 v6 v7 = 10:30 # increment is always 1 (or -1) v7 v8 = 4:(-4) v8 v8[1] v8[2] v8[7:9] v8[7:11] # many R-function automatically generate vectors as their # result: x = rnorm(10000) # 10000 standard-normal random numbers x plot(x)

hist(x)

hist(x,prob=TRUE) curve(dnorm(x),col="red",add=TRUE)

# R is pretty performant and offers lots of functionality # for calculations with random numbers; for every # probability distribution dist there are the 4 functions # # rdist (generates random numbers) # ddist (the prob-density) # pdist (integral over the density) # qdist (inverse of pdist) # Matrices: x = 1:20 x mat1 = matrix(x,nrow=4,ncol=5) mat1 mat2 = matrix(x,nrow=5,ncol=4) mat2 mat3 = matrix(x,nrow=5,ncol=4,byrow=TRUE) mat3 mat3[2,4] mat3[4:5,2] mat3[4:5,2:3] # R typically calculates element-wise: x x^2 x+0.33 1/x mat3 mat3^2 # this is not matrix-multiplication mat3+0.77 1/mat3 # this is not the inverse of a matrix # Matrix-Multiplication: mat4 = mat1 %*% mat2 mat4 det(mat4) # not invertible v1=c(1,1,1) v2=c(1,2,3) v3=c(1,4,9) mat5 = rbind(v1,v2,v3) # "rowbind" mat5 det(mat5) mat5inv = solve(mat5) # this is matrix-inverse mat5inv # check: mat5 %*% mat5inv zapsmall(mat5 %*% mat5inv) # Eigenvalues und Eigenvectors of a Matrix: n = 5 # we repeat this for n=1000 x = rnorm(n*n) # n^2 random numbers x mat = matrix(x,nrow=n,ncol=n) # nxn random matrix mat eigen(mat) res = eigen(mat) str(res) names(res) # the result of eigen(mat) is a list with 2 elements, # res[[1]]=res$values is of type vector and # res[[2]]=res$vectors is of type matrix. # the main use of lists is to collect objects of different # data types into one single object. Often the return type # of R-functions is of type list, since these functions # provide various typs of information. For example, the # return type of the function lm() ("lm" for "linear model") # which performs a linear regression, is a list with 13 # elements. res[[1]] res$values res[[2]] res$vectors n = 1000 # redo above, starting with code-line x = rnorm(n*n) # -> quite fast calculation of eigenvectors and values plot(res$values) # eigenvalues seem to be equally distributed # in a cicle with radius, probably, sqrt(n) # let's draw this circle and add to the plot of eigenvalues above: phi = seq(from=0,to=2*pi,length=101) r = sqrt(n) z = r*exp(1i*phi) # 1i is R-Syntax for complex i=sqrt(-1) points(z,col="red",type="l")

# apparently R is able to compute with complex numbers: z = 0.5 + sqrt(3)/2 *1i z Re(z) Im(z) abs(z) arg(z) Arg(z) # R is case-sensitive Arg(z)/pi # 60 degrees # Besides vectors, matrices and lists there is the data type # data frame. Typically, if some external data are imported # to R, they are stored in a data frame. # More on this in chapter2.

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