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STATS 380 THE UNIVERSITY OF AUCKLAND SEMESTER 2, 2020 Campus: City STATISTICS Statistical Computing (Time allowed: TWO Hours) INSTRUCTIONS • Attempt ALL questions. • Total marks are 100. • Calculators are permitted. • R Quick Reference is available in Attachment. Page 1 of 25 STATS 380 Part I: Programming For questions in Part I, avoid using explicit loops or anything equivalent as much as possible, unless the question asks to use them. 1. Use the colon operator (:), seq(), rep() and some other commonly used arithmetic operators/functions to create the sequences given below. Note that you must not use c() or any explicit loop to create the sequences. (a) 1 -4 3 -8 5 -12 7 -16 [3 marks] (b) 1 1 1 2 2 3 3 3 4 4 [3 marks] (c) 1.4 2.1 2.8 3.5 4.2 4.9 [3 marks] (d) 0 0 1 1 1 2 2 2 3 3 3 4 [3 marks] (e) 0 0 0 0 0 5 0 0 0 0 4 0 0 0 3 0 0 2 0 1 [3 marks] [15 marks] Page 2 of 25 STATS 380 2. Let X = (x1, x2, . . . , xn) be a non-zero real vector. For each part write suitable R code. Note that your code should handle NA values in X. (a) Find the sum of elements of X. [2 marks] (b) Find the mean of positive elements of X. [3 marks] (c) Find the first element of X which is less than the preceding value (e.g. if X = c(4, 2, 3) the answer is 2), or NA if no such element exists. [4 marks] (d) Write a function called index.n which determine the index of the element of X which is the n-th to satisfy a given condition. Note that n should be an argument and your function should handle exceptional cases such as follows. [5 marks] > X= 10:20 > index.n (X > 14 , 2) [1] 7 > index.n (X > 14 , 7) [1] NA [14 marks] Page 3 of 25 STATS 380 3. Assume that X is a matrix of 1s and 0s. Write an R function called maj.mat that creates a vector as follows: For each row of the matrix X, the corresponding element of the vector will be either 1 or 0, depending on whether the majority of the first d elements in that row is 1 or 0. Here, d will be a parameter that we may wish to vary. > X [,1] [,2] [,3] [,4] [,5] [1,] 1 0 1 1 0 [2,] 1 1 0 0 0 [3,] 1 0 0 0 1 [4,] 0 1 1 1 0 > apply(x, 1 , maj.mat , d = 3) [1] 1 1 0 1 > apply(x, 1 , maj.mat , d = 2) [1] 0 1 0 0 [6 marks] Page 4 of 25 STATS 380 4. Complete the following R code (replace the dashes with your code) to create the panels shown in Figure 1. > mat = matrix(c(—————),4 ,3 , byrow = ——-) > layout(mat, widths = ——-, heights = c(1,2,2,4)) > —-(——-) > box(“—–“) [5 marks] 1 2 3 4 5 6 Figure 1: A layout to display 6 plots. Page 5 of 25 STATS 380 5. Figure 2 shows the number of individual visits to a medical center during the last year. The data is collected from a group of 50 men and 50 women living in an Aged Care. This figure is produce with the following code: > plot(x, y , pch=2) # Male > points(x, z , pch=16) # Female 0 10 20 30 40 50 2 3 4 5 x y Figure 2: Simple Plot. Complete the following R code to improve it to Figure 3 (the improvments are a) the colour of character, b) the axes limits and labels , c) legend of the graph, and d) the blue dashed line in the middle of the graph). > plot(x, y , pch = 2, xlim = c(-2, 52), ylim = ——, ——, —–) > points(x, z , pch = 16, col = ‘red’) > legend( —– , c(‘Male’, ‘Female’), pch = —–, col = —–, cex = 0.75) > —(—–) > lines( x= —–, y = c(3, 3), —— , col = “blue”) [10 marks] Page 6 of 25 STATS 380 0 10 20 30 40 50 0 1 2 3 4 5 6 Individuals Visits Male Female Number of Individual Visits Figure 3: Revised Simple Plot. Page 7 of 25 STATS 380 Part II: Data Technology 6. Suppose text is a character vector in the current R session. Explain in thorough detail the purpose of the following R expressions. Your answer should include an explanation of the purpose of each expression, the meaning of each argument in the function calls, and what sort of data structure is produced by each expression and function call. (a) lapply(text, nchar) [2 marks] (b) paste(1:length(text), text, sep = “\\)”) [2 marks] (c) regmatches(text, gregexpr(“WORD”, text)) <- “word” [2 marks] (d) lapply(strsplit(text, ” |,|[.]”), length) [2 marks] (e) sub(“[(](.+?)[)]\\{(.+?)\\}”, “(\\2){\\1}”, text) [2 marks] [10 marks] Page 8 of 25 STATS 380 7. Suppose allData is a data frame in R. Each column of allData contains the information of the sales for different retail stores of an imaginary company for a specific month in a specific year. > head(colnames(allData)) [1] “jan2009” “feb2009” “jun2009” “aug2009” “dec2009” “jan2010” Complete the following R expression in such a way that it prints out the last few rows of the sales information during January. > tail(allData[, __________________ ]) [5 marks] 8. The symbol population is a data frame in R that contains the population information of different cities in a country. The first few rows and columns of this data frame are shown for your reference. > population[1:3, 1:4] name pop2010 pop2011 pop2012 1 city 1 100 100 105 2 city 2 110 111 110 3 city 3 120 125 110 > ncol(population) [1] 11 Complete the following R expression in such a way that it calculates the average population of each city over time. > apply(population, 1, function(x){ ____________________ }) [5 marks] Page 9 of 25 STATS 380 9. For each day of the year, the hourly strength of the wind in a place is recorded in a text file called “wind.txt”. The strength of wind is recorded as a number between 0–9. Each line of “wind.txt” corresponds to a specific day of the year, and it contains 24 numbers for each hour of that day. We use the readLines() function to read this file into R as a character vector and assign it to the symbol wind. > head(wind) [1] “435353365363344466336554” “574363365756553766353434” [3] “673655763567754773765635” “356366757774344767646565” [5] “637377343346735543676445” “006523701725174167555206” Write R code to process wind and create a matrix of the values of the strength throughout the year. Your code should assign the result to the symbol strength. > strength[1:3, 1:4] [,1] [,2] [,3] [,4] [1,] 4 3 5 3 [2,] 5 7 4 3 [3,] 6 7 3 6 > ncol(strength) [1] 24 [10 marks] Page 10 of 25 STATS 380 10. The information about a medical measurement for some individuals are recorded in a data frame in R called df. > head(df) id age gender measure 1 pe001 39 M 0.19 2 pe002 40 M 0.95 3 pe003 41 M 0.36 4 pe004 45 F 0.11 5 pe005 50 M 0.91 6 pe006 47 M 0.04 Complete the following code in such a way that it selects a random sample of size 5 from each level of gender. > selectedID = tapply(df$id, ________ , function(x){ ___________ }) > result = lapply(selectedID, function(x){ _______________ }) > result $F id age gender measure 26 pe0026 45 F 0.34 33 pe0033 30 F 0.91 35 pe0035 29 F 0.03 52 pe0052 26 F 0.85 80 pe0080 42 F 0.98 $M id age gender measure 20 pe0020 41 M 0.07 22 pe0022 50 M 0.83 56 pe0056 48 M 0.32 63 pe0063 22 M 0.03 83 pe0083 32 M 0.11 [10 marks] Page 11 of 25 STATS 380 11. The summary statistics for variables x1 to x100 are recorded in a data frame in R called sum.out. The first few rows of the first few columns of this data frame are shown for your reference. > sum.out[1:4, 1:4] name x1 x2 x3 1 mean 0.49090990 1.248966e-02 0.49860616 2 n 100.00000000 1.010000e+03 200.00000000 3 var 0.08825365 9.970411e-01 0.08952302 4 std 0.29707516 9.985195e-01 0.29920397 Use the functions in the reshape2 library to transpose this data frame and assign it to the symbol t.sum.out. The first few observations of the desired output are shown for your reference. > t.sum.out[1:4, 1:4] variable max mean min 1 x1 0.9932668 0.49090990 0.005427403 2 x2 4.1965327 0.01248966 -3.218583296 3 x3 0.9966696 0.49860616 0.002607813 4 x4 0.4664198 -0.45272577 -1.470112336 [10 marks] Page 12 of 25 ATTACHMENT FOLLOWS ATTACHMENT STATS 380 R QUICK REFERENCE Basic Data Representation TRUE, FALSE logical true and false 1, 2.5, 117.333 simple numbers 1.23e20 scientific notation, 1.23 × 1020 . 3+4i complex numbers “hello, world” a character string NA missing value (in any type of vector) NULL missing value indicator in lists NaN not a number Inf positive infinity -Inf negative infinity “var” quotation for special variable name (e.g. +, %*%, etc.) Creating Vectors c(a1, . . . , an) combine into a vector logical(n) logical vector of length n (containing falses) numeric(n) numeric vector of length n (containing zeros) complex(n) complex vector of length n (containing zeros) character(n) character vector of length n (containing empty strings) Creating Lists list(e1, . . . , ek) combine as a list vector(k, “list”) create a list of length k (the elements are all NULL) Basic Vector and List Properties length(x) the number of elements in x mode(x) the mode or type of x Tests for Types is.logical(x) true for logical vectors is.numeric(x) true for numeric vectors is.complex(x) true for complex vectors is.character(x) true for character vectors is.list(x) true for lists is.vector(x) true for both lists and vectors Page 13 of 25 ATTACHMENT STATS 380 Tests for Special Values is.na(x) true for elements which are NA or NaN is.nan(x) true for elements which are NaN is.null(x) tests whether x is NULL is.finite(x) true for finite elements (i.e. not NA, NaN, Inf or -Inf) is.infinite(x) true for elements equal to Inf or -Inf Explicit Type Coercion as.logical(x) coerces to a logical vector as.numeric(x) coerces to a numeric vector as.complex(x) coerces to a complex vector as.character(x) coerces to a character vector as.list(x) coerces to a list as.vector(x) coerces to a vector (lists remain lists) unlist(x) converts a list to a vector Vector and List Names c(n1=e1,. . .,nk=ek) combine as a named vector list(n1=e1,. . .,nk=ek) combine as a named list names(x) extract the names of x names(x) = v (re)set the names of x to v names(x) = NULL remove the names from x Vector Subsetting x[1:5] select elements by index x[-(1:5)] exclude elements by index x[c(TRUE, FALSE)] select elements corresponding to TRUE x[c(“a”, “b”)] select elements by name List Subsetting x[1:5] extract a sublist of the list x x[-(1:5)] extract a sublist by excluding elements x[c(TRUE, FALSE)] extract a sublist with logical subscripts x[c(“a”, “b”)] extract a sublist by name Extracting Elements from Lists x[[2]] extract an element of the list x x[[“a”]] extract the element with name “a” from x x$a extract the element with name name “a” from x Logical Selection ifelse(cond, yes, no) conditionally select elements from yes and no which(v) returns the indices of TRUE values in v List Manipulation lapply(X, FUN, …) apply FUN to the elements of X split(x, f) split x using the factor f Page 14 of 25 ATTACHMENT STATS 380 Sequences and Repetition a:b sequence from a to b in steps of size 1 seq(n) same as 1:n seq(a,b) same as a:b seq(a,b,by=s) a to b in steps of size s seq(a,b,length=n) sequence of length n from a to b seq(along=x) like 1:length(n), but works when x has zero length rep(x,n) x, repeated n times rep(x,v) elements of x with x[i] repeated v[i] times rep(x,each=n) elements of x, each repreated n times Sorting and Ordering sort(x) sort into ascending order sort(x, decreasing=TRUE) sort into descending order rev(x) reverse the elements in x order(x) get the ordering permutation for x Basic Arithmetic Operations x + y addition, “x plus y” x – y subtraction, “x minus y” x * y multiplication, “x times y” x / y division, “x divided by y” x ^ y exponentiation, “x raised to power y” x %% y remainder, “x modulo y” x %/% y integer division, “x divided by y, discard fractional part” Rounding round(x) round to nearest integer round(x,d) round x to d decimal places signif(x,d) round x to d significant digits floor(x) round down to next lowest integer ceiling(x) round up to next highest integer Common Mathematical Functions abs(x) absolute values sqrt(x) square root exp(x) exponential functiopn log(x) natural logarithms (base e) log10(x) common logarithms (base 10) log2(x) base 2 logarithms log(x,base=b) base b logarithms Page 15 of 25 ATTACHMENT STATS 380 Trigonometric and Hyperbolic Functions sin(x), cos(x), tan(x) trigonometric functions asin(x), acos(x), atan(x) inverse trigonometric functions atan2(x,y) arc tangent with two arguments sinh(x), cosh(x), tanh(x) hyperbolic functions asinh(x), acosh(x), atanh(x) inverse hyperbolic functions Combinatorics choose(n, k) binomial coefficients lchoose(n, k) log binomial coefficients factorial(x) factorials lfactorial(x) log factorials Special Mathematical Functions beta(x,y) the beta function lbeta(x,y) the log beta function gamma(x) the gamma function lgamma(x) the log gamma function psigamma(x,deriv=0) the psigamma function digamma(x) the digamma function trigamma(x) the trigamma function Bessel Functions besselI(x,nu) Bessel Functions of the first kind besselK(x,nu) Bessel Functions of the second kind besselJ(x,nu) modified Bessel Functions of the first kind besselY(x,nu) modified Bessel Functions of the third kind Special Floating-Point Values .Machine$double.xmax largest floating point value (1.797693 × 10308) .Machine$double.xmin smallest floating point value (2.225074 × 10−308) .Machine$double.eps machine epsilon (2.220446 × 10−16) Page 16 of 25 ATTACHMENT STATS 380 Basic Summaries sum(x1,x2,. . .) sum of values in arguments prod(x1,x2,. . .) product of values in arguments min(x1,x2,. . .) minimum of values in arguments max(x1,x2,. . .) maximum of values in arguments range(x1,x2,. . .) range (minimum and maximum) Cumulative Summaries cumsum(x) cumulative sum cumprod(x) cumulative product cummin(x) cumulative minimum cummax(x) cumulative maximum Parallel Summaries pmin(x1,x2,. . .) parallel minimum pmax(x1,x2,. . .) parallel maximum Statistical Summaries mean(x) mean of elements sd(x) standard deviation of elements var(x) variance of elements median(x) median of elements quantile(x) median, quartiles and extremes quantile(x, p) specified quantiles Page 17 of 25 ATTACHMENT STATS 380 Uniform Distribution runif(n) vector of n Uniform[0,1] random numbers runif(n,a,b) vector of n Uniform[a,b] random numbers punif(x,a,b) distribution function of Uniform[a,b] qunif(x,a,b) inverse distribution function of Uniform[a,b] dunif(x,a,b) density function of Uniform[a,b] Binomial Distribution rbinom(n,size,prob) a vector of n Bin(size,prob) random numbers pbinom(x,size,prob) Bin(size,prob) distribution function qbinom(x,size,prob) Bin(size,prob) inverse distribution function dbinom(x,size,prob) Bin(size,prob) density function Normal Distribution rnorm(n) a vector of n N(0, 1) random numbers pnorm(x) N(0, 1) distribution function qnorm(x) N(0, 1) inverse distribution function dnorm(x) N(0, 1) density function rnorm(n,mean,sd) a vector of n normal random numbers with given mean and s.d. pnorm(x,mean,sd) normal distribution function with given mean and s.d. qnorm(x,mean,sd) normal inverse distribution function with given mean and s.d. dnorm(x,mean,sd) normal density function with given mean and s.d. Chi-Squared Distribution rchisq(n,df) a vector of n χ 2 random numbers with degrees of freedom df pchisq(x,df) χ 2 distribution function with degrees of freedom df qchisq(x,df) χ 2 inverse distribution function with degrees of freedom df dchisq(x,df) χ 2 density function with degrees of freedom df t Distribution rt(n,df) a vector of n t random numbers with degrees of freedom df pt(x,df) t distribution function with degrees of freedom df qt(x,df) t inverse distribution function with degrees of freedom df dt(x,df) t density function with degrees of freedom df F Distribution rf(n,df1,df2) a vector of n F random numbers with degrees of freedom df1 & df2 pf(x,df1,df2) F distribution function with degrees of freedom df1 & df2 qf(x,df1,df2) F inverse distribution function with degrees of freedom df1 & df2 df(x,df1,df2) F density function with degrees of freedom df1 & df2 Page 18 of 25 ATTACHMENT STATS 380 Matrices matrix(x, nr=r, nc=c) create a matrix from x (column major order) matrix(x, nr=r, nc=c, create a matrix from x (row major order) byrow=TRUE) Matrix Dimensions nrow(x) number of rows in x ncol(x) number of columns in x dim(x) vector coltaining nrow(x) and ncol(x) Row and Column Indices row(x) matrix of row indices for matrix x col(x) matrix of column indices for matrix x Naming Rows and Columns rownames(x) get the row names of x rownames(x) = v set the row names of x to v colnames(x) get the column names of x colnames(x) = v set the column names of x to v dimnames(x) get both row and column names (in a list) dimnames(x) = list(rn,cn) set both row and column names Binding Rows and Columns rbind(v1,v2,. . .) assemble a matrix from rows cbind(v1,v2,. . .) assemble a matrix from columns rbind(n1=v1,n2=v2,. . .) assemble by rows, specifying row names cbind(n2=v1,n2=v2,. . .) assemble by columns, specifying column names Matrix Subsets x[i,j] submatrix, rows and columns specified by i and j x[i,j] = v reset a submatrix, rows and columns specified by i and j x[i,] submatrix, contains just the rows a specified by i x[i,] = v reset specified rows of a matrix x[,j] submatrix, contains just the columns specified by j x[,j] = v reset specified columns of a matrix x[i] subset as a vector x[i] = v reset elements (treated as a vector operation) Matrix Diagonals diag(A) extract the diagonal of the matrix A diag(v) diagonal matrix with elements in the vector v diag(n) the n×n identity matrix Applying Summaries over Rows and Columns apply(X,1,fun) apply fun to the rows of X apply(X,2,fun) apply fun to the columns of X Page 19 of 25 ATTACHMENT STATS 380 Basic Matrix Manipulation t(A) matrix transpose A %*% B matrix product outer(u, v) outer product of vectors outer(u, v, f) generalised outer product Linear Equations solve(A, b) solve a system of linear equations solve(A, B) same, with multiple right-hand sides solve(A) invert the square matrix A Matrix Decompositions chol(A) the Choleski decomposition qr(A) the QR decomposition svd(A) the singular-value decomposition eigen(A) eigenvalues and eigenvectors Least-Squares Fitting lsfit(X,y) least-squares fit with carriers X and response y Page 20 of 25 ATTACHMENT STATS 380 Factors and Ordered Factors factor(x) create a factor from the values in x factor(x,levels=l) create a factor with the given level set ordered(x) create an ordered factor with the given level set is.factor(x) true for factors and ordered factors is.ordered(x) true for ordered factors levels(x) the levels of a factor or ordered factor levels(x) = v reset the levels of a factor or ordered factor Tabulation and Cross-Tabulation table(x) tabulate the values in x table(f1,f2,. . .) cross tabulation of factors Summary over Factor Levels tapply(x,f,fun) apply summary fun to x broken down by f tapply(x,list(f1,f2,. . .),fun) apply summary fun to x broken down by several factors Data Frames data.frame(n1=x1,n2=x2,. . .) create a data frame row.names(df) extract the observation names from a data frame row.names(df) = v (re)set the observation names of a data frame names(df) extract the variable names from a data frame names(df) = v (re)set the variable names of a data frame Subsetting and Transforming Data Frames df[i,j] matrix subsetting of a data frame df[i,j] = dfv reset a subset of a data frame subset(df,subset=i) subset of the cases of a data frame subset(df,select=i) subset of the variables of a data frame subset(df,subset=i,select=j) subset of the cases and variables of a data frame transform(df,n1=e1,n2=e2,. . .) transform variables in a data frame merge(df1,df2,. . .) merge data frames based on common variables Page 21 of 25 ATTACHMENT STATS 380 Reading Lines readline(prompt=””) read a line of input readLines(file, n) read n lines from the specified file readLines(file) read all lines from the specified file Reading Vectors and Lists scan(file, what = numeric()) read a vector or list from a file Formatting and Printing format(x) format a vector in a common format sprintf(fmt, …) formatted printing of R objects cat(…) concatenate and print vectors print(x) print an R object Reading Data Frames read.table(file, header=FALSE) read a data frame from a file read.csv(file, header=FALSE) read a data frame from a csv file Options for read.table and read.csv header=true/false does first line contain variable names? row.names=· · · row names specification col.names=· · · variable names specification na.strings=”NA” entries indicating NA values colClasses=NA the types associated with columns nrows=· · · the number of rows to be read Writing Data Frames write.table(x, file) write a data frame to a file write.csv(x, file) write a data frame to a csv file String Handling paste(…, sep = ” “, collapse = NULL) paste strings together strsplit(x, split) split x on pattern split (returns a list) grep(pattern, x) return subscripts of matching elements grep(pattern, x, value = TRUE) return matching elements sub(pattern, replacement, x) replace pattern with given replacement gsub(pattern, replacement, x) globally replace Page 22 of 25 ATTACHMENT STATS 380 High-Level Graphics plot(x, y) scatter plot plot(x, y, type = “l”) line plot plot(x, y, type = “n”) empty plot Adding to Plots abline(a, b) line in intercept/slope form abline(h = yvals) horizontal lines abline(v = xvals) vertical lines points(x, y) add points lines(x, y) add connected polyline segments(x0, y0, x1, y1) add disconnected line segments arrows(x0, y0, x1, y1, code) add arrows rect(x0, y0, x1, y1, col) add rectangles filled with colours polygon(x, y) a polygon(s) Low-Level Graphics plot.new() start a new plot/figure/panel plot.window(xlim, ylim, …) set up plot coordinates Options to plot.window xaxs=”i” don’t expand x range by 8% yaxs=”i” don’t expand y range by 8% asp=1 equal-scale x and y axes Graphical Parameters par(. . . ) set/get graphical parameters Useful Graphical Parameters mfrow = c(m,n) set up an m by n array of figures, filled by row mfcol = c(m,n) set up an m by n array of figures, filled by column mar=c(m1,m2,m3,m4) set the plot margins (in lines) mai=c(m1,m2,m3,m4) set the plot margins (in inches) cex=m set the basic font magnification to m bg=col set the device background to col Measuring Text Size strwidth(x, “inches”, font, cex) widths of text strings in inches strheight(x, “inches”, font, cex) heights of text strings in inches Layouts layout(mat,heights,widths) set up a layout layout.show(n) show layout elements (up to n) lcm(x) size specification in cm Page 23 of 25 ATTACHMENT STATS 380 Compound Expressions { expr1, . . . , exprn} compound expressions Alternation if (cond) expr1 else expr1 conditional execution if (cond) expr conditional execution, no alternative Iteration for (var in vector) expr for loops while (cond) expr while loops repeat expr infinite repetition continue jump to end of enclosing loop break break out of enclosing loop Function Definition function(args) expr function definition var function argument with no default var=expr function argument with default value return(expr) return the given value from a function missing(a) true if argument a was not supplied Error Handling stop(message) terminate a computation with an error message warning(message) issue a warning message on.exit(expr) save an expression for execution on function return Language Computation quote(expr) returns the expression expr unevaluated substitute(arg) returns the expression passed as argument arg substitute(expr,subs) make the specified substitutions in the given expression Page 24 of 25 ATTACHMENT STATS 380 Interpolation approx(x, y, xout) linear interpolation at xout using x and y spline(x, y, xout) spline interpolation at xout using x and y approxfun(x, y, xout) interpolating linear function for x and y splinefun(x, y, xout) interpolating spline for x and y Root-Finding and Optimisation polyroot(coef) roots of polynomial with coefficients in coef uniroot(f,interval) find a root of the function f in the given interval optimize(f,interval) find an extreme of the function f in the given interval optim(x,f) find an extreme of the function f starting at the point x nlm(f,x) an alternative to optim nlminb(x,f) optimization subject to constraints Integration integrate(x,lower,upper) integrate the function f from lower to upper Page 25 of 25
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