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Chapter 4 HW
#1 – R code and plot
#Square plot par(pty = "s") #Set up some dummy values for plot a1<-c(-1,1) a2<-c(-1,1) plot(x = a1, y = a2, type = "n", main = "Problem 1", xlab = "X", , ylab = "Y") grid(nx = NULL, ny = NULL, col = "gray", lty = "dotted") abline(h = 0, lty = 1, lwd = 2) abline(v = 0, lty = 1, lwd = 2) arrows(x0 = 0, y0 = 0, x1 = 2/sqrt(5), y1 = 1/sqrt(5), col = 2, lty = 1) arrows(x0 = 0, y0 = 0, x1 = 1/sqrt(5), y1 = -2/sqrt(5), col = 2, lty = 1) -1.0 -0.5 0.0 0.5 1. -1. -0.
Problem 1 X Y
#2 – Below is code and output copied directly from
the R Console
> # NOTES: If you have problems seeing what a function is doing, > # type and run help(function) where "function" is replaced > # with the function name. > X<-matrix(data =c(2, 2, 3, 1, 2, 4, 3, 5, 2, 6, 3, 4, 4, 6, 8), nrow=5, ncol=3) > X [,1] [,2] [,3] [1,] 2 4 3 [2,] 2 3 4 [3,] 3 5 4 [4,] 1 2 6 [5,] 2 6 8 > ####### > #Part a > R<-cor(x = X) #Also could do just cor(X) > R [,1] [,2] [,3] [1,] 1.0000000 0.6708204 -0. [2,] 0.6708204 1.0000000 0. [3,] -0.3535534 0.3162278 1. > sigma.hat<-var(x = X) > sigma.hat [,1] [,2] [,3] [1,] 0.50 0.75 -0. [2,] 0.75 2.50 1. [3,] -0.50 1.00 4. > # diag( ) finds the diagonal values > sum(diag(R)) [1] 3 > sum(diag(sigma.hat)) [1] 7 > ####### > #Part b > # Determinant using the product of the eigenvalues > # eigen( ) has two components - the values and the vectors > prod(eigen(R)$values)
[1,] 1.046255e- [2,] 3.930233e- [3,] -9.864885e- > ####### > #Part d > sum(eigen(sigma.hat)$values) [1] 7 > sum(diag(sigma.hat)) [1] 7 > prod(eigen(sigma.hat)$values) [1] 0. > ####### > #Part e > #rbind combines into one row the results of the mean functions > # If you have problems seeing what the function is doing, examine the individual mean functions first > mu.hat<-rbind(mean(X[,1]), mean(X[,2]), mean(X[,3])) > #Another way to get the mu.hat vector - this tells R to "apply" the function "mean" to each column (MARGIN=2) of the X matrix > mu.hat<-apply(X = X, MARGIN = 2, FUN = mean) > #t() finds the transpose > sqrt(t(mu.hat)%%mu.hat) [,1] [1,] 6. > #solve() finds an inverse of a matrix > sqrt(t(mu.hat)%%solve(sigma.hat)%%mu.hat) [,1] [1,] 5. > sqrt(t(X[2,]-mu.hat)%%solve(sigma.hat)%%(X[2,]-mu.hat)) [,1] [1,] 0. > ####### > #Part f > a<-c(2, -1, 3) > t(a)%%X[1,] [,1] [1,] 9
