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R-Project for Statistical Computing

The R-Project is extensive and rigorous Open Source software for statistics and graphics. It has become the tool of choice for professional statistical analysis, and runs on UNIX, Microsoft-Windows and Mac operating systems.

Table of Contents

Introduction to R

Tutorials

Statistics, Spreadsheets and R

Examples of the Versatility of R

Bibliography

Introduction to R

R was initially written by Robert Gentleman and Ross Ihaka of the Statistics Department of the University of Auckland, New Zealand. It is freely distributable under the GNU general public license *.

There is a wide variety of contributed Library Packages associated with R, which are available from the Comprehensive R Archive Network (CRAN). These packages include routines for statistics, graphics, solving equations, signal processing, the analysis of complex surveys, statistical process control, and packages designed for use in spectroscopy and chemometrics. The availability of these packages is one of the major strengths of R, together with its extensibility and flexibility.

Tutorials

Flash video tutorial, released by Decision Science News.

Statistics, Spreadsheets and R

In order to be easy to use, a spreadsheet, like Excel, Open Office or Gnumeric, does not separate functions from data. Usually, this does not matter when the data is simple and straightforward.

When it comes to serious statistical calculations, however, a spreadsheet is not suitable. Numerical errors are likely to arise, and, furthermore, the user is often totally unaware that this has happened.

Spreadsheets are great if the data is simple - they are positively dangerous if the data is complex.

These issues do not apply to R since its functions and data are separated within the software.

Examples of the Versatility of R

To show what R can do, we give a few simple examples.

Two-Dimensional Brownian Motion Example

The first example plots a two-dimensional Brownian motion in R.

###### R Code for Two-Dimensional Brownian Motion ###### # Parameters to be specified in function call # # mu = constant drift # sigma = constant volatility > 0 # T = time horizon # N = number of time steps (number of points) # # A sample path of Brownian Motion in two spatial dimensions. Brownian.fn <- function(mu, sigma, T, N){ dt <- T/N t <- c(rep(NA, N)) B1 <- c(rep(NA, N)) B2 <- c(rep(NA, N)) t[1] <- 0 B1[1] <- 0 B2[1] <- 0 ### Simulating Two Independent Brownian Motions ### for(j in 2:N){ t[j] <- t[j-1] + dt B1[j] <- B1[j-1] + mu*dt + sigma*sqrt(dt)*rnorm(1, mean = 0, sd = 1) B2[j] <- B2[j-1] + mu*dt + sigma*sqrt(dt)*rnorm(1, mean = 0, sd = 1) } ### Plotting ### plot(y = B2, x = B1, xlim = c(-3, 3), ylim = c(-3, 3), main = "Two-Dimensional Brownian Motion", xlab = "B1(t)", ylab = "B2(t)", type ="l", lty = 1, col=4) } ### Sample paths of Standard BM starting at origin, time horizon = 1, 5000 time steps Brownian.fn(1, 2, 1, 1000) Brownian.fn(0, 1, 1, 5000)

Entered into R, this code outputs the Brownian motion as a graph, with B1(t) along the abscissa and B2(t) along the ordinate. Since we are modeling a radom process, each run of the code produces graphs of differing appearances. Here is one of mine.

Dirac Delta Function

The second example plots the Dirac delta function as a Fourier series

###### R Code for Fourier Series of Dirac's Delta Function ###### # # f(x) = 1/(2*pi) + (1/pi)*InfiniteSeries cos(kx) # # Parameters to be specified in function call # # N = choose a large number for accurate approximation # # Graphs the Nth Fourier Sum and the Dirac's Delta Function, # and also calculates their supremum (Chebyshev/L-infinty) distance # and their mean absloute error (L1-distance), and mean square error (L2-distance). # # Notes: This particular Fourier series does NOT converge uniformly, # nor converges in either the L1 or L2 norms. # # Since the Dirac's Delta Function has a single discontinuity at zero # where the height is Infinity, you will see the "ringing effect" # in the Nth Fourier Sum at zero, known as the "Gibbs Phenomenon." diracdelta.fn <- function(N){ t <- seq(from = -pi, to = pi, length = 10000) F <- c(rep(NA, 10000)) G <- c(rep(NA, 10000)) a <- c(rep(NA, N)) g <- c(rep(NA, N)) S <- c(rep(NA, N)) ### Fourier Coefficients ### for(k in 1:N){ a[k] <- NA } ### Dirac Delta Function for Supremum Distance Calculation ### for(j in 1:10000){ if(-0.001 <= t[j] && t[j] <= 0.001){ G[j] <- Inf } else{ G[j] <- 0 } } ### Linear Combination of Eigenfunctions ### S.fn <- function(t){ for(k in 1:N){ g[k] <- cos(k*t) } S <- g } ### Nth Fourier Sum ### for(j in 1:10000){ F[j] <- 1/(2*pi) + (1/pi)*sum(S.fn(t[j])) } ### Dirac Delta Function for Plotting ### Oh <- c(rep(0, 10000)) green <- seq(from = 0, to = 25, length = 10000) ### Calculate the Supremum Distance of the Nth Fourier Sum and Dirac Delta Function ### supremum.norm <- max(abs(F - G)) ### Calculate the L2-Distance (MSE) of the Nth Fourier Sum and Dirac Delta Function ### ### Note: sets of measure zero are negligible in calculation of integrals ### dx <- (2*pi)/10000 mean.square <- sum(dx*(F - 0)^2) ### Calculate the L1-Distance (Mean Absolute Error) of the Nth Fourier Sum and ### the Dirac Delta Function ### ### Note: sets of measure zero are negligible in calculation of integrals ### dx <- (2*pi)/10000 mean.absolute <- sum(dx*abs(F - 0)) print(cbind(supremum.norm, mean.square, mean.absolute)) ### Plotting ### plot(y = F, x = t, xlim = c(-pi, pi), ylim = c(-3, 15), main = "Dirac Delta Function as Fourier Series", xlab = "x", ylab = "F(x;n)", type ="l", lty = 1, lwd = 2, col = 4) lines(y = green, x = Oh, lty = 1, lwd = 2, col= 3) legend(y = 15, x = -pi, c("Nth Fourier Sum", "Dirac's Delta Function"), lty = c(1,1), col = c(4,3), lwd = c(2,2), cex = 0.8) abline(h = 0, lwd = 2, col = 3) abline(h = 0, lty = 3) } ### Nth Fourier Sum of Dirac's Delta Function diracdelta.fn(1) diracdelta.fn(2) diracdelta.fn(3) diracdelta.fn(4) diracdelta.fn(5) diracdelta.fn(10) diracdelta.fn(20) diracdelta.fn(30) diracdelta.fn(40)

The Fourier series for the Dirac delta function is rapidly built up and you end up with this graph.

Bibliography

[1] Nicholas J. Horton, Elizabeth R. Brown and Linjuan Qian, “Use of R as a Toolbox for Mathematical Statistics Exploration”, The American Statistician, 58, 343 (2004). DOI:10.1198/000313004X5572

[2] Katharine M. Mullen and Ivo H. M. van Stokkum, “An Introduction to the Special Volume ‘Spectroscopy and Chemometrics in R’”, Journal of Statistical Software, 18, 1 (2007). Download

*     A different version of R, called Arc has been proposed by the University of Minnesota. Arc stands for Applied Regression Including Computing and Graphics. The authors of Arc claim that their approach is more general than R and corresponds to the way people supposedly “think about data”. In my opinion, this is an unhelpful development. If R had been found wanting, as the authors of Arc claim, then they would have been better employed in developing R further, in the same spirit of the evolution of LaTeX, rather than producing different software.