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by: Alvena Marks IV


Alvena Marks IV

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This 75 page Class Notes was uploaded by Alvena Marks IV on Thursday October 29, 2015. The Class Notes belongs to FNCE 7550 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/231810/fnce-7550-university-of-colorado-at-boulder in Finance at University of Colorado at Boulder.

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Date Created: 10/29/15
An Introduction to R Notes on R A Programming Environment for Data Analysis and Graphics Version 272 2008 08 25 W N Venables D M Smith and the R Development Core Team Copyright 1990 W N Venables Copyright 1992 W N Venables amp D M Smith Copyright 1997 R Gentleman amp R lhaka Copyright 19977 1998 M Maechler Copyright 199972006 R Development Core Team Permission is granted to make and distribute verbatim copies of this manual provided the copy right notice and this permission notice are preserved on all copies Permission is granted to copy and distribute modi ed versions of this manual under the condi tions for verbatim copying7 provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one Permission is granted to copy and distribute translations of this manual into another language7 under the above conditions for modi ed versions7 except that this permission notice may be stated in a translation approved by the R Development Core Team ISBN 3 900051127 Chapter 1 Introduction and preliminaries 3 There is an important difference in philosophy between S and hence R and the other main statistical systems In S a statistical analysis is normally done as a series of steps with intermediate results being stored in objects Thus whereas SAS and SPSS will give copious output from a regression or discriminant analysis R will give minimal output and store the results in a t object for subsequent interrogation by further R functions 14 R and the window system The most convenient way to use R is at a graphics workstation running a windowing system This guide is aimed at users who have this facility In particular we will occasionally refer to the use of R on an X window system although the vast bulk of what is said applies generally to any implementation of the R environment Most users will nd it necessary to interact directly with the operating system on their computer from time to time In this guide we mainly discuss interaction with the operating system on UNIX machines If you are running R under Windows or MacOS you will need to make some small adjustments Setting up a workstation to take full advantage of the customizable features of R is a straight forward if somewhat tedious procedure and will not be considered further here Users in dif culty should seek local expert help 15 Using R interactively When you use the R program it issues a prompt when it expects input commands The default prompt is gt which on UNIX might be the same as the shell prompt and so it may appear that nothing is happening However as we shall see it is easy to change to a different R prompt if you wish We will assume that the UNIX shell prompt is In using R under UNIX the suggested procedure for the rst occasion is as follows 1 Create a separate sub directory say work to hold data les on which you will use R for this problem This will be the working directory whenever you use R for this particular problem mkdir work Cd work Start the R program with the command R At this point R commands may be issued see later to F To quit the R program the command is gt 10 At this point you will be asked whether you want to save the data from your R session On some systems this will bring up a dialog box and on others you will receive a text prompt to which you can respond yes no or cancel a single letter abbreviation will do to save the data before quitting quit without saving or return to the R session Data which is saved will be available in future R sessions Further R sessions are simple 1 Make work the working directory and start the program as before Cd work R 2 Use the R program terminating with the q command at the end of the session To use R under Windows the procedure to follow is basically the same Create a folder as the working directory and set that in the Start In7 eld in your R shortcut Then launch R by double clicking on the icon Chapter 1 Introduction and preliminaries 6 gt rmx y z ink junk temp foo bar All objects created during an R sessions can be stored permanently in a le for use in future R sessions At the end of each R session you are given the opportunity to save all the currently available objects If you indicate that you want to do this7 the objects are written to a le called RData75 in the current directory7 and the command lines used in the session are saved to a le called Rhistory When R is started at later time from the same directory it reloads the workspace from this le At the same time the associated commands history is reloaded It is recommended that you should use separate working directories for analyses conducted with R It is quite common for objects with names x and y to be created during an analysis Names like this are often meaningful in the context of a single analysis7 but it can be quite hard to decide what they might be when the several analyses have been conducted in the same directory 5 The leading dot in this le name makes it invisible in normal le listings in UNIX Chapter 2 Simple manipulations numbers and vectors 8 and so on all have their usual meaning max and min select the largest and smallest elements of a vector respectively range is a function whose value is a vector of length two namely 6 minx maxx lengthx is the number of elements in x sumx gives the total of the elements in x and prodx their product Two statistical functions are meanx which calculates the sample mean which is the same as sumxlengthx and varx which gives sum xmeanx quot2 lengthx 1 or sample variance If the argument to varO is an n by p matrix the value is a p by p sample covariance matrix got by regarding the rows as independent p variate sample vectors sort x returns a vector of the same size as x with the elements arranged in increasing order however there are other more exible sorting facilities available see order or sortlist which produce a permutation to do the sorting Note that max and min select the largest and smallest values in their arguments even if they are given several vectors The parallel maximum and minimum functions pmax and pmin return a vector of length equal to their longest argument that contains in each element the largest smallest element in that position in any of the input vectors For most purposes the user will not be concerned if the numbers in a numeric vector are integers reals or even complex lnternally calculations are done as double precision real numbers or double precision complex numbers if the input data are complex To work with complex numbers supply an explicit complex part Thus sqrt 17 will give NaN and a warning but sqrt 170i will do the computations as complex numbers 23 Generating regular sequences R has a number of facilities for generating commonly used sequences of numbers For example 130 is the vector Cl 2 29 30 The colon operator has high priority within an ex pression so for example 21 15 is the vector C2 4 28 30 Put n lt 10 and compare the sequences 1 n 1 and 1 n 1 The construction 30 1 may be used to generate a sequence backwards The function seq is a more general facility for generating sequences It has ve arguments only some of which may be speci ed in any one call The rst two arguments if given specify the beginning and end of the sequence and if these are the only two arguments given the result is the same as the colon operator That is seq2 10 is the same vector as 2 10 Parameters to ser and to many other R functions can also be given in named form in which case the order in which they appear is irrelevant The rst two parameters may be named fromva1ue and toVa1ue thus seq130 seqfrom1 to30 and seqto30 from1 are all the same as 130 The next two parameters to seq may be named byva1ue and lengthva1ue which specify a step size and a length for the sequence respectively If neither of these is given the default by1 is assumed For example gt seq5 5 by2 gt SS generates in 3 the vector c 50 48 46 46 48 50 Similarly gt 54 lt seqlength51 from5 by2 generates the same vector in s4 Chapter 2 Simple manipulations numbers and vectors 9 The fth parameter may be named alongvector which if used must be the only parameter and creates a sequence 1 2 lengthvector or the empty sequence if the vector is empty as it can be A related function is repO which can be used for replicating an object in various complicated ways The simplest form is gt 55 lt repx times5 which will put ve copies of x end to end in 55 Another useful version is gt 56 lt repx each5 which repeats each element of x ve times before moving on to the next 24 Logical vectors As well as numerical vectors R allows manipulation of logical quantities The elements of a logical vector can have the values TRUE FALSE and NA for not available see below The rst two are often abbreviated as T and F respectively Note however that T and F are just variables which are set to TRUE and FALSE by default but are not reserved words and hence can be overwritten by the user Hence you should always use TRUE and FALSE Logical vectors are generated by conditions For example gt temp lt x gt 13 sets temp as a vector of the same length as x with values FALSE corresponding to elements of x where the condition is not met and TRUE where it is The logical operators are lt lt gt gt for exact equality and for inequality In addition if c1 and c2 are logical expressions then c1 amp c2 is their intersection and cl c2 is their union or and c1 is the negation of c1 Logical vectors may be used in ordinary arithmetic in which case they are coerced into numeric vectors FALSE becoming O and TRUE becoming 1 However there are situations where logical vectors and their coerced numeric counterparts are not equivalent for example see the next subsection 25 Missing values In some cases the components of a vector may not be completely known When an element or value is not available or a missing value in the statistical sense a place within a vector may be reserved for it by assigning it the special value NA In general any operation on an NA becomes an NA The motivation for this rule is simply that if the speci cation of an operation is incomplete the result cannot be known and hence is not available The function isnax gives a logical vector of the same size as x with value TRUE if and only if the corresponding element in x is NA gt z lt c13NA ind lt isnaz Notice that the logical expression x NA is quite different from isnax since NA is not really a value but a marker for a quantity that is not available Thus x NA is a vector of the same length as x all of whose values are NA as the logical expression itself is incomplete and hence undecidable Note that there is a second kind of missing values which are produced by numerical com putation the so called Not a Number NaN values Examples are gt 00 Chapter 2 Simple manipulations numbers and vectors 10 gt Inf Inf which both give NaN since the result cannot be de ned sensibly In summary isnaxx is TRUE both for NA and NaN values To differentiate these is nanxx is only TRUE for NaNs Missing values are sometimes printed as ltNAgt when character vectors are printed without quotes 26 Character vectors Character quantities and character vectors are used frequently in R for example as plot labels Where needed they are denoted by a sequence of characters delimited by the double quote character eg quotxvaluesquot quotNew iteration resultsquot Character strings are entered using either matching double quot or single quotes but are printed using double quotes or sometimes without quotes They use C style escape sequences using as the escape character so is entered and printed as and inside double quotes quot is entered as quot Other useful escape sequences are n newline t tab and b backspaceisee Quotes for a full list Character vectors may be concatenated into a vector by the c function examples of their use will emerge frequently The paste function takes an arbitrary number of arguments and concatenates them one by one into character strings Any numbers given among the arguments are coerced into character strings in the evident way that is in the same way they would be if they were printed The arguments are by default separated in the result by a single blank character but this can be changed by the named parameter sepstring which changes it to string possibly empty For example gt labs lt pastecquotXquot quotYquot 110 sepquotquot makes labs into the character vector CquotXlquot quotY2quot quotX3quot quotY4quot quotX5quot quotY6quot quotX7quot quotY8quot quotX9quot quotY10quot Note particularly that recycling of short lists takes place here too thus CquotXquot quotYquot is repeated 5 times to match the sequence 1103 27 Index vectors selecting and modifying subsets of a data set Subsets of the elements of a vector may be selected by appending to the name of the vector an index vector in square brackets More generally any expression that evaluates to a vector may have subsets of its elements similarly selected by appending an index vector in square brackets immediately after the expression Such index vectors can be any of four distinct types 1 A logical vector In this case the index vector must be of the same length as the vector from which elements are to be selected Values corresponding to TRUE in the index vector are selected and those corresponding to FALSE are omitted For example gt y lt xisnax creates or recreates an object y which will contain the non missing values of x in the same order Note that if x has missing values y will be shorter than x Also gt x1 isnax amp xgt0 gt 2 creates an object 2 and places in it the values of the vector x1 for which the corresponding value in x was both non missing and positive 1 collapsess joins the arguments into a single Character string putting ss in between There are pas s more tools for character manipulation see the help for sub and substrlng Chapter 5 Arrays and matrices 19 53 Index matrices As well as an index vector in any subscript position a matrix may be used with a single index matrix in order either to assign a vector of quantities to an irregular collection of elements in the array or to extract an irregular collection as a vector A matrix example makes the process clear In the case of a doubly indexed array an index matrix may be given consisting of two columns and as many rows as desired The entries in the index matrix are the row and column indices for the doubly indexed array Suppose for example we have a 4 by 5 array X and we wish to do the following 0 Extract elements X 1 3 X 2 2 and X 3 1 as a vector structure and 0 Replace these entries in the array X by zeroes In this case we need a 3 by 2 subscript array as in the following example gt x lt array120 dimc 45 Generate a 4 by 5 array gt x 1 2 3 4 5 1 1 5 9 13 17 2 2 6 10 14 18 3 3 7 11 15 19 4 4 8 12 16 20 gt i lt arrayc1331 dimc32 gt i i is a 3 by 2 index array 1 2 1 1 3 2 2 2 3 3 1 gt xi Extract those elements 1 9 6 3 gt xi lt O Replace those elements by zeros gt x 1 2 3 4 5 1 1 5 o 13 17 2 2 o 10 14 18 3 o 7 11 15 19 4 4 8 12 16 20 gt Negative indices are not allowed in index matrices NA and zero values are allowed rows in the index matrix containing a zero are ignored and rows containing an NA produce an NA in the result As a less trivial example suppose we wish to generate an unreduced design matrix for a block design de ned by factors blocks b levels and varieties v levels Further suppose there are n plots in the experiment We could proceed as follows gt Xb lt matrix0 n b Xv lt matrix0 n v ib lt cbind1n blocks iv lt cbind1n varieties Xb ib lt 1 Xv iv lt 1 gt X lt cbindXb Xv VVVV V To construct the incidence matrix N say we could use gt N lt crossprodXb Xv Chapter 5 Arrays and matrices 22 57 Matrix facilities As noted above a matrix is just an array with two subscripts However it is such an important special case it needs a separate discussion R contains many operators and functions that are available only for matrices For example tX is the matrix transpose function as noted above The functions nrowA and ncolA give the number of rows and columns in the matrix A respectively 571 Matrix multiplication The operator o o is used for matrix multiplication An 71 by 1 or 1 by 71 matrix may of course be used as an n vector if in the context such is appropriate Conversely vectors which occur in matrix multiplication expressions are automatically promoted either to row or column vectors whichever is multiplicatively coherent if possible although this is not always unambiguously possible as we see later If for example A and B are square matrices of the same size then gt A B is the matrix of element by element products and gt A 796 B is the matrix product If x is a vector then gt x A x is a quadratic form1 The function crossprodO forms crossproducts meaning that crossprodX y is the same as tX o o y but the operation is more e icient If the second argument to crossprodO is omitted it is taken to be the same as the rst The meaning of diagO depends on its argument diagv where v is a vector gives a diagonal matrix with elements of the vector as the diagonal entries On the other hand di agM where M is a matrix gives the vector of main diagonal entries of M This is the same convention as that used for diagO in MATLAB Also somewhat confusingly if k is a single numeric value then diag k is the k by k identity matrix 572 Linear equations and inversion Solving linear equations is the inverse of matrix multiplication When after gt b lt A x only A and b are given the vector x is the solution of that linear equation system In R gt solveAb solves the system returning x up to some accuracy loss Note that in linear algebra formally x A lb where A 1 denotes the inverse of A which can be computed by solve A but rarely is needed Numerically it is both ine icient and potentially unstable to compute x lt solve A o o b instead of solveAb The quadratic form x A lx which is used in multivariate computations should be computed by something like x o o solve Ax rather than computing the inverse of A 1 Note that x 3939 x is ambiguous as it could mean either xx 0r xx where x is the column form In such cases the smaller matrix seems implicitly to be the interpretation adopted so the scalar xx is in this case the result The matrix xx may be calculated either by cbindx 3939 x or x 3939 rbindx since the result of rbindo 0r cbindo is always a matrix However the best way to compute xx 0r xx is crossprodx or x 39o39 K respectively 0 EVen better would be to form a matrix square root B with A BB and nd the squared length of the solution of By x perhaps using the Cholesky 0r eigendecomposition of A Chapter 5 Arrays and matrices 25 510 Frequency tables from factors Recall that a factor de nes a partition into groups Similarly a pair of factors de nes a two way cross classi cation and so on The function table allows frequency tables to be calcu lated from equal length factors If there are k factor arguments the result is a k Way array of frequencies Suppose for example that statef is a factor giving the state code for each entry in a data vector The assignment gt statefr lt tablestatef gives in statefr a table of frequencies of each state in the sample The frequencies are ordered and labelled by the levels attribute of the factor This simple case is equivalent to but more convenient than gt statefr lt tapplystatef statef length Further suppose that incomef is a factor giving a suitably de ned income class77 for each entry in the data vector for example with the cut function gt factorcutincomes breaks 351007 gt incomef Then to calculate a two way table of frequencies gt tableincomef statef statef incomef act nsw nt qld sa tas ViC wa 35 45 1 1 0 1 0 0 1 0 45 55 1 1 1 1 2 0 1 3 55 65 0 3 1 3 2 2 2 1 65 75 0 1 0 0 0 0 1 0 Extension to higher way frequency tables is immediate Chapter 6 Lists and data frames 26 6 Lists and data frames 61 Lists An R list is an object consisting of an ordered collection of objects known as its components There is no particular need for the components to be of the same mode or type and for example a list could consist of a numeric vector a logical value a matrix a complex vector a character array a function and so on Here is a simple example of how to make a list gt Lst lt list namequotFredquot wifequotMaryquot nochildren3 child agesc 4 7 9 Components are always numbered and may always be referred to as such Thus if Lst is the name of a list with four components these may be individually referred to as Lst1 Lst 2 L51 3 and L51 4 lf further L51 4 is a vector subscripted array then Lst4 1 is its rst entry If Lst is a list then the function lengthLst gives the number of top level components it has Components of lists may also be named and in this case the component may be referred to either by giving the component name as a character string in place of the number in double square brackets or more conveniently by giving an expression of the form gt namecomponentname for the same thing This is a very useful convention as it makes it easier to get the right component if you forget the number So in the simple example given above Lstname is the same as L51 1 and is the string quotFredquot Lstwife is the same as L51 2 and is the string quotMaryquot Lstchildages 1 is the same as Lst4 1 and is the number 4 Additionally one can also use the names of the list components in double square brackets ie Lst quotnamequot is the same as Lstname This is especially useful when the name of the component to be extracted is stored in another variable as in gt x lt quotnamequot Lstx It is very important to distinguish L51 1 from L51 1 is the operator used to select a single element whereas is a general subscripting operator Thus the former is the rst object in the list Lst and if it is a named list the name is not included The latter is a sublist of the list Lst consisting of the rst entry only If it is a named list the names are transferred to the sublist The names of components may be abbreviated down to the minimum number of letters needed to identify them uniquely Thus Lstcoefficients may be minimally speci ed as Lstcoe and Lstcovariance as Lstcov The vector of names is in fact simply an attribute of the list like any other and may be handled as such Other structures besides lists may of course similarly be given a names attribute also 62 Constructing and modifying lists New lists may be formed from existing objects by the function list An assignment of the form Chapter 6 Lists and data frames 28 The attachO function takes a database such as a list or data frame as its argument Thus suppose lentils is a data frame with three variables lentilsu lentilsv lentilsw The attach gt attachlentils places the data frame in the search path at position 2 and provided there are no variables 11 v or w in position 1 u v and w are available as variables from the data frame in their own right At this point an assignment such as gt u lt vw does not replace the component 11 of the data frame but rather masks it with another variable 11 in the working directory at position 1 on the search path To make a permanent change to the data frame itself the simplest way is to resort once again to the notation gt lentilsu lt vw However the new value of component 11 is not visible until the data frame is detached and attached again To detach a data frame use the function gt detachO More precisely this statement detaches from the search path the entity currently at position 2 Thus in the present context the variables 11 v and w would be no longer visible except under the list notation as lentilsu and so on Entities at positions greater than 2 on the search path can be detached by giving their number to detach but it is much safer to always use a name for example by detachlentils or detachquotlentilsquot Note In R lists and data frames can only be attached at position 2 or above and what is attached is a copy of the original object You can alter the attached values via assign but the original list or data frame is unchanged 633 Working With data frames A useful convention that allows you to work with many different problems comfortably together in the same working directory is o gather together all variables for any well de ned and separate problem in a data frame under a suitably informative name 0 when working with a problem attach the appropriate data frame at position 2 and use the working directory at level 1 for operational quantities and temporary variables before leaving a problem add any variables you wish to keep for future reference to the data frame using the form of assignment and then detachO nally remove all unwanted variables from the working directory and keep it as clean of left over temporary variables as possible In this way it is quite simple to work with many problems in the same directory all of which have variables named x y and z for example 634 Attaching arbitrary lists attachO is a generic function that allows not only directories and data frames to be attached to the search path but other classes of object as well In particular any object of mode quotlistquot may be attached in the same way gt attachany old list Anything that has been attached can be detached by detach by position number or prefer ably by name Chapter 6 Lists and data frames 29 635 Managing the search path The function search shows the current search path and so is a very useful way to keep track of which data frames and lists and packages have been attached and detached Initially it gives gt search 1 quot GlobalEnvquot quotAutoloadsquot quotpackagezbasequot where GlobalEnv is the workspace1 After lentils is attached we have gt search 1 quot GlobalEnvquot lentils quotAutoloadsquot quotpackage basequot gt ls2 quotuquot quotVquot quotwquot and as we see ls or obj ects can be used to examine the contents of any position on the search path Finally7 we detach the data frame and con rm it has been removed from the search path gt detachquotlentilsquot gt search 1 quot GlobalEnvquot quotAutoloadsquot quotpackagezbasequot 1 See the online help for autoload for the meaning of the second term Chapter 7 Reading data from les 30 7 Reading data from les Large data objects will usually be read as values from external les rather than entered during an R session at the keyboard R input facilities are simple and their requirements are fairly strict and even rather in exible There is a clear presumption by the designers of R that you will be able to modify your input les using other tools such as le editors or Perll to t in with the requirements of R Generally this is very simple If variables are to be held mainly in data frames as we strongly suggest they should be an entire data frame can be read directly with the readtable function There is also a more primitive input function scan that can be called directly For more details on importing data into R and also exporting data see the B Data Im portExport manual 71 The readtab1e function To read an entire data frame directly the external le will normally have a special form 0 The rst line of the le should have a name for each variable in the data frame 0 Each additional line of the le has as its rst item a row label and the values for each variable If the le has one fewer item in its rst line than in its second this arrangement is presumed to be in force So the rst few lines of a le to be read as a data frame might look as follows lnput le form with names and row labels Price Floor Area Rooms Age Centheat 01 5200 1110 830 5 62 no 02 5475 1280 710 5 75 no 03 5750 1010 1000 5 42 no 04 5750 1310 690 6 88 no 05 5975 930 900 5 19 yes By default numeric items except row labels are read as numeric variables and non numeric variables such as Centheat in the example as factors This can be changed if necessary The function readtable can then be used to read the data frame directly gt HousePrice lt read table quothouses dataquot Often you will want to omit including the row labels directly and use the default labels In this case the le may omit the row label column as in the following lnput le form without row labels Price Floor Area Rooms Age Cent heat 5200 1110 830 5 62 no 5475 1280 710 5 75 no 5750 1010 1000 5 42 no 5750 1310 690 6 88 no 5 1 9 yes 59 75 93 0 900 1 Under UNIX the utilities Sed or Awk can be used Chapter 7 Reading data from les 32 74 Editing data When invoked on a data frame or matrix7 edit brings up a separate spreadsheet like environment for editing This is useful for making small changes once a data set has been read The command gt xnew lt editxold will allow you to edit your data set xold7 and on completion the changed object is assigned to xnew If you want to alter the original dataset xold7 the simplest way is to use fixxold7 which is equivalent to xold lt editxold Use gt xnew lt edit data frame to enter new data via the spreadsheet interface Chapter 8 Probability distributions 33 8 Probability distributions 81 R as a set of statistical tables One convenient use of R is to provide a comprehensive set of statistical tables Functions are provided to evaluate the cumulative distribution function PX x the probability density function and the quantile function given q the smallest x such that PX z gt q and to simulate from the distribution Distribution R name additional arguments beta beta shapel shape2 ncp binomial binom size prob Cauchy cauchy location scale chi squared chisq df ncp exponential exp rate f df 1 df 2 ncp gamma gamma shape scale geometric geom prob hypergeometric hyper m n k log normal lnorm meanlog sdlog logistic logis location scale negative binomial nbinom size prob normal norm mean sd Poisson poi s lambda Student s t t df ncp uniform unif min max Weibull weibull shape scale Wilcoxon wilcox m n Pre x the name given here by d for the density p for the CDF q for the quantile function and r for simulation random deviates The rst argument is x for dxxx q for pxxx p for qxxx and n for rxxx except for rhyper and rwilcox for which it is ln not quite all cases is the non centrality parameter ncp are currently available see the on line help for details The pxxx and qxxx functions all have logical arguments lowertail and logp and the dxxx ones have log This allows eg getting the cumulative or integrated hazard function 11175 10g1 Ft7 by pXXXt lowertail FALSE logp TRUE or more accurate log likelihoods by dxxx log TRUE directly In addition there are functions ptukey and qtukey for the distribution of the studentized range of samples from a normal distribution Here are some examples gt 2 tailed p value for t distribution gt 2pt243 df 13 gt upper 1 point for an F2 7 distribution gt qf O 01 2 7 lower tail FALSE 82 Examining the distribution of a set of data Given a univariate set of data we can examine its distribution in a large number of ways The simplest is to examine the numbers Two slightly different summaries are given by summary and fivenum and a display of the numbers by stem a stem and leaf77 plot Chapter 8 Probability distributions 34 gt attachfaithful gt summaryeruptions Min lst Qu Median Mean 3rd Qu Max 1600 2163 4000 3488 4454 5100 gt fivenumeruptions 1 16000 21585 40000 44585 51000 gt stemeruptions The decimal point is 1 digits to the left of the I 16 I 070355555588 18 lllI l888 20 I 00002223378800035778 22 I 0002335578023578 24 I 00228 26 I 23 28 I 080 30 I 7 32 I 2337 34 I 250077 36 I 0000823577 38 I 2333335582225577 40 I 0000003357788888002233555577778 42 I 03335555778800233333555577778 44 I l8888 46 I 0000233357700000023578 48 I 00000022335800333 50 I 0370 A stem and leaf plot is like a histogram7 and R has a function hist to plot histograms gt histeruptions make the bins smaller7 make a plot of density gt hist eruptions seq1 6 5 2 O 2 probTRUE gt linesdensityeruptions bw0 1 gt rugeruptions show the actual data points More elegant density plots can be made by density and we added a line produced by density in this example The bandwidth bw was chosen by trial and error as the default gives Chapter 8 Probability distributions 35 too much smoothing it usually does for interesting densities Better automated methods of bandwidth choice are available7 and in this example bw quotSJquot gives a good result Histogram of eruptions R elawe Frequency m n l fi 15 2D 25 an 35 4D 45 an empnuns We can plot the empirical cumulative distribution function by using the function ecdf gt plot ecdf eruptions do pointsFALSE verticalsTRUE This distribution is obviously far from any standard distribution How about the right hand mode7 say eruptions of longer than 3 minutes Let us t a normal distribution and overlay the tted CDF gt long lt eruptions eruptions gt 3 gt plot ecdf long do pointsFALSE verticalsTRUE gt x lt seq3 54 001 gt linesx pnormx meanmeanlong sdsqrtvarlong lty3 ecdflong Quantile quantile Q Q plots can help us examine this more carefully parptyquotsquot arrange for a square figure region qqnormlong qqlinelong Chapter 8 Probability distributions 36 which shows a reasonable t but a shorter right tail than one would expect from a normal distribution Let us compare this with some simulated data from a 25 distribution Normal Q Q Plot Sample Ouanhles Theuveucal Ouarmles x lt rt250 df 5 qqnorm x qqlinex which will usually if it is a random sample show longer tails than expected for a normal We can make a Q Q plot against the generating distribution by qqplot qt ppoints250 df 5 x xlab quotQQ plot for t dsnquot qqline x Finally7 we might want a more formal test of agreement with normality or not R provides the Shapiro Wilk test gt shapiro test long ShapiroWilk normality test data long W 09793 pvalue 001052 and the Kolmogorov Smirnov test gt ks testlong quotpnormquot mean meanlong sd sqrtvarlong Onesample KolmogorovSmirnov test data long D 00661 pvalue 04284 alternative hypothesis twosided Note that the distribution theory is not valid here as we have estimated the parameters of the normal distribution from the same sample 83 One and twosample tests So far we have compared a single sample to a normal distribution A much more common operation is to compare aspects of two samples Note that in R7 all classical tests including the ones used below are in package stats which is normally loaded Consider the following sets of data on the latent heat of the fusion of ice calgm from Rice 19957 p490 Chapter 8 Probability distributions Method A 7998 8004 8002 8004 8003 8003 8004 7997 8005 8003 8002 8000 8002 Method B 8002 7994 7998 7997 7997 8003 7995 7997 Boxplots provide a simple graphical comparison of the two samples A lt scan 7998 8004 8002 8004 8003 8003 8004 7997 8005 8003 8002 8000 8002 B lt scan 8002 7994 7998 7997 7997 8003 7995 7997 boxplotA B which indicates that the rst group tends to give higher results than the second 79 9e 79 99 an an an 02 an m 79 94 To test for the equality of the means of the two examples7 we can use an unpaired t test by gt ttestA B Welch Two Sample ttest data A and B t 32499 df 12027 p value 000694 alternative hypothesis true difference in means is not equal to 0 95 percent confidence interval 001385526 007018320 sample estimates mean of x mean of y 8002077 7997875 which does indicate a signi cant difference7 assuming normality By default the R function does not assume equality of variances in the two samples in contrast to the similar S PLUS ttest function We can use the F test to test for equality in the variances7 provided that the two samples are from normal populations gt var test A B F test to compare two variances Chapter 8 Probability distributions 38 data A and B F 05837 num df 12 denom df 7 pvalue 03938 alternative hypothesis true ratio of variances is not equal to 1 95 percent confidence interval 01251097 21052687 sample estimates ratio of variances 05837405 which shows no evidence of a signi cant difference7 and so we can use the classical t test that assumes equality of the variances gt ttestA B varequalTRUE Two Sample ttest data A and B t 34722 df 19 pvalue 0002551 alternative hypothesis true difference in means is not equal to 0 95 percent confidence interval 001669058 006734788 sample estimates mean of x mean of y 8002077 7997875 All these tests assume normality of the two samples The two sample Wilcoxon or Mann Whitney test only assumes a common continuous distribution under the null hypothesis gt wilcoxtestA B Wilcoxon rank sum test with continuity correction data A and B W 89 pvalue 0007497 alternative hypothesis true location shift is not equal to 0 Warning message Cannot compute exact pvalue with ties in wilcoxtestA B Note the warning there are several ties in each sample7 which suggests strongly that these data are from a discrete distribution probably due to rounding There are several ways to compare graphically the two samples We have already seen a pair of boxplots The following gt plot ecdf A do pointsFALSE verticalsTRUE xlimrange A B gt plot ecdf B do pointsFALSE verticalsTRUE addTRUE will show the two empirical CDFs7 and qqplot will perform a Q Q plot of the two samples The Kolmogorov Smirnov test is of the maximal vertical distance between the two ecdf s7 assuming a common continuous distribution gt kstestA B Twosample KolmogorovSmirnov test data A and B D 05962 p value 005919 Chapter 8 Probability distributions alternative hypothesis twosided Warning message cannot compute correct pvalues with ties in kstestA B Chapter 9 Grouping loops and conditional execution 40 9 Grouping loops and conditional execution 91 Grouped expressions R is an expression language in the sense that its only command type is a function or expression which returns a result Even an assignment is an expression whose result is the value assigned and it may be used wherever any expression may be used in particular multiple assignments are possible Commands may be grouped together in braces expr1 exprm in which case the value of the group is the result of the last expression in the group evaluated Since such a group is also an expression it may for example be itself included in parentheses and used a part of an even larger expression and so on 92 Control statements 921 Conditional execution if statements The language has available a conditional construction of the form gt if expr1 expr2 else expr3 where exprJ must evaluate to a single logical value and the result of the entire expression is then evident The short circuit operators ampamp and I are often used as part of the condition in an if statement Whereas amp and apply element wise to vectors ampamp and apply to vectors of length one and only evaluate their second argument if necessary There is a vectorized version of the ifelse construct the ifelse function This has the form ifelsecondition a b and returns a vector of the length of its longest argument with elements ai if condition i is true otherwise bi 922 Repetitive execution for loops repeat and while There is also a for loop construction which has the form gt for name in expr1 expr2 where name is the loop variable expril is a vector expression often a sequence like 1 20 and expr2 is often a grouped expression with its sub expressions written in terms of the dummy name expri2 is repeatedly evaluated as name ranges through the values in the vector result of expril As an example suppose ind is a vector of class indicators and we wish to produce separate plots of y versus x within classes One possibility here is to use coplot1 which will produce an array of plots corresponding to each level of the factor Another way to do this now putting all plots on the one display is as follows gt xc lt splitx ind gt yc lt splity ind gt for i in 1lengthyc plotxc ill yc ill abline lsfit xc i yc i Note the function split which produces a list of vectors obtained by splitting a larger vector according to the classes speci ed by a factor This is a useful function mostly used in connection with boxplots See the help facility for further details 1 to be discussed later or use xyplot from package lattice Chapter 10 Writing your own functions 42 10 Writing your own functions As we have seen informally along the way the R language allows the user to create objects of mode function These are true R functions that are stored in a special internal form and may be used in further expressions and so on In the process the language gains enormously in power convenience and elegance and learning to write useful functions is one of the main ways to make your use of R comfortable and productive It should be emphasized that most of the functions supplied as part of the R system such as mean varO postscriptO and so on are themselves written in R and thus do not differ materially from user written functions A function is de ned by an assignment of the form gt name lt functionarg1 arg2 expression The expression is an R expression usually a grouped expression that uses the arguments argj to calculate a value The value of the expression is the value returned for the function A call to the function then usually takes the form name expr1 expr2 and may occur anywhere a function call is legitimate 101 Simple examples As a rst example consider a function to calculate the two sample t statistic showing all the steps This is an arti cial example of course since there are other simpler ways of achieving the same end The function is de ned as follows gt twosam lt functiony1 y n1 lt lengthy1 n2 lt lengthy2 ybl lt meany1 yb2 lt meany2 51 lt vary1 52 lt vary2 s lt n11sl n2152n1n22 tst lt ybl yb2sqrts1n1 1n2 tst With this function de ned you could perform two sample t tests using a call such as gt tstat lt twosamdatamale datafemale tstat As a second example consider a function to emulate directly the MATLAB backslash com mand which returns the coef cients of the orthogonal projection of the vector y onto the column space of the matrix X This is ordinarily called the least squares estimate of the regression coef cients This would ordinarily be done with the qu function however this is sometimes a bit tricky to use directly and it pays to have a simple function such as the following to use it safely Thus given a n by 1 vector y and an n by p matrix X then X y is de ned as X X X y where X X is a generalized inverse of X X gt bslash lt functionX y f X lt qrX qr coef X y After this object is created it may be used in statements such as gt regcoeff lt bslashXmat yvar and so on Chapter 10 Writing your own functions 45 It is numerically slightly better to work with the singular value decomposition on this occasion rather than the eigenvalue routines The result of the function is a list giving not only the ef ciency factors as the rst component but also the block and variety canonical contrasts since sometimes these give additional useful qualitative information 1062 Dropping all names in a printed array For printing purposes with large matrices or arrays it is often useful to print them in close block form without the array names or numbers Removing the dimnames attribute will not achieve this effect but rather the array must be given a dimnames attribute consisting of empty strings For example to print a matrix X gt temp lt X gt dimnamestemp lt listrepquotquot nrowX repquot quot ncolX gt temp rmtemp This can be much more conveniently done using a function nodimnames shown below as a wrap around77 to achieve the same result It also illustrates how some effective and useful user functions can be quite short nodimnames lt functiona Remove all dimension names from an array for compact printing d lt list 1 lt O fori in dima d1 lt 1 1 lt Iepquotquot i dimnamesa lt d a With this function de ned an array may be printed in close format using gt no dimnames X This is particularly useful for large integer arrays where patterns are the real interest rather than the values 1063 Recursive numerical integration Functions may be recursive and may themselves de ne functions within themselves Note however that such functions or indeed variables are not inherited by called functions in higher evaluation frames as they would be if they were on the search path The example below shows a naive way of performing one dimensional numerical integration The integrand is evaluated at the end points of the range and in the middle If the onepanel trapezium rule answer is close enough to the two panel then the latter is returned as the value Otherwise the same process is recursively applied to each panel The result is an adaptive integration process that concentrates function evaluations in regions where the integrand is farthest from linear There is however a heavy overhead and the function is only competitive with other algorithms when the integrand is both smooth and very dif cult to evaluate The example is also given partly as a little puzzle in R programming area lt functionf a b eps 1 Oe06 lim 10 funl lt functionf a b fa fb a0 eps lim fun function funl is only visible inside area d lt a b2 Chapter 10 Writing your own functions h lt b a4 fd lt fd a1 lt h fa fd 212 lt h fd fb ifabsa0 a1 a2 lt eps II lim 0 returna1 a2 else returnfunf a d fa fd a1 eps lim 1 fun funf d b fd fb a2 eps lim 1 fun fa lt fa fb lt fb 210 lt fa fb b a2 fun1f a b fa fb a0 eps lim funl 107 Scope The discussion in this section is somewhat more technical than in other parts of this document However it details one of the major differences between S PLUS and R The symbols which occur in the body of a function can be divided into three classes formal parameters local variables and free variables The formal parameters of a function are those occurring in the argument list of the function Their values are determined by the process of binding the actual function arguments to the formal parameters Local variables are those whose values are determined by the evaluation of expressions in the body of the functions Variables which are not formal parameters or local variables are called free variables Free variables become local variables if they are assigned to Consider the following function de nition f lt functionx y lt 2x printx printy printz In this function x is a formal parameter y is a local variable and z is a free variable In R the free variable bindings are resolved by rst looking in the environment in which the function was created This is called lexical scope First we de ne a function called cube cube lt functionn sq lt functionO nn nsq The variable 11 in the function sq is not an argument to that function Therefore it is a free variable and the scoping rules must be used to ascertain the value that is to be associated with it Under static scope S PLUS the value is that associated with a global variable named 11 Under lexical scope R it is the parameter to the function cube since that is the active binding for the variable 11 at the time the function sq was de ned The difference between evaluation in R and evaluation in S PLUS is that S PLUS looks for a global variable called 11 while R rst looks for a variable called 11 in the environment created when cube was invoked rst evaluation in S Sgt cube2 Error in sq Object quotnquot not found Chapter 10 Writing your own functions 47 Dumped Sgt n lt 3 Sgt cube2 1 18 then the same function evaluated in R Rgt cube2 1 8 Lexical scope can also be used to give functions mutable state In the following example we show how R can be used to mimic a bank account A functioning bank account needs to have a balance or total7 a function for making withdrawals7 a function for making deposits and a function for stating the current balance We achieve this by creating the three functions within account and then returning a list containing them When account is invoked it takes a numerical argument total and returns a list containing the three functions Because these functions are de ned in an environment which contains total7 they will have access to its value The special assignment operator7 ltlt 7 is used to change the value associated with total This operator looks back in enclosing environments for an environment that contains the symbol total and when it nds such an environment it replaces the value7 in that environment7 with the value of right hand side If the global or toplevel environment is reached without nding the symbol total then that variable is created and assigned to there For most users ltlt creates a global variable and assigns the value of the right hand side to it2 Only when ltlt has been used in a function that was returned as the value of another function will the special behavior described here occur openaccount lt functiontotal list deposit functionamount ifamount lt 0 stopquotDeposits must be positivenquot total ltlt total amount catamount quotdeposited Your balance isquot total quotnnquot withdraw functionamount ifamount gt total stopquotYou don t have that much moneynquot total ltlt total amount catamount quotwithdrawn Your balance isquot total quotnnquot balance function catquotYour balance isquot total quotnnquot ross lt openaccount100 robert lt openaccount200 rosswithdraw30 rossbalance robertbalance 2 In some sense this mimics the behavior in SePLUS since in SePLUS this operator always creates or assigns to a global Variable Chapter 10 Writing your own functions 48 rossdeposit 50 rossbalance0 rosswithdraw500 108 Customizing the environment Users can customize their environment in several different ways There is a site initialization le and every directory can have its own special initialization le Finally the special functions First and Last can be used The location ofthe site initialization le is taken from the value of the RPROFILE environment variable If that variable is unset the le Rprofilesite in the R home subdirectory etc is used This le should contain the commands that you want to execute every time R is started under your system A second personal pro le le named Rprofile73 can be placed in any directory If R is invoked in that directory then that le will be sourced This le gives individual users control over their workspace and allows for different startup procedures in different working directories If no Rprofile7 le is found in the startup directory then R looks for a Rprofile7 le in the user s home directory and uses that if it exists Any function named First in either of the two pro le les or in the RData7 image has a special status It is automatically performed at the beginning of an R session and may be used to initialize the environment For example the de nition in the example below alters the prompt to and sets up various other useful things that can then be taken for granted in the rest of the session Thus the sequence in which les are executed is Rprofilesite Rprofile RData7 and then First A de nition in later les will mask de nitions in earlier les gt First lt functionO optionspromptquot quot continuequottquot is the prompt optionsdigits5 length999 custom numbers and printout x110 for graphics parpch quotquot plotting character source file pathSys getenvquotHOMEquot quotRquot quotmystuff Rquot my personal functions libraryMASS attach a package Similarly a function Last if de ned is normally executed at the very end of the session An example is given below gt Last lt functionO graphics off a small safety measure cat pastedate quotnAdiosnquot Is it time for lunch 109 Classes generic functions and object orientation The class of an object determines how it will be treated by what are known as generic functions Put the other way round a generic function performs a task or action on its arguments speci c to the class of the argument itself If the argument lacks any class attribute or has a class not catered for speci cally by the generic function in question there is always a default action provided An example makes things clearer The class mechanism offers the user the facility of designing and writing generic functions for special purposes Among the other generic functions are plot 3 So it is hidden under UNIX Chapter 10 Writing your own functions 49 for displaying objects graphically summary for summarizing analyses of various types and anova for comparing statistical models The number of generic functions that can treat a class in a speci c way can be quite large For example the functions that can accommodate in some fashion objects of class quotdata frame include lt any asmatrix lt mean plot summary A currently complete list can be got by using the methods function gt methods class data frame Conversely the number of classes a generic function can handle can also be quite large For example the plot function has a default method and variants for objects of classes dataframe density factor and more A complete list can be got again by using the methods function gt methods plot For many generic functions the function body is quite short for example gt coef function object UseMethod coef The presence of UseMethod indicates this is a generic function To see what methods are available we can use methods gt methods coef 1 coefaov coefArima coefdefault coeflistof 5 coefnls coefsummarynls Nonvisible functions are asterisked In this example there are six methods none of which can be seen by typing its name We can read these by either of gt getAnywhere coef aov A single object matching coefaov was found It was found in the following places registered SS method for coef from namespace stats namespacezstats with value function object z lt objectcoef zisnaz gt getSBmethod coef aov function object z lt objectcoef z isnaz The reader is referred to the R Language De nition for a more complete discussion of this mechanism Chapter 11 Statistical models in R 50 11 Statistical models in R This section presumes the reader has some familiarity with statistical methodology in particular with regression analysis and the analysis of variance Later we make some rather more ambitious presumptions namely that something is known about generalized linear models and nonlinear regression The requirements for tting statistical models are suf ciently well de ned to make it possible to construct general tools that apply in a broad spectrum of problems R provides an interlocking suite of facilities that make tting statistical models very simple As we mention in the introduction the basic output is minimal and one needs to ask for the details by calling extractor functions 111 De ning statistical models formulae The template for a statistical model is a linear regression model with independent homoscedastic errors 7 yiZ jmljei ei NlD002 i1n j0 ln matrix terms this would be written yX e where the y is the response vector X is the model matrix or design matrix and has columns m0m1mp the determining variables Very often mo will be a column of ones de ning an intercept term Examples Before giving a formal speci cation a few examples may usefully set the picture Suppose y x x0 x1 x2 are numeric variables X is a matrix and A B C are factors The following formulae on the left side below specify statistical models as described on the right y quot x y quot 1 x Both imply the same simple linear regression model of y on m The rst has an implicit intercept term and the second an explicit one y quot O x y quot 1 x y quot x 1 Simple linear regression of y on x through the origin that is without an intercept term logy quot x1 x2 Multiple regression of the transformed variable logy on 1 and 2 with an implicit intercept term y quot polyx2 y 1xIxquot2 Polynomial regression of y on x of degree 2 The rst form uses orthogonal polyno mials and the second uses explicit powers as basis y quot X polyx2 Multiple regression y with model matrix consisting of the matrix X as well as polynomial terms in z to degree 2 Chapter 11 Statistical models in R 51 y quot A Single classi cation analysis of variance model of y with classes determined by A y quot A x Single classi cation analysis of covariance model of y with classes determined by A and with covariate z y AB y quot A B AzB y quot B oin o A y quot AB Two factor non additive model of y on A and B The rst two specify the same crossed classi cation and the second two specify the same nested classi cation In abstract terms all four specify the same model subspace y quot A B C quot2 y quot ABC ABC Three factor experiment but with a model containing main effects and two factor interactions only Both formulae specify the same model y quot A 1 x 1 Separate simple linear regression models of y on x within the levels of A with different codings The last form produces explicit estimates of as many different intercepts and slopes as there are levels in A y quot AB Error C An experiment with two treatment factors A and B and error strata determined by factor C For example a split plot experiment with whole plots and hence also subplots determined by factor C The operator is used to de ne a model formula in R The form for an ordinary linear model is response quot op1 tem1 op2 tem2 op3 term3 where response is a vector or matrix or expression evaluating to a vector or matrix de ning the response variables opi is an operator either or implying the inclusion or exclusion of a term in the model the rst is optional termj is either 0 a vector or matrix expression or 1 o a factor or o a formula expression consisting of factors vectors or matrices connected by formula operators In all cases each term de nes a collection of columns either to be added to or removed from the model matrix A 1 stands for an intercept column and is by default included in the model matrix unless explicitly removed The formula operators are similar in effect to the Wilkinson and Rogers notation used by such programs as Glim and Genstat One inevitable change is that the operator becomes z since the period is a valid name character in R The notation is summarized below based on Chambers amp Hastie 1992 p29 Y quot M Y is modeled as M M1 M2 lnclude MJ and M2 Chapter 11 Statistical models in R 52 M1 M2 Include MJ leaving out terms of M2 M1 M2 The tensor product of M1 and M2 If both terms are factors then the subclasses factor M1 oin o M2 Similar to M1 M2 but with a different coding M1 M2 M1 M2 M1M2 M1 M2 M1 M2 oin o M1 Mquotn All terms in M together with interactions up to order n I M lnsulate M lnside M all operators have their normal arithmetic meaning and that term appears in the model matrix Note that inside the parentheses that usually enclose function arguments all operators have their normal arithmetic meaning The function I is an identity function used to allow terms in model formulae to be de ned using arithmetic operators Note particularly that the model formulae specify the columns of the model matrix the speci cation of the parameters being implicit This is not the case in other contexts for example in specifying nonlinear models 1111 Contrasts We need at least some idea how the model formulae specify the columns of the model matrix This is easy if we have continuous variables as each provides one column of the model matrix and the intercept will provide a column of ones if included in the model What about a k level factor A The answer differs for unordered and ordered factors For unordered factors k 7 1 columns are generated for the indicators of the second kth levels of the factor Thus the implicit parameterization is to contrast the response at each level with that at the rst For ordered factors the k 7 1 columns are the orthogonal polynomials on 1 k omitting the constant term Although the answer is already complicated it is not the whole story First if the intercept is omitted in a model that contains a factor term the rst such term is encoded into k columns giving the indicators for all the levels Second the whole behavior can be changed by the options setting for contrasts The default setting in R is optionscontrasts c quotcontr treatment quot quotcontr polyquot The main reason for mentioning this is that R and S have different defaults for unordered factors S using Helmert contrasts So if you need to compare your results to those of a textbook or paper which used S PLUS you will need to set optionscontrasts c quotcontr helmertquot quotcontr polyquot This is a deliberate difference as treatment contrasts R s default are thought easier for new comers to interpret We have still not nished as the contrast scheme to be used can be set for each term in the model using the functions contrasts and C We have not yet considered interaction terms these generate the products of the columns introduced for their component terms Although the details are complicated model formulae in R will normally generate the models that an expert statistician would expect provided that marginality is preserved Fitting for example a model with an interaction but not the corresponding main effects will in general lead to surprising results and is for experts only Chapter 11 Statistical models in R 53 112 Linear models The basic function for tting ordinary multiple models is lm and a streamlined version of the call is as follows gt fittedmode1 lt lmformu1a data dataframe For example gt fm2 lt lmy quot x1 x2 data production would t a multiple regression model of y on 1 and 2 with implicit intercept term The important but technically optional parameter data production speci es that any variables needed to construct the model should come rst from the production data frame This is the case regardless of whether data frame production has been attached on the search path or not 113 Generic functions for extracting model information The value of lm is a tted model object technically a list of results of class quot1mquot Information about the tted model can then be displayed extracted plotted and so on by using generic functions that orient themselves to objects of class quot1mquot These include addl deviance formula predict step alias dropl kappa print summary anova effects labels proj vcov coef family plot residuals A brief description of the most commonly used ones is given below anovaobject1 object2 Compare a submodel with an outer model and produce an analysis of variance table coef object Extract the regression coef cient matrix Long form coefficientsobject devianceobject Residual sum of squares weighted if appropriate formulaobject Extract the model formula plotobject Produce four plots showing residuals tted values and some diagnostics predict object newdatadata frame The data frame supplied must have variables speci ed with the same labels as the original The value is a vector or matrix of predicted values corresponding to the determining variable values in dataframe printobject Print a concise version of the object Most often used implicitly residualsobject Extract the matrix of residuals weighted as appropriate Short form residobject stepobject Select a suitable model by adding or dropping terms and preserving hierarchies The model with the smallest value of A10 Akaike s An Information Criterion discovered in the stepwise search is returned Chapter 11 Statistical models in R 55 would t a ve variate multiple regression with variables presumably from the data frame production t an additional model including a sixth regressor variable and t a variant on the model where the response had a square root transform applied Note especially that if the data argument is speci ed on the original call to the model tting function this information is passed on through the tted model object to update and its allies The name can also be used in other contexts but with slightly different meaning For example gt fmfull lt lmy quot data production would t a model with response y and regressor variables all other variables in the data frame production Other functions for exploring incremental sequences of models are add1 drop1 and step The names of these give a good clue to their purpose but for full details see the on line help 116 Generalized linear models Generalized linear modeling is a development of linear models to accommodate both non normal response distributions and transformations to linearity in a clean and straightforward way A generalized linear model may be described in terms of the following sequence of assumptions 0 There is a response y of interest and stimulus variables m1 m2 whose values in uence the distribution of the response The stimulus variables in uence the distribution of y through a single linear function only This linear function is called the linear predictor and is usually written 7 511 22 p7 hence x has no in uence on the distribution of y if and only if B O The distribution of y is of the form A My 7 w exp MM v WW W w where 4p is a scale parameter possibly known and is constant for all observations A represents a prior weight assumed known but possibly varying with the observations and a is the mean of y So it is assumed that the distribution of y is determined by its mean and possibly a scale parameter as well 0 The mean ILL is a smooth invertible function of the linear predictor and this inverse function to is called the link function These assumptions are loose enough to encompass a wide class ofmodels useful in statistical practice but tight enough to allow the development of a uni ed methodology of estimation and inference at least approximately The reader is referred to any of the current reference works on the subject for full details such as McCullagh amp Nelder 1989 or Dobson 1990 Chapter 11 Statistical models in R On the Aegean island of Kalythos the male inhabitants suffer from a congenital eye disease the effects of which become more marked with increasing age Samples of islander males of various ages were tested for blindness and the results recorded The data is shown below Age 20 35 45 55 70 No tested 50 50 50 50 50 No blind 6 17 26 37 44 The problem we consider is to t both logistic and probit models to this data and to estimate for each model the LD50 that is the age at which the chance of blindness for a male inhabitant is 50 If y is the number of blind at age z and n the number tested both models have the form y N 13717 F o 5190 ltIgtz is the standard normal distribution function and in the 1 5 In both cases the LD50 is LD50 76061 that is the point at which the argument of the distribution function is zero The rst step is to set the data up as a data frame where for the probit case logit case the default 5 gt kalythos lt dataframex C20 35 45 5570 n rep50 5 y C617263744 To t a binomial model using glmO there are three possibilities for the response o If the response is a vector it is assumed to hold binary data and so must be a 01 vector o If the response is a two column matrix it is assumed that the rst column holds the number of successes for the trial and the second holds the number of failures o If the response is a factor its rst level is taken as failure 0 and all other levels as success Here we need the second of these conventions so we add a matrix to our data frame gt kalythosYmat lt cbindkalythosy kalythosn kalythosy To t the models we use gt fmp lt glmYmat quot x family binomiallinkprobit data kalythos gt fml lt glmYmat quot x family binomial data kalythos Since the logit link is the default the parameter may be omitted on the second call To see the results of each t we could use gt summaryfmp gt summaryfml Both models t all too well To nd the LD50 estimate we can use a simple function gt ld50 lt functionb b1 b 2 gt ldp lt ld50coef fmp ldl lt ld50coeffml Cldp ldl The actual estimates from this data are 43663 years and 43601 years respectively Poisson models With the Poisson family the default link is the log and in practice the major use of this family is to t surrogate Poisson log linear models to frequency data whose actual distribution is often multinomial This is a large and important subject we will not discuss further here It even forms a major part of the use of non gaussian generalized models overall Occasionally genuinely Poisson data arises in practice and in the past it was often analyzed as gaussian data after either a log or a squareroot transformation As a graceful alternative to the latter a Poisson generalized linear model may be tted as in the following example Chapter 11 Statistical models in R 58 gt fmod lt glmy quot A B x family poisson1inksqrt data wormcounts Quasilikelihood models For all families the variance of the response will depend on the mean and will have the scale parameter as a multiplier The form of dependence of the variance on the mean is a characteristic of the response distribution for example for the poisson distribution Vary 1 For quasi likelihood estimation and inference the precise response distribution is not speci ed but rather only a link function and the form of the variance function as it depends on the mean Since quasi likelihood estimation uses formally identical techniques to those for the gaussian distribution this family provides a way of tting gaussian models with non standard link functions or variance functions incidentally For example consider tting the non linear regression t9121 y2202e which may be written alternatively as 1 5 y B i 52 where ml 2221 2 7121 31 101 and 82 0201 Supposing a suitable data frame to be set up we could t this non linear regression as gt nlfit lt glmy quot x1 x2 1 family quasi1inkinverse varianceconstant data biochem The reader is referred to the manual and the help document for further information as needed 117 Nonlinear least squares and maximum likelihood models Certain forms of nonlinear model can be tted by Generalized Linear Models glmO But in the majority of cases we have to approach the nonlinear curve tting problem as one of nonlinear optimization R s nonlinear optimization routines are optim n1m and from R 220 n1minb which provide the functionality and more of S PLUS s ms and n1minb We seek the parameter values that minimize some index of lack of t and they do this by trying out various parameter values iteratively Unlike linear regression for example there is no guarantee that the procedure will converge on satisfactory estimates All the methods require initial guesses about what parameter values to try and convergence may depend critically upon the quality of the starting values 1171 Least squares One way to t a nonlinear model is by minimizing the sum of the squared errors SSE or residuals This method makes sense if the observed errors could have plausibly arisen from a normal distribution Here is an example from Bates amp Watts 1988 page 51 The data are gt x lt C002 002 006 006 011 011 022 022 056 056 1 10 1 10 gt y lt C76 47 97 107 123 139 159 152 191 201 207 200 The t criterion to be minimized is Chapter 11 Statistical models in R 59 gt fn lt functionp sumy p1 xp2 xquot2 In order to do the t we need initial estimates of the parameters One way to nd sensible starting values is to plot the data guess some parameter values and superimpose the model curve using those values gt plot x y gt xfit lt seq02 11 05 gt yfit lt 200 xfit01 xfit gt linessplinexfit yfit We could do better but these starting values of 200 and 01 seem adequate Now do the t gt out lt nlmfn p C200 01 hessian TRUE After the tting outminimum is the SSE and outestimate are the least squares estimates of the parameters To obtain the approximate standard errors SE of the estimates we do gt sqrt diag2outminimumlengthy 2 solveouthessian The 2 in the line above represents the number of parameters A 95 con dence interval would be the parameter estimate i 196 SE We can superimpose the least squares t on a new plot gt plot x y gt xfit lt seq02 11 05 gt yfit lt 21268384222 xfit006412146 xfit gt linessplinexfit yfit The standard package stats provides much more extensive facilities for tting non linear models by least squares The model we have just tted is the Michaelis Menten model so we can use gt df lt dataframexx yy gt fit lt nlsy quot SSmicmenx Vm K df gt fit Nonlinear regression model model y quot SSmicmenx Vm K data df K 21268370711 006412123 residual sumofsquares 1195449 gt summaryfit Formula y quot SSmicmenx Vm K Parameters Estimate Std Error t value Prgtt Vm 2127e02 6947e00 30615 324e11 K 6412e02 8281e03 7743 157e05 Residual standard error 1093 on 10 degrees of freedom Correlation of Parameter Estimates Vm K 07651 1172 Maximum likelihood Maximum likelihood is a method of nonlinear model tting that applies even if the errors are not normal The method nds the parameter values which maximize the log likelihood or Chapter 11 Statistical models in R 60 equivalently which minimize the negative log likelihood Here is an example from Dobson 1990 pp 1087111 This example ts a logistic model to dose response data which clearly could also be t by glmO The data are gt x lt c16907 17242 17552 17842 18113 1 8369 1 8610 1 8839 gt y lt c 6 13 18 28 52 53 61 60 gt n lt C59 60 62 56 63 59 62 60 The negative log likelihood to minimize is gt fn lt functionp sum yp1p2 x nlog1expp1p 2 x logchoosen y We pick sensible starting values and do the t gt out lt nlmfn p c5020 hessian TRUE After the tting outminimum is the negative log likelihood and outestimate are the maxi mum likelihood estimates of the parameters To obtain the approximate SEs of the estimates we do gt sqrt diagsolve outhessian A 95 con dence interval would be the parameter estimate i 196 SE 118 Some nonstandard models We conclude this chapter with just a brief mention of some of the other facilities available in R for special regression and data analysis problems 0 Mixed models The recommended nlme package provides functions lme and nlme for linear and non linear mixed effects models that is linear and non linear regressions in which some of the coef cients correspond to random effects These functions make heavy use of formulae to specify the models Local approximating regressions The loess function ts a nonparametric regression by using a locally weighted regression Such regressions are useful for highlighting a trend in messy data or for data reduction to give some insight into a large data set Function loess is in the standard package stats together with code for projection pursuit regression Robust regression There are several functions available for tting regression models in a way resistant to the in uence of extreme outliers in the data Function lqs in the rec ommended package MASS provides stateof art algorithms for highly resistant ts Less resistant but statistically more ef cient methods are available in packages for example function rlm in package MASS Additive models This technique aims to construct a regression function from smooth additive functions of the determining variables usually one for each determining variable Functions avas and ace in package acepack and functions bruto and mars in package mda provide some examples of these techniques in user contributed packages to R An extension is Generalized Additive Models implemented in user contributed packages gam and mgcv Treebased models Rather than seek an explicit global linear model for prediction or interpretation tree based models seek to bifurcate the data recursively at critical points of the determining variables in order to partition the data ultimately into groups that are as homogeneous as possible within and as heterogeneous as possible between The results often lead to insights that other data analysis methods tend not to yield Models are again speci ed in the ordinary linear model form The model tting function is tree but many other generic functions such as plot and text are well adapted to displaying the results of a treebased model t in a graphical way Chapter 11 Statistical models in R Tree models are available in R via the user contributed packages rpart and tree Chapter 12 Graphical procedures 62 1 2 Graphical procedures Graphical facilities are an important and extremely versatile component of the R environment It is possible to use the facilities to display a wide variety of statistical graphs and also to build entirely new types of graph The graphics facilities can be used in both interactive and batch modes but in most cases interactive use is more productive lnteractive use is also easy because at startup time R initiates a graphics device diiuer which opens a special graphics window for the display of interactive graphics Although this is done automatically it is useful to know that the command used is X11 under UNIX windows under Windows and quartz under Mac OS X Once the device driver is running R plotting commands can be used to produce a variety of graphical displays and to create entirely new kinds of display Plotting commands are divided into three basic groups 0 Highlevel plotting functions create a new plot on the graphics device possibly with axes labels titles and so on Lowlevel plotting functions add more information to an existing plot such as extra points lines and labels Interactive graphics functions allow you interactively add information to or extract infor mation from an existing plot using a pointing device such as a mouse In addition R maintains a list of graphical parameters which can be manipulated to customize your plots This manual only describes what are known as base graphics A separate graphics sub system in package grid coexists with base 7 it is more powerful but harder to use There is a recommended package lattice which builds on grid and provides ways to produce multi panel plots akin to those in the Trellis system in S 121 Highlevel plotting commands High level plotting functions are designed to generate a complete plot of the data passed as ar guments to the function Where appropriate axes labels and titles are automatically generated unless you request otherwise High level plotting commands always start a new plot erasing the current plot if necessary 1211 The plot function One of the most frequently used plotting functions in R is the plot function This is a generic function the type of plot produced is dependent on the type or class of the rst argument plotx y plotxy lfx and y are vectors plotx y produces a scatterplot ofy against x The same effect can be produced by supplying one argument second form as either a list containing two elements x and y or a two column matrix plotx If x is a time series this produces a time series plot If x is a numeric vector it produces a plot of the values in the vector against their index in the vector If x is a complex vector it produces a plot of imaginary versus real parts of the vector elements plot f plot f y f is a factor object y is a numeric vector The rst form generates a bar plot of f the second form produces boxplots of y for each level of f Chapter 12 Graphical procedures 63 plot df plot quot expr ploty quot expr df is a data frame y is any object expr is a list of object names separated by eg a b c The rst two forms produce distributional plots of the variables in a data frame rst form or of a number of named objects second form The third form plots y against every object named in expr 1212 Displaying multivariate data R provides two very useful functions for representing multivariate data If X is a numeric matrix or data frame the command gt pairsX produces a pairwise scatterplot matrix of the variables de ned by the columns of X that is every column of X is plotted against every other column of X and the resulting nn 7 1 plots are arranged in a matrix with plot scales constant over the rows and columns of the matrix When three or four variables are involved a coplot may be more enlightening If a and b are numeric vectors and c is a numeric vector or factor object all of the same length then the command gt coplota quot b I C produces a number of scatterplots of a against b for given values of c If c is a factor this simply means that a is plotted against b for every level of c When 6 is numeric it is divided into a number of conditioning intervals and for each interval a is plotted against b for values of 6 within the interval The number and position of intervals can be controlled with givenva1ues argument to coplotOithe function co intervals is useful for selecting intervals You can also use two given variables with a command like gt coplota b I C d which produces scatterplots of a against b for every joint conditioning interval of c and d The coplotO and pairsO function both take an argument panel which can be used to customize the type of plot which appears in each panel The default is points to produce a scatterplot but by supplying some other low level graphics function of two vectors x and y as the value of panel you can produce any type of plot you wish An example panel function useful for coplots is panel smooth 1213 Display graphics Other high level graphics functions produce different types of plots Some examples are qqnorm x qqline x qqplot x y Distribution comparison plots The rst form plots the numeric vector x against the expected Normal order scores a normal scores plot and the second adds a straight line to such a plot by drawing a line through the distribution and data quartiles The third form plots the quantiles of x against those of y to compare their respective distributions hi st x hist x nclass11 histx breaksb Produces a histogram of the numeric vector x A sensible number of classes is usually chosen but a recommendation can be given with the nclass argument Alternatively the breakpoints can be speci ed exactly with the breaks argument Chapter 12 Graphical procedures 64 If the probabilityTRUE argument is given7 the bars represent relative frequencies instead of counts dotchartx Constructs a dotchart of the data in x In a dotchart the y axis gives a labelling of the data in x and the m axis gives its value For example it allows easy visual selection of all data entries with values lying in speci ed ranges imagex y z contourx y z perspx y z Plots of three variables The image plot draws a grid of rectangles using different colours to represent the value of 27 the contour plot draws contour lines to represent the value of 27 and the persp plot draws a 3D surface 1214 Arguments t0 highlevel plotting functions There are a number of arguments which may be passed to high level graphics functions7 as follows addTRUE Forces the function to act as a low level graphics function7 superimposing the plot on the current plot some functions only axesFALSE Suppresses generation of axesiuseful for adding your own custom axes with the axis function The default7 axesTRUE7 means include axes lognxn lognyn logquotxyquot Causes the z y or both axes to be logarithmic This will work for many7 but not all7 types of plot type The type argument controls the type of plot produced7 as follows typequotpquot Plot individual points the default typequotlquot Plot lines typequotbquot Plot points connected by lines both typequotoquot Plot points overlaid by lines typequothquot Plot vertical lines from points to the zero axis high density typensn typequotSquot Step function plots In the rst form7 the top of the vertical de nes the point in the second7 the bottom typequotnquot No plotting at all However axes are still drawn by default and the coordinate system is set up according to the data Ideal for creating plots with subsequent low level graphics functions xlabstring ylabstring Axis labels for the z and y axes Use these arguments to change the default labels7 usually the names of the objects used in the call to the high level plotting function mainstring Figure title7 placed at the top of the plot in a large font substring Sub title7 placed just below the m axis in a smaller font Chapter 12 Graphical procedures 65 122 Lowlevel plotting commands Sometimes the high level plotting functions don t produce exactly the kind of plot you desire In this case low level plotting commands can be used to add extra information such as points lines or text to the current plot Some of the more useful low level plotting functions are pointsx y linesx y Adds points or connected lines to the current plot plotO s type argument can also be passed to these functions and defaults to quotpquot for points and quotlquot for lines textx y labels Add text to a plot at points given by x y Normally labels is an integer or character vector in which case labels i is plotted at point x i yi The default is 11engthx Note This function is often used in the sequence gt plotx y typequotnquot textx y names The graphics parameter typequotnquot suppresses the points but sets up the axes and the text function supplies special characters as speci ed by the character vector names for the points abline a b abline hy abline vX abline 1m obj Adds a line of slope b and intercept a to the current plot hy may be used to specify y coordinates for the heights of horizontal lines to go across a plot and vx similarly for the m coordinates for vertical lines Also lmobj may be list with a coefficients component of length 2 such as the result of model tting functions which are taken as an intercept and slope in that order polygonx y Draws a polygon de ned by the ordered vertices in x y and optionally shade it in with hatch lines or ll it if the graphics device allows the lling of gures legendx y legend Adds a legend to the current plot at the speci ed position Plotting characters line styles colors etc are identi ed with the labels in the character vector legend At least one other argument v a vector the same length as legend with the corre sponding values of the plotting unit must also be given as follows legend fillV Colors for lled boxes legend colV Colors in which points or lines will be drawn legend ltyV Line styles legend lwdV Line widths legend pchV Plotting characters character vector Chapter 12 Graphical procedures 66 titlemain sub Adds a title main to the top of the current plot in a large font and optionally a sub title sub at the bottom in a smaller font axisside Adds an axis to the current plot on the side given by the rst argument 1 to 4 counting clockwise from the bottom Other arguments control the positioning of the axis within or beside the plot and tick positions and labels Useful for adding custom axes after calling plot with the axesFALSE argument Low level plotting functions usually require some positioning information eg z and y co ordinates to determine where to place the new plot elements Coordinates are given in terms of user coordinates which are de ned by the previous high level graphics command and are chosen based on the supplied data Where x and y arguments are required it is also sufficient to supply a single argument being a list with elements named x and y Similarly a matrix with two columns is also valid input In this way functions such as locatorO see below may be used to specify positions on a plot interactively 1221 Mathematical annotation In some cases it is useful to add mathematical symbols and formulae to a plot This can be achieved in R by specifying an expression rather than a character string in any one of text mtext axis or title For example the following code draws the formula for the Binomial probability function gt textx y expressionpastebgroupquotquot atopn x quotquot pquotx qquotnx More information including a full listing of the features available can obtained from within R using the commands gt helpplotmath gt exampleplotmath gt demoplotmath 1222 Hershey vector fonts It is possible to specify Hershey vector fonts for rendering text when using the text and contour functions There are three reasons for using the Hershey fonts Hershey fonts can produce better output especially on a computer screen for rotated andor small text Hershey fonts provide certain symbols that may not be available in the standard fonts In particular there are zodiac signs cartographic symbols and astronomical symbols Hershey fonts provide cyrillic and japanese Kana and Kanji characters More information including tables of Hershey characters can be obtained from within R using the commands gt helpHershey gt demoHershey gt helpJapanese gt demoJapanese 123 Interacting with graphics R also provides functions which allow users to extract or add information to a plot using a mouse The simplest of these is the locatorO function Chapter 12 Graphical procedures 67 locatorn type Waits for the user to select locations on the current plot using the left mouse button This continues until 11 default 512 points have been selected or another mouse button is pressed The type argument allows for plotting at the selected points and has the same effect as for high level graphics commands the default is no plotting locatorO returns the locations of the points selected as a list with two components x and y locatorO is usually called with no arguments It is particularly useful for interactively selecting positions for graphic elements such as legends or labels when it is difficult to calculate in advance where the graphic should be placed For example to place some informative text near an outlying point the command gt textlocator1 quotOutlierquot adj0 may be useful locatorO will be ignored if the current device such as postscript does not support interactive pointing identifyx y labels Allow the user to highlight any of the points de ned by x and y using the left mouse button by plotting the corresponding component of labels nearby or the index number of the point if labels is absent Returns the indices of the selected points when another button is pressed Sometimes we want to identify particular points on a plot rather than their positions For example we may wish the user to select some observation of interest from a graphical display and then manipulate that observation in some way Given a number of z y coordinates in two numeric vectors x and y we could use the identifyO function as follows gt plot x y gt identifyx y The identifyO functions performs no plotting itself but simply allows the user to move the mouse pointer and click the left mouse button near a point If there is a point near the mouse pointer it will be marked with its index number that is its position in the xy vectors plotted nearby Alternatively you could use some informative string such as a case name as a highlight by using the labels argument to identify or disable marking altogether with the plot FALSE argument When the process is terminated see above identifyO returns the indices of the selected points you can use these indices to extract the selected points from the original vectors x and y 124 Using graphics parameters When creating graphics particularly for presentation or publication purposes R s defaults do not always produce exactly that which is required You can however customize almost every aspect of the display using graphics parameters R maintains a list of a large number of graphics parameters which control things such as line style colors gure arrangement and text justi ca tion among many others Every graphics parameter has a name such as col which controls colors and a value a color number for example A separate list of graphics parameters is maintained for each active device and each device has a default set of parameters when initialized Graphics parameters can be set in two ways either permanently affecting all graphics functions which access the current device or temporarily affecting only a single graphics function call 1241 Permanent changes The par function The par function is used to access and modify the list of graphics parameters for the current graphics device Chapter 12 Graphical procedures 68 par Without arguments returns a list of all graphics parameters and their values for the current device Parcncoln nltyn With a character vector argument returns only the named graphics parameters again as a list par col4 lty2 With named arguments or a single list argument sets the values of the named graphics parameters and returns the original values of the parameters as a list Setting graphics parameters with the par function changes the value of the parameters permanently in the sense that all future calls to graphics functions on the current device will be affected by the new value You can think of setting graphics parameters in this way as setting default values for the parameters which will be used by all graphics functions unless an alternative value is given Note that calls to par always affect the global values of graphics parameters even when parO is called from within a function This is often undesirable behavioriusually we want to set some graphics parameters do some plotting and then restore the original values so as not to affect the user s R session You can restore the initial values by saving the result of par when making changes and restoring the initial values when plotting is complete gt oldpar lt parcol4 lty2 plotting commands gt paroldpar To save and restore all settable l graphical parameters use gt oldpar lt parnoreadonlyTRUE plotting commands gt paroldpar 1242 Temporary changes Arguments to graphics functions Graphics parameters may also be passed to almost any graphics function as named arguments This has the same effect as passing the arguments to the par function except that the changes only last for the duration of the function call For example gt plot x y pchquotquot produces a scatterplot using a plus sign as the plotting character without changing the default plotting character for future plots Unfortunately this is not implemented entirely consistently and it is sometimes necessary to set and reset graphics parameters using parO 125 Graphics parameters list The following sections detail many of the commonly used graphical parameters The R help documentation for the par function provides a more concise summary this is provided as a somewhat more detailed alternative Graphics parameters will be presented in the following form nameva1ue A description of the parameters effect name is the name of the parameter that is the argument name to use in calls to par or a graphics function value is a typical value you might use when setting the parameter Note that axes is not a graphics parameter but an argument to a few plot methods see xaxt and yaxt 1 Some graphics parameters such as the size of the current device are for information only Chapter 12 Graphical procedures 1251 Graphical elements R plots are made up of points lines text and polygons lled regions Graphical parameters exist which control how these graphical elements are drawn as follows Pchnn pch4 lty2 lwd2 col2 colaxis collab colmain colsub font2 font axis font lab font main font sub adj O 1 cex15 Character to be used for plotting points The default varies with graphics drivers but it is usually 0 Plotted points tend to appear slightly above or below the appropriate position unless you use quot quot as the plotting character which produces centered points When pch is given as an integer between 0 and 25 inclusive a specialized plotting symbol is produced To see what the symbols are use the command gt legendlocator1 ascharacter025 pch 025 Those from 21 to 25 may appear to duplicate earlier symbols but can be coloured in different ways see the help on points and its examples In addition pch can be a character or a number in the range 32255 representing a character in the current font Line types Alternative line styles are not supported on all graphics devices and vary on those that do but line type 1 is always a solid line line type 0 is always invis ible and line types 2 and onwards are dotted or dashed lines or some combination of both Line widths Desired width of lines in multiples of the standard line width Affects axis lines as well as lines drawn with lines etc Not all devices support this and some have restrictions on the widths that can be used Colors to be used for points lines text lled regions and images A number from the current palette see palette or a named colour The color to be used for axis annotation z and y labels main and sub titles re spectively An integer which speci es which font to use for text If possible device drivers arrange so that 1 corresponds to plain text 2 to bold face 3 to italic 4 to bold italic and 5 to a symbol font which include Greek letters The font to be used for axis annotation z and y labels main and sub titles respec tively Justi cation of text relative to the plotting position 0 means left justify 1 means right justify and 05 means to center horizontally about the plotting position The actual value is the proportion of text that appears to the left of the plotting position so a value of O 1 leaves a gap of 10 of the text width between the text and the plotting position Character expansion The value is the desired size of text characters including plotting characters relative to the default text size Chapter 12 Graphical procedures 70 G ex axis cexlab G ex main cexsub The character expansion to be used for axis annotation z and y labels main and sub titles respectively 1252 Axes and tick marks Many of Rs high level plots have axes and you can construct axes yourself with the low level axisO graphics function Axes have three main components the axis line line style controlled by the lty graphics parameter the tick marks which mark off unit divisions along the axis line and the tick labels which mark the units These components can be customized with the following graphics parameters labc5 7 12 The rst two numbers are the desired number of tick intervals on the z and y axes respectively The third number is the desired length of axis labels in characters including the decimal point Choosing a too small value for this parameter may result in all tick labels being rounded to the same number las1 Orientation of axis labels 0 means always parallel to axis 1 means always horizon tal and 2 means always perpendicular to the axis mgpc3 1 0 Positions of axis components The rst component is the distance from the axis label to the axis position in text lines The second component is the distance to the tick labels and the nal component is the distance from the axis position to the axis line usually zero Positive numbers measure outside the plot region negative numbers inside tck001 Length of tick marks as a fraction of the size of the plotting region When tck is small less than 05 the tick marks on the z and y axes are forced to be the same size A value of 1 gives grid lines Negative values give tick marks outside the plotting region Use tck0 01 and mgpc 1 15 0 for internal tick marks xaxsquotrquot axsquotiquot Axis st les for the z and axes res ectivel With st les quotiquot internal and quotrquot Y y y 7 p y y the default tick marks always fall within the range of the data however style quotIquot leaves a small amount of space at the edges S has other styles not implemented in 1253 Figure margins A single plot in R is known as a figure and comprises a plot region surrounded by margins possibly containing axis labels titles etc and usually bounded by the axes themselves Chapter 12 Graphical procedures 71 A typical gure is mar3 Plot region mailll Margin Graphics parameters controlling gure layout include maic1 05 05 O Widths of the bottom7 left7 top and right margins7 respectively7 measured in inches marc4 2 2 1 Similar to mai7 except the measurement unit is text lines mar and mai are equivalent in the sense that setting one changes the value of the other The default values chosen for this parameter are often too large the right hand margin is rarely needed7 and neither is the top margin if no title is being used The bottom and left margins must be large enough to accommodate the axis and tick labels Furthermore7 the default is chosen without regard to the size of the device surface for example7 using the postscriptO driver with the height4 argument will result in a plot which is about 50 margin unless mar or mai are set explicitly When multiple gures are in use see below the margins are reduced7 however this may not be enough when many gures share the same page Chapter 12 Graphical procedures 72 1254 Multiple gure environment R allows you to create an n by m array of gures on a single page Each gure has its own margins and the array of gures is optionally surrounded by an outer margin as shown in the following gure oma3 omi4 ngwzm 4 omi1 mfrowc32 The graphical parameters relating to multiple gures are as follows mfcolc3 2 mfrowc 2 4 Set the size of a multiple gure array The rst value is the number of rows the second is the number of columns The only difference between these two parameters is that setting mfcol causes gures to be lled by column mfrow lls by rows The layout in the Figure could have been created by setting mfrowc32 the gure shows the page after four plots have been drawn Setting either of these can reduce the base size of symbols and text controlled by parquotcexquot and the pointsize of the device In a layout with exactly two rows and columns the base size is reduced by a factor of 083 if there are three or more of either rows or columns the reduction factor is 066 mfgc2 2 S 2 Position of the current gure in a multiple gure environment The rst two numbers are the row and column of the current gure the last two are the number of rows and columns in the multiple gure array Set this parameter to jump between gures in the array You can even use different values for the last two numbers than the true values for unequally sized gures on the same page figc4 9 1 410 Position of the current gure on the page Values are the positions of the left right bottom and top edges respectively as a percentage of the page measured from the bottom left corner The example value would be for a gure in the bottom right of the page Set this parameter for arbitrary positioning of gures within a page If you want to add a gure to a current page use newTRUE as well unlike S Chapter 12 Graphical procedures 73 omac2 O 3 O omic0 O 08 0 Size of outer margins Like mar and mai the rst measures in text lines and the second in inches starting with the bottom margin and working clockwise Outer margins are particularly useful for page wise titles etc Text can be added to the outer margins with the mtextO function with argument outerTRUE There are no outer margins by default however so you must create them explicitly using oma or omi More complicated arrangements of multiple gures can be produced by the split screen and layout functions as well as by the grid and lattice packages 126 Device drivers R can generate graphics of varying levels of quality on almost any type of display or printing device Before this can begin however R needs to be informed what type of device it is dealing with This is done by starting a device driver The purpose of a device driver is to convert graphical instructions from R draw a line77 for example into a form that the particular device can understand Device drivers are started by calling a device driver function There is one such function for every device driver type helpDevices for a list of them all For example issuing the command gt postscriptO causes all future graphics output to be sent to the printer in PostScript format Some commonly used device drivers are X11 For use with the X11 window system on Unix alikes windows 0 For use on Windows quartzO For use on MacOS X postscript O For printing on PostScript printers or creating PostScript graphics les pdf Produces a PDF le which can also be included into PDF les pngO Produces a bitmap PNG le Not always available see its help page jpegO Produces a bitmap JPEG le best used for image plots Not always available see its help page When you have nished with a device be sure to terminate the device driver by issuing the command gt dev of f O This ensures that the device nishes cleanly for example in the case of hardcopy devices this ensures that every page is completed and has been sent to the printer This will happen automatically at the normal end of a session 1261 PostScript diagrams for typeset documents By passing the file argument to the postscriptO device driver function you may store the graphics in PostScript format in a le of your choice The plot will be in landscape orientation unless the horizontalFALSE argument is given and you can control the size of the graphic with the width and height arguments the plot will be scaled as appropriate to t these dimensions For example the command Chapter 12 Graphical procedures 74 gt postscriptquotfile psquot horizontalFALSE height5 pointsize10 will produce a le containing PostScript code for a gure ve inches high perhaps for inclusion in a document It is important to note that if the le named in the command already exists it will be overwritten This is the case even if the le was only created earlier in the same R session Many usages of PostScript output will be to incorporate the gure in another document This works best when encapsulated PostScript is produced R always produces conformant output but only marks the output as such when the onefileFALSE argument is supplied This unusual notation stems from S compatibility it really means that the output will be a single page which is part of the EPSF speci cation Thus to produce a plot for inclusion use something like gt postscript quotplotl epsquot horizontalFALSE onefileFALSE height8 width6 pointsize10 1262 Multiple graphics devices In advanced use of R it is often useful to have several graphics devices in use at the same time Of course only one graphics device can accept graphics commands at any one time and this is known as the current deulce When multiple devices are open they form a numbered sequence with names giving the kind of device at any position The main commands used for operating with multiple devices and their meanings are as follows x110 UNIX windows win printer 0 win metafile 0 Windows quartz MacOS X postscript 0 pdf Each new call to a device driver function opens a new graphics device thus extending by one the device list This device becomes the current device to which graphics output will be sent Some platforms may have further devices available dev li st 0 Returns the number and name of all active devices The device at position 1 on the list is always the null deulce which does not accept graphics commands at all devnext devprev Returns the number and name of the graphics device next to or previous to the current device respectively dev set whichk Can be used to change the current graphics device to the one at position k of the device list Returns the number and label of the device dev of f k Terminate the graphics device at point k of the device list For some devices such as postscript devices this will either print the le immediately or correctly complete the le for later printing depending on how the device was initiated Chapter 13 Packages 77 For example 00 is the transpose function in R but users might de ne their own function named t Namespaces prevent the user s de nition from taking precedence and breaking every function that tries to transpose a matrix There are two operators that work with namespaces The double colon operator selects de nitions from a particular namespace In the example above the transpose function will always be available as base t because it is de ned in the base package Only functions that are exported from the package can be retrieved in this way The triplecolon operator z may be seen in a few places in R code it acts like the double colon operator but also allows access to hidden objects Users are more likely to use the getAnywhereO function which searches multiple packages Packages are often inter dependent and loading one may cause others to be automatically loaded The colon operators described above will also cause automatic loading of the associated package When packages with namespaces are loaded automatically they are not added to the search list Appendix A A sample session 78 Appendix A A sample session The following session is intended to introduce to you some features of the R environment by using them Many features of the system will be unfamiliar and puzzling at rst but this puzzlement will soon disappear Login start your windowing system R Start R as appropriate for your platform The R program begins with a banner Within R the prompt on the left hand side will not be shown to avoid confusion help start 0 Start the HTML interface to on line help using a web browser available at your machine You should brie y explore the features of this facility with the mouse lconify the help window and move on to the next part x lt rnorm50 y lt rnorm x Generate two pseudo random normal vectors of z and y coordinates plot x y Plot the points in the plane A graphics window will appear automatically 150 See which R objects are now in the R workspace rmx y Remove objects no longer needed Clean up x lt 120 Make m 1220 w lt 1 sqrtx2 A weight vector of standard deviations dummy lt dataframexx y x rnormxw dummy Make a data frame of two columns z and y and look at it fm lt lmy quot x datadummy summaryfm Fit a simple linear regression and look at the analysis With y to the left of the tilde we are modelling y dependent on m fml lt lmy quot x datadummy weight1wquot2 summaryfm1 Since we know the standard deviations we can do a weighted regression attachdummy Make the columns in the data frame visible as variables lrf lt lowessx y Make a nonparametric local regression function plot x y Standard point plot linesx lrfy Add in the local regression abline O 1 lty3 The true regression line intercept O slope 1 abline coef fm Unweighted regression line Appendix A A sample session 79 ablinecoeffm1 col quotredquot Weighted regression line detachO Remove data frame from the search path plotfittedfm residfm xlabquotFitted valuesquot ylabquotResidualsquot mainquotResiduals vs Fittedquot A standard regression diagnostic plot to check for heteroscedasticity Can you see it qqnormresidfm mainquotResiduals Rankit Plotquot normal scores plot to check for skewness7 kurtosis and outliers Not very useful here rmfm fml lrf x dummy Clean up again The next section will look at data from the classical experiment of Michaelson and Morley to measure the speed of light This dataset is available in the morley object7 but we will read it to illustrate the readtable function filepath lt systemfilequotdataquot quotmorleyta quot packagequotdatasetsquot filepath Get the path to the data le file showfilepath Optional Look at the le m lt readtablefilepath mm Read in the Michaelson and Morley data as a data frame7 and look at it There are ve experiments column Expt and each has 20 runs column Run and 1 is the recorded speed of light7 suitably coded mmExpt lt factor mmExpt mmRun lt factor mmRun Change Expt and Run into factors attachmm Make the data frame visible at position 3 the default plotExpt Speed mainquotSpeed of Light Dataquot xlabquotExperiment No quot Compare the ve experiments with simple boxplots fm lt aovSpeed quot Run Expt datamm summaryfm Analyze as a randomized block7 with runs and experiments as factors me lt updatefm quot Run anovafm0 fm Fit the submodel omitting runs and compare using a formal analysis of variance detachO rmfm me Clean up before moving on We now look at some more graphical features contour and image plots x lt seqpi pi len50 y lt x z is a vector of 50 equally spaced values in 77139 g z 3 7139 y is the same Appendix A A sample session 80 f lt outerx y functionx y cosy1 xquot2 f is a square matrix with rows and columns indexed by z and y respectively of values of the function cosy1 m2 oldpar lt par no readonly TRUE par ptyquotsquot Save the plotting parameters and set the plotting region to square contourx y f contourx y f nlevels15 addTRUE Make a contour map of 1 add in more lines for more detail fa lt f tf2 fa is the asymmetric part77 of f t is transpose contourx y fa nlevels15 Make a contour plot par oldpar and restore the old graphics parameters imagex y f imagex y fa Make some high density image plots of which you can get hardcopies if you wish objectsO rmx y f fa and clean up before moving on R can do complex arithmetic also th lt seqpi pi len100 z lt exp1ith ii is used for the complex number 2 par ptyquotsquot plot 2 typequotlquot Plotting complex arguments means plot imaginary versus real parts This should be a circle w lt rnorm100 rnorm1001i Suppose we want to sample points within the unit circle One method would be to take complex numbers with standard normal real and imaginary parts w lt ifelseModw gt 1 1w w and to map any outside the circle onto their reciprocal plot w xlimc 1 1 ylimc 1 1 pChquotquot Xlabquotxquot ylabquotyquot lines 2 All points are inside the unit circle but the distribution is not uniform w lt sqrtrunif 100exp2pirunif 1001i plot w xlimc 1 1 ylimc 1 1 pChquotquot Xlabquotxquot ylabquotyquot lines 2 The second method uses the uniform distribution The points should now look more evenly spaced over the disc rmth w 2 Clean up again 10 Quit the R program You will be asked if you want to save the R workspace and for an exploratory session like this you probably do not want to save it Appendix B Invoking R 81 Appendix B Invoking R B1 Invoking R from the command line When working in UNIX or at a command line in Windows the command R can be used both for starting the main R program in the form R options ltin le gtout le or via the R CMD interface as a wrapper to various R tools eg for processing les in R documentation format or manipulating add on packages which are not intended to be called directly You need to ensure that either the environment variable TMPDIR is unset or it points to a valid place to create temporary les and directories Most options control what happens at the beginning and at the end of an R session The startup mechanism is as follows see also the on line help for topic Startup for more informa tion and the section below for some Windows speci c details 0 Unless no environ was given R searches for user and site les to process for setting environment variables The name of the site le is the one pointed to by the environment variable RENVIRON if this is unset RHOMEetcRenvironsite is used if it exists The user le searched for is Renviron7 in the current or in the user s home directory in that order These les should contain lines of the form name Va1ue 7 See helpStartup for a precise description Variables you might want to set include RPAPERSIZE the default paper size RPRINTCMD the default print command and RLIBS speci es the list of R library trees searched for add on packages Then R searches for the site wide startup pro le unless the command line option no site file was given The name of this le is taken from the value of the RPROFILE environment variable If that variable is unset the default RHOMEetcRprofilesite is used if this exists Then unless no init file was given R searches for a le called Rprofile7 in the current directory or in the user s home directory in that order and sources it norestore7 or It also loads a saved image from RData7 if there is one unless no restore data was speci ed Finally if a function First exists it is executed This function as well as Last which is executed at the end of the R session can be de ned in the appropriate startup pro les or reside in RData In addition there are options for controlling the memory available to the R process see the on line help for topic Memory for more information Users will not normally need to use these unless they are trying to limit the amount of memory used by R R accepts the following command line options help h Print short help message to standard output and exit successfully version7 Print version information to standard output and exit successfully encodingenc Specify the encoding to be assumed for input from the console or stdin This needs to be an encoding known to iconv see its help page RHOME Print the path to the R home directory77 to standard output and exit success fully Apart from the front end shell script and the man page R installation puts everything executables packages etc into this directory Appendix B Invoking R 83 minnsizeN maxnsizeN Specify the amount of memory used for xed size objects by setting the number of cons cells77 to N See the previous option for details on N A cons cell takes 28 bytes on a 32 bit machine and usually 56 bytes on a 64 bit machine maxppsizeN Specify the maximum size of the pointer protection stack as N locations This defaults to 10000 but can be increased to allow large and complicated calculations to be done Currently the maximum value accepted is 100000 maxmemsizeN Windows only Specify a limit for the amount of memory to be used both for R objects and working areas This is set by default to the smaller of 15Gb1 and the amount of physical RAM in the machine and must be between 32Mb and the maximum allowed on that version of Windows quiet silent q Do not print out the initial copyright and welcome messages slave Make R run as quietly as possible This option is intended to support programs which use R to compute results for them It implies quiet and no save interactive UNIX only Assert that R really is being run interactively even if input has been redirected use if input is from a FIFO or pipe and fed from an interactive program verbose Print more information about progress and in particular set R s option verbose to TRUE R code uses this option to control the printing of diagnostic messages debuggername d name7 UNIX only Run R through debugger name For most debuggers the exceptions are valgrind and recent versions of gdb further command line options are disregarded and should instead be given when starting the R executable from inside the debugger guitype g type7 UNIX only Use type as graphical user interface note that this also includes in teractive graphics Currently possible values for type are X117 the default pro vided that TclTk support is available Tk and gnome provided that package gnomeGUI is installed For back compatibility xll tk and GNOME are ac cepted args7 This ag does nothing except cause the rest of the command line to be skipped this can be useful to retrieve values from it with commandArgsTRUE Note that input and output can be redirected in the usual way using lt and gt but the line length limit of 1024 bytes still applies Warning and error messages are sent to the error channel stderr The command R CMD allows the invocation of various tools which are useful in conjunction with R but not intended to be called directly The general form is R CMD command args where command is the name of the tool and args the arguments passed on to it Currently the following tools are available 1 25Gb on Versions of Windows that support 3Gb per process and have the support enabled see the rwFAQ Q29 35Gb on some 64bit versions of Windows Appendix B Invoking R 84 BATCH Run R in batch mode COMPILE UNIX only Compile les for use with R SHLIB Build shared library for dynamic loading INSTALL Install add on packages REMOVE Remove add on packages build Build that is package add on packages check Check add on packages LINK UNIX only Front end for creating executable programs Rprof Post process R pro ling les Rdconv Convert Rd format to various other formats including HTML Nroff ETEX plain text and S documentation format Rd2dvi Convert Rd format to DVIPDF Rd2txt Convert Rd format to text Sd2Rd Convert S documentation to Rd format config Obtain con guration information Use R CMD command help to obtain usage information for each of the tools accessible via the R CMD interface B2 Invoking R under Windows There are two ways to run R under Windows Within a terminal window eg cmdexe or commandcom or a more capable shell the methods described in the previous section may be used invoking by Rexe or more directly by Rtermexe These are principally intended for batch use For interactive use there is a consolebased GUI Rgui exe The startup procedure under Windows is very similar to that under UNIX but references to the home directory7 need to be clari ed as this is not always de ned on Windows If the environment variable RUSER is de ned that gives the home directory Next if the environment variable HOME is de ned that gives the home directory After those two user controllable settings R tries to nd system de ned home directories It rst tries to use the Windows personal directory typically CDocuments and SettingsusernameMy Documents in Windows XP If that fails and environment variables HOMEDRIVE and HOMEPATH are de ned and they normally are these de ne the home directory Failing all those the home directory is taken to be the starting directory You need to ensure that either the environment variables TMPDIR TMP and TEMP are either unset or one of them points to a valid place to create temporary les and directories Environment variables can be supplied as nameva1ue pairs on the command line If there is an argument ending RData7 in any case it is interpreted as the path to the workspace to be restored it implies restore and sets the working directory to the parent of the named le This mechanism is used for drag and drop and le association with RGui exe but also works for Rtermexe If the named le does not exist it sets the working directory if the parent directory exists The following additional command line options are available When invoking RGui exe Appendix B Invoking R 85 mdi sdi7 nomdi Control whether Rgui will operate as an MDI program with multiple child windows within one main window or an SDI application with multiple top level windows for the console graphics and pager The command line setting overrides the setting in the user s Rconsole le debug Enable the Break to debugger77 menu item in Rgui and trigger a break to the debugger during command line processing In Windows with R CMD you may also specify your own bat or exe7 le instead of one of the built in commands It will be run with the following environment variables set appropriately RHOMERVERSIONRCMDROSTYPEPATHPERL5LIBand TEXINPUTS Fbrexanipbify0u already have latexexe on your path then R CMD latex exe mydoc will run ETEX on mydoc tex with the path to Rs sharetexmf macros added to TEXINPUTS B3 Invoking R under Mac OS X There are two ways to run R under Mac OS X Within a Terminalapp window by invoking R the methods described in the previous sections apply There is also consolebased GUI R app that by default is installed in the Applications folder on your system It is a standard double clickable Mac OS X application The startup procedure under Mac OS X is very similar to that under UNIX The home directory7 is the one inside the Rframework but the startup and current working directory are set as the user s home directory unless a different startup directory is given in the Preferences window accessible from within the GUI B4 Scripting with R If you just want to run a le fooR of R commands the recommended way is to use R CMD BATCH fooR If you want to run this in the background or as a batch job use OS speci c facilities to do so for example in most shells on Unix alike OSes R CMD BATCH fooR amp runs a background job You can pass parameters to scripts via additional arguments on the command line for example R CMD BATCH args arg1 arg2 fooR amp will pass arguments to a script which can be retrieved as a character vector by args lt commandArgsTRUE This is made simpler by the alternative front end Rscript which can be invoked by Rscript fooR arg1 arg2 and this can also be used to write executable script les like at least on Unix alikes and in some Windows shells pathtoRscript args lt commandArgsTRUE qstatusltexit status codegt If this is entered into a text le runfoo and this is made executable by chmod 755 runfoo it can be invoked for different arguments by Appendix B Invoking R 86 runfoo arg1 arg2 For further options see helpquotRscriptquot If you do not wish to hardcode the path to Rscript but have it in your path which is normally the case for an installed R7 use usrbinenv Rscript At least in Bourne and bash shells7 the mechanism does not allow extra arguments like usrbinenv Rscript vanilla One thing to consider is what stdin refers to It is commonplace to write R scripts with segments like chem lt scann24 290 310 340 340 370 370 280 250 240 240 270 220 528 337 303 303 2895 377 340 220 350 360 370 370 and stdinO refers to the script le to allow such traditional usage If you want to refer to the process s stdin use quotstdinquot as a file connection7 eg scanquotstdinquot Another way to write executable script les suggested by Francois Pinard is to use a here document like binsh environment variables can be set here R slave other options ltltEOF R program goes here EOF but here stdin refers to the program source and quotstdinquot will not be usable Very short scripts can be passed to Rscript on the command line via the e ag Appendix C The command line editor 88 017 Cf Go back one character Go forward one character On most terminals7 you can also use the left and right arrow keys instead of 0 12 and C f7 respectively Editing and re submission text Cf text DEL C d M d C k CV C t M l MC RET Insert text at the cursor Append text after the cursor Delete the previous character left of the cursor Delete the character under the cursor Delete the rest of the word under the cursor7 and save it Delete from cursor to end of command7 and save it lnsert yank the last saved text here Transpose the character under the cursor with the next Change the rest of the word to lower case Change the rest of the word to upper case Re submit the command to R The nal RET terminates the command line editing sequence Appendix F References 94 Appendix F References D M Bates and D G Watts 19887 Nonlinear Regression Analysis and Its Applications John Wiley amp Sons7 New York Richard A Becker7 John M Chambers and Allan R Wilks 19887 The New S Language Chap man amp Hall7 New York This book is often called the Blue Book John M Chambers and Trevor J Hastie eds 19927 Statistical Models in S Chapman amp Hall7 New York This is also called the White Book John M Chambers 1998 Programming with Data Springer7 New York This is also called the Green Book A C Davison and D V Hinkley 19977 Bootstrap Methods and Their Applications7 Cambridge University Press Annette J Dobson 19907 An Introduction to Generalized Linear Models7 Chapman and Hall7 London Peter McCullagh and John A Nelder 19897 Generalized Linear Models Second edition7 Chap man and Hall7 London John A Rice 19957 Mathematical Statistics and Data Analysis Second edition Duxbury Press7 Belmont7 CA S D Silvey 19707 Statistical Inference Penguin7 London FNCE 7550 notes Garland B Durham Leeds School of Business University of Colorado September 8 2008 1 Kalman lter References 0 Hamilton 1994 Tsay 2005 o Durbin and Koopman 2001 o Shumway and Stoffer 2006 Construction of lter Adapted from Hamilton 1994 Let y G1 1 v iid N0 R n1 Art width wt iid N0 C Suppose that y is observed but not 1 One often refers to such a setup as a state space system and to I as the state variable The underlying idea we use is based on the following Suppose that given an information set 7 we have le N N02 2 Suppose that we then observe an additional signal Y where Y GX v and v N N0Ri Then Ele Yl X lY EWl l where B Var Ylffl COV XXL GEG R 1G2i Furthermore Var XLFY EX 7 EleYX 7 Ele YM E 7 22 139712 This is often referred to as the linear projection updating formula see Hamilton It is just a regression using population rather than sample moments The coef cient on Y tells us how much to weight the signal Now we will apply this idea to the problem of interest First7 we de ne some notation it E Et1zt t 3 E271yt at E yt Q E E Vart1zt Q E Var 11yt Var 11at We will use the following facts Fact 1 Q Gin Fact 2 y i Q CIz 3 1 Fact 3 Qt Var 17191 EKM 909 Q GEKI 7 2011 7 it G Evtv GEEG R Fact 4 COV 17111 in 9 El n 309 zyl ERIE 3 CIz in my 26quot The idea is to proceed iteratively given the distribution of Xt t17 we need to be able to advance the lter by constructing XHm That is7 given it and 2 we need to be able to construct 2amp1 and EH1 The lter is started by setting X0 equal to its marginal distribution typically First we show how to advance the mean iiei construct 2H1 given iv Begin by slightly restating the problem 9 GIL U I it 6 where e N N0 2 Then proceeding as above A 71 A ELI I COV 171127 92 Var 171ytl 9t ya it EtG Qflat And so iHl AEtzt Ail Kzaz where K AEEG Qili The matrix K is referred to as the Kalman gaini Now we need to show how to advance the variance iiei construct EH1 given Err Using the expression obtained immediately above we get n1 it1 n1 7 Aft 7 Klan AIt it wz1 Kt CIz it UL A KzG It i it wz1 szz Therefore Z3t1 E n1 3amp1 n1 it1 A 7 KEGEEA 7 KEG C KERK AEEA 7 AEECKGEEGHL R lGEEA C where the last equality follows from simply plugging in for KP For future reference this is used in the simulation smoother we also need Vartzti Using an expression above we get I 7 E411 I 7 it 7 EEG fe I i in EraQ1Gz it 11 I 7 EEG9463th 7 it 7 EEG flvti Vartzt 17 209mein EEG leG EEGQfR flGE Steady state distribution Solving for the steady state distribution of the state variable is a standard probleml First recall how we obtain the steady state or marginal distribution of a scalar AR1 process Zz1 W 6z17 2 where at has mean 0 and variance 0 l The mean of the marginal distribution is zero If lpl lt l the variance exists Taking the variance of both sides of the preceding equation we get Var 214A p2 Var 23 02 The steady state variance Var 20 must thus satify var 20gt 020 7 K We can do something similar in the multivariate case In this case the marginal steady state variance of the state variable is given by solving El AElA C which can be shown to be vec21 I 7 A A 1vecC see Hamilton edition 10213 The marginal variance of the observed variable is 91 G216 R The marginal expected value is 0 for both the state and observed variables Estimation One is often interested in estimating such models MLE is easy to do since ytlyl l l l yt1 N N 9 Given a sequence of observed data yl l l l yn one computes Q1 l l l n and 91 l l l 9 using the iteration abovel Typically one would initialize the iteration by setting Q1 and 21 equal to their marginal values as described above The log likelihood can thus be computed as IogL Zlogm 7 that 1 where is the normal density The MLE is obtained by maximizing over the parameter space Forecasting Suppose we are interested in forecasting the distribution of Ytk conditional on information available at time t We begin by deriving the forecast distribution of Mark conditional on information avail able at time t We know Etzt1 2H1 and Vartzt1 EHL Now we operate recur sively Given Etzti1 and VartzHZ1 we must construct E411 and Vartztiz ELILi AELlt L1gt1gt VarzIzi Var 2A1zi71 wzi A Var zti1A C Repeat the recursion for i 2 37 i i i kl Now that we know the forecast distribution of n1 we are ready to derive the forecast distribution of kaz Ezyzk GEzIzk Vartytk G Var tztkaV R Diagnostics One way to get some model diagnostics is based on a residual analysis similar to that used for standard time series models We will work with individual components of the data Let y be the ith component of y and similarly for Q Also7 let 91 be the ith diagonal element of Q If the model is correct7 the generalized residuals77 Mui gr zit7 tl in m should be iid standard normal Distributional characteristics can be tested by computing eigi Jarque Bera test statis tics or looking at QQplotsi Dynamics can be tested by computing eigi BoxPierce statis tics or looking at correlogramsi It is sometimes also useful to look at multistep generalized residualsi lie 2 k 7 yizk Ezyiwk i z z 7 7 l l Var 1yizk12 Where Etytk and Var 1ytk are computed as described above If the model is correctly speci ed these should be N0 1 But they Will exhibit correlation the rst k 7 l lags but not for lags k and greater A great deal of work has appeared recently dealing With more sophisticated approaches to speci cation testing based on generalized residualsi Papers include Bontemps and Med dahi 2005 Duan 2003 Bai 2002 Hong and Li 2002 and Diebold Gunther and Tay 1998 See SVmix paperi 2 Kalman smoother Construction of the simulation smoother The goal now is to simulate draws from ztlylp Wyn t ll l n Whereas the distri bution of ztlylp l y is referred to as the ltered distribution when we condition on the entire data set it is known as the smoothed distributionl When we speak here of the simulation smoother77 we are referring to the fact that we are simulating draws from the smoothed distribution not calculating the smoothed distribution itself which we will do in a moment First we need to de ne some more notation in 3 Edit it til E Et711t Em E Varzm Emil E Var 1111 Suppose that we have already computed itirm it EH1 and 2 Suppose addition ally that we have already drawn a value Il from zHllyl l l ynl We need to simulate a draw I from Itixt17 ylp l y Note that once we know n1 there is no additional infor mation in ytl7 l l ynl Therefore we will have effectively drawn from ztlztirl y1l l ynl If we are able to do this then by starting at In and following a backwards recursion through Inil l l 11 we will have accomplished our goal We will use a linear projection update as before But rst we need one preliminary resultl Fact 1 COV 417144 EWr imxzz it1ltl ElWr MXAI Uz1 7 Aim Elm MOW itlzyAl EmA Now making use of the linear projection updating formula once more Ellerz17917 A A A 7 ml 55m C0Vz1z7121lVarz1z1l711z1 itlltgt in EzleEJ1 EIt1 iz1lz in JzIz1 itlltgt where J Et tAEtlll ti For the variance we have It EIzl1217917 A A A 79 It i in JLIL1 iz1lz It i in JtAIz fun wt1gt 7 LAKE 7 it 7 thtlrli So Var ztlzt1y1 i yt 7 JEAEE EI7 JEAY JECJti Now we are ready to construct a draw ziiuz from the smoother ie from zliuznly1uiyni The rst step is to construct it kl it 21 171 and Em for t 1 i i i n using the Kalman ltering procedure described earlier Now we begin the back wards recursioni Initialize by drawing 1 N Ninn Enm Then for t n 7 l i i i 1 draw I from the normal distribution with mean 5 t JEI1 it1lt and variance 1 7 Muzma 7 JEA JECJ Smoothed distributions Rather than simulating from the smoother we may be interested in the smoothed distribu tion itself see Hamilton Section 136 for derivations Elzzlyn A A 7971 itln fun Jzit1ln itlltgt var llzlyip A A 7971 Em 22 Jz2z1n Et1lt t Diagnostics Using techniques similar to those described above one could also compute the smoothed dis tibution of ytlylp i yt1 yt17 i i yni In principle one could compute generalized residuals and use them for diagnostics similar to those based on the corresponding ltered distribu tions described above In practice however this does not turn out to be very useful The residuals are autocorrelated at all lags and distributional tests are weak Though people who don7t know better ocassionally try this Relationship between state space models and ARMA reference Meddahi 2002 There is an equivalent ARMA representation to linear Gaussian statespace models For example if y and I are both univariate there is an equivalent ARMA11 model In general if the observed variable is univariate and the state is p dimensional the model is equivalent to an ARMApp 1 if there is no noise in the observation process and ARMApp if there is noisei Example y 11 12 H In WILLA vi le39l lt Li 172 where Um v2 my is white noise Zr 1 YlL 1 72Lyz y 7 71 Wynn WWW 11 12 u Vilma 711171 7111271 7211 71 721224 V2ut71 W1W2I1 72 7172961272 W172uz72 11 711171 12 721171 7211t71 711172 7112z71 W2I2 72gt u i 71 V2u271 7172uz72 111 112 7211171 7111271 u i 71 72util 7172112 Thus 2 is MAO and y is ARMA22l This also works for other statespace models but see Meddahi 2002 for details HW 1 We will work with the simplest case with z and y univariatei In practice it is not much more dif cult to do this in the multivariate casei Suppose we have n 1000 observations from the model y G1 1 v iid N0R 1H1 Art wt1 wt iid N0 C where y y1 i yn is observed but not x 11 i i i The model parameters are known 0 1 A 99 R 1 and C 1 There are two data sets hw01data1dat and hw01data2dati The rst column of each is the observed values of yt and the second column is the unobserved values of 1 Repeat the following steps for each data set One of the data sets was really generated from the above model and one was not Which is which H Compute the Kalman lter Plot the data and ltered states both it and iqkl on the same set of axes Just plot the rst 100 observations to Compute the generalized residuals 3 Make a normalquantile plot of the residuals and interpret 4 Make a correlogram out to 30 lags of the residuals and interpret 5 Compute Jarque Bera and Ljung Box test statistics and interpret 533 Plot the smoothed states it n together With the data and ltered states it Q Plot the data and smoothed states together With 10 draws from the simulation smoother References BAI J 2002 Testing parametric conditional distributions of dynamic models Working paper Boston College BONTEMPS C AND N MEDDAHI 2005 Testing normality a GMM approach Journal of Econometrics 124 1497186 DIEBOLD F T GUNTH39ER AND A TAY 1998 Evaluating density forecasts with applications to nancial risk management International Economic Review 39 8637883 DUAN J C 2003 A speci cation test for timeseries models by a normality transformation Working paper Rotman School of Management University of Toronto DURBIN l AND S KOOPMAN 2001 Time Series Analysis by State Space Models Oxford University Press Oxford HAMILTON J 1994 Time Series Analysis Princeton University Press Princeton NJ HONG Y AND H L1 2002 Nonparametric speci cation testing for continuoustime models With applications to spot interest rates Working paper Cornell University MEDDAHI N 2002 ARMA representation of twofactor models Working paper University of Montreal SHUMWAY R EL AND D S STOFFER 2006 Time Series Analysis and Its Applications Springer New York second edn TSAY R S 2005 Analysis of Financial Time Series John Wiley amp Sons Inc Hoboken NJ second edn


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