Hands-On Matrix Algebra Using R: Active and - beck
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1. R snippet paste x 1 4 sep paste x 1 4 sep Note that x contains five elements including the period If we attach the sequence of only 4 numbers R is smart and recycles the numbers 1 4 again and again till needed The argument sep is designed to remove any space between them If sep R wil place the in between Almost anything can be placed between the characters by using the sep option 1 Ini God2 We3 Trust4 R q gt paste x 1 4 sep 1 In 1 God 2 We 3 Trust 4 1 Date objects Current date including time is obtained by the command date where the empty parentheses are needed date 1 Sat Jul 11 14 24 38 2009 Sys Date weekdays Sys Date 100 plural weekday weekdays as Date 1971 09 03 find day of week from birthdate The output of the above commands is given below Note that only the system date without the time is obtained by the command Sys Date It is given in the international format as YYYY mm dd for year month and day Find the date 30 days from the current date HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 12 xxvii 1 6 Hands on Matrix Algebra Using R If you want to have the date after 30 days use the command Sys Date 30 Find the day of wee2. R is really important to the point that it s hard to overvalue it R is numerically one of the most accurate languages perhaps because it is free with transparent code which can be checked for accuracy by almost anyone anywhere R is supported by volunteer experts from around the world and available anywhere one has access to high speed Internet R works equally well on Windows Linux and Mac OS X computers R is based on an earlier public domain language called S developed in Bell Labs in 1970 s I had personally used S when I was employed at Bell Laboratories At that time it was available only on UNIX operating system computers based on principles of object oriented languages S PLUS is the commercial version of S whereas R is the free version It is a very flexible object oriented language OOL in the sense that inputs data and outputs are all objects e g files inside the computer R is powerful and fun R may seem daunting at first to someone who has never thought of himself or herself as a programmer Just as anyone can use a calculator without being a programmer anyone can start using R as a calculator For example when balancing a checkbook R can help by adding a bunch of numbers 300 25 25 33 26 12 6 where the repeated number 12 need not be typed six times into R The advantage with R over a calculator is that one can see the numbers being added and correct any errors on the go The tip numbered xxvi among the tips
3. God We Trust x print x quote FALSE y 1 5 first 5 numbers Zz y 2 modulo division of numbers by 2 this yields 1 0 1 0 1 as five numbers a as logical z zf factor z create a categorical variable data frame x y z a Zf rename data frame as object mydata mydata data frame x y z a Zf summary mydata length mydata how many rows gt data frame x y z a zf xy Z a zf 1 In 1 1 TRUE 1 HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html R Preliminaries 19 2 God 2 0 FALSE 3 We 3 1 TRUE 4 Trust 4 0 FALSE 5 5 1 TRUE 1 gt mydata data frame x y z a zf or Oo gt summary mydata x y Zz a zf 1 Min 1 Min 0 0 Mode logical 0 2 God 1 ist Qu 2 ist Qu 0 0 FALSE 2 1 3 In 1 Median 3 Median 1 0 TRUE 3 Trust 1 Mean 3 Mean 0 6 NA s 0 We 21 3rd Qu 4 3rd Qu 1 0 Max io Max 21 0 gt length mydata 1 5 The summary function of R was encountered in the previous section Sec 1 5 It is particularly useful when applied to a data frame object The summary sensibly reports the number of times each word or symbol is repeated for character data the number of True and False values for logical variables and the number of observations in each category for categorical or factor variables Note that the length of the data frame object is report
4. by myfunction of the snippet 1 6 1 The R output follows gt myout reports the list output of myfunction outl 1 1 4 916 out2 1 1 gt myout out1 2 2 means raise to power 2 the output called out1 1 1 16 81 256 gt 13 myout out2 compute 13 times the second output out2 1 13 The rules of syntax in writing R functions are fairly simple and intuitive HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 16 Hands on Matrix Algebra Using R In any case tens of thousands of open source functions have already been written to do almost any task spread over two thousand R packages The reader can freely modify any of them giving proper credit to original authors Accessing outputs of R functions with dollar symbol conven tion The point to remember from the above illustration is that one need not write down the output of an R function on a piece of paper and type it in as an input to some other R function R offers a simple and standardized way by using the dollar symbol and output name as a suffix after the sym bol to cleanly access the outputs from any R function and then use them as inputs to another R function or manipulate collect and report them as needed The exponential growth and popularity of R can be attributed to various convenient facilities to build on the work of othe
5. 7 Generally speaking R ignores all spaces inside command lines For example x 1 3 is the same as x 1 3 Comments within R code are put anywhere starting with a hash mark such that everything from that point to the end of the line is a comment by the author of the code usually to oneself and completely ignored by the R processor is a very important and very basic function in R which combines its arguments One cannot really use R without learning to use the function For example x c 1 5 17 99 will create a vector or a list object called x with elements 1 2 3 4 5 17 99 The c stands for concatenate combine or catalog Generally each command should be on a separate line of code If one wants to type or have two or more commands on one line a semi colon must be used to separate them R is case sensitive similar to UNIX For example the lower case named object b is different from upper case named object B The object names usually start with a letter never with numbers Spe cial characters including underscores are not allowed in object names but periods are allowed R may be used as a calculator The usual arithmetic operators HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html xii xii
6. fol lowing section discusses the notion of attributes in greater detail It is particularly useful for matrix algebra R snippet x c 1 4 17 99 this x has 6 elements attr x dim c 3 2 this makes x a 3 by 2 matrix x Then R will place the elements in the object x column wise into a 3 by 2 matrix and report 1 2 1 1 4 2 2 17 3 3 99 xix R has extensive graphics capabilities The plot command is very pow erful and will be illustrated at several places in this book xx R can by tailor made for specific analyses xxi R is an Interactive Programming language but a set of R programs can be submitted in a batch mode also xxii R distinguishes between the following classes of objects nu meric logical character list matrix array factor and data frame The summary function is clever enough to do the right summarizing upon taking into account the class xxiii It is possible to convert objects from one class to another within limits Let us illustrate this idea Given a vector of numbers we can convert into a logical vector containing only two items TRUE and HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html xxiv R Preliminaries 9 FALSE by using the function as logical All numbers positive
7. format and have illustrative ex amples and user friendly vignettes The user simply has to know what the functions do and go ahead and use them at will for free Even if R is a language it is an interpreted language similar to a calculator and the language C not compiled language similar to FORTRAN GAUSS or similar older languages Similar to a calculator all R commands are implemented as they are received by R typed Instead of having subroutines similar to FORTRAN R has functions Calculations requiring hundreds of steps can be defined as functions and subsequently implemented by providing values for the suitable number of arguments to these functions A typical R package has several functions and data sets R is functional and interpreted computer language which can readily import and employ the code for functions writ ten in other languages including C C FORTRAN among others John M Chambers one of my colleagues at Bell Labs is credited with creating S and helping in the creation of R He has published an article entitled The Future of R in the first issue May 2009 of the R Journal available on the Internet at http journal r project org break2009 1 RJournal _2009 1 _Chambers pdf John lists six facets of R HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 781
8. listed in Sec 1 5 HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html R Preliminaries 3 shows how to get the birth day of the week from a birthday of a friend It can also give you the date and day of the week 100 days before today and fun things like that There are many data sets already loaded in R The dataset women contains Average Heights and Weights for American Women The simple R command summary women gives basic descriptive statistics for the data minimum maximum quartiles median and mean No calculator can do this so conveniently The dataset named co2 has Mauna Loa Atmospheric CO2 Concentration to check global warming The reader will soon discover that it is more fun to use R rather than any calculator Actually easy and powerful graphics capabilities of R make it fun for many of my students The command example points gives code examples showing all kinds of lines along points and creation of sophisticated symbols and shapes in R graphs A great Internet site for all kinds of R graphics will convince anyone how much fun one can have with R http AddictedToR free fr graphiques index php The reason why R a very powerful and useful calculator is that thou sands of functions and program packages are already written in R The packages are well documented in standard
9. row 3 and column 2 Its summary under column for second variable V2 below correctly recognizes that one item is missing gt summary x Vi v2 Min 21 0 Min 4 00 ist Qu 1 5 ist Qu 7 25 Median 2 0 Median 10 50 Mean 22 0 Mean 10 50 3rd Qu 2 5 3rd Qu 13 75 Max 23 0 Max 17 00 NA s 1 00 gt summary y Min st Qu Median Mean 3rd Qu Max 1 0 2 0 3 0 5 4 4 0 17 0 gt summary z Min ist Qu Median Mean 3rd Qu Max NA s 1 Recall that y picks out only the non missing values whereas z picks out the one missing value denoted by NA Note also that R does not foolishly try to find the descriptive statistics of the missing value xxv R admits character vectors For example HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html xxvi R Preliminaries 11 R snippet X C In i God MS We T Trust n b x print x quote FALSE We need the print command to have the options quote FALSE to prevent printing of all individual quotation marks as in the output below 1 In God We Trust n P n 1 In God We Trust Character vectors are distinguished by quotation marks in R One can combine character vectors with numbers without gaps by using the paste function of R and the argument separation with a blank as in the following snippet
10. 4 html 4 Hands on Matrix Algebra Using R i ii an interface to computational procedures of many kinds interactive hands on in real time iii functional in its model of programming iv object oriented everything is an object v modular built from standardized pieces and vi collaborative a world wide open source effort This list beautifully explains the enormous growth and power of R in recent years Chambers goes on to explain how R functions are themselves objects with their own functionality I ask my students to first type a series of commands into a text editor and then copy and paste them into R I recommend the text editor called Tinn R freely available at http www sciviews org Tinn R The reader should go to the bottom of the web page and download it to his or her computer Next click on Setup for Tinn R Be sure to use the old stable version 1 17 2 4 exe 5 2 Mb compatible with Rgui in SDI or MDI mode Tinn R color codes the R commands and provides helpful hints regarding R syntax in the left column entitled R card Microsoft Word is also a good text editor to use but care is needed to avoid smart or slanted quotation marks unknown to R 1 3 Obtaining R One can Google the word r project and get the correct Internet address The entire software is mirrored or repeated at several sites on the Internet around the world One can go to http www
11. Chapter 1 R Preliminaries 1 1 Matrix Defined Deeper Understanding Using Software Scientists and accountants often use a set of numbers arranged in rectangu lar arrays Scientists and mathematicians call rectangular arrays matrices For example a 2x3 matrix of six numbers is defined as Po k as E f 5 i wd Q21 Q22 Q23 6912 where the first subscript i 1 2 of aj refers to the row number and the second subscript 7 1 2 3 refers to the column number Such matrices and their generalizations will be studied in this book in considerable detail Rectangular arrays of numbers are called spreadsheets or workbooks in accounting parlance and Excel software has standardized them in recent decades The view of rectangular arrays of numbers by different professions can be unified The aim of this book is to provide tools for developing a deeper understanding of the reality behind rectangular arrays of numbers by applying many powerful results of matrix algebra along with certain software and graphics tools Exercise 1 1 1 Construct a 2x2 matrix with elements a i 7 Hint the first row will have numbers 2 3 and second row will have 3 4 Exercise 1 1 2 Given integers 1 to 9 construct a 3x3 matrix with first row having numbers 1 to 3 second row with numbers 4 to 6 and last row with numbers 7 to 9 The R software allows a unified treatment of all rectangular arrays as data frame objects Although matrix algebra theorems do n
12. E g 2 God 2 0 FALSE 0 3 We 3 1 TRUE 1 4 Trust 4 0 FALSE 0 5 4 gr pH t TRUE 1 gt as list mydata x 1 In God We Trust Levels God In Trust We y i 12345 z 1 10101 a 1 TRUE FALSE TRUE FALSE TRUE zf 1 10101 Levels O 1 Unfortunately having to know the various types of objects available in R may be discouraging or seem cumbersome to newcomers Most readers can safely postpone a study of these things and learn these concepts only as needed However an understanding of classes of objects and related concepts described in this chapter will be seen to be helpful for readers wishing to exploit the true power of R HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html
13. The setup does every thing including creating an Icon for R Gui graphical user interface It immediately lets you know what version is being used 1 4 Reference Manuals in R Starting at the R website http cran r project org left column click on manuals and access the first bullet point called An Introduction to R which is the basic R manual with about 100 pages It can be browsed in html format at http cran r project org doc manuals R intro html It can also be downloaded in the pdf format by clicking at a link within that bullet as http cran r project org doc manuals R intro pdf The reader is encouraged to read all chapters especially Chapter 5 of R intro dealing with arrays and Chapter 12 dealing with graphics There are dozens of free books available on the Internet about R at http cran r project org doc contrib The books are classified into 12 books having greater than 100 pages starting with Using R for Data Analysis and Graphics Introduction Examples and Commentary by John Maindonald Section 7 8 of the above pdf file deals specifically with matrices and arrays in R http cran r project org doc contrib usingR pdf There are additional links to some 18 as of July 2009 short books having less than 100 pages The list starts with R for Beginners by Emmanuel Paradis Social scientists and economists have found my own book Vinod 2008a containing several examples of R code for v
14. arious tasks a useful reference The website of the book also contains several solved problems among its ex ercises A website of a June 2009 conference on R in social sciences at Ford ham University in New York is http www cis fordham edu QR2009 The reader can view various presentations by distinguished researchers who HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 6 Hands on Matrix Algebra Using R use R 1 5 Basic R Language Tips Of course the direct method of learning any language is to start using it I recommend learning the assignment symbols or lt and the combine symbol c and start using R as a calculator A beginning user of R should consult one or more of the freely available R manuals mentioned in Sec 1 4 Nevertheless this section lists some practical points useful to beginning users as well as some experienced users of R i vii viii ix x Unlike paper and pencil mathematics means multiply either the hat symbol A or are used for raising to a power means the logical or and means logical negation If a command is inside simple parentheses not brackets it is printed to the screen Colon means generate a sequence For example x 3 7 creates an R list or vector object named x containing the numbers 3 4 5 6
15. ed as the number of rows in it Each object comprising the data frame also has the exact same length 5 Without this property we could not have constructed our data frame object by the function data frame If all objects are numerical R has the cbind function to bind the columns together into one matrix object If objects are mixed type with some character type columns then the cbind function will convert them all into character type before binding the columns together into one character matrix object We now illustrate the use of as matrix and as array functions using the data frame object called mydata inside the earlier snippet The following snippet will not work unless the earlier snippet is in the memory of R R program snippet 1 7 1 2 as matrix as list is next R snippet Place previous snippet into R memory mydata data frame x y z a zf as matrix mydata as list mydata HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 20 Hands on Matrix Algebra Using R Note from the following output that as matrix simply places quotes around the elements of the data frame object The as list converts the character vector x into a factor arranges them alphabetically as levels gt as matrix mydata x y Zz a zf 1 In IY t TRU
16. g functions and do all kinds of tasks including reading and writing HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html R Preliminaries 13 of data writing of citations data manipulations and plotting Typical R functions have some input information or objects which are then converted into some output information or objects The rules of syntax for creation of functions in R are specifically designed to facilitate collaboration and extension Since R is open source the underlying code however complicated for all R functions is readily available for extension and or modification For example typing matrix prints the internal R code for the R function matrix to the screen ready for modification Of course more useful idea is to type matrix no spaces lower case m to get the relevant user s manual page for using the function matrix The true power of R can be exploited if one knows some basic rules of syntax for R functions even if one may never write a new function After all hundreds of functions are already available for most numerical tasks in basic R and thousands more are available in various R packages It is useful to know i how to create an object containing the output of a funtion function already in the memory of R and ii the syntax for accessing the output
17. i R Preliminaries T and the hat symbol A for raising to a power are available R also has log exp sin cos tan sqrt min max with self explanatory meanings Expressions and assignments are distinct in R For example if you type the expression 2 3 4 2 R will proceed to evaluate it If you assign the expression a name x by typing xX 2 3 4 2 x The resulting object called x will contain the evaluation of the expres sion and will be available under the name x but not automatically printed to the screen Assuming that one wants to see R print it one needs to type x or print x as a separate command We have in cluded x after the semi colon to suggest a new command asking R to print the result to the screen The R language and its packages together have tens of thousands of functions for doing certain tasks on their arguments For example sqrt is a function in R which computes the square root of its argu ment It is important to use simple parentheses as in print x or sqrt x when specifying the arguments of R functions Using curly braces or brackets instead of parentheses will give syntax error or worse confusion in R Brackets are used to extract elements of an array For example x 2 extracts second element of a list x Similarly x 2 1 extracts the number along the second row and first column of a matrix x Again using curly braces or
18. iminaries 15 By choosing the name myout we send the output of the function to an object bearing that name inside R Self explanatory names are advisable More important the snippet shows how to access the objects created by any existing R function for further manipulation by anyone by using the dollar symbol as a suffix followed by the name of the output object to the name of the object created by the function R snippet 1 6 1b illustrates calling of R functions Assume myfunction is already in memory of R inp1 1 4 define first input as 1 2 3 4 inp2 pi 2 define second input as pi by 2 myout myfunction inp1 inp2 creates object myout from myfunction myout reports the list output of myfunction myout out1 2 2 means raise to power 2 the output called outi 13 myout out2 compute 13 times the second output out2 The snippet 1 6 1b illustrates how to create an object called myout by using the R function called myfunction The dollar symbol attached to myout then allows complete access to the outputs outl and out2 created by myfunction The snippet illustrates how to compute and print to the screen the square of the first output called outl It is accessed as an object under the name myout out1 for possible further manipulation where the dollar symbol is a suffix The last line of the snippet 1 6 1b shows how to compute and print to the screen 13 times second output object called out2 created
19. ix algebra in this book as a study of some relevant infor mation A part of the information can be studied in the form of a matrix The summary function is a very fundamental tool in R For example most people have to contend with stock market or financial data at some point in their lives Such data are best reported as data frame objects in R having matrix like structure to deal with any kind of data matrices In medical data the sex of the patient is a categorical vector variable patient name is a character vector whereas patient pulse is a numerical variable All such variables can be handled satisfactorily in R The matrix object in R is reserved for numerical data However many operations analogous to those on numerical matrices e g summarize the data can be suitably redefined for non numerical objects We can conveniently implement all such operations in R by using the concept of data frame objects invented in the S language which preceded R We illustrate the construction of a data frame object in R by using the function data frame and then using the summary function on the data frame object called mydata Note that R does a sensible job of summarizing the mixed data frame object containing character numerical categorical and logical variables It also uses the modulo function which finds the remainder of division of one number by another R program snippet 1 7 1 1 is next x c In
20. k 100 days before the current date If you want to know the day of week 100 days before today use the R command weekdays Sys Date 100 If you want to know the day of the week from a birth date which is on September 3 1971 then use the command weekdays as Date 1971 09 03 with quotes as shown and R will respond with the correct day of the week Friday gt date 1 Sat Jul 11 20 04 41 2009 gt Sys Date 1 2009 07 11 gt weekdays Sys Date 100 plural weekday 1 Thursday gt weekdays as Date 1971 09 03 find day of week from birth date 1 Friday These are some of the fun uses of R An R package called chron James and Hornik 2010 allows further manipulation of all kinds of date objects Complex number i 1 is denoted by 1i For example if x 1 2i and y 1 2i the product ry 1 2 5 as seen in the following snippet R program snippet 1 5 3 dot product is next x 1 2 1i number 1 followed by letter i is imaginary i in y 1 2 1i x y dot product of two imaginary numbers gt x y dot product of two imaginary numbers 1 5 0i Packages within R Recall from Sec 1 3 that John Chambers has noted that R is modular and collaborative The modules mostly contain so called functions in R which are blocks of R code which follow some rules of syntax Blocks of input code submitted to R are called snippets in this book These can call existin
21. or negative are made TRUE and the number zero is made FALSE Conversely given a logical vector we can convert it into numerical vector by using the function as numeric The reader should see that this converts TRUE to the number 1 and FALSE to the number 0 The R function as complex converts the numbers into complex num bers with the appropriate coefficient 0 for the imaginary part of the complex number identified by the symbol i y 1 The as character function places quotes around the elements of a vector R program snippet 1 5 1 as something is next x c 0 3 17 99 this x has 6 elements y as logical x y makes O FALSE all numbers TRUE as numeric y evaluate as 1 or 0 as complex x will insert i as character x will insert quotes The following illustrates how R converts between classes using the as something command The opeartion of as matrix and as array will be illustrated in the following Sec 1 7 gt y as logical x y 1 FALSE TRUE TRUE TRUE TRUE TRUE gt x c 0 3 17 99 this x has 6 elements gt y as logical x y makes O FALSE all numbers TRUE 1 FALSE TRUE TRUE TRUE TRUE TRUE gt as numeric y evaluate as 1 or 0 1 011111 gt as complex x will insert i 1 0 0i 1401 2 0i 3 0i 17 0i 99 0i gt as character x will insert quotes 1 oO TAY 2w g ona gg Missing data in R are denoted by NA without the quotation marks These are handled cor
22. ot directly apply to data frames they are a useful preliminary allowing matrix algebra to be a very practical and useful tool in everyday life not some esoteric subject for scientists and mathematicians For one thing the data frame 1 HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 2 Hands on Matrix Algebra Using R objects in R software allow us to name the rows and columns of matrices in a meaningful way We can of course strip away the row column names when treating them as matrices Matrix algebra is abstract in the sense that its theorems hold true irrespective of row column names However the deeper understanding of the reality behind those rectangular arrays of numbers requires us to have easy access to those names when we want to interpret the meaning of matrix algebra based conclusions 1 2 Introduction Why R The New York Times 6 January 2009 had an article by Daryl Pregibon a research scientist at Google entitled Data Analysts Captivated by R s Power It said that the software and graphics system called R is fast becoming the lingua franca of a growing number of data analysts inside corporations and academia Countless statisticians engineers and scientists without computer programming skills find R easy to use It is hard to believe that R is free The article also stated that
23. parentheses will not work for this purpose Curly braces and are used to combine several expressions into one procedure Re executing previous commands is done by using vertical arrow keys on the keyboard Modifications to such commands can be made by using horizontal arrows on the keyboard Typing history provides a list of all recent commands typed by the user These commands can then be copied into MS Word or any text editor suitably changed copied and pasted back into R R provides numerical summaries of data For example let us use the vector of seven numerical values mentioned above as x and let us issue certain commands in R especially the summary command collected in the following R software input snippet HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 8 Hands on Matriz Algebra Using R R snippet x c 1 5 17 99 summary x Then R will evaluate them and report to the screen the following out put Min ist Qu Median Mean 3rd Qu Max 1 00 2 50 4 00 18 71 11 00 99 00 Note that the function summary computes the minimum first quar tile 25 of data are below this and 75 are above this median third quartile 75 of data are below this and 25 are above this and the maximum xviii Attributes function attr can be used to create a matrix The
24. r project org and choose a geographically nearby mir ror For example US users can choose http cran us r project org Once arriving at the appropriate Internet site one looks on the left hand side under Download Next click on CRAN Link Sometimes the site asks you to pick a mirror closest to your location again Click on Windows if you have a Windows computer and then Click on base For example if you are using Windows XP computer and if the latest version of R is 2 9 1 then click on Download R 2 9 1 for Windows It is a good idea to download the setup program file to a temporary direc tory which usually takes about 7 8 minutes if you have a reasonably fast connection to the Internet Now double click on the setup file R 2 9 1 win32 exe to get R set up on any local computer When the setup wizard HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html R Preliminaries 5 will first ask you language choose English Then click Next and follow other on screen instructions If you have a Windows Vista computer among frequently asked ques tion they have answer to the question How do I install R when using Windows Vista One of the hints is Run R with Administrator privileges in sessions where you want to install packages Choose simple options Do not customize it
25. rectly by R In the following code we deliber ately make the x value along third row and second column as NA that is we specify that x 3 2 as missing The exclamation symbol is used as a logical not Hence if is na means is missing is na means is not missing The bracketed conditions in R with repetition of x outside and inside the brackets are very useful for subset selection For example x is na x and x is na x in the following snippet cre HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 10 Hands on Matrix Algebra Using R ate subsets to x containing only the non missing values and only the missing values respectively R program snippet 1 5 2 is next x c 1 4 17 99 this x has 6 elements attr x dim c 3 2 this makes x a 3 by 2 matrix x 3 2 NA y x is na x picks only non missing subset of x z xlis na x picks only the missing subset of x summary x summary y summary z Then R will recognize that x is a 3 by 2 matrix and the summary function in the snippet 1 5 2 will compute the summary statistics for the two columns separately Note that y converts x matrix into an array of non missing set of values and computes their summary statistics The vector z contains only the one missing value in the location at
26. rs The standardized access to function outputs by using the dollar symbol is one example of such facility in R A second example is how R has standardized the non trivial task of package writing Every author of a new R package must follow certain detailed and strict rules describing all important details of data inputs and outputs of all included functions with suitably standardized look and feel of all software manuals Of course important additional reasons facilitating world wide collaboration include the fact that R is OOL free and open source R packages consist of collections of such functions designed to achieve some specific tasks relevant in some scientific discipline We have already encountered base and contrib packages in the process of installation of R The base package has the collection of hundreds of basic R functions belonging to the package The contrib packages have other packages and add ons They are all available free and on demand as follows Use the R Gui menu called Packages and from the menu choose install packages R asks you to choose a local comprehensive R archive network CRAN mirror or CRAN mirror Then R lets you Install Packages chosen by name from the long alphabetical list of some two thousand names The great flexibility and power of R arises from the availability of these packages The users of packages are requested to give credit to the creators of packages b
27. s list outl out1 out2 out2 Note that it has strange repetition of names This is actually not strange but allows the author com plete freedom to name objects inside his function The outl out1l simply means that the author does not want to use an external name distinct from the internal name For example if the author had internally called the two outputs as ol and 02 the list command would look like list out1 ol out2 02 The last line of a function is usually a curly brace Any user of this function then calls the R function myfunction with the R command myout myfunction inl in2 This command will not work unless the function object myfunction and input objects inpl and inp2 are already in the current memory of R That is the snippet 1 6 1 must be loaded and R must give hack the prompt before we can use the function Note that the symbol myout on the left hand side of this R command is completely arbitrary and chosen by the user Short and unique names are advisable If no name is chosen the command myfunction inp1 inp2 will generally report certain outputs created by the function to the screen By simply typing the chosen name myout with carriage return sends the output to the screen HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html R Prel
28. s of the function by the dollar symbol suffix Hence I illustrate these basic rules using a rather trivial R function for illustration purposes R snippet 1 6 1 explains inputs outputs of R functions myfunction function inp1 inp2 verbos FALSE function code begins with a curly brace if verbos print inp1 print inp2 outi inp1 2 first output squares the first input out2 sin inp2 second output computes sin of second input if verbos print out1 print out2 list outi out1 out2 out2 syntax for listing outputs function code ENDS with a curly brace The reader should copy and paste all the lines of snippet 1 6 1 into R and wait for the R prompt Only if the function is logically consistent R will return the prompt If some errors are present R will try to report to the screen those errors The first line of the function code in snippet 1 6 1 declares its name as myfunction and lists the two objects called inpl and inp2 as required inputs The function also has an optional logical variable verbos FALSE as its third input to control the printing with a default setting indicated by the equal symbol as FALSE All input objects must be listed inside parentheses not brackets separated by commas HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 14 Hands on Matrix Algebra U
29. sing R Since a function typically has several lines of code we must ask R to treat all of them together as a set This is accomplished by placing them inside two curly braces This is why the second line has left curly brace The third line of the snippet 1 6 1 has the command if verbos print inp1 print inp2 This will optionally print the input objects to the screen if verbos is TRUE during a call to the function as myfunc tion inp1 inp2 verbos TRUE Now we enter the actual tasks of the function Our trivial function squares the first input inpl object and returns the squared values as the first output as an output object named outl The writer of the function has full freedom to name the objects within reason so as not to conflict with other R names The code outl inp12 creates the outl for our function The function also needs to find the sine of second inp2 object and return the result as the second output named out2 here This calculation is done here still within the curly braces by the code on the fifth line of the snippet 1 6 1 line out2 sin inp2 Once the outputs are created the author of the R function must decide what names they will have as output objects The author has the option to choose an output name different from the name internal to the function The last but one line of a function usually has a list of output objects In our example it i
30. two dimensions The array objects in R can be 3 dimensional matrices iii factors are categorical data objects Unlike some languages R allows object to be mixed type with some vectors numerical and other vectors with characters For example in stock market data the Ticker symbol is a character vector and stock price is a numerical vector iv Complex number objects In mathematics numerical vectors can be complex numbers with i 1 attached to the imaginary part v R objects called lists permit mixtures of all these types of objects vi Logical vectors contain only two values TRUE and FALSE R does understand the abbreviation T and F for them How does R distinguish between different types of objects Each object has certain distinguishing attributes such as mode and length It is interesting that R can convert some objects by converting their attributes For example R can convert the numbers 1 and 0 into logical values TRUE and FALSE and vice versa The as logical function of R applied to a bunch of zeros and ones makes them a corresponding bunch of TRUE and FALSE values This is illustrated in Sec 1 7 1 HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html 18 Hands on Matrix Algebra Using R 1 7 1 Dataframe Matrix and Its Summary We think of matr
31. y citing the authors in all publications After all the developers of free R packages do not get any financial reward for their efforts The citation function of the base package of R reports to the screen detailed citation information about citing any package Since there are over two thousand R packages available at the R website HANDS ON MATRIX ALGEBRA USING R Active and Motivated Learning with Applications World Scientific Publishing Co Pte Ltd http www worldscibooks com mathematics 7814 html R Preliminaries 17 it is not possible to summarize what they do Some R packages of interest to me are described at my web page at http www fordham edu economics vinod r lang doc My students find this file containing my personal notes about R MS Word file some 88 pages long as a time saving devise They use the search facility of MS Word on it One can use the standard Google search to get answers to R queries but this can be inefficient since it catches the letter R from anywhere in the document Instead I find that it is much more efficient to search answers to any and all R related questions at http www rseek org 1 7 R Object Types and Their Attributes R is OOL with various types of objects We note some types of objects in this section i vector objects created by combining numbers separated by commas are already encountered above ii matrices are objects in R which generalize vectors to have
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