The -biplot- command and software development at StataCorp.

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1 Outline The -biplot- command and software development at StataCorp. June 26, 2009

2 1 Properties Mathematical Background 2 Biplot now Forthcomming biplot 3

3 Outline Properties Mathematical Background 1 Properties Mathematical Background 2 Biplot now Forthcomming biplot 3

4 Biplot s properties Outline Properties Mathematical Background Biplot multivariate analysis feature graphical two-dimensional representation of dataset: arrows: variables marker symbols: observations Dimension 2 variable2 3 4 variable variable3 Dimension 1 Variables Observations Biplot of 8 observations and 3 variables Explained variance by component Explained variance by component Total explained variance

5 Biplot s properties Outline Properties Mathematical Background Helpfull in understanding the relationship between variables and observations separately and jointly: the distance between observations is approximately preserved the cosine of the angle between arrows approximates the correlation between variables relation of observations to variables Dimension 2 variable2 Biplot variable3 variable1 Dimension 1 Variables Observations

6 Example: students Outline Properties Mathematical Background Students scores in mathematics, literature, and sports (1-10). list stud sex math literat sport 1. Beth female Cindy female Kathy female Shari female Bill male David male John male Kevin male Dimension 2 literature Biplot Beth Kathy Cindy Bill David Shari Kevin John sport mathematics Dimension 1 Variables Students

7 Methods and formulas Properties Mathematical Background 1 Singular value decomposition of the centered data matrix X : X = U obs L V var X = U obs L α L 1 α V var for α [0, 1] 2 Coordinates for Observations and variables: X = G H G = U obs L α H = L 1 α V var 3 Coefficient α: columns preserving biplot for α = 0 rows preserving biplot for α = 1 symmetric biplot for α = 0.5

8 Outline Biplot now Forthcomming biplot 1 Properties Mathematical Background 2 Biplot now Forthcomming biplot 3

9 Biplot now Forthcomming biplot Biplot s syntax biplot varlist [ if ][ in ][, options ]

10 Biplot now Forthcomming biplot Example of options: rowopts() affects rendition of observations. biplot math literat sport, rowopts(mlabel(stud) name(students) msymbol(t) mcolor(red)) Biplot Dimension 2 literat Kathy Cindy Shari math Beth Kevin David John sport Bill Dimension 1 Variables Students

11 Biplot now Forthcomming biplot Example of options: dim() affects dimensions table displays table showing biplot coordinates. biplot math literat sport, table dim(2 3) Biplot coordinates Observations dim3 dim Variables dim3 dim2 Dimension 2 2 literat sport 6 5 Biplot 3 4 math Dimension 3 Variables Observations math literat sport Biplot of 8 observations and 3 variables Explained variance by component Explained variance by component Total explained variance

12 What will be new in biplot? Biplot now Forthcomming biplot No limits for number of observations: matrix computations implemented in Mata New options: rowover() and row#opts() rowlabel generate()

13 Example Outline Biplot now Forthcomming biplot. biplot math literat sport, rowover(sex) norowlabel row1opts(msymbol(x) mcolor(red)) row2opts(msymbol(t) mcolor(green)) generate(coord1 coord2) Biplot Dimension 2 literat math sport Dimension 1 Variables female male

14 generate(coord1 coord2) Biplot now Forthcomming biplot. list coord1 coord2 coord1 coord Dimension 2 literat 3 4 math 2 Biplot sport Dimension 1 Variables male female

15 Outline 1 Properties Mathematical Background 2 Biplot now Forthcomming biplot 3

16 The story of one option rowover()

17 The story of one option rowover() Before Biplot Dimension literat sport math Dimension 1 Variables Observations

18 The story of one option rowover() Before After Biplot Biplot Dimension literat sport math Dimension 1 Variables Observations Dimension literat sport math Dimension 1 Variables female male

19 Name rowover The choice of the new option s name : "by" generic in context of graphs means separate graph for every group

20 Name rowover The choice of the new option s name : "by" generic in context of graphs means separate graph for every group "over" over different groups overlay graphs still too general

21 Name rowover The choice of the new option s name : "by" generic in context of graphs means separate graph for every group "over" over different groups overlay graphs still too general "rowover" over different groups in rows overlay graphs concerns rows

22 Name rowover The choice of the new option s name : "by" generic in context of graphs means separate graph for every group "over" over different groups overlay graphs still too general "rowover" over different groups in rows overlay graphs concerns rows 1st lesson learned Names used in Stata must suit the whole system of commands

23 Input Outline New option has to accept all syntactically correct input. For example: unbalanced parentheses unbalanced quotes special characters. biplot math literat sport, rowover(sex) row1opts(name( ")"" ))

24 Input Outline New option has to accept all syntactically correct input. For example: unbalanced parentheses unbalanced quotes special characters. biplot math literat sport, rowover(sex) row1opts(name( ")"" )) 2nd lesson learned Stata commands must be robust

25 Certification of the command New option has to pass certification. Certification script tests the quality of program: covers many possible use cases contains the prediction of the correct output checks the conformity between obtained and expected results Example. local expectedresult = biplot math literat sport, generate(coord1 coord2). assert expectedresult == coord1[1]

26 Certification of the command New option has to pass certification. Certification script tests the quality of program: covers many possible use cases contains the prediction of the correct output checks the conformity between obtained and expected results Example. local expectedresult = biplot math literat sport, generate(coord1 coord2). assert expectedresult == coord1[1] 3rd lesson learned The quality of the program must be sufficiently tested

27 Documentation The new option has to be documented: information for users (help files and manuals) information for developers (internal documentation)

28 Documentation The new option has to be documented: information for users (help files and manuals) information for developers (internal documentation) 4th lesson learned Good documentation is as important as a good program

29 Graphical User Interface New option has to be added to the dialog box in the proper way.

30 Graphical User Interface New option has to be added to the dialog box in the proper way. 5th lesson learned Do not forget about usability

31 Certification of the command in system Certification script tests the quality of program: Proof that the command works as a part of the system Proof that the command does not have any undesirable side effects on the system Proof that the comand works on different platforms and in different editions of Stata

32 Certification of the command in system Certification script tests the quality of program: Proof that the command works as a part of the system Proof that the command does not have any undesirable side effects on the system Proof that the comand works on different platforms and in different editions of Stata 6th lesson learned A command is always a part of the system

33 Outline Conclusions Lessons learned: 1 Names used in Stata must suit the whole system of commands 2 Stata commands must be robust 3 The quality of the program must be sufficiently tested 4 Good documentation is as important as a good program 5 Do not forget about usability 6 A command is always a part of the system

Outline Conclusions Lessons learned: 1 Names used in Stata must suit the whole system of commands 2 Stata commands must be robust 3 The quality of the

34 Outline Conclusions Lessons learned: 1 Names used in Stata must suit the whole system of commands 2 Stata commands must be robust 3 The quality of the program must be sufficiently tested 4 Good documentation is as important as a good program 5 Do not forget about usability 6 A command is always a part of the system

Solving Systems of Equations Using Matrices With the TI-83 or TI-84

Solving Systems of Equations Using Matrices With the TI-83 or TI-84 Dimensions of a matrix: The dimensions of a matrix are the number of rows by the number of columns in the matrix. rows x columns *rows