New pictures for correlation structure Jan Graffelman 1 1 Department - PowerPoint PPT Presentation

1 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation New pictures for correlation structure Jan Graffelman 1 1 Department of Statistics and Operations Research Universitat Polit` ecnica de Catalunya jan.graffelman@upc.edu 6 th Carme conference, Rennes, 10 th of February, 2011 6 t h Carme Conference, Rennes, February 2011 Graffelman

2 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation Outline 1 Correlations and cosines 6 t h Carme Conference, Rennes, February 2011 Graffelman

3 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation Outline 1 Correlations and cosines 2 The interpretation function 6 t h Carme Conference, Rennes, February 2011 Graffelman

4 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation Outline 1 Correlations and cosines 2 The interpretation function 3 Biplots and interpretation function ( p = 2) 6 t h Carme Conference, Rennes, February 2011 Graffelman

5 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation Outline 1 Correlations and cosines 2 The interpretation function 3 Biplots and interpretation function ( p = 2) 4 Biplots and interpretation function ( p > 2) 6 t h Carme Conference, Rennes, February 2011 Graffelman

6 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation Outline 1 Correlations and cosines 2 The interpretation function 3 Biplots and interpretation function ( p = 2) 4 Biplots and interpretation function ( p > 2) 5 Final comments 6 t h Carme Conference, Rennes, February 2011 Graffelman

7 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation Correlations are cosines Sample geometry of X : in R n the cosines of the angles between p variable vectors equal their correlations. � ( x i − x )( y i − y ) x ′ y √ � ( x i − x ) 2 √ � ( y i − y ) 2 = r ( x , y ) = � = cos ( α ). � x �� y In full space PCA biplots the cosine of the angle between two biplot vector equals the sample correlation coefficient of the corresponding variables. In CCA the canonical correlation is the cosine of the angle between two subspaces. ... It seems natural to represent correlations by cosines. 6 t h Carme Conference, Rennes, February 2011 Graffelman

8 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation Correlations are not necessarily cosines (a) PCA biplot (r = 0.5) ● ● ● ● 3 ● ● ● −2 ● ● ● 2 ● ● ● −1 ● 2 ● ● ● 1 ● ● ● ● ● ● 0 ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● 1 ● ● 1 ● ● ● −1 ● ● ● ● 2 ● ● −2 ● ● 3 ● ● 6 t h Carme Conference, Rennes, February 2011 Graffelman

9 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation Correlations are not necessarily cosines (b) Scatterplot (r = 0.5) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 6 t h Carme Conference, Rennes, February 2011 Graffelman

10 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation The interpretation function f ( α ) You can relate r to the angle in the way you like: 1.0 0.5 Correlation 0.0 −0.5 −1.0 0 1 2 3 4 5 6 Angle (in radians) 6 t h Carme Conference, Rennes, February 2011 Graffelman

11 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation The interpretation function f ( α ) You can relate r to the angle in the way you like: 1.0 0.5 Correlation 0.0 −0.5 −1.0 0 1 2 3 4 5 6 Angle (in radians) 6 t h Carme Conference, Rennes, February 2011 Graffelman

12 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation The interpretation function f ( α ) You can relate r to the angle in the way you like: 1.0 0.5 Correlation 0.0 −0.5 −1.0 0 1 2 3 4 5 6 Angle (in radians) 6 t h Carme Conference, Rennes, February 2011 Graffelman

13 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation The interpretation function f ( α ) You can relate r to the angle in the way you like: 1.0 0.5 Correlation 0.0 −0.5 −1.0 0 1 2 3 4 5 6 Angle (in radians) 6 t h Carme Conference, Rennes, February 2011 Graffelman

14 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation The interpretation function f ( α ) The number of possible ways to represent a correlation is infinite. Which interpretation function makes most sense? 6 t h Carme Conference, Rennes, February 2011 Graffelman

15 Outline Correlations and cosines The interpretation function Biplots and interpretation function ( p = 2 ) Biplots and interpretation A biplot questionnaire a) r(x1,x2)=........ b) r(x1,x2)=........ c) r(x1,x2)=........ 4 ● ● 2 2 ● ● ● ● ● ● ● ● x1 ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● x1 ● ● ● ● ● ● ● ● ● ● x1 ● ● ● ● ● ● ● 0 ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● x2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● x2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 ● −2 ● ● x2 −2 ● ● ● ● ● ● ● ● ● −4 −4 −4 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 −3 −2 −1 0 1 2 d) r(x1,x2)=........ e) r(x1,x2)=........ f) r(x1,x2)=........ 4 4 4 ● ● ● ● 2 ● ● ● ● 2 ● x1 ● ● ● ● x2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● x2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● x1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● x1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● x2 ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 ● ● ● ● ● ● ● ● ● −2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 ● ● −4 −4 −2 −1 0 1 2 3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 6 t h Carme Conference, Rennes, February 2011 Graffelman