2. Plan for today An example with 3 variables Face ratings 1: age, gender, and attractiveness – Histograms and scatter plots – Using symbols and colors to visually segment data – Full and partial correlations – 3D scatter plots Understanding intransitive correlations Face ratings 2: dominance, neoteny, and attractiveness – 3D scatter plots – Stepwise and full regression Fully crossed data – Surface plots Time permitting: simplifying high-dimensional data – Principal components analysis (PCA)

3. Face ratings Gender? Age?Attractiveness? 76 raters x 276 faces x 3 characteristics (with Corinne Olafsen ’14)

4. Face ratings Mean data across 76 raters 279 faces : 3 characteristics (with Corinne Olafsen ’14)

5. Exploratory look at the data: histograms – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714];

6. Exploratory look at the data: histograms – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; >> figure(101) >> hist(age2);

7. Exploratory look at the data: histograms – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; >> figure(102) >> hist(gender2);

8. Exploratory look at the data: histograms – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; >> figure(103) >> hist(attr2);

9. 3D Histograms! – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; >> figure(104) >> hist(age2, gender2); >> xlabel(‘Age’); ylabel(‘Gender’);

10. Looking at 2 variables at a time Pairwise scatter plots – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714];

11. Looking at 2 variables at a time Pairwise scatter plots – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; (1) Attractiveness versus Gender In Matlab: >> figure(1); set(gca,'fontsize',16); >> plot(gender2, attr2, '.k'); >> xlabel('Gender (1=f, 4=m)') >> ylabel('Attractiveness (1-4)’)

12. Looking at 2 variables at a time Pairwise scatter plots – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; (1) Attractiveness versus Gender In Matlab: >> figure(1); set(gca,'fontsize',16); >> plot(gender2, attr2, '.k'); >> xlabel('Gender (1=f, 4=m)') >> ylabel('Attractiveness (1-4)’)

13. Looking at 2 variables at a time Pairwise scatter plots – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; (2) Attractiveness versus Age In Matlab: >> figure(2); set(gca,'fontsize',16); >> plot(age2, attr2, '.k'); >> xlabel(’Age (years)') >> ylabel('Attractiveness (1-4)’)

14. Looking at 2 variables at a time Pairwise scatter plots – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; (2) Attractiveness versus Age In Matlab: >> figure(2); set(gca,'fontsize',16); >> plot(age2, attr2, '.k'); >> xlabel(’Age (years)') >> ylabel('Attractiveness (1-4)’)

15. Looking at 2 variables at a time Pairwise scatter plots – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; (3) Age versus Gender In Matlab: >> figure(3); set(gca,'fontsize',16); >> plot(gender2, age2, '.k'); >> xlabel(’Gender (1=f, 4=m)’) >> ylabel(’Age (years)')

16. Looking at 2 variables at a time Pairwise scatter plots – In Matlab: >> age2 = [2.0517 1.6724 2.0517 … 3.9138]; >> gender2 = [38.1034 28.2759 37.7586 … 38.9655]; >> attr2 = [2.1429 2.3143 1.6571 … 1.7714]; (3) Age versus Gender In Matlab: >> figure(3); set(gca,'fontsize',16); >> plot(gender2, age2, '.k'); >> xlabel(’Gender (1=f, 4=m)’) >> ylabel(’Age (years)')

17. Breaking it down Attractiveness versus age (using different symbols for gender) In Matlab: >> figure(801); set(gca,'fontsize',16); >> m2 = find(gender2>2.5); >> plot(age2(m2),attr2(m2),'.b', 'markersize',25); >> hold on >> f2 = find(gender2<=2.5); >> plot(age2(f2),attr2(f2),'.r', 'markersize',25); >> xlabel('Age'); ylabel('Attractiveness (1-4)'); >> legend({'male' 'female'});

18. Breaking it down Attractiveness versus age (using different symbols for gender) In Matlab: >> figure(801); set(gca,'fontsize',16); >> m2 = find(gender2>2.5); >> plot(age2(m2),attr2(m2),'.b', 'markersize',25); >> hold on >> f2 = find(gender2<=2.5); >> plot(age2(f2),attr2(f2),'.r', 'markersize',25); >> xlabel('Age'); ylabel('Attractiveness (1-4)'); >> legend({'male' 'female'});

19. Breaking it down Attractiveness versus gender (using different symbols for gender) In Matlab: >> figure(802); set(gca,'fontsize',16); >> plot(gender2(m2),attr2(m2),'.b', 'markersize',25); >> hold on >> plot(gender2(f2),attr2(f2),'.r', 'markersize',25); >> xlabel(’Gender'); ylabel('Attractiveness (1-4)');

20. Breaking it down Attractiveness versus gender (using different symbols for gender) In Matlab: >> figure(802); set(gca,'fontsize',16); >> plot(gender2(m2),attr2(m2),'.b', 'markersize',25); >> hold on >> plot(gender2(f2),attr2(f2),'.r', 'markersize',25); >> xlabel(’Gender'); ylabel('Attractiveness (1-4)');

21. Breaking it down Correlations between gender and attractiveness? Overall: >> [r p] = corr(gender2, attr2) r = -0.2797 p = 2.3604e-06

22. Breaking it down Correlations between gender and attractiveness? Overall: >> [r p] = corr(gender2, attr2) r = -0.2797 p = 2.3604e-06 Just males: >> [r p] = corr(gender2(m2), attr2(m2)) r = 0.0307 p = 0.7028

23. Breaking it down Correlations between gender and attractiveness? Overall: >> [r p] = corr(gender2, attr2) r = -0.2797 p = 2.3604e-06 Just males: >> [r p] = corr(gender2(m2), attr2(m2)) r = 0.0307 p = 0.7028 Just females: >> [r p] = corr(gender2(f2), attr2(f2)) r = -0.5560 p = 5.2062e-11

24. Putting it all together with color Hue  Gender Brightness  Age >> figure(804); clf; set(gcf,'color','w'); set(gca,'fontsize',16); >> for i=1:length(m2) markercolor = [0 0 1-age2(m2(i))-min(age2(m2)))/(max(age2(m2))-min(age2(m2)))]; plot(gender2(m2(i)),attr2(m2(i)),'.', 'markersize',35,'color’,markercolor); hold on >> end >> for i=1:length(f2) markercolor = [1-age2(f2(i))-min(age2(f2)))/(max(age2(f2))-min(age2(f2))) 0 0]; plot(gender2(f2(i)),attr2(f2(i)),'.', 'markersize',35,'color',’markercolor’); >> end >> xlabel('Gender');ylabel('Attractiveness');

25. Putting it all together with color Blue = Male Red = Female Brighter = younger Darker = older

26. 3D scatter plots >> figure(803); clf; set(gcf,'color','w'); set(gca,'fontsize',16); >> plot3(age2,gender2,attr2, '.k’) >> xlabel('Age'); ylabel('Gender');zlabel('Attractiveness'); Figure 803

27. 3D scatter plots with colors >> figure(805); clf; set(gcf,'color','w'); set(gca,'fontsize',16); for i=1:length(m2) markercolor = [0 0 1-(age2(m2(i))-min(age2(m2)))/(max(age2(m2))-min(age2(m2)))]; plot3(age2(m2(i)),gender2(m2(i)),attr2(m2(i)),'.', 'markersize',25,'color',markercolor); hold on end for i=1:length(f2) markercolor = [1-(age2(f2(i))-min(age2(f2)))/(max(age2(f2))-min(age2(f2))) 0 0]; plot3(age2(f2(i)),gender2(f2(i)),attr2(f2(i)),'.', 'markersize',25, 'color',markercolor) end xlabel('Age'); ylabel('Gender');zlabel('Attractiveness'); Figure 805

28. New example: Dominance, neoteny, and attractiveness (with Brianna Jeska ’15)

29. Predictions: Dominance  attractiveness Neoteny  attractiveness But dominance is negatively related to neoteny (???) New example: Dominance, neoteny, and attractiveness (with Brianna Jeska ’15)

30. Dominance, neoteny, and attractiveness Data: 13 raters x 39 faces x 3 characteristics Mean data across 13 raters 39 faces : 3 characteristics

31. Dominance, neoteny, and attractiveness In Matlab: >> dom = [4.2308 4.4615 3.0769 … 4.3077]; >> neot = [3.6154 2.9231 2.7692 … 3.6923]; >> attr = [3.7692 2.9231 2.6923 … 3.0769];

32. Pairwise scatter plots and correlations Attractiveness vs. neoteny In Matlab: >> figure(1); set(gca,'fontsize',16); >> plot(dom,attr,'.k'); >> xlabel('Neoteny') >> ylabel('Attractiveness'); >> [r p] = corr(neot,attr) r = 0.2359 p = 0.1483 marginally correlated

33. Pairwise scatter plots and correlations Attractiveness vs. dominance In Matlab: >> figure(2); set(gca,'fontsize',16); >> plot(neot,attr,'.k'); >> xlabel('Dominance') >> ylabel('Attractiveness'); >> [r p] = corr(dom,attr) r = 0.5251 p = 5.9848e-04 strongly correlated

34. Pairwise scatter plots and correlations Neoteny vs. dominance? In Matlab: >> figure(3); set(gca,'fontsize',16); >> plot(dom,neot,'.k'); >> xlabel('Dominance') >> ylabel('Neoteny'); >> [r p] = corr(dom,neot) r = -0.5032 p = 0.0011 negatively correlated!

35. 3D scatter plots >> figure(4); set(gca,'fontsize',16); >> plot3(dom,neot,attr,'.k'); >> xlabel('Dominance'); ylabel('Neoteny'); zlabel('Attractiveness'); Figure 4

36. 3D scatter plots >> figure(4); set(gca,'fontsize',16); >> plot3(dom,neot,attr,'.k'); >> xlabel('Dominance'); ylabel('Neoteny'); zlabel('Attractiveness'); Figure 4 >> view([0 0]) attr vs. dom

37. 3D scatter plots >> figure(4); set(gca,'fontsize',16); >> plot3(dom,neot,attr,'.k'); >> xlabel('Dominance'); ylabel('Neoteny'); zlabel('Attractiveness'); Figure 4 >> view([90 0]) attr vs. neot

38. 3D scatter plots >> figure(4); set(gca,'fontsize',16); >> plot3(dom,neot,attr,'.k'); >> xlabel('Dominance'); ylabel('Neoteny'); zlabel('Attractiveness'); Figure 4 >> view([0 90]) neot vs. dom

39. Stepwise and full regression Stepwise regression model: >> stepwise([dom neot],attr)

40. Stepwise and full regression Stepwise regression model: >> stepwise([dom neot],attr) Full regression model: >> c = regress(attr,[dom neot]) c = 0.6460 0.3284 (coefficients of dom and neot)

41. Stepwise and full regression Stepwise regression model: >> stepwise([dom neot],attr) Full regression model: >> c = regress(attr,[dom neot]) c = 0.6460 0.3284 (coefficients of dom and neot) >> y = c(1)*dom + c(2)*neot; >> figure(5); set(gca,'fontsize',16); >> plot(y,attr,'.'); >> xlabel('Attractiveness predictor'); >> ylabel('Attractiveness');

42. Surface plots Requirements: – 2 independent variables – Data for very combination of values on the independent variables For example: a lexical decision task – IVs: 1.Orientation of the string (0°, 45°, 90°, 135°, 180°) 2.String length (3 letters, 4 letters, 5 letters, 6 letters) – DV: Reaction time

43. Surface plots Made up data:

44. Surface plots Made up data: In Matlab: >> v = [1.25 1.23 1.32 1.4 1.42 1.41 1.45 1.48 1.43 1.53 1.54 1.59 1.78 1.81 1.91 2.03 2.15 2.33 1.6 1.68 1.63 1.81 2.32 2.71 2.89 2.98 2.9 1.86 1.89 1.83 2.01 2.41 2.99 3.25 3.49 3.78 2.01 1.98 2.07 2.48 3.02 3.79 4.08 4.3 4.28 2.34 2.45 2.6 3.15 3.87 4.03 4.52 4.86 4.69]; >> figure(11) >> surf(v)

45. Surface plots Figure 11 Made up data: In Matlab: >> v = [1.25 1.23 1.32 1.4 1.42 1.41 1.45 1.48 1.43 1.53 1.54 1.59 1.78 1.81 1.91 2.03 2.15 2.33 1.6 1.68 1.63 1.81 2.32 2.71 2.89 2.98 2.9 1.86 1.89 1.83 2.01 2.41 2.99 3.25 3.49 3.78 2.01 1.98 2.07 2.48 3.02 3.79 4.08 4.3 4.28 2.34 2.45 2.6 3.15 3.87 4.03 4.52 4.86 4.69]; >> figure(11) >> surf(v)

46. Surface plots Figure 12 Made up data: In Matlab: >> v = [1.25 1.23 1.32 1.4 1.42 1.41 1.45 1.48 1.43 1.53 1.54 1.59 1.78 1.81 1.91 2.03 2.15 2.33 1.6 1.68 1.63 1.81 2.32 2.71 2.89 2.98 2.9 1.86 1.89 1.83 2.01 2.41 2.99 3.25 3.49 3.78 2.01 1.98 2.07 2.48 3.02 3.79 4.08 4.3 4.28 2.34 2.45 2.6 3.15 3.87 4.03 4.52 4.86 4.69]; >> figure(11) >> surf(v) OR >> figure(12); set(gca,'fontsize',16) >> a = [0:22.5:180]; >> wl = [3:8]; >> surf(a,wl,v); >> xlabel('Angle') >> ylabel('Word length') >> zlabel('Reaction time')

47. Dimensionality reduction Principal components analysis – A type of factor analysis, assuming normally distributed variables – Reduces high-dimensional data into a manageable number of dimensions

48. Dimensionality reduction Principal components analysis – A type of factor analysis, assuming normally distributed variables – Reduces high-dimensional data into a manageable number of dimensions Example: face space

49. Parameterizing silhouettes Davidenko, Journal of Vision, 2007

50. Parameterizing silhouettes Davidenko, Journal of Vision, 2007

51. Parameterizing silhouettes Davidenko, Journal of Vision, 2007 1

52. Parameterizing silhouettes Davidenko, Journal of Vision, 2007 1 2

53. Parameterizing silhouettes Davidenko, Journal of Vision, 2007 1 2 480

54. Parameterizing silhouettes Davidenko, Journal of Vision, 2007 1 2 480 36 x 480 matrix is the basis for PCA

55. PCA There is a lot of inter-correlation among the 36 original parameters. By using Principal Components Analysis, one can more efficiently represent the underlying data (in this case, silhouette faces) with fewer than 36 dimensions.

56. Principal Components Analysis x-y representation

57. Principal Components Analysis x-y representation

58. Principal Components Analysis x-y representationPC representation

59. PCA in Matlab In Matlab: e.g. X is an n by m matrix, where n = number of data points m = number of dimensions >> [pc score latent tsquare] = princomp(x); Note 1: n needs to be greater than m Note 2: it is useful to use zcore(x) instead of x

60. A slice of silhouette face space Davidenko, Journal of Vision, 2007

61. The average of 480 faces Davidenko, Journal of Vision, 2007 A slice of silhouette face space

62. Traveling along PC 1 Davidenko, Journal of Vision, 2007 A slice of silhouette face space

63. Traveling along PC 2 Davidenko, Journal of Vision, 2007 A slice of silhouette face space

64. Thank you! Slides, data, and Matlab code will be on the CSASS website: csass.ucsc.edu/short courses/index.html Email me with any questions or if you would like help analyzing and/or visualizing your multivariate data: ndaviden@ucsc.edu