(Analyse des Correspondances (AFC) (Simple) Correspondence Analysis (CA) Hervé Abdi CNAM STA201

Transcription

1 u u (Analyse des Correspondances (AFC) (Simple) Correspondence Analysis (CA) How to Analye the... Hervé Abdi CNAM-04. STA0 herve herve@utdallas.edu Page of 85

2 . u u Refresher: Weight and Masses χ Chi-squared Independence Start with a small eample (punctuation) New: What is a profile? Masses and weights for CA Centroïd, barycenter, etc Distributional equivalence (big property) Row profile analysis What to do with the columns try What to do with the columns try Duality: (rows to columns) = (column to rows) How to Analye the... Page of 85

3 . Eample: Punctuation... u u Data from Brunet (989): The punctuation marks of si great French writers, Rousseau, Chateaubriand, Hugo, Zola, Proust, and Giraudou. Data: Number of times each writer uses three punctuation marks: the period, the comma, and all the other marks (i.e., interrogation mark, eclamation mark, colon, and semicolon). How to Analye the... Page 3 of 85

4 u u Table : 989). The punctuation marks of si French writers (from Brunet, All the other marks How to Analye the... Rousseau Chateaubriand Hugo Zola Proust Giraudou Page 4 of 85

5 Table with total and frequencies P (Author) Author s name i+ i+ ++ Rousseau Chateaubriand Hugo Zola Proust Giraudou u u How to Analye the j P (Mark) +j Page 5 of 85

6 3. Chi-Square Refresher: Independence Rows and Columns Data: A contingency table. Question: Do the writers differ in their punctuation? Statistical answer: test for independence Rows and Columns Chi-Square for Independence The Data Table Under Independence: Step one find the probabilities under independence. The Product Rule: Probability of independent events = product of each event. Probability of using the punctuation marks for an author: Probability of the mark the author u u How to Analye the... Page 6 of 85

7 Probability under Independence u u All the other marks Rousseau Chateaubriand Hugo Zola Proust Giraudou How to Analye the... Page 7 of 85

8 Real Probability u u All the other marks Rousseau Chateaubriand Hugo Zola Proust Giraudou How to Analye the... Page 8 of 85

9 Real Probability Independence u u All the other marks Rousseau Chateaubriand Hugo Zola Proust Giraudou Square everything, sum and multiply by grand total: χ = 57.6 So it is awfully significant! (so what?) How to Analye the... Page 9 of 85

10 Official formula for χ u u χ = (O i,j E i,j ) E = E i,j (O i,j E i,j ) = md Remember a sum of square distances is called Inertia (e.g. variance) So: Chi-Square is an Inertia! How to Analye the... Page 0 of 85

11 4. How to Analye the Table u u First idea: PCA right away! Not a good idea! To see why: Do it with the hidden alias of Zola: Alo. [ ] How to Analye the... (Just like Zola but wrote only a short novel) Page of 85

12 The PCA: Centered Data! u u Rousseau Giraudou Proust Chateaubriand Hugo Zola How to Analye the... Figure : PCA analysis of the Punctuation. Centered Data. Page of 85

13 The PCA: Centered Data! u u Alo Rousseau Giraudou Proust Chateaubriand Hugo Zola How to Analye the... Figure : PCA analysis of the Punctuation. Centered Data. Alo is a supplementary element. Page 3 of 85

14 The PCA: Centered & Normalied Data! u u Chateaubriand Hugo Rousseau Giraudou Proust Zola How to Analye the... Figure 3: PCA analysis of the Punctuation. Centered & Normalied Data. Page 4 of 85

15 The PCA: Centered & Normalied Data! u u Chateaubriand Hugo Rousseau Giraudou Alo Proust Zola How to Analye the... Figure 4: PCA analysis of the Punctuation. Centered & Normalied Data. Alo is a supplementary element. Page 5 of 85

16 The Profiles u u All the other marks Rousseau Chateaubriand Hugo Zola Proust Giraudou How to Analye the... Page 6 of 85

17 Plot them! u u Rousseau.5 How to Analye the... Page 7 of Figure 5: Plotting a Profile.

18 .5 Plot them! [0,0,] Rousseau u u How to Analye the... Page 8 of 85.0 y [0,,0] [,0,0] Figure 6: Plotting a Profile. The Simple

19 Plot them! 0 u u How to Analye the... 0 Figure 7: The Row Profile Space in D. 0 Page 9 of 85

20 Plot them! 0 u u.486 How to Analye the Rousseau 0.34 Figure 8: The Row Profile Space in D. Plotting Rousseau 0 Page 0 of 85

21 Plot them! 0 u u H R C G Z P How to Analye the Figure 9: The Row Profile Space in D. Plotting Them All 0 Page of 85

22 Plot them! 0 u u H R C G Z P How to Analye the Figure 0: The Row Profile Space in D. Plotting Them All. With Center of Gravity. Page of 85

23 We want the space to epress the Information of the variables u u Information? Rare variables provide a lot of information Frequent variables provide little information How to Analye the... How to do that? Stretch the space dimensions (variables). with the inverse of the frequency Page 3 of 85

24 Plot them! u u [0,0,].0 [0,0,] How to Analye the y [0,,0] [,0,0] [,0,0] [0,,0] y Figure : What the weights are doing to the row space..0 Page 4 of 85

25 Plot them! u u G H R Z C P G H R Z C P 0 0 Figure : What the weights are doing to the row space. How to Analye the... Page 5 of 85

26 Plot them! u u 0 0 R C H G Z P How to Analye the... 0 Figure 3: What the weights are doing to the row space. A closer view. Page 6 of 85

27 Plot them! u u 0 0 R C G H Z P How to Analye the... 0 Figure 4: Zooming in. Page 7 of 85

28 Plot them! u u 0 0 R C G H Z P How to Analye the... 0 Figure 5: Zooming in. Page 8 of 85

29 Plot them! u u R C H G Z P How to Analye the... Figure 6: Zooming in. Page 9 of 85

30 Plot them! u u R C H G Z P How to Analye the... Figure 7: Zooming in. Page 30 of 85

31 Plot them! u u Giraudou Hugo Rousseau Chateaubriand Zola Proust How to Analye the... Figure 8: Zooming in. Page 3 of 85

32 Pause Inertia u u I R = i,j m i [ wj (r i,j c j ) ] How to Analye the... = i,j m i w j (r i,j c j ) but: w j = c j = i,j m i c j (r i,j c j ) = i,j m i (r i,j c j ) c j Page 3 of 85 = χ ++

33 So Pause Inertia I R = χ ++ The inertia of the row profiles is proportional to χ. Factors are χ orthogonal slices! u u How to Analye the... Page 33 of 85

34 Pause Properties Distributional Equivalence If two rows have the same profiles: We can add them up u u How to Analye the... And nothing is changed! Page 34 of 85

35 u u The Rows: Correspondence Analysis m m m F F ctr ctr F F d cos cos Rousseau Chateaubriand Hugo Zola Proust Giraudou λ λ I R 76% 4% τ τ How to Analye the... Page 35 of 85 Table : The Projections, masse Squared projections, Contributions, inertia to barycenter, and (Squared) Cosines for the punctuation eample

36 The columns point are too far away u u How to Keep Them? Idea: Compute their Inertia and Normalie How to Analye the... Page 36 of 85

37 The columns point are too far away u u A bit of Magic here The Inertia of the column points/vertices is equal to for each dimension! How to Analye the... Gasp!!!!! Why? Intuition: Etreme points have maimum correlation! For more, see end of slides: Mathematical digression Page 37 of 85

38 Plot them! u u 0 0 R C G H Z P How to Analye the... 0 Figure 9: Zooming in. Page 38 of 85

39 Plot them! u u 0 0 R C G H Z P How to Analye the... 0 Figure 0: Zooming in. Page 39 of 85

40 Plot them! u u R C G H Z P How to Analye the... 0 Figure : Zooming in. Page 40 of 85

41 Plot them! u u R C G H Z P How to Analye the... Figure : Zooming in. Page 4 of 85

42 Plot them! u u R C G H Z P How to Analye the... Figure 3: Zooming in. Page 4 of 85

43 .49 u u.37 G H R C.05 Z. P How to Analye the... Figure 4: Zooming in. Page 43 of 85

44 .49 u u.9 R C H G Z P.09 How to Analye the... Figure 5: Zooming in. Page 44 of 85

45 u u.0 G H R C.04 Z P How to Analye the... Figure 6: Zooming in. Page 45 of 85

46 u u R C H G Z P How to Analye the... Figure 7: Zooming in. Page 46 of 85

47 u u R C H G Z P How to Analye the... Figure 8: Zooming in. Page 47 of 85

48 u u How to Analye the... R C H G Z P Page 48 of 85 Figure 9: Zooming in.

49 u u Giraudou Hugo Zola Rousseau Proust Chateaubriand How to Analye the... Figure 30: Zooming in. Page 49 of 85

50 u u m m m F F ctr ctr F F d cos cos PERIOD COMMA OTHER λ λ I C 76% 4% τ τ How to Analye the... Table 3: Columns: The Projections, masse Squared projections, Contributions, inertia to barycenter, and Cosines for the punctuation eample Page 50 of 85

51 The dual analysis: the Column Profiles How to Analye the... Page 5 of 85

52 The Column Profiles u u All the other marks Rousseau Chateaubriand Hugo Zola Proust Giraudou How to Analye the... Page 5 of 85

53 Giraudou u u How to Analye the... Rousseau Hugo Zola Proust Page 53 of 85

54 Giraudou u u How to Analye the... Rousseau Hugo Giraudou Zola Proust Page 54 of 85

55 Giraudou u u How to Analye the... Rousseau Hugo Giraudou Zola Proust Page 55 of 85

56 Giraudou u u How to Analye the... Rousseau Hugo Giraudou Rousseau Hugo Zola Proust Chateaubriand Zola Proust Page 56 of 85

57 Giraudou u u How to Analye the... Rousseau Hugo Giraudou Rousseau Hugo Zola Proust Chateaubriand Zola Proust Page 57 of 85

58 Giraudou u u How to Analye the... Rousseau Hugo Giraudou Rousseau Hugo Zola Proust Chateaubriand Zola Proust Page 58 of 85

59 u u Hugo Rousseau Giraudou Chateaubriand Zola Proust How to Analye the... Page 59 of 85 Figure 37: Zooming in.

60 Hugo Rousseau Giraudou Chateaubriand Zola Proust u u How to Analye the... Figure 38: Zooming in. Page 60 of 85

61 Supplementary Elements One new Author: Abdi (994) Chapter The punctuation marks (colon, semicolon, interrogation, eclamation). u u How to Analye the... Page 6 of 85

62 COLON u u SEMICOLON Chateaubriand Proust OTHER COMMA MARKS Rousseau Zola Hugo PERIOD Giraudou INTERROGATION EXCLAMATION How to Analye the... Page 6 of 85 Abdi

63 Another Eample: The Sound of Colors from: Abdi & Béra (04). paper # C8 u u A science project (independent and dependent variables): Twenty-two participants (4 children and 0 adults) were presented with nine pieces of music and asked to associate one of ten colors to each piece of music. The pieces of music: How to Analye the... ) the music of a video game (video), ) a Ja song (ja), 3) a country and western song (country), 4) a rap song (rap), 5) a pop song (pop), 6) an etract of the opera Carmen (opera), 7) the low F note played on a piano (low F), 8) the middle F note played on the same piano 9) the high F note still played on the same piano. Page 63 of 85

64 u u Table 4: Twenty-two participants associated one of ten colors to nine pieces of music. The column labeled i+ gives the total number of choices made for each color. N is the grand total of the data table. The vector of mass for the rows, r, is the proportion of choices made for each color (r i = i+ /N). The row labeled +j gives the total number of times each piece of music was presented (i.e., it is equal to the number of participants) The centroid row, gives these values as proportions. Color Video Ja Country Rap Pop Opera Low F High F Middle F i+ red orange yellow green blue purple white black pink brown j N = 98 c T How to Analye the... Page 64 of 85

65 u u Color Key Value Number of Participants Choosing a Color (white is none) red orange How to Analye the... yellow green blue purple white black pink brown Page 65 of 85 Video Ja Country Rap Pop Opera Low F High F Middle F Figure 40: CA The Colors of Music. A nicer way to look at the data. A heat map of the data.

66 u u Table 5: CA The Color of Music. Factor scores, contributions, mass, mass squared factor scores, inertia to barycenter, and squared cosines for the rows. For convenience, squared cosines and contributions have been multiplied by 000 and rounded. r i r i r i F F ctr ctr r i F F d r,i cos cos How to Analye the... red orange yellow green blue purple white black pink brown λ λ I 39% 6% τ τ Page 66 of 85

67 u u Table 6: CA The Colors of Music. Factor scores, contributions, mass, mass squared factor scores, inertia to barycenter, and squared cosines for the columns. For convenience, squared cosines and contributions have been multiplied by 000 and rounded. c j c j c j G G G G ctr ctr c j G G d c,j cos cos Video Ja Country Rap Pop Opera Low.F High.F Middle.F λ λ I 39% 6% τ τ How to Analye the... Page 67 of 85

68 u u Percentage of Eplained Variance Eplained Variance per Dimension How to Analye the... Page 68 of 85 Dimensions Figure 4: CA The Colors of Music. Scree Test of the percentage of eplained variance. Three dimensions are larger than 8.

69 purple Video pink High.F orange yellow white Opera Ja red Rap black u u How to Analye the... blue Low.F Country green Pop Middle.F Page 69 of 85 brown Figure 4: CA The Colors of Music. Symmetric Plot: The projections of the rows and the columns are displayed in the same map. λ =.87, τ = 39; λ =.9, τ = 6. In this plot the proimity between rows and columns cannot be directly interpreted.

70 High.F Rap u u Video Opera Ja purple pink orange yellow white red black How to Analye the... blue Low.F green Country brown Page 70 of 85 Pop Middle.F Figure 43: CA The Colors of Music. Asymmetric Plot: The projections of the rows and the columns are displayed in the same map. The inertia of the projections of the column factor scores is equal to one for each dimension and the inertia of the projections of the row factor scores are λ =.87, τ = 39; λ =.9, τ = 6.

71 bonus Zola u u How to Analye the... Page 7 of 85

72 Bonus: Zola u u Table 7: the punctuation of the Novels of Zola. Original Data from Brunet (996??).....!?, ; : It Raquin Ferrat Fortune Curée Ventre Conquête Faute Ecellence Assommoir Page Nana Pot Bouille Bonheur joie Germinal Oeuvre Terre Rêve Bête Argent Débacle Pascal How to Analye the... Page 7 of 85

73 Table 8: The novels of Zola and their punctuation. The Projections, Contributions, inertia to barycenter, and Cosines for the punctuation eample. First Analysis. All punctuation marks as active elements u u F F ctr ctr m d cos cos Qual Raquin Ferrat Fortune Curée Ventre Conquête Faute Ecellence Assommoir Page Nana Pot Bouille Bonheur Joie Germinal L œuvre Terre Rêve Bête Argent Débacle Pascal How to Analye the... Page 73 of 85

74 u u Table 9: Zola s punctuation. Columns: The Projections, Contributions, inertia to barycenter, and Cosines F F ctr ctr m d cos cos Qual ! ? , ; : It How to Analye the... Page 74 of 85

75 Le rêve λ =.0055 τ = 0% Thérèse Raquin Le ventre de Paris Quote u u La bête humaine L argent La débacle Le docteur Pascal Germinal L oeuvre La faute de l abbé Mouret La terre L assommoir La joie de vivre Eugène Rougon Au bonheur des dames Hyphen Nana Une page d amour Interrogation Pot bouille Colon Italic La conquête de Plassans Eclamation Dots Hyphen La curée La Fortune des Rougon Madeleine Ferrat SemiColon λ =.05 τ = 55% How to Analye the... Figure 44: The punctuation of Zola. First Pass. Rows and Columns. Page 75 of 85

76 Le rêve La bête humaine La débacle L argent Germinal Le docteur Pascal L oeuvre La joie de vivre La terre λ =.0055 τ = 0% Le ventre de Paris La curée La faute de l abbé Mouret L assommoir Eugène Rougon Au bonheur des dames Thérèse Raquin La Fortune des Rougon Madeleine Ferrat λ =.05 τ = 55% u u How to Analye the... Nana Pot bouille Une page d amour La conquête de Plassans Figure 45: The punctuation of Zola. First Pass. Rows. Page 76 of 85

77 Le rêve 8 La bête humaine La débacle 9 L argent 0 Le docteur Pascal Germinal 5 6 L oeuvre La joie de vivre 7 4 La terre 3 λ = τ = % 5 Le ventre de Paris 4 La curée La Fortune des Rougon 9 7 La faute de l abbé Mouret L assommoir Eugène Rougon Au bonheur des dames 8 Nana Une page d amour Pot bouille La conquête de Plassans Thérèse Raquin Madeleine Ferrat λ =.05 τ = 55% u u How to Analye the... Figure 46: The punctuation of Zola. First Pass. Rows with # of the Novels. Page 77 of 85

78 Recode Italic and Quote = supplementary Regroup! and? Regroup and u u How to Analye the... Page 78 of 85

79 Zola. Recoded Data Table 0: the punctuation of the Novels of Zola. Recoded Data: Italic and quote are supplementary elements, Eclamation and Interrogation are summed, Hyphen and are summed. Data from Brunet (996??).....!&?, ; : Raquin Ferat Fortune Curée Ventre Conquête Faute Ecellence Assommoir Page Nana Pot Bouille Bonheur joie Germinal Oeuvre Terre Rêve Bête Argent Débacle Pascal u u How to Analye the... Page 79 of 85

80 Table : Zola s punctuation. Columns: The Projections, Contributions, inertia to barycenter, and Cosines. Recoded Data: Italic and quote are supplementary elements, Eclamation and Interrogation are summed, Hyphen and are summed. F F ctr ctr m d cos cos Qual Raquin Ferat Fortune Curée Ventre Conquête Faute Ecellence Assommoir Page Nana Pot Bouille Bonheur joie Germinal Oeuvre Terre Rêve Bête Argent Débacle Pascal u u How to Analye the... Page 80 of 85

81 u u F F ctr ctr m d cos cos Qual !&? , ; : Table : Zola s punctuation. Columns: The Projections, Contributions, inertia to barycenter, and Cosines. Recoded Data: Italic and quote are supplementary elements, Eclamation and Interrogation are summed, Hyphen and are summed. How to Analye the... Page 8 of 85

82 Ecl Inter L oeuvre Le docteur Pascal L argent La terre λ = τ = % Pot bouille Au bonheur des dames Germinal La débacle La bête humaine Dots Nana Hyphen Colon La joie de vivre L assommoir Une page d amour La faute de l abbé Mouret La curée La conquête de Plassans Eugène Rougon SemiColon λ = τ = % Madeleine Ferrat u u How to Analye the... La Fortune des Rougon Le rêve Le ventre de Paris Thérèse Raquin Page 8 of 85 Figure 47: The punctuation of Zola. Second Pass. Rows and Columns.

83 Ecl Inter 6 L oeuvre L argent 7 La terre Le docteur Pascal λ = τ = % Pot bouille 3 Au bonheur des dames Germinal 5 La débacle La bête humaine Le rêve Dots 4 Hyphen Nana Une page d amour Colon Eugène Rougon La joie de vivre 9 L assommoir 8 La faute de l abbé Mouret La curée Le ventre de Paris La conquête de Plassans SemiColon 3 λ = τ = La Fortune des Rougon % Madeleine Ferrat Thérèse Raquin u u How to Analye the... Page 83 of 85 Figure 48: The punctuation of Zola. Second Pass. Rows.

84 The Math you thought you had escaped! u u See Papers # C8 & #C69 for Details How to Analye the... Page 84 of 85

85 With R! u u Use the EPosition package: (the punctuation eample comes with the package) library( EPosition ) data(authors) How to Analye the... # load the punctuation eample # to get the results of CA (with graphs): ca.authors.res = epca(authors$ca$data) Page 85 of 85