Data analysis. Univariate analysis. 1 variable. Bivariate analysis. 2 variables. Multivariate analysis. 3 or more variables

Transcription

1 Data analysis variable variables 3 or more variables Univariate analysis Bivariate analysis Multivariate analysis

2 Multivariate analysis Methods aiming at visualize and interprete huge data sets structure looking for the latent structure of data through the dimension reduction of the space spanned by variables or by individuals

3 Multidimensional data and factorial methods Bidimensional data x X 3 6 X x Cartesian plane Multidimensional data n X x x x n X x x x n X m x m x m x nm Factorial plane

4 n X x x x n X x x x n X m x m x m x nm Individuals space Variables space Clusters of individuals Associations among variables

5 Data matrices Input for multidimensional analysis Kind of matrices: Units by variables Units by units Varables by variables

6 Units variables matrix Prov. Reddito pro capite (MM) Tasso inflazione Veicoli circolanti Omicidi vol. Densità Quoz. Mortalità Spesa per sp. cinem. Num. Librerie (x00m ab.) AG 5,3 999,0 4,5 3,8 56,0 9,0, 3,0 AL 6,5 4,0 57,6 4,4,0 5, 3,5 7,4 AN 5,5 4, 57,8, 7,4 0,5 4, 9, AO 8,5 3,8 73,8,5 36,6,,5 5, AP,5 3,7 56,8 0,5 76,0 0,0 3,5 7,6 AQ 0,4,5 49,5 3,9 60,5 0,6,6,5 AR 3,5 3,8 59,6, 98,3,6 4,4 6,9 AT 3,7 4, 57,5,4 39, 3,9 4,3 5,7 AV 8,5 999,0 39,9 6,8 58,4 9,4 0,7 5,9 : : : : : : : : : : : : : : : : : : VI 5,8 4, 58, 0,8 8,8 8,6 3,8 6,5 VR 6,7 3,5 58,5,0 59,4 9,7,7 7,8 VT 0,9,6 56,7,4 80,5 0,8 3, 9,6 VV 4,3 999,0 36,9 6, 57, 8,7,,8

7 Variable x variable (modalities x modalities) matrices: contingency tables Totale Saggi culturali Burattini e marion. Musica leggera Rivista e Comm. Operetta Musica classica Lirica e balletti Teatro dialettale Prosa Piem onte Valle d'aosta Lom bardia Trentino-A.A Veneto Friuli-V.G Liguria Em ilia Rom agna Toscana Um bria Marche Lazio Abruzzo Molise Cam pania Puglia Basilicata Calabria Sicilia Sardegna Totale

8 Disjunctve coding (presence/absence) ANIMAL HAIR FEATHERS EGGS. TAIL DOMESTIC CATSIZE aardvark antelope bass bear boar buffalo calf carp wolf worm wren

9 Units x variables: questionnaires Id. Gender Age Marital status Education Income male 4 unmarried middle school.400 female 3 married degree female 8 unmarried high school male 7 widower middle school male 43 married degree male 5 married high school female 65 married degree female 46 diverced degree : : : : : n male married middle school.700

10 Structured matrices p q x x x p y y q n x x x p T p i n n 3 n n 4 n By column By row Three-way matrix

11 Methods for MDA Factorial methods Variables space.. M M M 3 M 4 Matrice dei Dati Cluster analysis Units space Searching the best subspace to visualize data ( dimensions) Serching groups pf units latte zucchero -0.5 cereali riso patate verdure vino burro uova carne

12 Main factorial methods PCA n units; p continuous variables. Principal Components Analysis CA categorical variables (k k categories) observed on n individuals (contingency table). Correspondence Analysis MCA Multiple Correspondence Analysis p categorical variables observed on n individuals. (Survey data from questionnaires).

13 Aims of MDA A) Searching relationships among variables: factorial methods - Aim: finding sintheses of variables = factors = dimensions = latent variables = not observed variables - Methods: - Principal Components Analysis (PCA) for continuous variables - Correspondence Analysis (CA) for contingency tables - Multiple Correspondence Analysis (MCA) for categorical or mixed variables x 3 x m x k x x j x p x

14 Aims of MDA B) Classification/clustering - Aim: finding subgroups of units with high internal homogeneity and high external heterogeneity - Methods: - Cluster Analysis: for continuous and categorical variables (non supervised analysis) - Segmentation (decision trees), for continuous and categorical variables (supervised analysis). a b c d e f g

15 Example of ACP: food consumption X : 6 EU countries, 0 continuous variables Cereals Rice Potatos Sugar Vegetab Wine Meat Milk Butter Eggs Belgium Denmark Germany Greece Spain France Ireland Italy Netherland Portugal UK Austria Finland Island Norway Sweden points (individuals) in the 0 dimensions space R 0 0 points (variables) in the 6 dimensions space R 6

16 Correlation circle axes F e F: 6,89 % 0.75 burro uova carne F (,97 %) zucchero latte patate vino riso cereali verdura F (38,9 %)

17 Interpretazione degli assi fattoriali assi F e F: 6,89 % High protein content uo va carne 0,75 burro 0,5 vino F (,97 %) Fat diets 0,5 0-0,5-0,5 zucchero latte patate riso cereali verdura Light diets -0, ,75-0,5-0,5 0 0,5 0,5 0,75 F (38,9 %) Low protein content

18 Il Piano degli individui assi F e F: 6,89 % 3 F rancia Germania B elgio D animarca F (,97 %) 0 - Islanda A ustria Olanda Svezia Irla nda R egno Unito F inlandia Spagna Italia Grecia - N o rvegia P o rto gallo F (38,9 %)

19 Rappresentazione congiunta unità-variabili assi F e F: 6,89 % 3 F rancia Germania B elgio D animarca F (,97 %) 0 - Islanda A ustria uo vacarne burro vino zucchero verdura Olanda latte Svezia Irla nda R egno Unito F inlandia patate cereali riso Spagna Italia Grecia - N o rvegia P o rto gallo F (38,9 %)