Analyzing spatial multivariate structures St ephane Dray Univ. - PowerPoint PPT Presentation

Analyzing spatial multivariate structures St´ ephane Dray Univ. Lyon 1 CARME 2011, Rennes St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 1 / 33

Introduction variables individuals Multivariate analysis : identifying multivariate structures St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 2 / 33

Introduction variables individuals Multivariate analysis : identifying multivariate structures Spatial component : identifying multivariate spatial structures St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 2 / 33

Introduction Community ecology species sites St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 3 / 33

Introduction Community ecology species env sites St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 3 / 33

Introduction Community ecology xy species env sites St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 3 / 33

Introduction Abundance of p species for n sites n sites in the geographical space St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 4 / 33

Introduction Environmental Control Model St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 5 / 33

Introduction Environmental Control Model St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 5 / 33

Introduction Environmental Control Model Biotic Control Model St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 5 / 33

Introduction Environmental Control Model Biotic Control Model St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 5 / 33

Introduction Environmental Control Model Biotic Control Model Spatial patterns in communities may originate from these two sources St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 5 / 33

Introduction Environmental Control Model Biotic Control Model Spatial patterns in communities may originate from these two sources Tools integrating both multivariate and spatial aspects are needed to distangle them St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 5 / 33

Introduction Friendly, M. (2007) A.-M. Guerry’s moral statistics of France : challenges for multivariable spatial analysis. Statistical Science , 22 :368-399. Two approaches to analyze datasets with both multivariate and geographical aspects : data-centric display map-centric display St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 6 / 33

Introduction Friendly, M. (2007) A.-M. Guerry’s moral statistics of France : challenges for multivariable spatial analysis. Statistical Science , 22 :368-399. Two approaches to analyze datasets with both multivariate and geographical aspects : data-centric display map-centric display ” the integration of these data-centric and map-centric visualization and analysis is still incomplete” St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 6 / 33

Introduction Friendly, M. (2007) A.-M. Guerry’s moral statistics of France : challenges for multivariable spatial analysis. Statistical Science , 22 :368-399. Two approaches to analyze datasets with both multivariate and geographical aspects : data-centric display map-centric display ” the integration of these data-centric and map-centric visualization and analysis is still incomplete” Spatial Multivariate Analysis St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 6 / 33

Standard approaches Guerry’s dataset Guerry, A.-M. (1833) Essai sur la Statistique Morale de la France. Crochard, Paris. 85 d´ epartements (counties) 6 variables : Label Description Crime pers Population per crime against persons Crime prop Population per crime against property Literacy Percent of military conscripts who can read and write Donations Donations to the poor Infants Population per illegitimate birth Suicides Population per suicide more (larger numbers) is ” morally”better Friendly (2007) Statistical Science St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 7 / 33

Standard approaches Multivariate analysis A multivariate analysis corresponds to a triplet : X ( n × p ) : the (transformed) data table Q ( p × p ) : metric for the individuals D ( n × n ) : metrics for the variables XQX T DK = KΛ X T DXQA = AΛ K contains the principal components ( K T DK = I r ). A contains the principal axis ( A T QA = I r ). Maximization of : Q ( a ) = a T Q T X T DXQa and S ( k ) = k T D T XQX T Dk St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 8 / 33

Standard approaches Multivariate analysis PCA of Guerry’s data x j ) / s j ] , Q = I p , D = 1 PCA on correlation matrix : X = [( x ij − ¯ n I n . d = 2 Eigenvalues Donations Crime_pers 26 12 33 53 74 14 80 20 15 55 19 75 6 58 1 82 16 57 17 Infants 69 37 67 32 64 56 46 24 81 2 0 18 47 68 34 59 38 41 71 72 30 50 51 52 25 49 21 4 44 54 39 8 42 36 84 63 35 48 Crime_prop 66 73 70 83 77 60 27 Literacy 11 3 78 28 29 23 79 76 65 45 13 61 5 9 31 43 22 10 62 40 7 Suicides St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 9 / 33

Standard approaches Multivariate analysis Adding geographical information Donations Crime_pers d = 1 Eigenvalues A B Finistere Infants Crime_prop Literacy Calvados Suicides Indre Morbihan Deux−Sevres Charente Vendee Nievre Cotes−du−Nord Charente−Inferieure Ardennes Creuse Orne Haute−Vienne Aisne Somme Oise Cher Correze Landes Haute−Saone Ille−et−Vilaine Nord Bas−Rhin Eure Sarthe Vienne Maine−et−Loire Allier N Cote−d'Or Manche Pas−de−Calais Loire−Inferieure Loir−et−Cher Ain Saone−et−Loire Seine−Inferieure Gironde Indre−et−Loire Meurthe Seine−et−Marne Mayenne E W C Haute−Marne Lot−et−Garonne Eure−et−Loir Hautes−Alpes Loire Meuse Dordogne Moselle Aube Loiret Jura Pyrenees−Orientales Yonne Isere S Marne Seine−et−Oise Rhone Vosges Seine Tarn−et−Garonne Gard Basses−Alpes Var Puy−de−Dome Bouches−du−Rhone Gers Drome Haut−Rhin Haute−Garonne Vaucluse Tarn Ardeche Lozere Basses−Pyrenees Cantal Aude Herault Lot Doubs Aveyron Haute−Loire Hautes−Pyrenees C Ariege St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 10 / 33

Standard approaches Multivariate analysis Adding geographical information Donations Crime_pers d = 1 Eigenvalues A B Finistere Infants Crime_prop Literacy Calvados Suicides Indre Morbihan Deux−Sevres Charente Vendee Nievre Cotes−du−Nord Charente−Inferieure Ardennes Creuse Orne Haute−Vienne Aisne Somme Oise Cher Correze Landes Haute−Saone Ille−et−Vilaine Nord Bas−Rhin Eure Sarthe Vienne Maine−et−Loire Allier N Cote−d'Or Manche Pas−de−Calais Loire−Inferieure Loir−et−Cher Ain Saone−et−Loire Seine−Inferieure Gironde Indre−et−Loire Meurthe Seine−et−Marne Mayenne E W C Haute−Marne Lot−et−Garonne Eure−et−Loir Hautes−Alpes Loire Meuse Dordogne Moselle Aube Loiret Jura Pyrenees−Orientales Yonne Isere S Marne Seine−et−Oise Rhone Vosges Seine Tarn−et−Garonne Gard Basses−Alpes Var Puy−de−Dome Bouches−du−Rhone Gers Drome Haut−Rhin Haute−Garonne Vaucluse Tarn Ardeche Lozere Basses−Pyrenees Cantal Aude Herault Lot Doubs Aveyron Haute−Loire Hautes−Pyrenees C Ariege Analysis of the multivariate structures, spatial information considered a posteriori St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 10 / 33

Standard approaches Spatial Autocorrelation Spatial weighting matrix W ( n × n ) : mathematical representation of the geographical layout of the region under study St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 11 / 33

Standard approaches Spatial Autocorrelation Spatial weighting matrix W ( n × n ) : mathematical representation of the geographical layout of the region under study connectivity matrix C : c ij = 1 if spatial units i and j are neighbors c ij = 0 otherwise. St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 11 / 33

Standard approaches Spatial Autocorrelation Spatial weighting matrix W ( n × n ) : mathematical representation of the geographical layout of the region under study connectivity matrix C : c ij = 1 if spatial units i and j are neighbors c ij = 0 otherwise. St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 11 / 33

Standard approaches Spatial Autocorrelation Spatial weighting matrix W ( n × n ) : mathematical representation of the geographical layout of the region under study connectivity matrix C : c ij = 1 if spatial units i and j are neighbors c ij = 0 otherwise. Row-sum standardization : n � w ij = c ij / c ij j =1 St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 11 / 33

Standard approaches Spatial Autocorrelation Spatial autocorrelation Moran’s coefficient : n � (2) w ij ( x i − ¯ x )( x j − ¯ x ) MC ( x ) = � � n x ) 2 (2) w ij i =1 ( x i − ¯ Geary’s ratio : � (2) w ij ( x i − x j ) 2 GR ( x ) = 2 � � n x ) 2 / ( n − 1) (2) w ij i =1 ( x i − ¯ n n with � � � (2) = for i � = j i =1 j =1 Moran (1948) JRSSB Geary (1954) The incorporated statistician St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 12 / 33