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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

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Analyzing spatial multivariate structures

St´ ephane Dray

  • Univ. Lyon 1

CARME 2011, Rennes

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

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Introduction

variables individuals

Multivariate analysis : identifying multivariate structures

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 2 / 33

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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

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Introduction

Community ecology species sites

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 3 / 33

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Introduction

Community ecology species sites env

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 3 / 33

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Introduction

Community ecology species sites env xy

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 3 / 33

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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

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Introduction

Environmental Control Model

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

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Introduction

Environmental Control Model

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

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Introduction

Environmental Control Model Biotic Control Model

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

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Introduction

Environmental Control Model Biotic Control Model

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

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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

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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

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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

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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

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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

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Standard approaches Guerry’s dataset

Guerry, A.-M. (1833) Essai sur la Statistique Morale de la France. Crochard, Paris.

Friendly (2007) Statistical Science

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

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 7 / 33

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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 XQXTDK = KΛ XTDXQA = AΛ K contains the principal components (KTDK = Ir). A contains the principal axis (ATQA = Ir). Maximization of : Q(a) = aTQTXTDXQa and S(k) = kTDTXQXTDk

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 8 / 33

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Standard approaches Multivariate analysis

PCA of Guerry’s data

PCA on correlation matrix : X = [(xij − ¯ xj )/sj ], Q = Ip, D = 1

n In.

d = 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

Crime_pers Crime_prop Literacy Donations Infants Suicides

Eigenvalues

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 9 / 33

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Standard approaches Multivariate analysis

Adding geographical information

d = 1

Ain Aisne Allier Basses−Alpes Hautes−Alpes Ardeche Ardennes Ariege Aube Aude Aveyron Bouches−du−Rhone Calvados Cantal Charente Charente−Inferieure Cher Correze Cote−d'Or Cotes−du−Nord Creuse Dordogne Doubs Drome Eure Eure−et−Loir Finistere Gard Haute−Garonne Gers Gironde Herault Ille−et−Vilaine Indre Indre−et−Loire Isere Jura Landes Loir−et−Cher Loire Haute−Loire Loire−Inferieure Loiret Lot Lot−et−Garonne Lozere Maine−et−Loire Manche Marne Haute−Marne Mayenne Meurthe Meuse Morbihan Moselle Nievre Nord Oise Orne Pas−de−Calais Puy−de−Dome Basses−Pyrenees Hautes−Pyrenees Pyrenees−Orientales Bas−Rhin Haut−Rhin Rhone Haute−Saone Saone−et−Loire Sarthe Seine Seine−Inferieure Seine−et−Marne Seine−et−Oise Deux−Sevres Somme Tarn Tarn−et−Garonne Var Vaucluse Vendee Vienne Haute−Vienne Vosges Yonne

Eigenvalues Crime_pers Crime_prop Literacy Donations Infants Suicides

C E N S W

A B C

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 10 / 33

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Standard approaches Multivariate analysis

Adding geographical information

d = 1

Ain Aisne Allier Basses−Alpes Hautes−Alpes Ardeche Ardennes Ariege Aube Aude Aveyron Bouches−du−Rhone Calvados Cantal Charente Charente−Inferieure Cher Correze Cote−d'Or Cotes−du−Nord Creuse Dordogne Doubs Drome Eure Eure−et−Loir Finistere Gard Haute−Garonne Gers Gironde Herault Ille−et−Vilaine Indre Indre−et−Loire Isere Jura Landes Loir−et−Cher Loire Haute−Loire Loire−Inferieure Loiret Lot Lot−et−Garonne Lozere Maine−et−Loire Manche Marne Haute−Marne Mayenne Meurthe Meuse Morbihan Moselle Nievre Nord Oise Orne Pas−de−Calais Puy−de−Dome Basses−Pyrenees Hautes−Pyrenees Pyrenees−Orientales Bas−Rhin Haut−Rhin Rhone Haute−Saone Saone−et−Loire Sarthe Seine Seine−Inferieure Seine−et−Marne Seine−et−Oise Deux−Sevres Somme Tarn Tarn−et−Garonne Var Vaucluse Vendee Vienne Haute−Vienne Vosges Yonne

Eigenvalues Crime_pers Crime_prop Literacy Donations Infants Suicides

C E N S W

A B C

Analysis of the multivariate structures, spatial information considered a posteriori

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 10 / 33

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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

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Standard approaches Spatial Autocorrelation

Spatial weighting matrix

W (n × n) : mathematical representation of the geographical layout of the region under study connectivity matrix C : cij = 1 if spatial units i and j are neighbors cij = 0 otherwise.

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 11 / 33

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Standard approaches Spatial Autocorrelation

Spatial weighting matrix

W (n × n) : mathematical representation of the geographical layout of the region under study connectivity matrix C : cij = 1 if spatial units i and j are neighbors cij = 0 otherwise.

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 11 / 33

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Standard approaches Spatial Autocorrelation

Spatial weighting matrix

W (n × n) : mathematical representation of the geographical layout of the region under study connectivity matrix C : cij = 1 if spatial units i and j are neighbors cij = 0 otherwise. Row-sum standardization : wij = cij /

n

  • j=1

cij

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 11 / 33

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Standard approaches Spatial Autocorrelation

Spatial autocorrelation

Moran’s coefficient : MC(x) = n

(2) wij (xi − ¯

x)(xj − ¯ x)

  • (2) wij

n

i=1 (xi − ¯

x)2 Geary’s ratio : GR(x) =

  • (2) wij (xi − xj )2

2

(2) wij

n

i=1 (xi − ¯

x)2/(n − 1) with

(2) = n

  • i=1

n

  • j=1

for i = j

Moran (1948) JRSSB Geary (1954) The incorporated statistician St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 12 / 33

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Standard approaches Spatial Autocorrelation

Spatial autocorrelation (MC)

MC(x) ≈

−1 n−1 : no structure

MC(x) >

−1 n−1 : positive

autocorrelation MC(x) <

−1 n−1 : negative

autocorrelation Randomization testing procedure

Histogram of sim sim Frequency −4 −2 2 4 50 100 150 200

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 13 / 33

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Standard approaches Spatial Autocorrelation

Spatial autocorrelation (MC)

MC(x) ≈

−1 n−1 : no structure

MC(x) >

−1 n−1 : positive

autocorrelation MC(x) <

−1 n−1 : negative

autocorrelation Randomization testing procedure

Histogram of sim sim Frequency −4 −2 2 4 50 100 150 200

MC Crime pers 0.411 (0.001) Crime prop 0.264 (0.001) Literacy 0.718 (0.001) Donations 0.353 (0.001) Infants 0.229 (0.001) Suicides 0.402 (0.001)

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 13 / 33

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Standard approaches Spatial Autocorrelation

Moran’s scatterplot

If W is row-standardized, we define the lag vector ˜ z = Wz. ˜ zi =

n

  • j=1

wij xj is the average of the neighbors of the i-th individual. MC(x) = n 1T

nW1n

zTWz zTz = zT˜ z zTz where z =

  • In − 1n1T

n/n

  • x

Anselin (1996) Spatial analytical perspectives on GIS

  • 10

20 30 40 50 60 70 20 30 40 50 60 70 Literacy Lagged Literacy

Hautes−Alpes

Lagged Literacy = 10.497 + 0.718 * Literacy

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 14 / 33

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Standard approaches Spatial Autocorrelation

Toward an integration of multivariate and geographical aspects

Two-step approach :

Goodall (1954) Australian Journal of Botany St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 15 / 33

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Standard approaches Spatial Autocorrelation

Toward an integration of multivariate and geographical aspects

Two-step approach : Very simple but the spatial pattern is considered a posteriori

Goodall (1954) Australian Journal of Botany St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 15 / 33

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Spatial multivariate analysis

Considering both aspects simultaneously

Multivariate : techniques of dimensionality reduction (e.g., PCA)

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 16 / 33

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Spatial multivariate analysis

Considering both aspects simultaneously

Multivariate : techniques of dimensionality reduction (e.g., PCA) Different alternatives to integrate the spatial information :

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 16 / 33

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Spatial multivariate analysis

Considering both aspects simultaneously

Multivariate : techniques of dimensionality reduction (e.g., PCA) Different alternatives to integrate the spatial information :

Partition Explanatory variables Graph and weighting matrix

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 16 / 33

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Spatial multivariate analysis Space as Partition St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 17 / 33

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Spatial multivariate analysis Space as Partition St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 17 / 33

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Spatial multivariate analysis Space as Partition St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 17 / 33

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Spatial multivariate analysis Space as Partition

Study of spatial patterns = Maximizing the differences between regions

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 17 / 33

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Spatial multivariate analysis Space as Partition

Study of spatial patterns = Maximizing the differences between regions Between-class analysis (X, Q, D) Y : n × g dummy variables

  • (A, Q, DY)

where : A = (YTDY)−1YTDX : matrix with ave- rages per region DY = (YTDY) : region weights

Dol´ edec and Chessel (1987) Acta Oecologica St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 17 / 33

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Spatial multivariate analysis Space as Partition

BCA explains 28.8 % of the total inertia (PCA) 59 % for axis 1, 30.2 % for axis 2

d = 2

Ain Aisne Allier Basses−Alpes Hautes−Alpes Ardeche Ardennes Ariege Aube Aude Aveyron Bouches−du−Rhone Calvados Cantal Charente Charente−Inferieure Cher Correze Cote−d'Or Cotes−du−Nord Creuse Dordogne Doubs Drome Eure Eure−et−Loir Finistere Gard Haute−Garonne Gers Gironde Herault Ille−et−Vilaine Indre Indre−et−Loire Isere Jura Landes Loir−et−Cher Loire Haute−Loire Loire−Inferieure Loiret Lot Lot−et−Garonne Lozere Maine−et−Loire Manche Marne Haute−Marne Mayenne Meurthe Meuse Morbihan Moselle Nievre Nord Oise Orne Pas−de−Calais Puy−de−Dome Basses−Pyrenees Hautes−Pyrenees Pyrenees−Orientales Bas−Rhin Haut−Rhin Rhone Haute−Saone Saone−et−Loire Sarthe Seine Seine−Inferieure Seine−et−Marne Seine−et−Oise Deux−Sevres Somme Tarn Tarn−et−Garonne Var Vaucluse Vendee Vienne Haute−Vienne Vosges Yonne

C E N S W

Eigenvalues −3 −1 1 3 5 −3 −1 1 3

C

d = 0.2 d = 0.2

Crime_pers Crime_prop Literacy Donations Infants Suicides

B D E A

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 18 / 33

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Spatial multivariate analysis Space as Partition

BCA explains 28.8 % of the total inertia (PCA) 59 % for axis 1, 30.2 % for axis 2

d = 2

Ain Aisne Allier Basses−Alpes Hautes−Alpes Ardeche Ardennes Ariege Aube Aude Aveyron Bouches−du−Rhone Calvados Cantal Charente Charente−Inferieure Cher Correze Cote−d'Or Cotes−du−Nord Creuse Dordogne Doubs Drome Eure Eure−et−Loir Finistere Gard Haute−Garonne Gers Gironde Herault Ille−et−Vilaine Indre Indre−et−Loire Isere Jura Landes Loir−et−Cher Loire Haute−Loire Loire−Inferieure Loiret Lot Lot−et−Garonne Lozere Maine−et−Loire Manche Marne Haute−Marne Mayenne Meurthe Meuse Morbihan Moselle Nievre Nord Oise Orne Pas−de−Calais Puy−de−Dome Basses−Pyrenees Hautes−Pyrenees Pyrenees−Orientales Bas−Rhin Haut−Rhin Rhone Haute−Saone Saone−et−Loire Sarthe Seine Seine−Inferieure Seine−et−Marne Seine−et−Oise Deux−Sevres Somme Tarn Tarn−et−Garonne Var Vaucluse Vendee Vienne Haute−Vienne Vosges Yonne

C E N S W

Eigenvalues −3 −1 1 3 5 −3 −1 1 3

C

d = 0.2 d = 0.2

Crime_pers Crime_prop Literacy Donations Infants Suicides

B D E A But... need an a priori partition of space

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 18 / 33

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Spatial multivariate analysis Space as explantory variables

Exploratory Data Analysis

Tukey (1977) Exploratory data analysis

DATA = SMOOTH + ROUGH

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 19 / 33

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Spatial multivariate analysis Space as explantory variables

Exploratory Data Analysis

Tukey (1977) Exploratory data analysis

DATA = SMOOTH + ROUGH

Student (1914) Biometrika

y = f (t, t2, · · · ) + ǫ

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 19 / 33

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Spatial multivariate analysis Space as explantory variables

Space as polynomials

compute a polynomial x, y, xy, x 2, y2, · · · of geographical coordinates

x x2 y xy y2

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 20 / 33

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Spatial multivariate analysis Space as explantory variables

Space as polynomials

compute a polynomial x, y, xy, x 2, y2, · · · of geographical coordinates

x x2 y xy y2

multivariate response

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 20 / 33

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Spatial multivariate analysis Space as explantory variables

Principal component analysis on instrumental variables (PCAIV)

(X, Q, D) Z a n × q matrix of explanatory variables

Rao (1964) Sankhya A St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 21 / 33

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Spatial multivariate analysis Space as explantory variables

Principal component analysis on instrumental variables (PCAIV)

(X, Q, D) Z a n × q matrix of explanatory variables

  • ˆ

X, Q, D

  • where :

ˆ X = PZX = Z(ZTDZ)−1ZTDX Particular cases : Redundancy analysis van den Wollenberg (1977) Psychometrika Canonical correspondence analysis ter Braak (1986) Vegetatio Between-class analysis

Rao (1964) Sankhya A St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 21 / 33

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Spatial multivariate analysis Space as explantory variables

Space as polynomials

Borcard et al. (1992) Ecology

compute a polynomial x, y, xy, x 2, y2, · · · of geographical coordinates PCAIV with polynomials as explanatory variables Analysis of the structures explained by ” space”

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 22 / 33

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Spatial multivariate analysis Space as explantory variables

Space as polynomials

Borcard et al. (1992) Ecology

compute a polynomial x, y, xy, x 2, y2, · · · of geographical coordinates PCAIV with polynomials as explanatory variables Analysis of the structures explained by ” space” Application to Guerry’s data set : PCAIV-POLY explains 32.4 % of the total inertia (PCA) 51.4 % for axis 1, 35.2 % for axis 2

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 22 / 33

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Spatial multivariate analysis Space as explantory variables

Space as polynomials

Borcard et al. (1992) Ecology

compute a polynomial x, y, xy, x 2, y2, · · · of geographical coordinates PCAIV with polynomials as explanatory variables Analysis of the structures explained by ” space” Application to Guerry’s data set : PCAIV-POLY explains 32.4 % of the total inertia (PCA) 51.4 % for axis 1, 35.2 % for axis 2 But...

  • nly broad scales → simple spatial structures

sampling area homogeneous and sampling design quite regular choice of degree

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 22 / 33

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Spatial multivariate analysis Space as explantory variables

Moran’s Eigenvector Maps (MEM)

Moran’s coefficient : MC(x) = n 1TW1 zTWz zTz

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 23 / 33

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Spatial multivariate analysis Space as explantory variables

Moran’s Eigenvector Maps (MEM)

Moran’s coefficient : MC(x) = n 1TW1 zTWz zTz Eigendecomposition of the spatial weighting matrix (symmetric W) : Ω = HWH = U ΛU T =

n

  • i=1

λiuiuT

i

MCmax = λ1(n/1TW1) and MCmin = λn(n/1TW1)

de Jong et al. (1984) Geographical Analysis St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 23 / 33

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Spatial multivariate analysis Space as explantory variables

Moran’s Eigenvector Maps (MEM)

Moran’s coefficient : MC(x) = n 1TW1 zTWz zTz Eigendecomposition of the spatial weighting matrix (symmetric W) : Ω = HWH = U ΛU T =

n

  • i=1

λiuiuT

i

MCmax = λ1(n/1TW1) and MCmin = λn(n/1TW1)

de Jong et al. (1984) Geographical Analysis

Vectors ui maximizes Moran’s coefficient under the constraint of

  • rthogonality

Griffith (1996) Canadian Geographer St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 23 / 33

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Spatial multivariate analysis Space as explantory variables

Explicit modeling of space in PCAIV

Dray et al. (2006) Ecological Modelling St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 24 / 33

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Spatial multivariate analysis Space as explantory variables

PCAIV-MEM

compute MEMs PCAIV with a subset of MEMs as explanatory variables Analysis of the structures explained by ” space”

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 25 / 33

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Spatial multivariate analysis Space as explantory variables

PCAIV-MEM

compute MEMs PCAIV with a subset of MEMs as explanatory variables Analysis of the structures explained by ” space” Application to Guerry’s data set (first 10 MEMs) : PCAIV-POLY explains 44.1 % of the total inertia (PCA) 54.9 % for axis 1, 26.3 % for axis 2

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 25 / 33

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Spatial multivariate analysis Space as explantory variables

PCAIV-MEM

compute MEMs PCAIV with a subset of MEMs as explanatory variables Analysis of the structures explained by ” space” Application to Guerry’s data set (first 10 MEMs) : PCAIV-POLY explains 44.1 % of the total inertia (PCA) 54.9 % for axis 1, 26.3 % for axis 2 But a variable selection step is required...

  • nly a part of the spatial information is considered

based on the fit of the model Blanchet et al. (2008) Ecology based on the minimization of the autocorrelation in residuals

Tiefelsdorf and Griffith (2007) Environment and Planning A St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 25 / 33

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Spatial multivariate analysis Space as spatial graph and weighting matrix

Multivariate extensions based on Geary’s ratio

C is a symmetric connectivity matrix DC is a diagonal matrix of neighbor weights (DC = diag(

n

  • j=1

cij )) Geary’s ratio : GR(x) = (n − 1) 1TC1 zT(C − DC)z zTz

Lebart (1969) Publication de l’Institut de Statistiques de l’Universit´ e de Paris Banet and Lebart (1984) COMPSTAT 84 St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 26 / 33

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SLIDE 59

Spatial multivariate analysis Space as spatial graph and weighting matrix

Multivariate extensions based on Geary’s ratio

C is a symmetric connectivity matrix DC is a diagonal matrix of neighbor weights (DC = diag(

n

  • j=1

cij )) Geary’s ratio : GR(x) = (n − 1) 1TC1 zT(C − DC)z zTz Local PCA : Diagonalization of XT(C − DC)X

Lebart (1969) Publication de l’Institut de Statistiques de l’Universit´ e de Paris Banet and Lebart (1984) COMPSTAT 84 St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 26 / 33

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SLIDE 60

Spatial multivariate analysis Space as spatial graph and weighting matrix

Multivariate extensions based on Geary’s ratio

C is a symmetric connectivity matrix DC is a diagonal matrix of neighbor weights (DC = diag(

n

  • j=1

cij )) Geary’s ratio : GR(x) = (n − 1) 1TC1 zT(C − DC)z zTz Local PCA : Diagonalization of XT(C − DC)X First integration of a graph structure in multivariate analysis Diagonalization of a positive definite matrix Extension to CA ... but optimization of the local differences

Lebart (1969) Publication de l’Institut de Statistiques de l’Universit´ e de Paris Banet and Lebart (1984) COMPSTAT 84 St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 26 / 33

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Spatial multivariate analysis Space as spatial graph and weighting matrix

Multivariate extensions based on Moran’s coefficient

Mulivariate Spatial Correlation Diagonalization of M = XTCX An important difference between this approach and PCA must be pointed

  • ut. Unlike R, the product-moment correlation matrix that is decomposed

in PCA, M is not positive definite. That is, M can have negative eigenvalues, which R cannot.

Wartenberg (1985) Geographical Analysis St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 27 / 33

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SLIDE 62

Spatial multivariate analysis Space as spatial graph and weighting matrix

Multivariate extensions based on Moran’s coefficient

Mulivariate Spatial Correlation Diagonalization of M = XTCX An important difference between this approach and PCA must be pointed

  • ut. Unlike R, the product-moment correlation matrix that is decomposed

in PCA, M is not positive definite. That is, M can have negative eigenvalues, which R cannot. Only symmetric and binary connectivity matrix Only normed-PCA

Wartenberg (1985) Geographical Analysis St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 27 / 33

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Spatial multivariate analysis Space as spatial graph and weighting matrix

Multivariate extensions : Moran with row standardisation

I (x) = n

i=1 (xi − ¯

x)(˜ xi − ¯ x) n

i=1 (xi − ¯

x)2 ∼ = n

i=1 (xi − ¯

x)2 n

i=1 (˜

xi − ¯ x)2 · rx,˜

x =

  • SSSx · rx,˜

x

Lee (2001) Geographical Analysis St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 28 / 33

slide-64
SLIDE 64

Spatial multivariate analysis Space as spatial graph and weighting matrix

Multivariate extensions : Moran with row standardisation

I (x) = n

i=1 (xi − ¯

x)(˜ xi − ¯ x) n

i=1 (xi − ¯

x)2 ∼ = n

i=1 (xi − ¯

x)2 n

i=1 (˜

xi − ¯ x)2 · rx,˜

x =

  • SSSx · rx,˜

x

a bivariate spatial association should include a point-to-point association between two variables, which requires the inclusion of a certain form of Pearson’s correlation between the two variables and should reflect the degrees of spatial autocorrelation for both variables under investigation. In

  • ther words, it should respond to the collective effect of the SSSs of the

variables

Lee (2001) Geographical Analysis St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 28 / 33

slide-65
SLIDE 65

Spatial multivariate analysis Space as spatial graph and weighting matrix

Multivariate extensions : Moran with row standardisation

I (x) = n

i=1 (xi − ¯

x)(˜ xi − ¯ x) n

i=1 (xi − ¯

x)2 ∼ = n

i=1 (xi − ¯

x)2 n

i=1 (˜

xi − ¯ x)2 · rx,˜

x =

  • SSSx · rx,˜

x

Wartenberg’s MSC : I (x, y) ∼ =

  • SSSy · rx,˜

y

Lee (2001) Geographical Analysis St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 28 / 33

slide-66
SLIDE 66

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTIvariate SPATial analysis based on Moran’s I

Generalization of Wartenberg’s MSC to : row-standardized spatial weighting matrix W any 1-table ordination method (e.g. CA) (X, Q, D)

Dray, Sa¨ ıd & D´ ebias (2008) Journal of Vegetation Science St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 29 / 33

slide-67
SLIDE 67

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTIvariate SPATial analysis based on Moran’s I

Generalization of Wartenberg’s MSC to : row-standardized spatial weighting matrix W any 1-table ordination method (e.g. CA) (X, Q, D)

  • X, Q, 1/2(WTD + DW)
  • Dray, Sa¨

ıd & D´ ebias (2008) Journal of Vegetation Science St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 29 / 33

slide-68
SLIDE 68

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTIvariate SPATial analysis based on Moran’s I

The lag matrix : ˜ X = WX

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 30 / 33

slide-69
SLIDE 69

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTIvariate SPATial analysis based on Moran’s I

The lag matrix : ˜ X = WX MULTISPATI is a coinertia analysis between the fully matched tables X and ˜ X.

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 30 / 33

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SLIDE 70

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTIvariate SPATial analysis based on Moran’s I

The lag matrix : ˜ X = WX MULTISPATI is a coinertia analysis between the fully matched tables X and ˜ X. For a given analysis corresponding to a triplet (X, Q, D), it finds coefficients (a) to obtain a linear combination of variables u = XQa which maximizes the quantity : Q(a) = uTD˜ u

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 30 / 33

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SLIDE 71

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTIvariate SPATial analysis based on Moran’s I

The quantity Q(a) = uTD˜ u can be rewritten as : Q(a) = ID(a)

spatial autocorrelation

· a2

D multivariate analysis

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 31 / 33

slide-72
SLIDE 72

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTIvariate SPATial analysis based on Moran’s I

The quantity Q(a) = uTD˜ u can be rewritten as : Q(a) = ID(a)

spatial autocorrelation

· a2

D multivariate analysis

Diagonalization of (1/2)XT(WTD + DW)XQ

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 31 / 33

slide-73
SLIDE 73

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTIvariate SPATial analysis based on Moran’s I

The quantity Q(a) = uTD˜ u can be rewritten as : Q(a) = ID(a)

spatial autocorrelation

· a2

D multivariate analysis

Diagonalization of (1/2)XT(WTD + DW)XQ In the case of normed PCA, Hjk ∼ = (1/2)

  • SSSyj · ryk,˜

yj + (1/2)

  • SSSyk · ryj ,˜

yk

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 31 / 33

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SLIDE 74

Spatial multivariate analysis Space as spatial graph and weighting matrix

MULTISPATI-PCA results

d = 2

Aude Haute−Loire Finistere Eigenvalues Crime_pers Crime_prop Literacy Donations Infants Suicides

A C B D E F G

Axis 1

λ=1.286 var=2.017

Axis 2

λ=0.694 var=1.177 MC=0.637 MC=0.590

−3 −1 1 3 5 −1.5 −0.5 0.5 1.5 2.5 −3 −1 1 3 −1.5 −0.5 0.5 1.5 2.5

St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 32 / 33

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SLIDE 75

Conclusions

One data structure, various methods

Dray and Jombart (in press) Annals of Applied Statistics

package Guerry on CRAN

In practice, spatial patterns are usually revealed by simple analysis

Dray et al. (2008) Journal of Vegetation Science St´ ephane Dray (Univ. Lyon 1) CARME 2011, Rennes 33 / 33

slide-76
SLIDE 76

Conclusions

One data structure, various methods

Dray and Jombart (in press) Annals of Applied Statistics

package Guerry on CRAN

In practice, spatial patterns are usually revealed by simple analysis

Dray et al. (2008) Journal of Vegetation Science

Some generalizations/extensions any similarities between individuals (e.g. spatial, temporal or phylogenetic) double constraint (e.g. spatial and phylogenetic) two or k data tables

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