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Bidimensional regression beyond goodness-of-fit

Measuring geometric distortions in urban mental maps produced by blind people, wheelchair users and people without disabilities

Jason Borioli1,2

1 Ph.D student, Institute of Geography, Faculty of Geosciences

and Environment, University of Lausanne, Switzerland

2 Graduate assistant, Laboratory for Experimental Research

• n Behavior (LERB), Institute of Psychology, Faculty, Of

Social and Political Sciences, University

• f

Lausanne, Switzerland

T he 15th E me r ging Ne w Re se ar c he r s in the Ge ogr aphy

• f He alth and Impair

me nt Confe r e nc e 10- 11 June 2010 - Par is – F r anc e http:/ / www.ir de s.fr / E nr ghi2010 e nr ghi2010@ir de s.fr

Outline

• Bidimensional regression: general presentation
• Conceptual bases
• Goodness-of-fit vs geometric transformations
• Geographical applications: urban mental maps
• Discussion

• Assess the similarity between two-dimensional data sets
• Regression

analysis and two-dimensional coordinate transformation models

Bidimensional regression

• Tobler introduced it to the geography literature [1965, 1966,

1978, 1994]. Was later introduced to the psychology [Friedman & Kohler, 2003] and computer-science literatures [Kare et al., 2008]

Conceptual bases

Y X

yi = axi +b+ei

ei

ei = yi − axi − b

S = ei

( )

2 i =1 n

∑

= yi − axi − b

( )

2 i =1 n

∑

15 4 8 11 3 7 12 2 13 9 6 1 14 10 5 15 4 8 11 3 7 12 2 13 9 6 1 14 10 5

ei fi

Conceptual bases

S2 = ei

( )2 + fi ( )2

     

i=1 n

∑

15 4 8 11 3 7 12 2 13 9 6 1 14 10 5

→ ui * vi *         = Α xi yi      

ui

*,vi * = predicted coordinates

A = coordinate transformation matrix

ui vi       = Α xi yi       + ei fi      

Conceptual bases

ui,vi = observed coordinates (i.e. dependent coordinates)

General definition of bidimensional regression

xi,yi = reference coordinates (i.e. independent coordinates) ei, fi = residuals

Projective transformation: A=8 parameters Affine transformation: A=6 parameters Curvilinear transformation: A=x parameters Curvilinear transformation: A=x parameters Projective transformation: A=8 parameters Affine transformation: A=6 parameters

Conceptual bases

Euclidean transformation: A=4 parameters

Conceptual bases

→ u* v*       = a1 −a2 a2 a1       x y       + b

1

b2      

Euclidean transformation: A=4 parameters

→ u* v*         = scosθ −ssinθ ssinθ scosθ       x y       + b1 b2      

Geometric transformations

1 5 10 15 1 5 10

• Scale 2
• 45°
• tx=2

ty=3 45° tx=2 ty=3 Scale 2

Geometric transformations

• 1. Scale - rotation - translation = rotation - scale - translation
• 2. Rotation - translation - scale
• 3. Scale - translation - rotation
• 4. Translation - scale - rotation = translation - rotation - scale

a1 −a2 b1 a2 a1 b2 1           → scosθ −ssinθ tx ssinθ scosθ ty 1           a1 −a2 b1 a2 a1 b2 1           → scosθ −ssinθ stx ssinθ scosθ sty 1           a1 −a2 b1 a2 a1 b2 1           → scosθ −ssinθ txcosθ − tysinθ ssinθ scosθ txsinθ +tycosθ 1           a1 −a2 b1 a2 a1 b2 1           → scosθ −ssinθ stxcosθ − stysinθ ssinθ scosθ stxsinθ +stycosθ 1          

Urban mental maps

Urban mental maps

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

a1=0.86, p=.000 a2=0.14, p=.348 b1=-5.32, p=.837 b2=36.63, p=.164

ui * vi *         = a1 −a2 a2 a1       xi yi       + b1 b2      

R2=0.57, p=.000

Urban mental maps

• 1. Scale - rotation - translation = rotation - scale - translation
• 2. Rotation - translation - scale
• 3. Scale - translation - rotation
• 4. Translation - scale - rotation = translation - rotation - scale

scale=0.87, p=0.364; rotation=-9.31° , p=.000; tx=-5.32, p=.823; ty=36.63, p=.124 scale=0.87, p=0.364; rotation=-9.31° , p=.000; tx=-6.09, p=.817; ty=41.88, p=.164 scale=0.87, p=0.364; rotation=-9.31° , p=.000; tx=-11.18, p=.677; ty=35.28, p=.115 scale=0.87, p=0.364; rotation=-9.31° , p=.000; tx=-12.78, p=.662; ty=40.33, p=.168

Urban mental maps

Transformation order

RST(SRT) RTS STR TRS(TSR)

Group

Blind people (n =14)

3 2 3 6

Wheelchair users (n =14)

5 3 3 3

Non-impaired (n =14)

3 4 5 2

χ

2 6

( ) = 9.394, p = .153

• Algebraic vs geometric parameters
• No a priori order
• “Preferential“ transformation orders

Discussion

• http://spatial-modelling.info/Darcy-2-module-de-comparaison

Thank you for your attention