1

Preferred citation style for this presentation

Axhausen, K.W. (2007) Problems with errors: A brief introduction, Englishseminar 2007, Schliersee, February 2007.

Problems with errors: A brief introduction

KW Axhausen

IVTETHZürich

February 2007

3

Example: Population growth of Swiss municipalities

4

Example: Link speeds (Kanton Zürich)

0-1920-3940-5960-7980-99

100-119>120

Km/h

5

Software

Hierarchial (multilevel) models: • MLWin• Any software, which estimates “mixed models” (SAS, GLIM,

LIMDEP, etc.)

Spatial error and lag models:• S or R (integrated into ArcGIS)• GeoDA• LeSage’s Econometric Toolbox for MatLab

6

Starting point

OLS assumes:

y Dependent variableβ Vector of parametersX Matrix of independent variablesε Errorσ Variance of the error

εβ += Xy~ (0, )iid Nε σ

7

What can go wrong ?

Heteroscedacity 1

Heteroscedacity 2

Collinearity

ˆ~ yε

~ xε1 0 00 1 0

cov( , )0

0 0 1

i jx x

⎧ ⎫⎪ ⎪⎪ ⎪≠ ⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

K

K

M M O

K

8

What else can go wrong ?

1 0 00 1 0

cov( , )0

0 0 1

n mε ε

⎧ ⎫⎪ ⎪⎪ ⎪≠ ⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

K

K

M M O

K

Spatial or temporal vicinity

9

Hierarchical regression (Simplest 2-level model)

with:

and:

ijijijij xxy 1100 ββ +=

0 0 0 0ij j ijuβ β ε= + +

1 1 1 1ij j ijuβ β ε= + +

fixed part random part

random partfixed part

Example:

y Relative population growthβ0,1 Parameterx0 Constantx1 Change in accessibilityu Systematic error (departure of the

j-th Cantons intercept (slope) from the overall value)

ε Error (departure of the i-th municipality‘s actual score from the predicted score)

i Level 1 (Municipality)j Level 2 (Kanton)

~ (0, )iid Nε σ

10

Example: Swiss population growth

Accessibility change

Rel

ativ

e po

pula

tion

grow

th

11

Example: “Systematic errors”

inte

rcep

tS

lope

Rank

Rank

12

Example: Neighbourhoods in Swiss population growth

OtherHighest interceptsHighest slopes

13

Spatial regression models

Spatial autoregressive model (SAR):

Spatial error model (SEM)

Spatial autoregressive and spatial error model combined (SAC):

with W: neighborhood matrix (contiguity matrix) with row sum=1ρ: influence factor of spatial autoregressive dependenceλ: influence factor of spatial dependence of error

εβρ ++= XyWy A

uXyWy A ++= βρ ελ += uWu E

uXy += β ελ += uWu E

~ (0, )iid Nε σ

14

Neighbors: Euclidean distance vs. network distance

Five spatially symmetric nearest neighbors by Euclidean distance

Five neighbors within a network distance of up to two intersections

15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

SE-neighbors (λ)

AR-n

eigh

bors

-17283--17250

-17317--17283

-17350--17317

-17383--17350

-17417--17383

-17450--17417

Log-Likelihood

AR-n

eigh

bors

(ρ)

0.0 2.0 5.9 12.5 21.1 31.9 45.0 60.4 78.20.0

2.0

5.9

12.5

21.1

31.9

45.0

60.4

78.2

SE-neighbors (λ)

AR

-nei

ghbo

rs

-17150--17117

-17183--17150

-17217--17183

-17250--17217

-17283--17250

-17317--17283

-17350--17317

-17383--17350

-17417--17383

-17450--17417

Log-Likelihood

Nearest neighbors byEuclidean distance

Nearest neighbors bynetwork distance of intersections

SAR: ρ4=0.249 -SEM: - λ7=0.374SAC: ρ11=0.242 λ3=0.173

SAR: ρ5.9=0.298 -SEM: - λ21.1=0.632SAC: ρ12.5=0.299 λ2.0 =0.142

Log-Likelihood Measure: Choosing Best Model (W-Matrix)

16

1 intersection (~2 neighbors) 5 intersections (~32 neighbors)

The CPU memory problem of the W-Matrices

17

Predictions

Ordinary least squares and weighted least squares (OLS, WLS):

Spatial autoregressive model (SAR):

Spatial error model (SEM):

Spatial autoregressive and spatial error model combined (SAC):

βXy =ˆ

βXy =ˆ

βρ XWIy A1)(ˆ −−=

βρ XWIy A1)(ˆ −−=

18

Do you always check ?

• Collinearity (Covariance matrix of the X) ?

• Correlations between y and ε ?

• Correlations between X and ε ? • Presence of structural units (e.g. organisational membership,

social networks and groups, cohorts, life style groups, residential clustering) ?

• Temporal correlations (DW – Test) ?• Spatial correlations (Moran’s I) ?

19

spatially correlated error?

auto-regressive correlation?

hetero- scedastic residuals?

homo- scedastic residuals?

spatially correlated error?

Spatial autoregression Spatially correlated error

OLS

WLS

SAR

SAC

No spatial correlation

SEM

Choosing appropriate regression model

20

Thanks to

Martin Tschopp (Hierarchical models)

Michael Bernard and Jeremy Hackney (Spatial lag and error models)