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Preferred citation style for this presentation
Axhausen, K.W. (2007) Problems with errors: A brief introduction, Englishseminar 2007, Schliersee, February 2007.
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Problems with errors: A brief introduction
KW Axhausen
IVTETHZürich
February 2007
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Example: Population growth of Swiss municipalities
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Example: Link speeds (Kanton Zürich)
0-1920-3940-5960-7980-99
100-119>120
Km/h
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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
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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
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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
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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
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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
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Example: Swiss population growth
Accessibility change
Rel
ativ
e po
pula
tion
grow
th
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Example: “Systematic errors”
inte
rcep
tS
lope
Rank
Rank
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Example: Neighbourhoods in Swiss population growth
OtherHighest interceptsHighest slopes
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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
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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
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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
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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)
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1 intersection (~2 neighbors) 5 intersections (~32 neighbors)
The CPU memory problem of the W-Matrices
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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 −−=
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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) ?
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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
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Thanks to
Martin Tschopp (Hierarchical models)
Michael Bernard and Jeremy Hackney (Spatial lag and error models)