Spatial Data Analysis Areas I: Rate Smoothing and the MAUP

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Spatial Data Analysis Areas I Rate Smoothing
and the MAUP
Ifgi, Muenster, Fall School 2005

• Gilberto Câmara
• INPE, Brazil

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Areal data

• Study region is partitioned in disjoint areas
• The region is the union of the areas
• Each map has one or more associated measures
• Treated as random variables
• Examples
• Map of Germany divided in municipalities. For
  each area, we measure the unemployment rate and
  the literacy rate.
• Is unemployment correlated with years of school?
• What about Brazil?

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Violence in Minas Gerais
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Violence in Minas Gerais
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Violence in Minas Gerais
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Attributes in areal data

• As a general rule, each measure is a sum, count
  or a similar aggregated function over all the
  area
• Each value is associated to all the corresponding
  area
• If we need to choose a single location, usually
  we take the polygon centroid
• There are no intermediate values

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What is mapped in areal data?

• Typical values are rates or proportions
• Numerator events
• Denominador pop at risk
• Log maps?

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Log rate of motor vehicle accident death per
100.000 residents, 1990-92
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Log ratio of homicide death of males 15-49 per
100.000 residents of same group age, 1990-92
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Models of Discrete Spatial Variation
Random variable in area i

• n of ill people
• n of newborn babies
• per capita income

Source Renato Assunção (UFMG/Brasil)
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Dealing with rates and proportions
When the study variable is a rate or a
proportion, mapping those rates is the first
obvious step in any analysis. However, the use of
raw observed rates might be misleading, since the
variability of those rates will be a function of
the population counts, which differs widely
between the areas. Bailey,1995
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Source Fred Ramos (CEDEST/Brasil)
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Model-Driven Approaches

• Model of discrete spatial variation
• Each subregion is described by is a statistical
  distribution Zi
• e.g., homicides numbers are Poisson (?, ?).
• The main objective of the analysis is to estimate
  the joint distribution of random variables Z
  Z1,,Zn
• We use a model-driven approach to correct the
  missing data
• It is called the Empirical Bayes method...
• We could also use the Full Bayes method (but
  that is another story...)

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(measured rate)
i
In Bayesian statistics, the best estimate
of the true and unknown rate is

where
Source Fred Ramos (CEDEST/Brasil)
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Empirical Bayes
Simplifying assumptions for estimating means and
variances for all random variables of all areas
(Marshall, 1991)
Source Fred Ramos (CEDEST/Brasil)
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Source Fred Ramos (CEDEST/Brasil)
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Infant Mortality Rate São Paulo (Raw)
Source Fred Ramos (CEDEST/Brasil)
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Infant Mortality Rate São Paulo (Corrected)
Source Fred Ramos (CEDEST/Brasil)
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Some Important Questions

• How does scale matter?
• How do the spatial partitions matter?
• How does proximity matter?
• What can we learn by studing how multiple data
  vary in space?
• How much prior assumptions can we impose in our
  spatial data?

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A Question of Scale
Problema das Unidades de Área Modificáveis - MAUP

• A basic problem with areal data
• The spatial definition of the frontiers of the
  areas impacts the results
• Different results can be obtained by just
  changing the frontiers of these zones.
• This problem is known as the the modifiable area
  unit problem

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Scale Effects
Per capita income
Jobs/ population
Illiterate / population
Source Fred Ramos (CEDEST/Brasil)
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Scale Effects
Per capita income
Jobs/ population
Illiterate / population
Source Fred Ramos (CEDEST/Brasil)
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Scale Effects Figthing the MAUP
Population gt60 years
Illiterates
per capita income
270 ZONES OD97
Source Fred Ramos (CEDEST/Brasil)
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Scale Effects Figthing the MAUP
Population gt60 years
Illiterates
per capita income
96 DISTRICTS OF SÃO PAULO
Source Fred Ramos (CEDEST/Brasil)
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Scale Effects Figthing the MAUP
Source Fred Ramos (CEDEST/Brasil)
Population gt60 years
Illiterates
per capita income
96 INCOME-HOMOGENOUS ZONES IN SÃO PAULO
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Correlation matrices
270 ZONES OD97
VARIABLES
A) Percentage of population 60 year-old or
more B) Percentage of illiterate population C)
Per capita individual income
96 DISTRICTS
96 INCOME-AGGREGATED
Source Fred Ramos (CEDEST/Brasil)
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A Questão da Escala
Get census data
Adaptation
Identify inter-tract variation
Reduce data variability
Minimize the outlier effect
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Regionalization

• Reagregate N small areas (finest scale available)
  into M bigger regions to reduce scale effects.
• A possible solution constrained clustering

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Regionalization Maps as graphs
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Regionalization Maps as graphs
Simple aggregation
Population-constrained aggregation