From spatial complexity to real estate prices: statistical and stochastic models" by Sarah Soleiman-Halevy, PhD Candidate @MAEngineers

1. From spatial complexity to real estate prices:
statistical and stochastic models
Talk WiMLDS

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MeilleursAgents : real estate platform
MeilleursAgents connects two populations that need each other but
whose relationship is complicated
Individuals
with a sale or
purchase
project
Estate agents
who want to take
advantage of
digitalization

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Produce very accurate information
● Very heterogeneous market
● Scarce information
○ in Paris, 30 000 transactions a year for 1.3M accommodations
Set up innovative models based on heterogeneous data
Mathematical models in the field of spatial statistics and machine learning

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Highlighting the spatial structure of prices

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An example of spatial analysis in Paris
● Multidimensional urban segregation: an exploratory case study (Cottrell et al.,
2018)
● Self-Organizing Maps algorithm based on the income distribution (INSEE)
● Spatial organisation is very similar to the real estate price heat map

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PhD Goals
● Relevant to highlight a spatial structure of the space in order to modelize
real estate market
● Describe socio-spatial patterns present in the residential areas in France
(Kohonen Algorithm for example)
● Design and adaptation of mathematical models to describe spatial
organization
○ Stochastic and multi-agent models such as the Schelling model
● Prove that spatial analysis is relevant to modelize real estate prices

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PhD Goals
● Relevant to highlight a spatial structure of the space in order to modelize
real estate market
● Describe socio-spatial patterns present in the residential areas in France
(Kohonen Algorithm for example)
● Design and adaptation of mathematical models to describe spatial
organization
○ Stochastic and multi-agent models such as the Schelling model
● Prove that spatial analysis is relevant to modelize real estate prices

8. Understanding the SOM algorithm

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Description of the Algorithm
● Self-Organizing Maps introduced by Teuvo Kohonen in 1984
● Supervised SOM first used for speech recognition
● Unsupervised SOM used as a clustering, visualization method
● Very close to k-Mean but benefits from the topology conservation property
● Good visualization tool

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Some more details
● SOM : a "nonlinear projection" of the
probability density function p(𝔁) of the
high-dimensional input data vector 𝔁 onto
the two-dimensional grid
● Grid composed of neurons
● Rectangular, hexagonal, or even irregular
● A neuron a reference vector
● Structure of the grid makes the difference
● Distance between clusters

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Best Matching Node
● Reference vectors are randomly initialized
● At each step, observation 𝔁 from the input
space is chosen randomly
● 𝔁 is compared to all reference vectors
● The smallest of the Euclidean distances is
used to define the best-matching node
denoted ｃ (the response)

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Best Matching Node
● Reference vectors are randomly initialized
● At each step, observation 𝔁 from the input
space is chosen randomly
● 𝔁 is compared to all reference vectors
● The smallest of the Euclidean distances is
used to define the best-matching node
denoted ｃ (the response)
● At this stage, none of the neurons are
modified in the network

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Update process
● Nodes that are topographically close
in the gird will activate each other to
learn something from the same input
𝔁.
● The update process is described by
● Neighborhood function 𝒉 has a
central role

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Two examples of neighborhood functions

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Sum-up
● Clustering algorithm that maps an
input space onto a two-dimensional
grid
● Grid is composed of neurons that have
neighboring properties
● Preserves the topology
● Neighboring observations in the input
space are located next to each other
on the grid

16. Let’s try

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Very accurate results at a specific geographical level
Results of Kohonen's algorithm in Les Lilas

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Kohonen at the IRIS level, based on the income

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Conclusion / Next steps
● Describe socio-spatial patterns using other algorithms than Kohonen
● Study the spatial organisation using clustering results
● Prove that it helps better modelize real estate prices

20. Conclusion et perspectives
Thank you!