Voronoi Data Linkage

# Voronoi Data Linkage
## Extracting Information from Polygons to Points
### Lucas da Cunha Godoy Luis Gustavo Silva e Silva Douglas Roberto Mesquita 1st Conference on Statistics and Data Science Salvador - Bahia, Brasil 
### 2018/10/16

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* Introduction
* Method
* Application
* Conclusion
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# What is .text-hl[Data Linkage]?

## It is a method used to gather .text-hl[information] from 
## .text-hl[different sources], giving rise to .text-hl[richer datasets]

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# What is .text-hl[Spatial Join]?

## It is a .text-hl[GIS] operation used to .text-hl[combine information] 
## from different sources of .text-hl[spatial data] based on a given
## relationship between their spatial features

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# Example 1
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# Example 3
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# Example 1
## Polygons Polygons
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# Example 2
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# Example 3
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# Example 1
## Polygons Polygons
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# Example 2
## Polygons Points 
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# Example 3
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![](index_files/figure-html/int_im3-1.svg)
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# Example 1
## Polygons Polygons
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# Example 2
## Polygons Points 
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# Example 3
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![](index_files/figure-html/int_im4-1.svg)
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.column.bg-main1[
.split-three[
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# Example 1
## Polygons Polygons
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# Example 2
## Polygons Points 
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# Example 3
## Points Polygons
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![](index_files/figure-html/int_im5-1.svg)
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.split-three[
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# Example 1
## Polygons Polygons
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# Example 2
## Polygons Points 
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# Example 3
## Points Polygons
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# .text-hl[Definitions]

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### `$$Y = \{Y_1, ..., Y_{n_y} \}$$`, where `$Y_i \in A \forall i$`.
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### `$Z = \{Z_1, ..., Z_{n_z} \}$`, where `$\cup_{i = 1}^{n_z} Z_i = A$`.

### `$X_k = \{X_{k,1}, ..., X_{k,p} \}$` is a vector of continous variables belonging
### to `$Z_k$`.
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# How can we .text-hl[extract data] from Spatial .text-hl[Polygons] to Spatial .text-hl[Points]?

Consider that we want to extract, or estimate, the variable `$X_{., 1}$` observed in
the spatial polygons `$\{Z\}$` for each point `$\{Y\}$`.

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# .text-hl[Naive Approach]

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### `$$E[X^*_{k, 1}] = E[\{ X_{j, 1} : Y_k \subset Z_j \}]$$`

### `$$Var[X^*_{k, 1}] = Var[\{ X_{j, 1} : Y_k \subset Z_j \}]$$`

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# .text-hl[Voronoi Data Linkage]

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# Voronoi Tessellation

![](img/vor_ex.gif)

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Now, we have a voronoi cell `$V_k$` associated to each point `$Y_k$`.
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### `$X^*_{k, 1} = \sum_{i = 1}^{n_z} p_{i, k} X_{i, 1}$`

### `$E[X^*_{k, 1}] = \sum_{i = 1}^{n_z} p_{i, k} E[X_{i, 1}]$`

### `$Var[X^*_{k, 1}] = \sum_{i = 1}^{n_z} p^2_{i, k} Var[X_{i, 1}] + 2 \mathop{\sum \sum}_{i < j} p_{j, k} p_{i, k} Cov(X_{j, 1}, X_{i, 1})$`

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# .text-hl[Recap]

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### Naive Approach

* `$Var[X^*_{k, 1}] = Var[\{ X_{j, 1} : Y_k \subset Z_j \}]$`
* Does not take advantage of all available data;
* More variability;
* Does not inherit the autocovariance structure from data;

### Voronoi Data Linkage

* `$Var[X^*_{k, 1}] = \sum_{i = 1}^{n_z} p^2_{i, k} Var[X_{i, 1}] + 2 \mathop{\sum \sum}_{i < j} p_{j, k} p_{i, k} Cov(X_{j, 1}, X_{i, 1})$`
* Takes advatange of all available data;
* Less variability;
* Inherits the autocovariance structure from data;

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.font5[Application: Brazil Elections Data]
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* Data from .text-hl[electoral sections] for President in the city of .text-hl[São Paulo] (2014) - .text-hl[Spatial Points];
      * Number of electors;
      * percent of votes for each candidate in the second round
* Socio demographic data from .text-hl[IBGE census sectros] (2010) - .text-hl[Spatial Polygons];
      * population, average income, household density, illiteracy rate, proportion of white people, proportion of women, and several variables about the proportion of people in different ages groups.

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# Prediction

For more analysis on this dataset, see [this link](https://lcgodoy.github.io/rbras/#27)

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# lucasdac.godoy@gmail.com
# [](https:://github.com/lcgodoy) github.com/lcgodoy
# [](https:://lcgodoy.github.io) lcgodoy.github.io