# Spatiotemporal Modeling of Urban Growth Using Machine Learning

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Input Variables

#### 2.2. Data Pre-Processing

- Using images from Landsat missions to obtain a set of initial estimates of the urban footprint in RGB color and the binary urban footprint. This is covered in Section 2.2.1.
- Performing a content-aware spatial resampling of all input variables for getting digital images with a common coordinate reference system, geographic extent, and spatial resolution. This is explained in Section 2.2.2.
- Applying a first-order temporal interpolation to the population distribution for getting annual estimates, and optionally if available, applying an adjustment of the annual population distribution estimates to match the total population defined by the corresponding National Bureau of Statistics (see Section 2.2.3). Notice that, if the official population values are only available for a subset of the years of interest, then a temporal regression is applied to estimate the values in missing years.
- Estimating the binary urban footprint for missing years by harmonizing it with the historical population distribution. This process is detailed in Section 2.2.4;
- Using a semantic-inpainting algorithm for estimating the urban footprint in RGB color for missing years. The block in this pre-processing stage is the same to the one explained in Section 2.6.
- Applying a zero-order hold to get annual estimates of the other input variables in missing years.

#### 2.2.1. Pre-Processing of Landsat Images

#### 2.2.2. Content-Aware Spatial Resampling of Images

#### 2.2.3. Temporal Interpolation of the Population Distribution and Official Adjustment

#### 2.2.4. Harmonization Between the Binary Urban Footprint and the Population Distribution

#### 2.3. Urban Growth Model

#### 2.4. Spatiotemporal Regression Model for the Population Distribution

#### 2.5. Binary Urban Footprint Estimation

#### 2.6. Urban Footprint Estimation

#### 2.7. Training, Model Selection, and Testing Strategies

#### 2.8. Implementation Details

#### 2.9. Case Studies

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GHSL | Global human settlement layer |

BUF | Binary urban footprint |

POP | Population distribution |

SAD | Sum of absolute differences |

SSD | Sum of square differences |

FP | False positive |

ZNCC | Zero-mean normalized cross correlation |

IoU | Intersection over Union |

KDE | Kernel density estimation |

Probability density function | |

LAC | Latin American cities |

ML | Machine learning |

## Appendix A. Saturation Functions for the Population Distribution

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**Figure 1.**The urban growth framework predicts population distribution, binary urban footprint, and urban footprint in color from these and other spatially distributed variables, if they are available.

**Figure 2.**A high-level overview of the data pre-processing applied to all the input variables of the urban growth framework.

**Figure 3.**Flow diagram with the pre-processing of Landsat images. There are three main stages highlighted in distinct colors in the diagram. The first stage removes clouds and NoData values and applies the result to the other two stages in parallel. The second stage extracts the urban footprint in RGB color for selected years. The third stage uses classification processes and morphological operations to obtain the binary urban footprint for available years.

**Figure 6.**Flow diagram with the harmonization process of the binary urban footprint using the population distribution. It has three main sub-processes that have been color-coded. The first sub-process finds the optimal threshold after which an area is considered urban. The second sub-process estimates the binary urban footprint for missing years using available information, a temporal correction, and the optimal threshold. The third sub-process cleans the binary urban footprint and only leaves urban pixels that are likely to be part of a city.

**Figure 7.**Interaction among the three key variables of the urban growth framework: population distribution, binary urban footprint, and urban footprint.

**Figure 8.**Example of required data for modeling a multiple input single output spatio-temporal dynamic system of a selected city, assuming a temporal lag of two years for the input variables, and selecting the population distribution as the output variable. These data can be seen as part of a single temporal window of a larger dataset.

**Figure 9.**Automated training, validation, and testing of the regression model to estimate the population distribution.

**Figure 11.**Tested functions for saturating the predicted population given a maximum population capacity.

**Figure 12.**Converting population distribution predictions into estimates of binary urban footprint during inference.

**Figure 13.**Semantic inpainting to estimate the urban footprint in color at a final time from the value at an initial time and the the binary urban footprints at the initial and final times.

**Figure 14.**Left: location of Valledupar and Rionegro in Colombia. Right: administrative divisions within the selected geographic extents for Valledupar and Rionegro.

**Figure 15.**Results of the classification process of Landsat images of Valledupar and Rionegro into urban and non-urban areas for the year 2015. The yellow regions highlight the urban classification results.

**Figure 16.**Estimated probability density function of a population threshold after which a non-urban pixel becomes urban in Valledupar.

**Figure 17.**Estimated probability density function of a population threshold after which a non-urban pixel becomes urban in Rionegro.

**Figure 19.**Diagnostic graphs of the geographic area of interest masked by the administrative division of Valledupar. (

**Left**) total urban population. (

**Center**) total urban area. (

**Right**) urban density.

**Figure 20.**Diagnostic graphs of the geographic area of interest masked by the administrative division of Rionegro. (

**Left**) total urban population. (

**Center**) total urban area. (

**Right**) urban density.

**Figure 21.**Estimated probability density functions of the mean square error in the spatiotemporal regression of the population distribution for different families of models as returned by the autotuning program in Valledupar.

**Figure 22.**Estimated probability density functions of the mean square error in the spatiotemporal regression of the population distribution for different families of models as returned by the autotuning program in Rionegro.

**Figure 23.**Framework performance in the geographic areas of interest. From left to right, the columns correspond to: (1) population distribution error (i.e., real value - predicted value); (2) histogram of the population distribution error; (3) binary urban footprint error (i.e., real value - predicted value); (4) histogram of the binary urban footprint error. POP = predicted population distributions; BUF = binary urban footprints; ZNCC = zero-mean normalized cross-correlation; SAD = sum of absolute differences; SSD = sum of squared differences.

**Figure 24.**Assessment of the urban predictions in the geographic area of interest masked by the administrative division of Valledupar with the test set.

**Figure 25.**Assessment of the urban predictions in the geographic area of interest masked by the administrative division of Rionegro with the test set.

**Table 1.**Examples of variables that the urban-growth framework can handle. GHSL = Global Human Settlement Layer; SRTM = Shuttle Radar Topography Mission; OSM = Open Street Map; IGAC = Agustín Codazzi Geographic Institute; WDPA = World Database on Protected Areas.

Variable Name | Data Source | Digital Format | Resolution | Availability | Importance |
---|---|---|---|---|---|

Population distribution. | GHSL pop [45]. | Raster. | 250 m × 250 m. | Global. | Essential. |

Urban footprint. | Landsat [47]. | Raster. | 30 m × 30 m. | Global. | Essential. |

Binary urban footprint. | Derived from the urban footprint through a binary classifier. | Raster. | 30 m × 30 m. | Global. | Essential. |

Official population projections. | National Bureau of Statistics [48]. | Tabular. | Administrative units. | National. | Optional. |

Maximum population capacity. | Master plan [49,50]. | Vector. | - | Local. | Optional. |

Land use (residential, industrial, commercial, official, and special). | Master plan [49,50]. | Vector. | - | Local. | Optional. |

Built-up urban ratio. | GHSL built-up [51]. | Raster. | 250 m × 250 m. | Global. | Optional. |

Terrain slope. | SRTM [52]. | Raster. | 90 m × 90 m. | Global. | Optional. |

Distances to nearest populated towns. | Derived from binary urban footprint. | Raster. | 30 m × 30 m. | Global. | Optional. |

Roads. | OSM [53], IGAC [54]. | Vector. | - | Global, National. | Optional. |

Natural hazard (flooding, landslide, fire, volcanic eruption, earthquake.). | Master plan [49,50]. | Vector. | - | Local. | Optional. |

Water bodies. | Master plan [49,50], IGAC [54]. | Vector. | - | Local. | Optional. |

Protected areas. | Master plan [49,50], WDPA [55]. | Vector. | - | Global, Local. | Optional. |

Urban development projects. | Master plan [49,50], others. | Vector. | - | Local. | Optional. |

Variables | Description |
---|---|

t | Time in years. |

r | Number of consecutive years of historical data records available for training. |

$x,y$ | Spatial coordinates along the East and North directions, respectively. |

$w,h$ | Width and height of the spatial extent under study in meters. |

${\Delta}_{x},{\Delta}_{y}$ | Spatial sampling period in meters along the x and y directions, respectively. |

$m,n$ | Number of rows and columns of each digital image. |

$i,j,k,l$ | Auxiliary variables. |

$pop$ | Population distribution. |

$buf$ | Binary urban footprint (black and white). |

$uf$ | Urban footprint (color). |

${v}_{i}$ | i-th input variable. |

${p}_{i},{q}_{i}$ | Number of rows and columns of the spatial window for ${v}_{i}$. |

${\varphi}_{i}$ | Number of consecutive temporal lags in years for ${v}_{i}$. |

${f}_{i}$ | Mathematical function applied to each possible spatial window of $\left[{p}_{i},{q}_{i}\right]$ pixels in ${v}_{i}$. In the naïve feature sampling, it corresponds to a reshape operation to convert the data dimensions from $\left[{p}_{i},{q}_{i}\right]$ to $\left[1,{p}_{i}\times {q}_{i}\right]$. In other scenarios it can be a spatial-filtering function that processes and reduces the number of features (e.g., an element-wise multiplication of the spatial window by a fixed spatial kernel of the same dimensions followed by a sum of its elements). |

${\eta}_{tw}$ | Number of temporal windows that can be extracted from the historical data records for all variables given the maximum consecutive temporal lag in the regression model. |

${m}_{o},{n}_{f}$ | Number of rows (i.e., observations) and columns (i.e., input features) of the resulting tabular dataset for training the population distribution growth model based on machine learning. |

$\gamma $ | Population threshold after which a region can be considered as urbanized. |

**Table 3.**Confusion matrices for the classification process of Landsat images in Valledupar. The top table corresponds to the initial classification process of pixels into four categories, and the bottom table corresponds to the final re-classification process into two categories.

Predicted Class | |||||
---|---|---|---|---|---|

built-up | bare soil | vegetation | water | ||

True class | built-up | 2094 | 45 | 3 | 0 |

bare soil | 28 | 9339 | 7 | 0 | |

vegetation | 0 | 24 | 17,504 | 6 | |

water | 0 | 6 | 43 | 89 | |

Predicted Class | |||||

urban | non-urban | ||||

True class | urban | 2094 | 48 | ||

non-urban | 28 | 27,018 |

**Table 4.**Confusion matrices for the classification process of Landsat images in Rionegro. The top table corresponds to the initial classification process of pixels into four categories and the bottom table corresponds to the final re-classification process into two categories.

Predicted Class | |||||
---|---|---|---|---|---|

built-up | bare soil | vegetation | water | ||

True class | built-up | 1355 | 29 | 11 | 1 |

bare soil | 61 | 535 | 15 | 1 | |

vegetation | 0 | 10 | 5557 | 0 | |

water | 1 | 7 | 3 | 73 | |

Predicted Class | |||||

urban | non-urban | ||||

True class | urban | 1355 | 41 | ||

non-urban | 62 | 6201 |

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**MDPI and ACS Style**

Gómez, J.A.; Patiño, J.E.; Duque, J.C.; Passos, S.
Spatiotemporal Modeling of Urban Growth Using Machine Learning. *Remote Sens.* **2020**, *12*, 109.
https://doi.org/10.3390/rs12010109

**AMA Style**

Gómez JA, Patiño JE, Duque JC, Passos S.
Spatiotemporal Modeling of Urban Growth Using Machine Learning. *Remote Sensing*. 2020; 12(1):109.
https://doi.org/10.3390/rs12010109

**Chicago/Turabian Style**

Gómez, Jairo A., Jorge E. Patiño, Juan C. Duque, and Santiago Passos.
2020. "Spatiotemporal Modeling of Urban Growth Using Machine Learning" *Remote Sensing* 12, no. 1: 109.
https://doi.org/10.3390/rs12010109