# Estimation of Gridded Population and GDP Scenarios with Spatially Explicit Statistical Downscaling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) emissions, disaster risks, and other factors affecting sustainability from a long-term perspective. The Intergovernmental Panel on Climate Change (IPCC) published Shared Socioeconomic Pathways (SSPs) [1,2] that describe future socioeconomic conditions under various scenarios, including SSP1-3. SSP1 makes relatively good progress toward sustainability under an open and globalized world. SSP2 is a middle-of-the-road scenario assuming that the typical trends in the last decades will continue, and in SSP3, the world is closed and fragmented into regions, but it fails to achieve sustainability.

## 2. Downscale Approach

#### 2.1. Overview

#### 2.2. City Growth Model: Estimation with Current Data

#### 2.3. Overview

#### 2.4. Projection of Urban Area

_{g}

_{,2000}denotes disturbance. Urban area

_{g}

_{,2000}is the urban area in the g-th grid in 2000 (see Table 1). q

_{g}

_{,2000}(r) represents the urbanization potential, where p

_{c}

_{,2000}is the population in the c-th city in 2000, d

_{c}

_{,g}is the arc distance between the c-th city and the center of the g-th grid. a, b, and r are parameters. This model describes urbanization due to city population increase, and urban shrinkage due to city population decrease.

^{2}of Equation (1). The estimate of r is 16.4, which implies that the distance at which 95% of the influence from city population change disappears is 49.2 (= 16.4 × 3) km. r = 16.4 is assumed for SSP2. On the other hand, r = 8.2 (= 0.5 × 16.4) is assumed for SSP1 to model compact urban growth, while r = 32.8 (= 2.0 × 16.4) is assumed in SSP3 to model dispersed growth. Figure 2 displays urbanization potentials estimated using Equation (2) in Europe in 2080. Because of the r values, potentials in SSP1 are the most compactly distributed while those in SSP3 are the most dispersed.

_{c}

_{,2000}with ${\widehat{p}}_{c,2005}$,…${\widehat{p}}_{c,2100}$, which are city populations projected by the city growth model.

#### 2.5. Downscale Approach

## 3. Result

#### 3.1. Parameter Estimation Result

_{g}

_{,t,k}= (baseline variables) × (control variables) whose control variables equal Road (i.e., 48% = 3% + 3% + 41%; see Table A2). It might be because cities strongly interact in SSP1, and small cities emerge in between these cities. On the other hand, ocean is more important than principal road in SSP3.

#### 3.2. Downscaling Result

## 4. Concluding Remarks

_{2}emissions, disaster risks, energy demand, and other variables determining future sustainability and resiliency.

_{2}emissions and amount of wasted heat. To combine our downscaled populations and GDPs with LCZs might be an interesting topic to devise appropriate policies.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Details of the Downscaling Approach

#### A.1. Projection of Urban Population and Urban Expansion

#### City Growth Model: Model

_{c}

_{,t}is the population of city c in year t.

**p**

_{t}

^{(log)}and Δ

**p**

_{t}

^{(log)}are N × 1 vectors whose c-th elements are log(p

_{c}

_{,t}) and log(p

_{c}

_{,t}/p

_{c}

_{,t-5}), respectively.

**X**

_{t}is an N × K matrix of explanatory variables,

**ε**

_{t}is an N × 1 vector of disturbance with variance σ

^{2},

**0**is an N × 1 vector of zeros,

**I**is an N × N identity matrix, α is a coefficient (scalar), and

**β**is a K × 1 coefficient vector.

**W**

_{geo},

**W**

^{e}

^{1}, and

**W**

^{e}

^{2}are given by row-standardizing (i.e., row sums are scaled to one)

**W**

^{0}

_{geo}

_{,}

**W**

^{0}

_{e}

_{1}, and

**W**

^{0}

_{e}

_{2}, which describe connectivity among cities.

**W**

^{0}

_{geo}is a spatial connectivity matrix whose (c, c’)-th element is exp(-d

_{c}

_{,c’}/h), where d

_{c}

_{,c’}is the arc distance between cities c and c’, and r is a range parameter. For instance, if h = 100 km, 95% of the spill over effects disappear within 300 km (=3 × 100 km; [25]). In other words, a large h implies global spill over from cities whereas a small h implies local spill over.

**W**

^{0}

_{e}

_{1}and

**W**

^{0}

_{e}

_{2}describe economic connectivity. Since we could not find any data on economic connectivity among cities, we approximated it with Equation (A2), which represents an estimate of trade amount between cities c and c’:

_{C}is the population of the country, including the c-th city, and T

_{C}

_{,C’}is the amount of trade between countries C and C’ (source: CoW data set; see Table 1). Equation (A2) simply distributes the amount of trade, T

_{C}

_{,C’}, in proportion to city populations. The (c, c’)-th element of

**W**

^{0}

_{e}

_{1}is given by ${\widehat{t}}_{c,{c}^{\prime}}$ if cities c and c’ are in different countries (i.e., C $\ne $ C’), and 0 otherwise. By contrast, the (c, c’)-th elements of

**W**

^{0}

_{e}

_{2}are given by ${\widehat{t}}_{c,{c}^{\prime}}$ if these cities are in the same country (i.e., C = C’), and 0 otherwise. Finally,

**W**

_{e}

_{1}and

**W**

_{e}

_{2}describe international and national economic connectivity, respectively.

_{geo}is positive, population growth in a city increases the populations in its neighboring cities. When ρ

_{e}

_{1}and/or ρ

_{e}

_{2}is positive, population growth in a city increases the populations in foreign cities with strong economic connectivity. Intuitively speaking, ρ

_{geo}and ρ

_{e}

_{2}capture local interactions, and ρ

_{e}

_{1}captures global interactions.

#### City Growth Model: Estimation

**W**

_{geo}, 2SLS is iterated while varying r values, and the optimal r value, which maximizes the adjusted R2, is identified.) The explanatory variables are road density (Road dens), distance to the nearest airport (Airport dist), and distance to the nearest ocean (Ocean dist; see Table 1), whose coefficients are denoted by β

_{road}, β

_{ocean}, and β

_{airport}, respectively.

_{ocean}suggests that city growth in inland areas is faster than that in coastal cities. This might be because coastal cities are already matured, and their populations are more stable than those of inland cities.

^{geo}has a statistically significant positive effect, whereas β

^{e}

^{2}does not. Thus, geographic proximity is a significant factor determining local-scale city interactions. On the other hand, β

^{e}

^{1}, which quantifies global-scale interactions, is statistically significant. It is suggested that consideration of both local and global-scale interactions is important in city growth modeling.

^{2}for the population change in 5 years, Δ

**p**

_{t}

_{+5}, is 0.401, which is not very accurate. However, the value of R

^{2}for the population after 5 years,

**p**

_{t}

_{+5}, is 0.998. Since we focus on the latter, the accuracy of the model is sufficient.

Estimate | t-value | |||
---|---|---|---|---|

Intercept | −6.19×10^{-4} | −8.12 | ^{***} | |

α | 1.87×10^{-3} | 8.98 | ^{***} | |

ρ^{geo} | 9.56×10^{-1} | 188.57 | ^{***} | |

ρ^{e}^{1} | 1.83×10^{-3} | 24.95 | ^{***} | |

ρ^{e}^{2} | 4.10×10^{-4} | 0.84 | ||

β_{road} | 1.21×10^{-3} | 3.46 | ^{***} | |

β_{ocean} | 2.10×10^{-4} | 2.19 | ^{***} | |

β_{airport} | −1.66×10^{-4} | −0.47 | ||

r | 209 | |||

Quasi-adjusted R^{2} | for Δp_{t}_{+5} | 0.405 | ||

for p_{t}_{+5} | 0.998 |

^{***}Statistical significance at the 1 % level.

#### City Growth Model: Application for City Population Projections

^{e}

^{1}doubles by 2100 in comparison with 2000 in SSP1, ρ

^{e}

^{1}is constant in SSP2, and ρ

^{e}

^{1}becomes half the value of 2000 by 2100 in SSP3. In each scenario, the values for ρ

^{e}

^{1}between 2000 and 2100 are linearly interpolated.

_{e}

_{1}values, city populations in 2005, 2010, … 2100 are estimated by sequentially applying the city growth model, Equation (A1), which projects the 5-year-after populations.

#### Projection of Urban Potentials

_{c}

_{,g}is the arc distance between the c-th city and the center of the g-th grid. The potential q

_{g}

_{,t}(r) increases nearby cities with large population.

**W**

^{0}

_{geo}, r represents the range of spill over around each city, whereas h (= 209 km; see Table A1) represents the range of spill over across cities. Thus, r must be smaller than h. Considering the consistency with the subsequent urban area projection in Section 2.4, r is given by a value maximizing the explanatory power of urban potential, q

_{g}

_{,t}(r’), on urban expansion. In other words, r is estimated by maximizing the adjusted R-squares (adj-R

^{2}) of the following model, Equation (2), which is estimated using the GRUMPS city population data in 2000. The estimated parameters in 2000 are $\widehat{r}=16.4$, ${\widehat{b}}_{0}=21.89$, and ${\widehat{b}}_{q}=0.126$. r = 16.4 is assumed for SSP2. On the other hand, r = 8.2 (= 0.5 × 16.4) is assumed for SSP1 to model compact urban growth, while r = 32.8 (= 2.0 × 16.4) is assumed in SSP3 to model dispersed growth.

#### Projection of Urban Area

^{2}maximization of Equation (1) whose Urban Area

_{g}

_{,2000}is replaced with Agri Area

_{g}

_{,2000}(Equation (A5) is obtained from Equation (4) after the replacement). The estimated values are ${\widehat{r}}^{A}=12.1$ and ${\widehat{b}}_{q}^{A}=0.129$. While b

_{q}

^{A}= 0.129 is assumed across scenarios, r

^{A}values in SSP1-3 are given by 6.05, 12.1, and 24.2, respectively, just like r.

_{g}

_{,t+5}+ Agri Area

_{g}

_{,t+5}) exceeds the area of the grid, Agri area

_{g}

_{,t+5}is reduced. Urban Area

_{g}

_{,2000}and Agri Area

_{g}

_{,2000}are used as baseline areas. Thus, each grid can have both urban and agricultural areas.

^{A}control the share of populations and gross productivity nearby cities. For instance, if r is very small as in SSP1, most people and gross productivity are concentrated nearby cities. As such, the proportional distribution can describe both urban expansion and shrinkage depending on the range parameter values. Similarly, r

^{A}controls the nonurban population distribution. In case of SSP1, the small r

^{A}concentrates nonurban populations into grids with greater Agri Area with greater potentials. The populations are dispersed in SSP3 whose r

^{A}value is large.

#### A.2. Downscale Approach

_{C}

_{,t}is population or GDP in country C including the g-th grid in year t. ${\tilde{a}}_{g}^{ssp}$ is a baseline variable to control urban expansion/shrinkage assumed in each scenario. Urban area

_{g}

_{,t}, Agri area

_{g}

_{,t}, and UAgri area

_{g}

_{,t}(=Urban area

_{g}

_{,t}+Agri area

_{g}

_{,t}; see Table 2), which are projected under each SSP, are used to downscale urban population, nonurban population, and GDP, respectively.

_{g}

_{,t,k}is a control variable capturing influence from auxiliary variables, where k is the index of control variables. We are not sure which auxiliary variables are appropriate for a

_{g}

_{,t,k}. Hence, this study downscales population/gross productivity in g-th grid at year t, y

_{g}

_{,t}, using a weighted average of dasymetric mapping models, which is formulated as follows

_{k,t}measures the importance of the k-th submodel, f(a

_{g}

_{,t,k}). The following country level model is obtained by aggregating the grid-level model presented by Equation (A8).

_{k}

_{,t}in the downscale model Equation (A7) is estimated by gradient boosting, which is an ensemble learning technique, for Equation (A8). As explained in Section 2.5, the gradient boosting takes a weighted ensemble mean of 12 submodels in the urban and nonurban population downscaling, while 16 submodels exist in the GDP downscaling. Meanwhile, our ensemble learning means averaging of the submodels based on the weights optimized by the gradient boosting.

_{k}

_{,t}: (i) the weights for the submodels are equally set by ω

_{k}

_{,t}= 1/K; (ii) residuals are evaluated using Equation (A8); (iii) samples (e.g., Y

_{C}

_{(g),t}values) are weighted according to the size of the residuals; (iv) the ω

_{k}

_{,t}values are updated so that model accuracy is improved for samples with larger weights (i.e., larger residuals in step (ii)); and (v) steps (ii), (iii), and (iv) are iterated until convergence. The gradient boosting procedure is known to be robust even if the submodels are collinear.

_{k}

_{,t}value is replaced with ${\stackrel{-}{\omega}}_{k,t}$ = (ω

_{k}

_{,t−1}+ ω

_{k}

_{,t}+ ω

_{k}

_{,t+1})/3, which is their temporal moving average. Finally, the submodels in year t is averaged by the gradient boosting first, and the resulting models at time t−1, t, and t+1 are temporally averaged subsequently. Table A2 summarizes estimated ω

_{k,t}parameters in 2080. Section 3.1 discusses the parameter estimates.

**Table A2.**Estimated importance of auxiliary variables in 2080 (a

_{g}

_{,k}= baseline variables × control variables).

Baseline Variables | Urban Area | Urban Pop | Urban Potential | ||||||||||||||

Control Variables | 1 | Road | Air | Ocean | 1 | Road | Air | Ocean | 1 | Road | Air | Ocean | |||||

Urban Population | SSP1 | 0.02 | 0.10 | 0.07 | 0.11 | 0.02 | 0.01 | 0.03 | 0.11 | 0.05 | 0.19 | 0.15 | 0.16 | ||||

SSP2 | 0.09 | 0.05 | 0.05 | 0.10 | 0.02 | 0.02 | 0.03 | 0.11 | 0.03 | 0.13 | 0.12 | 0.26 | |||||

SSP3 | 0.07 | 0.03 | 0.05 | 0.10 | 0.08 | 0.07 | 0.06 | 0.08 | 0.03 | 0.04 | 0.13 | 0.28 | |||||

Baseline Variables | Agri Area | Urban Pop | Urban Potential | ||||||||||||||

Control Variables | 1 | Road | Air | Ocean | 1 | Road | Air | Ocean | 1 | Road | Air | Ocean | |||||

Nonurban Population | SSP1 | 0.03 | 0.04 | 0.07 | 0.03 | 0.02 | 0.03 | 0.03 | 0.06 | 0.08 | 0.41 | 0.13 | 0.07 | ||||

SSP2 | 0.04 | 0.03 | 0.09 | 0.03 | 0.02 | 0.01 | 0.03 | 0.07 | 0.08 | 0.37 | 0.12 | 0.11 | |||||

SSP3 | 0.07 | 0.02 | 0.10 | 0.04 | 0.01 | 0.01 | 0.03 | 0.09 | 0.05 | 0.17 | 0.19 | 0.23 | |||||

Baseline Variables | Urban + Agri Area | Urban Pop | Urban Potential | SSP Pop | |||||||||||||

Control Variables | 1 | Road | Air | Ocean | 1 | Road | Air | Ocean | 1 | Road | Air | Ocean | 1 | Road | Air | Ocean | |

GDP | SSP1 | 0.07 | 0.01 | 0.04 | 0.05 | 0.18 | 0.10 | 0.14 | 0.04 | 0.06 | 0.03 | 0.03 | 0.04 | 0.02 | 0.06 | 0.09 | 0.04 |

SSP2 | 0.10 | 0.02 | 0.03 | 0.05 | 0.14 | 0.09 | 0.08 | 0.03 | 0.08 | 0.08 | 0.05 | 0.06 | 0.08 | 0.03 | 0.07 | 0.03 | |

SSP3 | 0.01 | 0.05 | 0.01 | 0.05 | 0.10 | 0.09 | 0.01 | 0.05 | 0.09 | 0.17 | 0.17 | 0.01 | 0.09 | 0.02 | 0.07 | 0.01 |

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**Figure 1.**Procedure for population and gross domestic product (GDP) downscaling. Variables by countries, cities, and grids are coloured by green, yellow, and red, respectively. The black arrows represent the downscaling procedure while the blue arrows represent subprocessing to consider auxiliary variables. As this figure shows, urban population is downscaled from countries to cities to grids, while nonurban population is downscaled from countries to grids. GDP is downscaled from countries to grids by utilizing downscaled populations. (a), (b), and (c) in this figure correspond to (a), (b), and (c) described in Section 2.1.

**Figure 5.**Average population densities/gross productivities in the grids, whose geometric centers are within 0–10 km, 10–20 km, … 190–200 km from the nearest city (GRUMP settlement point). Solid line: SSP1; Dashed line: SSP2; Dotted line: SSP3. For comparison, the evaluated values are standardized so that the sum becomes 1.

**Figure 6.**Downscaled gross productivities in 2080 (South and West Asia and Europe). Europe: (

**a**), SSP1 (

**b**), SSP2, and (

**c**) SSP3. South-West Asia: (

**d**) SSP1, (

**e**) SSP2, and (

**f**) SSP3.

**Figure 7.**Comparison of estimated populations in South-West Asia in 2080 (SSP2). (

**a**) our result, (

**b**) Jones and O’Neill (2015) [10].

Variables | Description | Unit | Source | Year |
---|---|---|---|---|

City pop | City population | 67,934 cities | GRUMP ^{1} | 1990, 1995, 2000 |

Urban area | Urban area [km^{2}] | 0.5-degree grids | Schneider et al., 2009 ^{2} | 2001–2002 |

Agri area | Agricultural area [km^{2}] | |||

Road dens | Total length [km] of principal roads | Natural Earth ^{3} | 2012 | |

Airport dist | Distance [km] to the nearest airport | N.A. | ||

Ocean dist | Distance [km] to the nearest ocean | 2010 | ||

Trade amount | Amount of bilateral trade [current US dollars] | Country | CoW ^{4} | 2009 |

^{1}Settlement Points, v1 (http://sedac.ciesin.columbia.edu/data/set/grump-v1-settlement-points; [19]) of Global Rural-Urban Mapping Project (GRUMP), SEDAC (Socioeconomic Data and Applications Center; http://sedac.ciesin.columbia.edu/).

^{2}Global maps of urban extent from satellite data (https://nelson.wisc.edu/sage/data-and-models/schneider.php), which is estimated from MODIS (MODerate resolution Imaging Spectroradiometer; https://modis.gsfc.nasa.gov/). See [20] for further details.

^{3}Natural Earth (http://www.naturalearthdata.com/).

^{4}CoW (The Correlates of War project; http://www.correlatesofwar.org/).

**Table 2.**Baseline and control variables for the urban population, nonurban population, and GDP downscaling. Baseline variables are projected under each shared socioeconomic pathways (SSP). Control variables are constant across years.

Baseline | × | Control | ||

Urban population | Nonurban population | GDP | (common) | |

City pop_{ag} | City pop_{ag} | City pop_{ag} | Constant | |

Urban pot | Urban pot | Urban pot | Road dens | |

Urban area | Agri area | UAgri Area | Airport dist | |

SSP pop | Ocean dist |

_{ag}: city populations, which are projected and aggregated into grids; Urban pot: urban potential; Agri area: agricultural area; SSP pop.: downscaled urban + nonurban SSP populations; UAgri area: urban area + agricultural area. For control variables, see Table 1.

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## Share and Cite

**MDPI and ACS Style**

Murakami, D.; Yamagata, Y. Estimation of Gridded Population and GDP Scenarios with Spatially Explicit Statistical Downscaling. *Sustainability* **2019**, *11*, 2106.
https://doi.org/10.3390/su11072106

**AMA Style**

Murakami D, Yamagata Y. Estimation of Gridded Population and GDP Scenarios with Spatially Explicit Statistical Downscaling. *Sustainability*. 2019; 11(7):2106.
https://doi.org/10.3390/su11072106

**Chicago/Turabian Style**

Murakami, Daisuke, and Yoshiki Yamagata. 2019. "Estimation of Gridded Population and GDP Scenarios with Spatially Explicit Statistical Downscaling" *Sustainability* 11, no. 7: 2106.
https://doi.org/10.3390/su11072106