 Regular article
 Open Access
 Published:
Accurate intercensal estimates of energy access to track Sustainable Development Goal 7
EPJ Data Science volume 11, Article number: 60 (2022)
Abstract
Intercensal estimates of access to electricity and clean cooking fuels at policy planning microregions in a country are essential for understanding their evolution and tracking progress towards Sustainable Development Goals (SDG) 7. Surveys are prohibitively expensive to get such intercensal microestimates. Existing works, mainly, focus on electrification rates, make predictions at the coarse spatial granularity, and generalize poorly to intercensal periods. Limited works focus on estimating clean cooking fuel access, which is one of the crucial indicators for measuring progress towards SDG 7. We propose a novel spatiotemporal multitarget Bayesian regression model that provides accurate intercensal microestimates for household electrification and clean cooking fuel access by combining multiple types of earthobservation data, census, and surveys. Our model’s estimates are produced for Senegal for 2020 at policy planning microregions, and they explain 77% and 86% of variation in regional aggregates for electrification and clean fuels, respectively, when validated against the most recent survey. The diagnostic nature of our microestimates reveals a slow evolution and significant lack of clean cooking fuel access in both urban and rural areas in Senegal. It underscores the challenge of expanding energy access even in urban areas owing to their rapid population growth. Owing to the timeliness and accuracy of our microestimates, they can help plan interventions by local governments or track the attainment of SDGs when no groundtruth data are available.
1 Introduction
Access to energy directly translates into a multitude of factors affecting human development that includes education, health, gender equality, clear air and water [1]. Yet, globally 840 million people live without electricity and 3 billion people cook using traditional fuels [2]. Even before the COVID19 crisis, it was projected that around 620 million people would still lack access to electricity in 2030 with 85% percent of them in SubSaharan Africa and 2.3 billion people would still not have access to clean cooking fuel [3]. The COVID19 pandemic threatens the progress that has been made towards the United Nations Sustainable Development Goal (SDG) 7 on affordable and clean energy access [4]. Access to electrification and clean cooking fuels are the two main indicators instrumental for measuring progress towards SDG 7, and is the focus of this work. Accurate tracking of SDG 7 is dependent on frequent and detailed microregional data, with special focus on clean cooking fuel access [5, 6].
Since census and surveys are laborintensive, cost millions of dollars and involve a lag of multiple years to get updated results, researchers are studying earth observation (EO) data owing to its high revisit rate to understand various facets of energy accessibility, grid structure, supply and demand [7–11]. Most extant works on household energy access have studied the status of electrification and are limited to coarse spatial granularity of countries or for sparse villages at continent scale for a single timepoint [12–14]. These estimates are validated for timepoints coinciding with surveys, when training and validation data are readily available. There seems to be scant work in studying the temporal evolution of these estimates beyond the survey years, paradoxically when these are most needed, except a study by [7] which reports poor results for electrification access. Therefore, there is an imminent need for methods that can accurately measure, track and nowcast population wide energy access during intercensal periods in a costeffective manner. Nowcasting is defined as the process of getting intercensal estimates of energy access. Importantly, extant works on energy access focus on binary variables related to electrification. There seems to be scant works in understanding the cooking fuel accessibility at microregional scale, as current studies [15, 16] focus on global and countrywide access of clean cooking fuels.
We propose a novel spatiotemporal multitarget Bayesian regression framework that reliably nowcasts household energy access for both the lighting and cooking needs, at microregions using multiple types of publicly available EO datasets, namely nighttime lights, aerosol optical depth data and Landsat8 satellite imagery, and census and surveys. We focus on indicators critical for tracking SDG 7 – access to electrification and clean cooking fuel for a household. Our model learns the complex relationship between features derived from EO and energy access targets for the censal year and, also, leverages data from georeferenced surveys conducted in subsequent years, to provide reliable nowcasts for intercensal periods.
We observe a positive correlation between a household’s access to electricity for lighting and liquefied petroleum gas (LPG) for cooking, and a negative correlation between its electrification and use of lamp for lighting or wood for cooking (see Additional file 1 Figure 1b). We exploit these correlations among the energy access indicators by formulating our problem as a multitarget regression, where the goal is to simultaneously learn multiple targets given a single input observation [17, 18]. Learning multiple targets (outputs) is shown to be beneficial when the outputs are multivariate and when complex interdependencies exist among them. In these scenarios, multitarget regression is shown to provide better predictive performance, robustness to noise and missing data, and computational efficiency [19].
To facilitate insights into inequities, we model the delineation of energy access along the urbanrural divide in our Bayesian framework. As a measure of our model’s generalization in time, we validate our intercensal microestimates using the temporally closest DHS surveys.
Our model’s microestimates are produced at policy planning microregions, called communes in Senegal, for intercensal years, 2015, 2017 and 2020, and are validated using concurrent DHS data.^{Footnote 1} On average, our nowcasts can explain >77% and >71% of variation in regional aggregates for electricity and clean cooking fuel access, respectively. For 2020, we report a Pearson’s r correlation of 0.88 for electricity and 0.92 for clean cooking fuel access between our estimates and DHS data. Our results expose stark disparities in energy attainment for communes delineated along the urbanrural divide as well as juxtaposed against their population growth. Our model simultaneously quantifies the evolution of all household energy access indicators in Senegal, e.g., wood, coal and kerosene lamps, thus providing the policy makers with a complete spectrum of energy accessibility.
In summary, our contributions are as follows:

1
We propose a novel spatiotemporal multitarget Bayesian regression model that accurately estimates the entire spectrum of household energy access at microregions using multiple types of publicly available EO datasets in Senegal for the intercensal periods. Two important distinctions of our model compared to existing works are as follows: Understanding a multispectrum access for household energy (compared to a mostly binary notion of presence/absence of electricity) by proposing a multitarget regression model and second, the use of aerosol data for energy access has not been explored yet.

2
We validate the reliability of the microestimates of our model for several intercensal years and report highly accurate results at regional levels both for spatial crossvalidation and for intercensal years. For 2020, our model’s errors are consistently better than the existing bestperforming model’s estimates for electricity and clean cooking fuel access.

3
We demonstrate the significant disparities in energy access for urban and rural areas, and juxtapose them against the population growth and provide insights for policy makers into the evolution and the challenges in household energy access in Senegal. Such insights are possible because we built a specialized kernel that explicitly models the urbanrural delineation along with spatial and temporal effects.

4
We model the accessibility of cooking fuel, which is a critical indicator for SDG 7.1 using multiple disparate data sources, and there seems to be scant work in understanding its access. Lack of clean fuel for cooking disproportionally impacts women and children, their educational attainment and their indoor air pollution, and the problem is exacerbated for poorer and vulnerable communities.
A note on definition of energy accessibility
Most existing studies have focused on a binary definition of energy access, i.e., if a household (or village) has access to electricity or not, by measuring binary responses to questions like, “does the household have electricity connection?” or “cooking with nonsolid fuels?”. However, this approach fails to capture the full spectrum of lighting and cooking fuel access for a household and recent works calls to move beyond such monodimensionality [7]. Hence, we adopt a multidimensional view of energy access at microregional level as determined by the census of that country, thus providing policy makers with more nuanced information about diverse sources of energy employed by the population for lighting and cooking in their homes. For Senegal, the prominent modes of lighting are candle, electricity, lamp and for cooking are coal, gas, wood. Therefore, rather than a single value, the energy access for a microregion is defined to be a vector whose length corresponds to the number of prominent categories of lighting and cooking modes, and each entry contains the fraction of households (in that microregion) that use that particular category of lighting and cooking.
The rest of the paper is organized as follows: Sect. 2 describes existing works that deal with estimated energy access using multiple data sets. Section 3 describes details of the data used in this study and Sect. 4 details the model and the inference procedure. Section 5 describes the validation and results for the target country of Senegal. Section 6 provides further discussions, including the limitations of this study and future directions.
2 Related works
Extant works on household energy access are, mostly, limited to studying binary notion of electrification [12–14, 20] using predominantly nighttime light data and producing estimates at a given time point. Most of these works provide promising estimates at timepoints coinciding with surveys, but it is unclear how they will generalize to intercensal time periods. There are very few studies to determine if the inferential relationships learnt will be robust over time – a need that has been highlighted by recent surveys on using satellite imagery for sustainable development [11, 21]. A recent work [7] maps the spatial heterogeneity of national electricity access from 20142019 for the Africa, but yields poor temporal generalization. While there are longitudinal studies mapping the evolution of electrification over time, but these are retrospective in nature, rather than a nowcasting model [12, 22, 23].
The existing models mostly study the binary access to electrification as this metric is easily interpretable. However, owing to issues of reliability of connection and affordability, binary metrics may obfuscate the nuanced ways in which households have access to energy [7].
The existing works to understand energy access from EO data mostly employ generalized linear models as these models provide interpretability [7, 22, 24]. Some top performing machine learning methods for electrification prediction task are gradient boosting classifiers [25, 26], logistic regression [27], Gaussian Process (GP) classification [12, 27]. We compare our proposed model with each of these existing works.
Besides electrification, researchers have explored different satellite data products, like Landsat8, Sentinel data, population data, for predicting developmental indicators, such as roof types [26], drinking water [14], poverty mapping [28–30] etc. Recent works [12, 14] have employed such data sets for electricity infrastructure prediction and household energy electrification prediction (again a binary notion) at continent wide scale and provide promising results using deep learning based approaches, based on a convolutional neural network(CNN). These approaches are not directly applicable for our problem setting, as they require substantial amounts of training data and we deal with only a handful of microregions for a given country, instead of thousands of villages spread across the entire continent of Africa.
Owing to the better predictive power of CNNbased features extracted from satellite images over simpler features, we employ the stateofart deep learning model based on the ResNet18 architecture [31], as our choice for feature extraction from satellite imagery. While researchers have pointed to the tradeoff between performance and interpretability with deep learning models, by using them as feature extractors in our Bayesian model, we weave interpretability into our modeling framework and provide insights useful to policy planners.
Additionally, most of these works provide estimates at coarse spatial granularity e.g. for villages spread across the entire African continent [31]. However for SDG monitoring countries need such estimates at policyplanning level as explored in this work [1]. A critical challenge, here, remains the unavailability of disaggregated statistical data from census and surveys.
3 Data
In this section, we describe the EO data sets used in this study, starting with a description of the target country. We then outline the procedure of extracting the various covariates from the EO data sets and calculating the energy access targets from census and survey data sets.
3.1 Country details
The study is conducted for Senegal, a SubSaharan country which is categorized with low human development. According to the 2022 Tracking SDG7 report, Senegal has an electrification access rate of 71% and national access to clean cooking solutions at 31%. It ranks 170 out of 191 countries on the Human Development Index in 2021 [32]. Lack of electricity supply is one of the main constraints hindering Senegal’s socioeconomic development. The remote and rural areas lack access to modern energy services, face frequent power cuts that lower the quality of life of the poor and vulnerable communities and reduce business efficiency [33]. Regarding cooking fuel access, rural areas are highly dependent on wood, while urban populations mostly use coal and, less frequently, gas.
3.2 Data
The following data sets were used for this study. See Table 1 for details regarding data procurement, and the details for feature extraction are given below:

1
Census data: We use a 10% sample of the most recent census (called RGPHAE (Recensement General de la Population de l’Habitat de l’Agriculture et de l’Elevage)), provided by Agence Nationale de la Statistique et de la Demographie (ANSD), which is the National Statistics Office of Senegal. It was conducted in 2013 and was made available in 2015. The data is evenly sampled across the entire population of Senegal, with data from 1.4 million individuals, spread across 150,000 households. It represents the most spatially detailed and comprehensive coverage of national statistics and has information about household features including mode of lighting and type of cooking fuel.

2
Demographic and Household survey data (DHS): These surveys collect a multitude of information across varied topics of interest for a population sample that participates in the DHS program. These are based on sampling clusters, which collect information for individuals or household records. For privacy reasons, cluster locations are displaced up to 2 km for urban areas and up to 5 km for rural areas, about 1% of which can be displaced up to 10 km [34]. The cluster locations for DHS corresponding to 2015, 2017 and 2019 are shown in Additional file 1 Fig. 1.

3
Nighttime lights (NTL) capture the radiance associated with lights at night and is often used in studying electrification access at various spatial heterogeneity [10, 35–37]. We use an integrated publicly available NTL dataset across the years [38].

4
Aerosol Optical Depth (AOD) is extracted from the Moderate Resolution Imaging Spectroradiometer (MODIS) on NASA’s Terra and Aqua satellites, and captures the aerosol content over a spatial location. A median composite of the annual AOD data is taken to mitigate the effect of seasonal dust storms.

5
Landsat8 satellite data: This data has been shown to predict infrastructural qualities, especially those related to electrification [14] in Africa, when compared to nightlights and Sentinel 1 satellite data. To extract features from this data, we employed a pretrained deep neural network, based on ResNet18 architecture and adapted for multispectral satellite imagery. This model has been shown to outperform other models in extracting features to predict asset wealth (that includes household indicators including electrification and possession of assets like television, phone, etc.) [31]. We use the intermediate activations from the penultimate layer in the deep neural network as features that, likely, correspond to information related to urban infrastructure, agricultural land and other land forms (like desert and water bodies). We use 1year median composite images for Senegal. The input to the deep neural network is the 7 band image of size \((224 \times 224)\) and the output is a 512 length vector corresponding to the activations from the penultimate layer. The composite Landsat8 image, in which each pixel corresponds to a 30 sq. m. area on ground, is divided into “tiles” of size \((224 \times 224)\), each corresponding to an area of 6.72 sq. km. Each tile is fed into the deep neural network and transformed into a 512 length feature vector.
3.3 Creating microregional covariates from EO data
The raw covariates for each of NTL, AOD and L8 data sets listed in Table 1 are aggregated to microregions, using a population weighted aggregation scheme to capture the perhousehold behavior in that microregion, which is empirically shown to provide better performance in estimating household energy access compared to extant works. Our scheme is outlined here. Note that for each EO data set, the geographical area corresponding to each pixel is different.
The covariates from EO data are extracted at the granularity of pixels, while our analysis is performed at policy planning microregions. Spatially, a microregion is composed of a number of pixels. While some pixels lie entirely within the spatial extents of a microregion, others may fall at its boundary with neighboring microregions. We follow a specific aggregation scheme to get the EO covariates for a microregion, outlined below.
For each EO covariate (f), we calculate the weighted mean \(\mu _{fc}\) and weighted variance \(\sigma ^{2}_{fc}\) for a given microregion, c, as follows:
where \(f_{i}\) corresponds to the covariates for a pixel (indexed by i). \(\bar{p}_{ic}\) is an areaadjusted population of the pixel, calculated as \(\bar{p}_{ic} = p_{i}\frac{a_{ic}}{a_{i}}\), where \(p_{i}\) is the population for the pixel, \(a_{i}\) is the geographical area of the pixel and \(a_{ic}\) is the geographical area of the pixel contained within the microregion c. Note that \(\bar{p}_{ic}\) is 0 for pixels that do not have any overlap with microregion c. The population count, \(p_{i}\), is obtained by resampling the gridded population data to the appropriate spatial resolution for the feature f.
While the numerator in (1) weights the EO covariates for pixels by underlying population, dividing it by the total population of the microregion, ensures that the features capture the perhousehold behavior for that microregion. We also estimate the variance corresponding to each aggregated EO covariates for a microregion as given in (2). It captures the noise that is attributed when EO covariates are aggregated to microregions.
Finally, there are 2 covariates corresponding to mean and variance of NTL and 2 for AOD. For the Landsat8 high dimensional feature vector, we use Principal Component Analysis (PCA) to reduce them to 20 covariates by mapping the data to the top 20 principal components that retains 95% of the data variance. Dimensionality reduction is often done for computational efficiency and to prevent overfitting in small datasets. The corresponding variances associated with each of these features is mapped in the same manner, giving 40 covariates (mean and variances) from L8 imagery.
3.4 Creating energy access targets from census and DHS data
Energy access targets from census
In Senegal’s 2013 census, the major categories for lighting, in order of popularity, are electricity, rechargeable lamp, candles and others; while those for cooking fuel are wood, coal, gas and others. Each household identifies as using a specific category of lighting and cooking fuel. We create an 8 length accessibility vector for each microregion corresponding to 4 categories each of lighting and cooking fuel. The exact mapping of each census response to this vector is detailed in Additional file 1 Section “Mapping of census responses”. Each entry in the vector contains the fraction of households using that particular category of lighting/cooking fuel within the microregion. The household responses are weighted by their sampling coefficients provided in the census to make them representative of the population.
Energy targets from DHS
DHS data occur for select clusters throughout the country, whose locations change for every new survey. For each DHS survey, the geocoded clusters are assigned to their spatially nearest microregion and a 8 length accessibility vector is created by consolidating the household responses related to lighting and cooking fuel access for all clusters that fall within that microregion, using the similar approach described for census above. We weight these responses using the provided sampling weights to account for the selection biases.
4 Model description
This section describes the proposed Bayesian model, and details on model training and inference. Since Gaussian Processes (GP) form the basis of our model, a brief background is provided.
4.1 Model intuition
We propose a semiparametric model given as: \({\textbf{y}} = {\mathbf{B}}{\textbf{x}} + f(\textbf{x},{\mathbf{s}},u, t) + \epsilon \). The first term models the linear relationship between EO covariates (x) and the targets (y), where B is the coefficient matrix for the linear model. The multiple targets of regression correspond to household energy access indicators (e.g., electricity, gas etc.) The second term employs a nonlinear functional mapping based on GP between an augmented covariate vector and y. The augmented covariate vector includes x, the spatiotemporal coordinates (s, t), and an urbanrural indicator (u).
GPs belong to the class of Bayesian models, where the choice of kernel functions enables one to learn highly nonlinear relationships between the covariates and target variables [40]. GPs can be made more flexible and interpretable by combining (adding or multiplying or convolving) different kernels, where each kernel models a certain effect within individual covariates.
We propose a specialized kernel for our GP model, with the following form:
The first kernel in (3) models three types of effects in an additive form: a EO covariate effect \({\mathbf{K}}_{c}\), a spatial autocorrelation effect with urbanrural delineation \({\mathbf{K}}_{sp} * {{\mathbf{K}}}_{ur}\) which assigns more weight to spatially proximal and similar microregions (i.e. in EO data, an urban location might derive some similarity from nearby rural locations and also from nearby other urban locations), and a temporal recency effect which assigns more weight to recent observations \({\mathbf{K}}_{t}\). The second kernel \({\mathbf{K}}_{\ell}\) provides the multitarget formalism by exploiting correlations across different targets.
The rationale for using such specialized kernel is that additive kernels are known to extrapolate well to unseen test data [41, 42], and we empirically demonstrate better performance of our model compared to existing works.
Model training involves estimating the optimal values for the coefficient matrix, B, and the hyperparameters associated with the kernel \({\mathbf{K}}_{mo}\) in (3), and is done by maximizing the marginalized loglikelihood of the training data. Elasticnet regularization is employed on the linear model to prevent learning from spurious features and to avoid overfitting on limited training data [43]. We perform out of sample spatial and temporal validation to test our model’s generalizability.
4.2 Model details
Notation
For a given microregion, indexed by c, the covariate vector, target vector, spatial coordinates, and the urbanrural indicator, are denoted by \({\mathbf{x}}_{t}^{(c)}\), \({\mathbf{y}}_{t}^{(c)}\), \({\mathbf{s}}^{(c)}\), and \(u^{(c)}\), respectively, and are collectively denoted as \({\mathbf{z}}^{(c)}_{t}\). Note that the covariate vectors and target vectors are also indexed by time t, denoting the corresponding years. Each individual target will be denoted by \(y^{(c)}_{to}\). For notational simplicity, we will drop the superscript c to denote a typical microregion, unless needed. In general, we will use a lowercase bold symbol to denote a vector, uppercase bold symbol to denote a matrix, and a lowercase normal symbol to denote a scalar value. Collections (or sets) of entities will denoted using calligraphic symbols, e.g., \(\mathcal{X}\), \(\mathcal{Y}\). The oth entry of a vector, e.g., x, will be denoted as \(x_{o}\).
4.2.1 Model description
The proposed semiparametric model is written as:
where B is the coefficient matrix for the linear component and ϵ denotes the unexplained noise and is modeled as a zeromean Gaussian random variable, i.e., \(\epsilon \sim N(0,\sigma ^{2}_{n})\). The function \(f()\) captures the nonlinear dependencies between the covariates and the residual vector, \(\boldsymbol{\delta}_{t}\), where \(\boldsymbol{\delta}_{t} = ({\mathbf{y}}_{t}  {\mathbf{B}}{\mathbf{x}}_{t})\), and is modeled using a Gaussian Process.
Background on Gaussian Processes (GP)
GP is a Bayesian formulation to learn nonparametric, nonlinear functions, through the use of kernels. A GP allows placing a stochastic prior on the function \(f({\mathbf{z}}_{t})\), where \({\mathbf{z}}_{t} \equiv ({\mathbf{x}}_{t},{\mathbf{s}},u, t)\). The GP prior is completely specified by a mean function, \(m(\cdot )\), and a positivedefinite kernel function \(k(\cdot ,\cdot )\). The mean function represents the expected value of \(f()\), i.e., \(m({\mathbf{z}}_{t}) = \mathbb{E}[f({\mathbf{z}})]\), and is often set to 0, i.e., \(m({\mathbf{z}}_{t}) = 0\). The kernel function defines the covariance between any two realizations of \(f()\), i.e.,
assuming a zero mean function.
The definition of GP specifies that for any finite collection of inputs, \(\mathcal{Z} = ({\mathbf{z}}^{c_{1}}_{t_{1}},{\mathbf{z}}^{c_{2}}_{t_{2}}, \ldots ,{\mathbf{z}}^{c_{n}}_{t_{n}})\) the vector of function values, \({\mathbf{f}}(\mathcal{Z}) = (f({\mathbf{z}}^{c_{1}}_{t_{1}}),f({\mathbf{z}}^{c_{2}}_{t_{2}}), \ldots , f({\mathbf{z}}^{c_{n}}_{t_{n}}))\), follow a multivariate Gaussian distribution, i.e.,
where \({\mathbf{K}}_{\mathcal{Z},\mathcal{Z}}\) is a \((n \times n)\) covariance matrix, such that the ijth entry is equal to \(k({\mathbf{z}}^{c_{i}}_{t_{i}},{\mathbf{z}}^{c_{j}}_{t_{j}})\).
For a single output, indexed by o, a GP regression model (GPR) can be defined by assuming that the targets are modeled as:
where I is the \((n \times n)\) identity matrix. Using (6) and (7), one can marginalize out \({\mathbf{f}}(\mathcal{Z})\), such that:
4.2.2 Choice of kernel function
Our kernel function is formulated as follows:
where \(k_{f}\), \(k_{sp}\), \(k_{ur}\) and \(k_{t}\) denote the kernels that capture the similarity in covariates, spatial autocorrelation, urbanrural delineation and temporal recency. We use squared exponential kernel function for \(k_{f}\), \(k_{sp}\) and \(k_{t}\), which is the most widely used kernel function because of its ability to learn smooth nonlinear functional relationships [40]. The individual kernel specifications are given as follows:
The urbanrural delineation is modeled by \(k_{ur}\), which is specified as the following categorical kernel:,
The scalars \(\sigma _{f}\), \(\ell _{f}\), \(\sigma _{sp}\), \(\ell _{sp}\), \(\sigma _{t}\), \(\ell _{t}\) are the hyperparameters of the kernel functions and are estimated from the data, as described later.
Feature selection
To perform feature selection on EO data, we employ Automatic Relevance Determination kernel (ARD) on our model. ARD kernels are effective in selecting a smaller explanatory subset of features from a large set of irrelevant features by regularizing the solution space using a datadependent prior [44]. Note that the feature kernel in (10) uses a single global characteristic length scale (\(\ell _{f}\)). However, for ARD each feature has a different characteristic length scale, denoted by \(\ell _{fr}\) for the rth feature. The feature kernel for ARD is given as:
The inverse of the length scales of each feature, i.e., \(\frac{1}{\ell _{fr}}\), is used a proxy for feature relevance [40].
4.2.3 Handling multiple targets
The GP regression model described above can only handle a single target. Since the problem studied in this paper involves multiple targets, we present the following scheme, adopted from [45], to exploit the correlations among the targets in the regression model. In this formulation, each instance consisting of a covariate vector and m length target vector is converted into m instances with a scalar target value. We introduce an additional discrete covariate, ℓ, which corresponds to the index of the target. For example, a covariate and a m length target vector pair given as \(\langle{\mathbf{z}}^{(c)}_{t}; \boldsymbol{\delta}^{(c)}_{t}\rangle \) is transformed into m pairs as follows:
Note that the target is transformed into a scalar. We denote the augmented covariate vector as \(\bar{\mathbf{z}}^{(c)}_{to} \equiv ({\mathbf{z}}^{(c)}_{t}, o)\). The extra covariate is handled by multiplying the kernel function, \(k()\), in (9) with a targetspecific kernel function, \(k_{\ell}()\), to obtain the final kernel function:
Note that the resulting covariance matrix for an augmented singletarget data set can be expressed as:
where ⊗ denotes the Kronecker product between the \((n \times n)\) covariance matrix, \({\mathbf{K}}_{\mathcal{Z},\mathcal{Z}}\) and the \((m \times m)\) matrix \({\mathbf{K}}_{\ell}\), such that \(k_{\ell}(o_{i},o_{j}) = {\mathbf{K}}_{\ell}[o_{i},o_{j}]\). For GP, \({\mathbf{K}}_{\bar{\mathcal{Z}},\bar{\mathcal{Z}}}\) needs to be a positivedefinite, which means that \({\mathbf{K}}_{\ell}\) should also be positivedefinite.
The \(m^{2}\) entries in \({\mathbf{K}}_{\ell}\) can be thought of as the hyperparameters of the kernel function in (16) and can be learnt from the training data. However, instead of treating each entry as a hyperparameter, we consider a parameterization of \({\mathbf{K}}_{\ell}\) using fewer hyperparameters. In particular, we consider a spherical parameterization [46] of \({\mathbf{K}}_{\ell}\), given as follows:
where S is an upper triangular matrix of size \((m \times m)\), whose oth column contains the spherical coordinates in \(\mathbb{R}^{o}\) of a point on the hypersphere, \(\mathbb{R}^{(o  1)}\), followed by \((mo)\) zeros. For example, for \(m = 4\):
Here, \(\phi ^{(1)}, \phi ^{(1)}, \ldots \) are the hyperparameters that parameterize the matrix S. For m targets, one would require \(\frac{m(m  1)}{2}\) hyperparameters to specify S. The spherical parameterization has three advantages. First, it allows us to parameterize a \((m \times m)\) matrix using only \(\frac{m(m  1)}{2}\) hyperparameters. Second, it ensures that the resulting matrix \({\mathbf{K}}_{\ell}\) is positivedefinite. And finally, the offdiagonal entries of \({\mathbf{K}}_{\ell}\) encode the correlation among the targets and can be interpreted as such after training the model.
4.2.4 Model training
The parameters of the proposed model consist of the coefficient matrix for the linear model, B, the variance term for the observational likelihood in (7), \(\sigma _{n}\), the kernel hyperparameters, \(\ell _{f}\), \(\sigma _{f}\), \(\ell _{sp}\), \(\sigma _{sp}\), \(\ell _{t}\), \(\sigma _{t}\) (see (10), (11), (12)), and the spherical coordinates in the uppertriangular entries of S.
We assume that the training data consists of n instances, \(\mathcal{Z} = ({\mathbf{z}}^{(c_{1})}_{t_{1}},{\mathbf{z}}^{(c_{2})}_{t_{2}}, \ldots ,{\mathbf{z}}^{(c_{n})}_{t_{n}})\), where each \({\mathbf{z}}^{(c_{i})}_{t_{i}} \equiv ({\mathbf{x}}^{(c_{i})}_{t_{i}},{\mathbf{s}}^{(c_{i})},u^{(c_{i})},t_{i})\), and the corresponding targets \(\mathcal{Y} = ({\mathbf{y}}^{(c_{1})}_{t_{1}},{\mathbf{y}}^{(c_{2})}_{t_{2}}, \ldots ,{\mathbf{y}}^{(c_{n})}_{t_{n}})\). The linear coefficient matrix B is first estimated using a regularized least squares estimation procedure, with the loss function defined as:
where \(\Vert \cdot \Vert ^{2}_{F}\) and \(\vert \cdot \vert \) denote the square of the Frobenius norm and the \(l_{1}\) norm of a matrix, respectively. X is the covariate matrix consisting of the covariate vectors, i.e., \({\mathbf{X}} = ({\mathbf{x}}^{(c_{1})}_{t_{1}}, {\mathbf{x}}^{(c_{2})}_{t_{2}}, \ldots , {\mathbf{x}}^{(c_{n})}_{t_{n}})^{\top}\), and Y is the target matrix consisting of the target vectors, i.e., \({\mathbf{Y}} = ({\mathbf{y}}^{(c_{1})}_{t_{1}}, {\mathbf{y}}^{(c_{2})}_{t_{2}}, \ldots , {\mathbf{y}}^{(c_{n})}_{t_{n}})^{\top}\). While the first term in (20) is the standard least squares loss, the second and third terms act as an elasticnet regularizer on the coefficients, which is employed to reduce the impact of spurious features and to avoid overfitting [47], where a model performs well for insample data, but does poorly for outofsample points. The scalars α and λ are known as the regularization parameters and are tuned using crossvalidation on the training data. In this study, the tuned values for α and λ are 0.1 and 0.5, respectively. The optimization of the loss function in (20) is done using a coordinate descent algorithm.
After estimating the optimal coefficients in B, the hyperparameters associated with the GP are estimated by maximizing the marginal loglikelihood of the residuals, using the marginal likelihood in (8). For each training instance, the residual vector is defined as \(\boldsymbol{\delta}^{(c_{i})}_{t_{i}} = {\mathbf{y}}^{(c_{i})}_{t_{i}}  { \mathbf{B}}^{\top}{\mathbf{x}}^{(c_{i})}_{t_{i}}\). Let \(\bar{\mathcal{Z}}\) denote the training data set in which every training instance is augmented according to (15). Let \(\bar{\boldsymbol{\delta}}\) be the vector containing all the scalar targets. Given that the marginalized conditional probability distribution, \((\bar{\boldsymbol{\delta}}\vert \bar{\mathcal{Z}})\) is a multivariate Gaussian with zero mean and covariance as \(({\mathbf{K}}_{\bar{\mathcal{Z}},\bar{\mathcal{Z}}} + \sigma _{n}^{2}I)\) (see (8)), the marginalized loglikelihood can be expressed as:
The marginalized loglikelihood is maximized with respect to the kernel hyperparameters and \(\sigma _{n}\), using stochastic gradient descent [48].
4.2.5 Model inference
To infer any target for a microregion at a new time instance, we use the GP formulation to estimate the posterior distribution for the target, conditioned on the training data set, \((\mathcal{Z}, \mathcal{Y})\). Let the covariates for the test instance be denoted as \({\mathbf{z}}_{*} = ({\mathbf{x}}_{*},{\mathbf{s}}_{*},u_{*},t_{*})\). For the oth target, the posterior distribution of \(y_{*o}\) is a Gaussian distribution, whose mean, \(\bar{y}_{*o}\), and variance, \(\operatorname{var}[y_{*o}]\) are given by the following expressions [40]:
where \({\mathbf{b}}_{o}\) corresponds to the oth column of the coefficient matrix, B. The vector \({\mathbf{k}}_{*}\) contains the kernel function evaluation between every augmented training instance and the test instance, and the scalar \(k_{**}\) is the kernel function evaluation for the test instance with itself.
5 Results
We describe two sets of experimental results: first, validation results for spatial and temporal generalizability, and second, insights provided by our model. We also provide through comparison of our model’s performance with the class models, namely linear, Gaussian Process Regression (GPR) and Gradient Boosted Regression (GBR). While none of the previous works have used the all the datasets as described here, for the comparison, here, we use our feature set and their models. Regarding the insights, we provide three details: energy access estimates for 2020 for the entire country; energy access delineated by urbanrural divide and juxtaposed against the population growth
5.1 Validation results
Spatial crossvalidation
During each run of spatial crossvalidation, the training and test sets are sampled from geographically distinct regions to mitigate the effect of spatial auto correlation and this procedure is shown to produce robust results [49]. The specific strategy for Senegal used in this study is described in [28], and ensures that during the multiple runs of CV, all microregions are represented in training and test samples.
Table 2(a) depicts the results of spatial crossvalidation procedure performed for censal year (2013) and emphasizes the efficacy of our model in predicting energy access at microregions with highly significant correlations and low errors when compared to competing methods. Spearman’s correlation of > 0.6 indicates that rank correlations are preserved, which is important as the correct ordering of microregions is, at times, sufficient to identify the most deprived ones. The values of Pearson’s r correlation are much higher than rank correlation indicating the linear correspondence of the targets and model estimates. Our model predicts electricity access better than gas access. However we notice low RMSE errors in gas access for rural microregions than urban ones. Detailed results for all energy indicators are given in Additional file 1 Table 2.
Temporal validation
We test the validity of our nowcasts by using the concurrent DHS survey. For countrywide spatial coverage, this validation is performed at regional level. The geocoded clusters from DHS are assigned to their respective regions (this mapping is already provided in DHS data). Our nowcasts are also aggregated to region level for comparison and rsquares are reported. To nowcast for 2020, our model is trained on EO data and targets for censal year (2013), as well as EO data for the years when subsequent DHS surveys are available, which are 2015 and 2017 for Senegal. Figure 1 shows that our model can explain 77% and 86% of the variation in the regional aggregates for electric and gas access, respectively.
Table 2(b) shows high Pearson’s r and rank correlations for both electric and gas access for temporal validation. The errors are also lower for rural areas, than urban ones. We also experiment for intercensal years, 2015 and 2017, whose estimates are validated with the DHS derived indices concurrent to those years. The details of the experimental setup and scatter plots in each case are given in Additional file 1 Table 1 and Additional file 1 Fig. 2 respectively. For both these years, we report a rsquared of >0.78 and >0.71 for electricity and gas access, respectively. These results state the accuracy of nowcasting abilities of our model for datascarce situations. Our study also provides accurate nowcasts for other prominent modes of lighting and cooking, namely wood, coal and lamp, see Additional file 1 Table 3, which could help policy makers to target appropriate interventions.
Focusing on our model’s errors for 2020 for electrification access, we see that our model marginally underpredicts the electricity access for most regions irrespective of these regions being urban/rural or with high/low electricity access. The most underpredicted regions for electricity access are Kolda and Kaffrine. Doing a similar error analysis for clean cooking fuel access, we again note that our model marginally underpredicts for most regions irrespective of their urban/rural status. The urban microregions of Saint Louis, Kaolack, Thies, Louga, Fatick and rural areas of Dakar are the most underpredicted, while the urban areas of Dakar are slightly overpredicted. We would like to note that these results are data dependent, with various factors affecting the model performance with prominent ones being the noise in the EO data that is input to our model, and the quality of the surveys (temporal and spatial coverage).
5.2 Insights into model’s intercensal estimates
Estimates of electricity and clean cooking fuel access for microregions in Senegal in 2020
Our model’s estimates for 2020 are depicted in Fig. 2.^{Footnote 2} In 2013, about 57% of households were electrified, which were mostly concentrated in the capital region of Dakar and the nearby urban area of Thies. Compared to 7% in 2013, about 11% of all rural microregions have more than half of their households electrified in 2020. The number of electrified households in urban microregions has remained the same (which amounts to about more than 85% of those microregions), even while accounting for rapid population growth in these areas.
The change in electrification between these years is depicted in Fig. 2(e). While, several rural areas in Kedougou and Sedhiou report positive change, it seems that electrification in some urban areas in Dakar has not kept up in 2020. We attribute it, mainly, to rapid growth of urban population in recent years, causing the electrification rate to lag or stay stagnant (further results are detailed below).
Focusing on gas access in Fig. 2(c), (d), (f), we notice that it was concentrated only in the urban regions of Dakar in 2013. Nationally, 67% of the households had no access to clean cooking fuels in 2013. The 2020 nowcasts show that gas access is slowly spreading to other urban centers in the country. Focusing on the urban regions, we find that about 47% of these areas have more than a quarter of their households with gas access. However, the picture looks dismal in rural microregions. Even in 2020 most of them have more than 75% of their households without access to clean cooking fuel. Figure 2(f) depicts this change and highlights the disparity between urban and rural areas, which is described in detail below.
Highlighting the urbanrural disparity in energy access
Disaggregated energy access along urbanrural divide from 20132020 is depicted in Fig. 3, where wide heterogeneity becomes evident not only between urbanrural microregions, but also within each of the urban (or rural) categories. Urban areas usually exhibit much wider energy inequities, with some of them having lower household electrification than select rural ones. However, on average, urban areas have markedly higher access to electricity than rural areas through the years. Our model reveals stark disparities in energy disparities even in 2020.
The spread of gas accessibility has a wider disparity among urban areas in 2013, with very few areas (mostly in Dakar) boasting high access to gas, while rural areas had hardly any access. It corroborates with national numbers which allocate very few households with the income to purchase clean stoves to burn gas and the recurring purchase of gas cylinders, as well as the lack of distribution outlets in farflung rural areas [50, 51]. Our analysis reveals a very marginal increase in the gas access in rural areas in 2020.
Analyzing the dynamics of energy access and population growth
Most urban and rural areas report a positive change in electrification at regional level, see Fig. 4(c), despite their population growth, which puts Senegal in an optimistic growth curve. Contrasting the regional plots with microregional ones elucidates the point that several heterogenieties are lost when data is aggregated to subnational levels. For microregions, we notice that urban areas have a broader horizontal spread, in both electricity and gas access, highlighting the existence of disparities within these areas, even with similar percentage population growth.
Figure 4(b) depicts that urban microregions show a negative percentage point change in gas access, highlighting that it has not kept up with population growth in urban areas. Most rural areas show a positive percentage point change in energy access highlighting that gas access is beginning to pick up in these areas even with the increased population growth. Note that most of the rural areas had no access to gas as cooking fuel in 2013, and thus they exhibit marked percentage point change in Figs. 4(c) and (d).
6 Discussion, limitations and conclusions
The objective of our EOdata based modeling approach is to provide microestimates when surveys are unavailable, e.g. during intercensal periods or in regions of conflict or those recovering from natural disasters or political upheaval and, thus, to augment the existing surveying efforts on the ground.
The basic premise of using heterogeneous satellite data is the assumption that they can capture the heterogenieties in energy access on the ground, possibly via nighttime luminosity data or urbanbuildup. To identify which input features are most useful, we perform feature selection using ARD kernel, and the top features deemed important are, indeed, nighttime lights, selected Landsat8 features and aerosol data, as shown in Fig. 5.
Visualizations of the features extracted using Landsat8 imagery point to semantically meaningful ones, likely capturing urban areas, sparse rural settlements, agricultural and presence of water, shown in Fig. 6. Though these selected features are specific to the EO data and country analyzed here, they conform to the broader consensus of existing research with nightlights as the most important feature [10, 35–37].
Our model leverages EO and DHS data for all past years when available. This involves allocating yearly DHS clusters to nearest microregions. Our allocation mechanism is robust to the inherent noise in spatial locations of DHS clusters, whose geocoordinates are moved to protect privacy (clusters in urban areas are moved by up to 2 km and those in rural by 10 km). Contrasting, most existing works rely on extracting satellite imagery (which is usually at 30 sq. m resolution) around the DHS clusters and are, thus, susceptible to learning from noisy or misaligned data [7, 14, 31].
Validation at intercensal microestimates and nowcasts remain a challenge, given the lack of finegrained ground truth data. We mitigated it by providing validation of our nowcasts at regional level using temporally closest DHS data. The next census of Senegal (likely scheduled for 2023) or more local data will likely provide a good validation point.
Regarding the generalizability of our model to other countries, we do believe that our methodology can be replicated to our countries, given the availability of their EO data and targets for training purposes (so that the model will learn countryspecific relationships). We understand that significant gaps in both temporal and spatial coverage of surveys do exist for many countries, however our methodology is not dependent on availability of surveys with uniform temporal regularity and spatial coverage. Our model can be trained with the existing survey data and EO data for a country to nowcast. The kernel function, that lies at the core of our modeling approach, is designed to appropriately weight the temporal and spatial information in the surveys (i.e., more weight to more recent survey data).
Another research avenue that is worth exploring regarding generalizability is how well does our model that is trained on one country, perform for another country, especially neighboring countries.
Limitations
There are limitations to employing nighttime light data to accurately measure aspects of human development, including access to electrification, which was predicted less successfully for some countries than others, as shown in [52]. Researchers have highlighted the limitations of existing models that learn solely from nighttime imagery, particularly their tendency to generally underperform in differentiating deprived (or poor) from the critically deprived (or ultrapoor) regions, as shown in the context of SubSaharan Africa [53]. Researchers have also demonstrated the susceptibility of such models to inherent noise in the data [54].
While our model leverages additional input data besides nighttime lights, more concerted research efforts are needed to comprehensively understand its performance and generalizability. Satellite imagery, especially at the resolution analyzed in this work, might not be able to distinguish between subtle nuances of urban and periurban areas (e.g. the presence of slums or unauthorized settlements), as highlighted by [7] and, thus, would be weak in distinguishing energy access in such microregions.
Recent works also highlight important concerns related to the presence of bias when human developmental indicators, notably poverty and electrification are predicted using nighttime lights [55–57]. We are currently working to understand the fairness aspects of our model, so that our microestimates can be trusted and used by policymakers.
Future directions
While this paper focuses on Senegal, the proposed framework can also be developed for other countries, by training the model using the EO data and energy targets for that country, so that it can learn countryspecific mappings and produce the desired microestimates. With many geolocated household surveys being conducted regularly and cheap availability of EO data, our framework has the potential to provide a cheap and good approximation, and continuous monitoring for intercensal statistics at the microregional level and can supplement the surveying tasks for better tracking of SDG 7.
Availability of data and materials
The census data can be obtained by contacting Dr. Emmanuel Letouze (eletouze@datapopalliance.org). The DHS data used, in this work, can be obtained by registering at https://dhsprogram.com/. The links to downloading EO data are given in Methods. The associated code to generate the results in this manuscript is available for review at https://github.com/neetip/energy_access.
Notes
Last census for Senegal was done in 2013
Here, clean cooking fuel refers to use of liquefied petroleum gas or electricity used for cooking at household level, and is, henceforth, referred as gas in our paper.
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Acknowledgements
We gratefully acknowledge the financial support of the Irving Institute for Energy and Society’s seed grant program. We thank Dr. Stephen Doig for valuable guidance while conducting this study.
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NP and SV conceived the study. NP analyzed the data, performed experiments and interpreted results. EL procured the census data for Senegal. NP, EL and SV wrote the manuscript. All authors read and approved the final manuscript.
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Supplementary information. Provides additional information about extracting targets from census data and additional results (PDF 1.6 MB)
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Pokhriyal, N., Letouzé, E. & Vosoughi, S. Accurate intercensal estimates of energy access to track Sustainable Development Goal 7. EPJ Data Sci. 11, 60 (2022). https://doi.org/10.1140/epjds/s13688022003715
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DOI: https://doi.org/10.1140/epjds/s13688022003715
Keywords
 Clean energy access
 Gaussian processes
 Earthobservation data
 Sustainable Development Goals