Neural Network for Sky Darkness Level Prediction in Rural Areas

Abstract

A neural network was developed using the Multilayer Perceptron (MLP) model to predict the darkness value of the night sky in rural areas. For data collection, a photometer was placed in three different rural locations in the province of Cáceres, Spain, recording darkness values over a period of 23 months. The recorded data were processed, debugged, and used as a training set (75%) and validation set (25%) in the development of an MLP capable of predicting the darkness level for a given date. The network had a single hidden layer of 10 neurons and hyperbolic activation function, obtaining a coefficient of determination (R²) of 0.85 and a mean absolute percentage error (MAPE) of 6.8%. The developed model could be employed in unpopulated rural areas for the promotion of sustainable astronomical tourism.

1. Introduction

During the industrial revolution, there was a significant exodus of population from rural regions to urban areas, where the main centres of production were concentrated. Today, due to the technological and digital revolution, a similar migration is taking place, resulting in the depopulation of rural zones [1]. This causes a series of disadvantages, as these areas concentrate an important part of agricultural and livestock production, as well as being historically and culturally relevant. To revitalise these areas, various government initiatives are being implemented [2], including the promotion of rural tourism. According to several authors [3,4,5], sustainable tourism development has a positive impact on both local residents and visitors, increasing their satisfaction level.

A promising alternative for this sustainable revitalisation is the promotion of astronomical tourism or “astro-tourism” [6,7,8,9], as rural zones tend to have low levels of light pollution, making them ideal for stargazing. There are several case studies that link this type of tourism to sustainable tourism development [10,11,12,13]. Investment in infrastructure and services is of key importance to allow the arrival of astro-tourists throughout the year, especially the digitisation of these areas, which also promotes new employment opportunities [14]. Tobari et al. [15] examined the influence of various attributes of smart tourism technologies (STTs) in emerging rural tourism destinations, concluding that tourists with higher STT skills and knowledge had a higher intention to revisit the destination in the future. Therefore, it can be concluded that the development of this type of technology is of great importance for tourism development. To clearly appreciate celestial phenomena, adequate night sky brightness (NSB) conditions are required [16], so having a predictive mathematical model to determine this darkness level can be very useful to establish the observing conditions, facilitating the arrival of astro-tourists. For this reason, a neural network has been developed based on NSB data recorded in rural areas. This tool has been developed with the objective of being able to predict the NSB level in advance. This could be useful for astro-tourists, allowing them to consult the predicted NSB level in advance to avoid dates when observation conditions are inadequate, saving time and money on travel. This tool could be used to enhance the value of starry sky resources in rural areas, encouraging the arrival of astro-tourists, which could help to revitalise these areas.

Conceptual Framework

One of the first NSB models was developed by Garstang et al. [17], allowing the calculation of the night sky’s darkness in the centre and surroundings of a city, considering the brightness generated by the city as proportional to its population. These authors also developed another model to predict the variation in NSB associated with seasonal changes [18]. This work is considered one of the most prominent studies in this field [19], being successfully verified by multiple comparisons between theoretical and empirical data [20,21,22,23] and used by other authors such as Cinzano et al. [24,25]. Subsequently, another series of more complex models emerged. The one developed by Kyba et al. [26], where the factors of altitude and cloud reflectance were introduced, allowed the simulation to be carried out under overcast conditions. It should be noted that there are several factors that can influence NSB, such as weather conditions [27,28], the presence of the Moon [29,30] or the level of light pollution [31,32]. Small changes in these variables can considerably modify NSB, so statistical predictive models tend to fit reality better than deterministic ones [33,34]. This is where we find a major knowledge gap, as the existing literature presents a deterministic approach to calculating the NSB level. There is only one article in which a probabilistic approach to NSB prediction is performed, specifically the work by C-Sánchez et al. “Astrotourism and Night Sky Brightness Forecast: First Probabilistic Model Approach” [34], where a Neural Network Autoregression model (NNAR) was developed. In this work, measurements taken at a single measurement point located at the National Astronomical Observatory of Japan, located in Tokyo, were used. The authors employed the sliding window method (using different time ranges in this window, specifically, 5, 10 and 15 days) to train an NNAR. It should be noted that, as the data collection site is located in an urban area, there will presumably be a high amount of light pollution due to lighting, which can exert a high influence on the darkness level of the night sky and therefore on the NSB level recorded. This is where the necessity of developing a predictive NSB model oriented towards rural areas arises, since it is precisely in these locations where higher values [35] of nighttime darkness are reached (due to lower urban illumination), and, therefore, a greater ability to perceive the starry night sky is achieved. As a reference, values of around 22 mag/arcsec² are usually achieved in rural areas with low levels of light pollution, and 16 mag/arcsec² in urban areas [34]. These higher NSB levels are closely linked to an increased capacity to observe the stars in the night sky. This represents an important novelty, as there is no predictive model available in the scientific literature specifically designed for these rural areas with low light pollution levels.

In this study, a neural network was developed using the Multilayer Perceptron (MLP) model for the NSB prediction in rural areas, presenting multiple data recording locations. It should be noted that NSB prediction is a non-linear time series prediction problem. Neural networks with MLP structure have been shown [36,37,38,39] to work much more precisely than other more complex models in this type of non-linear time series prediction problem such as this one, which is why this structure was chosen. This study is part of the GLOBALTOUR Euroace project [40], which focused its research on the sustainable tourism development of rural areas by enhancing the value of starry sky resources. The great potential of these remote rural areas for astronomical observation, which is often underestimated, should be pointed out. The low levels of light pollution in these areas make them ideal locations for night sky observation. The promotion of astronomical tourism represents a sustainable and environmentally friendly way for economic and social development in these areas, as pointed out by a variety of leading authors in the field [6,8,10,12]. This work is the continuation of another series of previous articles carried out by the authors in this line of research. First, a study [35] was carried out to determine the lighting conditions that should be established in the vicinity of the observation site for minimising the light pollution level. At this point, it should be noted that light pollution has an important influence on the observation of the sky [11,16,41,42]. From this first analysis, it was possible to determine a series of three key locations where NSB measurements could be carried out under optimal conditions. From there, a second study [43] analysed the influence of climatic variables on the NSB level using weather stations and identified those critical for observation. In this last work, the variation in nocturnal NSB throughout the year was studied, seeking the development of a predictive model. Finally, based on the conclusions acquired in these previous works, the present study was proposed, the objective of which was the generation of a neural network that could predict in advance the NSB level reached in rural areas.

2. Data Collection

2.1. Location

For the experimental development, three rural locations with a low light pollution level were selected in the province of Cáceres, Spain, specifically, Santiago de Alcántara (39°36′22.11″ N, 7°14′40.28″ W), Zarza la Mayor (39°52′38″ N, 6°51′44″ W), and Valencia de Alcántara (39°24′47.72″ N, 7°14′36.93″ W). Data collection points were chosen away from urban centres to avoid local lighting interfering with the measurements. The process used for selecting these municipalities, together with relevant data on the measurement site, is detailed in the cited article [43]. At each of these locations, a photometer was installed, taking NSB measurements over approximately 23 months (from May 2022 to April 2024). The data collection period was set at 15 min, following the same criteria used by other authors [34]. Subsequently, all recorded data were compiled. Records made during daylight hours were eliminated, and those between 0 and 6:45 a.m. remained. A total of 21,069 data were recorded.

2.2. Equipment

To quantify the sky darkness level, a “Sky Quality Meter” (SQM) was used, a light-sensitive optical sensor oriented towards the sky capable of registering the amount of light coming from different parts of the spectrum. This equipment is commonly used in astronomical observations, measurements of light pollution impact, and in the protection of protected natural areas. In astronomy, the apparent magnitudes per square arcsecond (mag/arcsec²) are usually used as the unit of measurement for NSB, with a maximum value of 24 mag/arcsec² for observations of the sky in total darkness. As the measurement points were located in remote areas, the SG-WAS (SkyGlow Wireless Autonomous Sensor) photometer [44] was chosen, as it is self-contained with a solar panel for charging and a wireless system for data transfer. The equipment is also highly robust in adverse weather conditions, making it ideal for this application. In the article “SG-WAS: A New Wireless Autonomous Night Sky Brightness Sensor” [45], all specifications and technical features of the equipment can be found in extensive detail. It is important to highlight that the equipment has shown calibration errors of less than 0.02 mag/arcsec². Differences between simultaneous measurements of multiple SG-WAS photometers have been found to have a standard deviation of 0.01 mag/arcsec².

To enter the data collected into the neural network, the format of the recording date was transformed from “day/month” to its corresponding day within the year, with 1 January corresponding to day 1 and 31 December to day 365. For the hour recording, a change of scale to decimal format was made, with each 15-min section corresponding to a fraction of 0.25. In this way, the measurements started at time 0 (00:00 a.m.) until time 6.75 (6:45 a.m.). All 21,069 measurements can be found in the following link [46]. A summary of the statistics of the recorded experimental dataset is shown in

Table 1. Statistics of experimental measurements.

Statistic	Value (mag/arcsec2)
Minimum	9.5
Maximum	22.91
Mean	17.93
Standard deviation	3.99

.

3. Multilayer Perceptron Development

A neural network is a computational model used for solving complex pattern recognition, classification, regression, and machine learning problems. Among the various neural network models, one of the most widely used for time series prediction is the Multilayer Perceptron (MLP). It is made up of an input layer (input data), one or several hidden layers (where there is a set of neurons in charge of processing this information), and an output layer (network response). This model was employed thanks to its simplicity and high accuracy in the prediction of non-linear systems [37,38]. Despite their simplicity, MLPs have been shown to achieve higher forecasting accuracy than other complex statistical models [39]. Since NSB prediction had no evident time dependency that required the use of long-term memory, the information was processed as it flowed in one direction only, from input to output, without loops or backward connections. This type of information processing is referred to as Feedforward Propagation Neural Networks (FNNs). Although other models also employ FNNs, the MLP is the most popular. The ability of a neural network to approximate any system depends on its learning ability. A training process with a specific set of data, organised into pairs of inputs and desired outputs, is necessary to learn the behaviour of the system it intends to replicate. In this way, each time one of these inputs is fed into the network, an output response is obtained, which is compared with the desired one to calculate an error. The errors obtained for all outputs are added up to calculate a global error, the value of which is minimised using the Levenberg–Marquardt algorithm, which adjusts the weights of the connections of each of the neurons appropriately, ensuring that the network has learned the different patterns of behaviour of the system to be replicated. Once the network has been trained, it is necessary to verify whether this learning was carried out correctly. This means checking if the network is capable of recognizing the different patterns and variations present in the dataset. For this purpose, a completely different “validation” dataset was introduced, distinct from the training data. In this case, only the input data were fed into the network, resulting in a set of predictions at the network’s output. These predicted values were then compared with the actual values to evaluate the performance of the predictions made.

The recorded dataset described in Section 2 was used as a training and validation set for the developed neural network. The objective was to create a model capable of predicting the sky darkness value for a given date. For the development of the neural network, the software Matlab R2024a [47] was used, and the toolboxes Statistics and Machine Learning and Deep Learning were installed. The Levenberg–Marquardt algorithm is implemented in Matlab using the “train” function. First, the total recorded dataset was divided completely randomly into two different parts, training (75%) and validation (25%), following percentages similar to those used by other authors [37]. The reason for this type of random division was to ensure a homogeneous distribution of the data, avoiding, for example, that data from a particular month would be concentrated in one of the two divisions. This ensured that in each of the two sets (training and validation), there were data for all months, days, and hours, ensuring the correct functioning of the network (otherwise, it could be the case that the training set only had data for a specific time of the year, for example, summer, which would make it impossible to predict for the winter months).

The inputs were composed of a matrix of two rows and 21,069 columns, where the first row contained the values of the recording day (d_j) and the second row its corresponding hour (h_j). For this reason, the network had two neurons in the input layer, one for the “day” variable and the other for the “hour” variable. The output was composed of a vector of one row and 21,069 columns containing the NSB values recorded in the experimental phase (NSB_j), thus presenting a single neuron in the output layer. A schematic of the inputs and outputs of the system is shown in Figure 1.

An MLP is composed of an input data layer, one or more hidden layers (where the information is processed), and an output layer (system response). There is no exact procedure for setting the neural network parameters, so the model configuration is usually performed through an iterative process of calculating the global error, adjusting the network parameters, and evaluating its behaviour [34,48]. Therefore, one of the main advantages of using an MLP is that its development ensures that the prediction error is minimised. It should be noted that the success of MLPs in providing predictions with a high degree of accuracy is due to the fact that they have been shown to be universal approximators [49,50], i.e., they can approximate any continuous function using a single hidden layer, as long as it contains a sufficient number of neurons. For this reason, a single hidden layer with 10 neurons was used. Other arrangements with different numbers of neurons were tested during the simulation, with the above configuration achieving a balance between modelling capability and simplicity. The use of fewer neurons did not allow the generation of a model that correctly fitted the experimental data, increasing the error committed. Configurations with a larger number of neurons (up to 15) did not significantly reduce the error level achieved but increased the computational time. Settings with a number of neurons higher than 15 showed a significant increase in the error due to overfitting.

The functioning of the MLP is explained below. The “j” neurons in one layer process the information received from the “i” neurons in the previous layer. The output of neuron “y_j” is modelled as shown in Equation (1), where “x_i” represents the inputs from the neuron in the previous layer, “w_ji” the weight of the connection between the two neurons, “b_j” the bias constant of the neuron, and “σ” the activation function, which allows the introduction of the non-linearity necessary for the neural network to model complex behaviour.

$${\mathrm{y}}_{\mathrm{j}}=\mathsf{\sigma } \left(\sum {\mathrm{w}}_{\mathrm{ji}}{\mathrm{x}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{j}}\right)$$

(1)

In the hidden layer, the hyperbolic tangent is usually used as the activation function (Equation (2)). The value of the different bias weights and constants is optimised during the learning process using the Levenberg–Marquardt algorithm, minimising the overall error made in the predictions.

$${\mathrm{y}}_{\mathrm{j}}={\displaystyle \frac{1}{1+{\mathrm{e}}^{-\left(\sum {\mathrm{w}}_{\mathrm{ji}}{\mathrm{x}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{j}}\right)}}}$$

(2)

In

Table 2. Weights of the input layer to hidden layer connection and hidden layer biases.

wj,1	wj,2	bj
2.36	−0.21	−2.86
4.03	1.30	−3.18
−3.54	3.84	0.11
3.05	−3.13	0.01
4.53	4.77	0.40
−3.94	−3.93	−0.22
−6.91	−1.32	−2.38
0.25	2.86	0.80
2.73	3.03	4.09
3.67	2.65	4.49

, the value calculated for the weights between the two input layer neurons and the hidden layer neurons (where w_j,1 corresponds to the weight of the connection with the input neuron of the variable “day” and w_j,2 to the weight of the connection with the input neuron of the variable “hour”) is shown. The bias constant “b_j” values are also shown for each neuron in the hidden layer.

Finally, the data are passed from the hidden layer to the output layer by means of a linear activation function (Equation (3)), since in this case its function is only to adjust the response to the data structure.

$${\mathrm{y}}_{\mathrm{j}}={\mathrm{w}}_{\mathrm{ji}}{\mathrm{x}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{j}}$$

(3)

The value calculated for the weights of the connections between the neurons in the hidden layer and the neuron in the output layer is shown below (

Table 3. Weights of the connection from the hidden layer to the output layer.

wOutput,j
1.32
−0.32
4.02
4.93
1.90
2.37
0.39
0.52
1.50
−1.43

). The bias constant of the output neuron had a value of 1.07.

To summarise, a schematic of the neural network configuration used is shown in Figure 2.

4. Results and Discussion

The script with the developed code, including explanatory comments about its design, as well as the set of the 21,069 recorded data used in the simulation, can be consulted in the following link [46]. A line of code was added that generates a graphical window so that the user can enter the specific date to be predicted. In the case of no data record for the chosen day and time, the program scans the entire existing dataset for the closest available time record, whose data will be used in the prediction.

Figure 3 shows the graphical representation of all 21,069 measurements taken (black) versus their predicted value (red). A high degree of agreement between the actual values and those predicted by the network can be seen. This can be observed in the model statistics. The neural network developed showed a standard deviation of 1.54 mag/arcsec² and a coefficient of determination R² of 0.85. A value of R² close to 1 (as obtained in the developed model) implies a high capacity of the model to explain the variability of the response, i.e., most of the variations of the recorded experimental data are predicted by the network. The formula used to calculate this coefficient is shown in Equation (4).

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\mathrm{SS}}_{\mathrm{Res}}}{{\mathrm{SS}}_{\mathrm{Tot}}}}$$

(4)

where SS_Res represents the sum of squares of the errors, and SS_Tot the total sum of squares:

$${\mathrm{SS}}_{\mathrm{Res}}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-{\mathrm{ŷ}}_{\mathrm{i}}\right)}^2$$

$${\mathrm{SS}}_{\mathrm{Tot}}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-\overline{\mathrm{y}}\right)}^2$$

where “n” is the number of samples, “y_i” is the recorded NSB value, “ŷ_i” is the predicted NSB value, and “ȳ_i” is the average value of the data.

Next, the mean absolute error (MAE) was calculated, being the average of the absolute difference between the actual experimentally measured values and the values predicted by the network (Equation (5)). Lower values of this parameter reflect a lower prediction error of the model. The predictions showed an MAE of 1.21 mag/arcsec². To interpret this error in relative terms, the mean absolute percentage error (MAPE) was calculated (Equation (6)), obtaining a value of 6.8%.

$$\mathrm{MAE}={\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{y}}_{\mathrm{i}}-{\mathrm{ŷ}}_{\mathrm{i}}\right|}{\mathrm{n}}}$$

(5)

$$\mathrm{MAPE}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|{\displaystyle \frac{{\mathrm{y}}_{\mathrm{i}}-{\mathrm{ŷ}}_{\mathrm{i}}}{{\mathrm{y}}_{\mathrm{i}}}}\right|\times 100$$

(6)

The root mean square error (RMSE) was also calculated, yielding a value of 1.51 mag/arcsec² (Equation (7)). This can be interpreted as the standard deviation of the residual values between the predicted and actual values, i.e., the dispersion of the model prediction errors. Lower values of RMSE imply a higher accuracy of the model.

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-{\mathrm{ŷ}}_{\mathrm{i}}\right)}^2}$$

(7)

As an example, the last 20 predicted NSB values were plotted together with their respective recorded values, which makes it easier to appreciate the degree of adjustment of the system (Figure 4).

A sensitivity analysis was performed to determine which input variables had the greatest impact on the model output. The sensitivity of the two input variables “day” and “hour” to a small perturbation was assessed. An average sensitivity of 0.03 was obtained for the day variable and 1.85 for the hour variable. Given this analysis, it can be noted that changes in the variable “day” have little influence (0.03) on the predicted NSB level. In contrast, changes in the variable “hour” do have a high impact on NSB. This is reasonable, since as the variable “hour” increases, i.e., as we move further into the night, the level of darkness increases, thus also increasing NSB. Figure 5 shows the NSB predictions made for the 365 days of the year at different times, with day 1 corresponding to 1 January and day 365 to 31 December. A significant decrease in NSB can be seen in hours 5 and 6 during the summer months due to sunrise. The highest NSB levels were reached in hour 3 of the period between days 125 and 135 (5–15 May), reaching values above 21.7 mag/arcsec², optimum values for sky observation.

As mentioned above, after a literature search, only one article was found in which a probabilistic approach to NSB prediction is performed, highlighting the novelty of applying this technology to this field. The work carried out by C-Sánchez et al. [34], where an NNAR with sliding window was developed, represents the closest model conceptually to the one presented in this paper, so both can be compared objectively despite the differences in terms of data collection and development.

Table 4. Comparison between the statistics obtained in the models of C-Sánchez et al. and Martínez-Martín et al.

Model	Sd	R2	MAE	RMSE
Martínez-Martín et al. (this study)	1.54	0.85	1.21	1.51
C-Sánchez et al. [34]	4.77	0.87	1.57	2.09

shows the statistics obtained in both models. The values provided by C-Sánchez et al. are given for the six-neuron neural network and 5-day sliding window, with this configuration being the one that obtained the most favourable statistics of those developed by this author. This author used NSB data for 22 months (2016 and 2017, except May); the length of the data collection period was very similar to that of the present study (approximately two years). However, data from a single photometer were used, whereas in the present study, measurements were taken at three different locations simultaneously, thus recording three times as much data, which could be one of the reasons why the statistics achieved in this study are slightly better than those of C-Sánchez et al. On the other hand, the data collection location of C-Sánchez et al. was in an urban area (Tokyo, Japan) with abundant and constant artificial lighting, which means that the fluctuations in NSB throughout the night might be much smaller than in a rural area and therefore easier to predict.

5. Conclusions

The development of a predictive model to determine night sky darkness level in rural areas could help to promote astro-tourism in these areas, allowing the development of sustainable tourism to alleviate the problems of depopulation in these areas. NSB level strongly affects the observing capacity, so knowing its value in advance is critical to be able to observe under optimal conditions. For this reason, a neural network capable of predicting NSB for a specific date was developed. The network was created using the Multilayer Perceptron (MLP) model, featuring a single hidden layer with 10 neurons. The Levenberg–Marquardt algorithm was used for connection weight optimisation. Other configurations with different numbers of neurons were used during programming, optimising the performance for the 10-neuron structure. The predictions made by the network showed an MAE of 1.21 mag/arcsec², an MAPE of 6.8%, and an RMSE of 1.51 mag/arcsec², values close to those reported by other authors in similar studies [34]. The network was developed using MATLAB 2024 software. The generated code along with the NSB data recorded in the experimental phase were uploaded to an open access platform for consultation [43]. The results of the work are oriented towards the prediction of NSB in rural areas, which could be useful for the promotion of sustainable astronomical tourism in these areas.

5.1. Limitations/Future Lines of Research

One of the limitations to be highlighted is the presence of unfavourable meteorological conditions. These exert an important influence on the NSB level, especially the presence of clouds [27,28,29], which can distort the reliability of the recording made. This fact was also pointed out by C-Sánchez et al. [34] as one of the main limitations of their study. As a possible future line of research, meteorological stations could be set up to record weather conditions simultaneously with NSB level. In the previous work carried out by the authors [40], the environmental variables influencing NSB were already determined, so it would be relevant to include the recording of these variables as an input in the neural network. In this way, a higher degree of accuracy could be achieved in the forecasts made, especially in adverse weather conditions. Additionally, in the future, it is intended to implement this neural network developed on an online website or in an app for smartphones so that it can be used by users in a simple way.

5.2. Contribution to the Academic and Local Community

This work represents a contribution at an academic level as there is no predictive model of these characteristics in the available bibliography. The script developed [46], as well as the procedures followed for the collection of experimental data, can be applied in a similar way to other rural destinations, allowing the development of a predictive model applied to this area. This work also opens the door to the future development of other predictive models for the prediction of NSB, a field that is little developed at the scientific level. In terms of local communities, those destinations with night skies free of light pollution (such as those in remote rural areas) have a high potential for the development of astro-tourism, allowing a positive ecological and socio-cultural impact to be generated in the area [51]. Several international organisations such as the International Dark-Sky Association (IDA) have highlighted the importance of the preservation and valorisation of starry sky resources. In this sense, the neural network developed in this paper could be used in these areas, promoting the arrival of astro-tourists. They would have a new tool to know in advance the level of darkness of the sky at the observation site, thus generating an opportunity thanks to the use of digital technologies, adding value to the tourist activity. This would have a doubly positive impact. On the one hand, it would help the economic development of the area, increasing economic activity with the arrival of new tourists, also avoiding the problem of depopulation that rural areas often suffer. On the other hand, being respectful to the environment, it would be promoting the generation of ecological and sustainable tourism.