# EDISON: An Edge-Native Method and Architecture for Distributed Interpolation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- We present an edge-native, distributed interpolation architecture for the smart city networking environment, characterized by spatio-temporal nature and large-scale communications.
- We present a distributed learning and inference method for our architecture, to make edge-native interpolations with spatio-temporally distributed data.
- We evaluate our solution with a controlled environment of simulations, enforcing the natural phenomena observed in our previous work [23].

## 2. Related Work

**Edge computing.**The terminology and the definitions of concepts in edge computing are not fully agreed upon. For example, proponents of the fog computing model consider fog to be a continuum of computing resources along the path from devices to the cloud, and identify edge computing with the devices and their users [26]. On the other hand, edge computing proponents consider edge to comprise resources for communication, computation, control and storage, in close proximity to the devices and end-users, with those resources ranging from light devices to small-scale edge data centers [15]. In this article, we follow the terminology of the edge computing proponents, and consider a three-layer model with a remote cloud, local edge computing servers, and finally the devices.

**City-scale computing.**The highlighted challenges of city-scale computing can coarsely be summarised as (1) large-scale data quantity and how to process it efficiently [3], (2) heterogeneous data providers in terms of data quality, source (varying from private citizens to vehicles and industrial applications), and sampling frequency [6,7], (3) mobile and low-capacity devices as a part of the system, especially private carry-on devices and vehicles [10,27], and (4) real-time requirements to produce efficient recommendations, situational awareness, and other big-picture services and applications [10]. Some platforms are suggested for city-scale computation activities, including both “traditional” elastic cloud services [28] and data lakes [29].

**Edge computing for smart cities.**Today, the current research trend agrees that whenever cloud-only architectures are not feasible anymore, edge computing paradigm needs to emerge into the city-scale environment. For instance, Hossain et al. [32] present an edge computing framework for situation awareness in an IoT-based smart city. Their first experiments consider latency and situation awareness when raw IoT data is processed at the edge devices, with a multi-layer architecture. However, they utilize the edge only for data processing, demanding the cloud services for the final combination of the data and running learning models. This is, by our understanding, not meeting the real-time requirements when not only processing but also the delivery of results should be considered in a timely manner. On contrary, Barthélemy et al. [10] utilize a local computational board of a camera to fulfill real-time requirements in a local context, but the applicability over a widely spread system (and other verticals) is still left as an open question.

**City-scale sensing and data analytics.**Smart cities rely on IoT, big data, cyber-physical systems, and edge-cloud computing continuum technologies [28,33] to provide data not only for novel applications but also other key functionalities of the enlarging urban spaces, such as increasing urban sustainability [2]. However, sensor data collection is rarely enough to provide timely feedback, decision support and situational awareness. Rather, multi-phased data processing is required. Pre-processing, cleaning, data fusion, and interpolation techniques can need to be widely considered before even the first steps of ML/AI learning phases can be run. Considering these pre-pocessing steps alone—add to the actual model building and evaluation—makes analyzing the huge amount of urban data both challenging and time-consuming.

**Interpolation.**There is a long history of interpolation based on Gaussian process (GP) regression [24]. A fundamental problem, however, is the method’s computational complexity, relative to ${N}^{3}$ in processing time and ${N}^{2}$ in memory capacity, where N is the number of observations [24,36,37]. A few recent studies address this issue.

**Edge-native distributed learning.**A number of recent surveys (see e.g., [22,42,43,44,45]) review edge-native machine learning and EdgeAI approaches. While most approaches focus on distributed learning of neural networks (see e.g., federated distillation by Jeong et al. [46], a variant of Google’s federated learning [47] approach), there are currently no neural network-based approaches for interpolation which can cope with mobile sensors [37]. Further, we have found no approaches which consider spatial covariance structures of the data in the distribution of learning.

## 3. EDISON

- Calibration
- (a)
- CLOUD: Estimate calibration parameters for mobile sensors. Calibrate the collected sensor training set.
- (b)
- CLOUD: Transmit estimate calibration parameters to edge servers.
- (c)
- EDGE SERVERS: Transmit calibration parameters to IoT gateways passing by.
- (d)
- IOT GATEWAYS: Transmit calibration parameters to mobile sensors.
- (e)
- MOBILE SENSORS: Apply calibration.

- Distributed learning
- (a)
- CLOUD: Partition the training set into subsets of observations around each edge server. Aim for subsets whose observations are maximally independent of the observations in other subsets.
- (b)
- CLOUD: Send the partitioned training set to all edge servers, rasterized to reduce transmission burden.
- (c)
- EDGE SERVERS: Fit a local, spatio-temporal interpolation model for the observations in the edge server’s subset of the training set.

- Distributed inference
- (a)
- MOBILE SENSORS: Send all observations immediately to the IoT gateway in the vehicle.
- (b)
- IOT GATEWAY: Store observations. Send stored observations to an edge server when passing by.
- (c)
- FIXED SENSORS: Send all observations immediately to edge servers.
- (d)
- EDGE SERVERS: Every time interval, find the right edge server (i.e., the right cluster) for each new mobile observation from IoT gateways that have passed by.
- (e)
- EDGE SERVERS: Send new mobile observations to selected edge servers.
- (f)
- EDGE SERVERS: Every time interval, apply the local interpolation model with the data collected by the sensors.

#### 3.1. Distributed Learning

- Independence: each subset should be as independent as possible from the others.
- Spatial connectedness: each resulting subset should be a spatially connected set of points.

Algorithm 1:Distributed learning |

#### 3.2. Clustering

- ${y}_{lo}\in [0,1]\phantom{\rule{1.em}{0ex}}\forall l,o$
- ${\sum}_{l=1}^{K}{y}_{lo}=1\phantom{\rule{1.em}{0ex}}\forall i$

#### 3.3. Distributed Inference

Algorithm 2:Distributed inference |

## 4. Evaluation

- Generate artificial ground truth data comprising complex spatio-temporal dependency structures.
- Simulate sensor data.
- (a)
- Simulate static sensor locations.
- (b)
- Simulate mobile sensor trajectories.
- (c)
- Collect observations from the static sensor locations and along the mobile sensor trajectories.

- Run EDISON.
- (a)
- Split the observations into training and test sets.
- (b)
- Conduct EDISON distributed learning on the training set.
- (c)
- Conduct EDISON distributed inference on the test set.

- Calculate results.
- (a)
- Compare EDISON results to ground truth with RMSE.
- (b)
- Compare reference results to ground truth with RMSE.

#### 4.1. Data Generation

`fields`[53].

`stat::rnorm`function), and multiply those with a Cholesky factor (with, e.g., the R

`base::chol`function) of $\mathsf{\Sigma}$.

#### 4.2. Sensor Simulation

`trajectories`[55].

#### 4.3. EDISON

`gstat`[56,57] and

`spacetime`[58,59] packages.

#### 4.4. Results

- global: unclustered interpolation over the whole map
- oracle: interpolation with pre-knowledge of the borders between the four data-generating processes
- baseline: each observation is assigned to the closest edge server
- E2: EDISON algorithm whose proximity part of the distance function (i.e., the spatial distance part) is squared, $d(\{{x}_{i},{\theta}_{i}\},\{{x}_{j},{\theta}_{j}\})=\lambda {\sum}_{a=1}^{2}{({x}_{ia}-{x}_{ja})}^{2}+(1-\lambda ){\sum}_{b=1}^{Q}{({\theta}_{ib}-{\theta}_{jb})}^{2},$ instead of cubed (see Section 3.2)

## 5. Discussion

**Results.**The main motivation of EDISON is the distribution of the large-scale sensor data and the heavy computations related to spatio-temporal interpolation of the data. However, the RMSE results (Table 3 and Figure 8) show that in fact, even if global modelling were possible, EDISON would improve on the global interpolation by, at best, ca. 10%. Indeed, fitting a single variogram over the whole area and subsequently using that variogram for interpolation loses the detail of the local spatial processes and leads to worse overall performance.

**Limitations.**As evidenced by the evaluation results (see Table 3 and Figure 8), EDISON shines when data is generated by a number of complex, relatively independent, spatially distributed processes. Such processes arguably include, for example, short-term surface temperatures in urban environments with a number of independent heat sources as well as varying surface materials and densities. However, as a result of the distributed nature of EDISON, the interpolated values often have sharp edges between the different clusters (see Figure 7, EDISON row). If the data-generating processes vary smoothly over long distances, such sharp edges may not be desirable.

**Future considerations.**There are a number of possible avenues for mitigating the above limitations. The sharp edges between cluster interpolations, if undesired in the application, could be addressed by modifying the interpolation method. For example, the patchwork Kriging method by Park and Apley [38] could replace the ordinary Kriging approach used here. Patchwork Kriging generates pseudo-observations along the boundaries between neighbouring clusters to tie their results smoothly together. The resulting communication burden between the edge servers would, however, need to be closely considered for such a change.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AP | Access point |

EDISON | Edge-native distributed interpolation |

ES | Edge server |

GP | Gaussian process |

MDPI | Multidisciplinary Digital Publishing Institute |

## References

- United Nations, Department of Economic and Social Affairs, Population Division. World Urbanization Prospects: The 2018 Revision (ST/ESA/SER.A/420); United Nations: New York, NY, USA, 2019. [Google Scholar]
- Meijer, A.; Bolívar, M.P.R. Governing the smart city: A review of the literature on smart urban governance. Int. Rev. Adm. Sci.
**2016**, 82, 392–408. [Google Scholar] - Gaur, A.; Scotney, B.; Parr, G.; McClean, S. Smart city architecture and its applications based on IoT. Procedia Comput. Sci.
**2015**, 52, 1089–1094. [Google Scholar] [CrossRef] - Strohbach, M.; Ziekow, H.; Gazis, V.; Akiva, N. Towards a big data analytics framework for IoT and smart city applications. In Modeling and Processing for Next-Generation Big-Data Technologies; Springer: Berlin/Heidelberg, Germany, 2015; pp. 257–282. [Google Scholar]
- Angelidou, M.; Psaltoglou, A.; Komninos, N.; Kakderi, C.; Tsarchopoulos, P.; Panori, A. Enhancing sustainable urban development through smart city applications. J. Sci. Technol. Policy Manag.
**2018**, 9. [Google Scholar] [CrossRef] - Naphade, M.; Banavar, G.; Harrison, C.; Paraszczak, J.; Morris, R. Smarter cities and their innovation challenges. Computer
**2011**, 44, 32–39. [Google Scholar] [CrossRef] - Lau, B.P.L.; Marakkalage, S.H.; Zhou, Y.; Hassan, N.U.; Yuen, C.; Zhang, M.; Tan, U.X. A survey of data fusion in smart city applications. Inf. Fusion
**2019**, 52, 357–374. [Google Scholar] [CrossRef] - Bokolo, A.J.; Majid, M.A.; Romli, A. A trivial approach for achieving Smart City: A way forward towards a sustainable society. In Proceedings of the 2018 21st Saudi Computer Society National Computer Conference (NCC), Riyadh, Saudi Arabia, 25–26 April 2018; pp. 1–6. [Google Scholar]
- Jararweh, Y.; Otoum, S.; Al Ridhawi, I. Trustworthy and sustainable smart city services at the edge. Sustain. Cities Soc.
**2020**, 62, 102394. [Google Scholar] [CrossRef] - Barthélemy, J.; Verstaevel, N.; Forehead, H.; Perez, P. Edge-computing video analytics for real-time traffic monitoring in a smart city. Sensors
**2019**, 19, 2048. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cicirelli, F.; Guerrieri, A.; Spezzano, G.; Vinci, A. An edge-based platform for dynamic Smart City applications. Future Gener. Comput. Syst.
**2017**, 76, 106–118. [Google Scholar] [CrossRef] - Taleb, T.; Dutta, S.; Ksentini, A.; Iqbal, M.; Flinck, H. Mobile edge computing potential in making cities smarter. IEEE Commun. Mag.
**2017**, 55, 38–43. [Google Scholar] [CrossRef] [Green Version] - Giordano, A.; Spezzano, G.; Vinci, A. Smart agents and fog computing for smart city applications. In Proceedings of the International Conference on Smart Cities, Malaga, Spain, 15–17 June 2016; Springer: Berlin/Heidelberg, Germany, 2016; pp. 137–146. [Google Scholar]
- Deng, Y.; Chen, Z.; Yao, X.; Hassan, S.; Wu, J. Task scheduling for smart city applications based on multi-server mobile edge computing. IEEE Access
**2019**, 7, 14410–14421. [Google Scholar] [CrossRef] - Chiang, M.; Shi, W. Grand Challenges in Edge Computing; Technical Report; National Science Foundation: Washington, DC, USA, 2017.
- Shi, W.; Cao, J.; Zhang, Q.; Li, Y.; Xu, L. Edge Computing: Vision and Challenges. IEEE Internet Things J.
**2016**, 3, 637–646. [Google Scholar] [CrossRef] - Kitchin, R. Making sense of smart cities: Addressing present shortcomings. Camb. J. Reg. Econ. Soc.
**2015**, 8, 131–136. [Google Scholar] [CrossRef] [Green Version] - He, Y.; Yu, F.R.; Zhao, N.; Leung, V.C.; Yin, H. Software-defined networks with mobile edge computing and caching for smart cities: A big data deep reinforcement learning approach. IEEE Commun. Mag.
**2017**, 55, 31–37. [Google Scholar] [CrossRef] - Li, M.; Si, P.; Zhang, Y. Delay-tolerant data traffic to software-defined vehicular networks with mobile edge computing in smart city. IEEE Trans. Veh. Technol.
**2018**, 67, 9073–9086. [Google Scholar] [CrossRef] - Lovén, L.; Leppänen, T.; Peltonen, E.; Partala, J.; Harjula, E.; Porambage, P.; Ylianttila, M.; Riekki, J. EdgeAI: A vision for distributed, edge-native artificial intelligence in future 6G networks. In Proceedings of the 1st 6G Wireless Summit, Levi, Finland, 24–26 March 2019; pp. 1–2. [Google Scholar]
- Partala, J.; Lovén, L.; Peltonen, E.; Porambage, P.; Ylianttila, M.; Seppänen, T. EdgeAI: A vision for privacy-preserving machine learning on the edge. In Proceedings of the 10th Nordic Workshop on System and Network Optimization for Wireless (SNOW), Ruka, Finland, 1–4 April 2019. [Google Scholar]
- Park, J.; Samarakoon, S.; Bennis, M.; Debbah, M.M. Wireless network intelligence at the edge. Proc. IEEE
**2019**, 107, 2204–2239. [Google Scholar] [CrossRef] [Green Version] - Lovén, L.; Karsisto, V.; Järvinen, H.; Sillanpää, M.J.; Leppänen, T.; Peltonen, E.; Pirttikangas, S.; Riekki, J. Mobile road weather sensor calibration by sensor fusion and linear mixed models. PLoS ONE
**2019**, 14, 1–17. [Google Scholar] [CrossRef] [Green Version] - Rasmussen, C.E.; Williams, C.K. Gaussian Processes for Machine Learning; The MIT Press: Cambridge, MA, USA, 2006. [Google Scholar] [CrossRef] [Green Version]
- Lovén, L.; Peltonen, E.; Pandya, A.; Leppänen, T.; Gilman, E.; Pirttikangas, S.; Riekki, J. Towards EDISON: An edge-native approach to distributed interpolation of environmental data. In Proceedings of the 28th International Conference on Computer Communications and Networks (ICCCN2019), 1st Edge of Things Workshop 2019 (EoT2019), Valencia, Spain, 29 July–1 August 2019. [Google Scholar]
- Iorga, M.; Feldman, L.; Barton, R.; Martin, M.J.; Goren, N.; Mahmoudi, C. Fog Computing Conceptual Model; Technical Report; National Institute of Standards and Technology: Gaithersburg, MD, USA, 2018. [CrossRef]
- Walravens, N. Mobile city applications for Brussels citizens: Smart City trends, challenges and a reality check. Telemat. Inform.
**2015**, 32, 282–299. [Google Scholar] [CrossRef] - Santana, E.F.Z.; Chaves, A.P.; Gerosa, M.A.; Kon, F.; Milojicic, D. Software platforms for smart cities: Concepts, requirements, challenges, and a unified reference architecture. ACM Comput. Surv.
**2016**, 50, 1–37. [Google Scholar] [CrossRef] - Mehmood, H.; Gilman, E.; Cortes, M. Implementing big data lake for heterogeneous data sources. In Proceedings of the 1st International Workshop on Data-Driven Smart Cities, in Conjunction with 35th IEEE International Conference on Data Engineering (ICDE 2019), Macao, China, 8–12 April 2019. [Google Scholar]
- Raza, U.; Camerra, A.; Murphy, A.L.; Palpanas, T.; Picco, G.P. What does model-driven data acquisition really achieve in wireless sensor networks? In Proceedings of the 2012 IEEE International Conference on Pervasive Computing and Communications, PerCom 2012, Lugano, Switzerland, 19–23 March 2012; pp. 85–94. [Google Scholar] [CrossRef] [Green Version]
- Peltonen, E.; Leppänen, T.; Lovén, L. EdgeAI: Edge-native distributed platform for artificial intelligence. In Proceedings of the 1st 6G Wireless Summit, Levi, Finland, 24–26 March 2019; pp. 1–2. [Google Scholar]
- Hossain, S.K.A.; Rahman, M.A.; Hossain, M.A. Edge computing framework for enabling situation awareness in IoT based smart city. J. Parallel Distrib. Comput.
**2018**, 122, 226–237. [Google Scholar] [CrossRef] - Fortino, G.; Russo, W.; Savaglio, C.; Viroli, M.; Zhou, M. Modeling opportunistic IoT services in open IoT ecosystems. In Proceedings of the XVIII Workshop “From Objects to Agents”, Scilla, Italy, 15–17 June 2017; pp. 90–95. [Google Scholar]
- Baker, T.; Aldawsari, B.; Asim, M.; Tawfik, H.; Maamar, Z.; Buyya, R. Cloud-SEnergy: A bin-packing based multi-cloud service broker for energy efficient composition and execution of data-intensive applications. Sustain. Comput. Inform. Syst.
**2018**, 19, 242–252. [Google Scholar] [CrossRef] [Green Version] - Lagerspetz, E.; Varjonen, S.; Concas, F.; Mineraud, J.; Tarkoma, S. Demo: MegaSense: Megacity-scale accurate air quality sensing with the edge. In Proceedings of the 24th Annual International Conference on Mobile Computing and Networking (MobiCom ’18), New Delhi, India, 29 October–2 November 2018; ACM: New York, NY, USA, 2018; pp. 843–845. [Google Scholar]
- Van Stein, B.; Wang, H.; Kowalczyk, W.; Emmerich, M.; Bäck, T. Cluster-based kriging approximation algorithms for complexity reduction. Appl. Intell.
**2020**, 50, 778–791. [Google Scholar] [CrossRef] [Green Version] - Amato, F.; Guignard, F.; Robert, S.; Kanevski, M. A novel framework for spatio-temporal prediction of environmental data using deep learning. Sci. Rep.
**2020**, 10, 1–11. [Google Scholar] [CrossRef] [PubMed] - Park, C.; Apley, D. Patchwork kriging for large-scale Gaussian process regression. J. Mach. Learn. Res.
**2018**, 19, 1–43. [Google Scholar] - Yasojima, C.; Protázio, J.; Meiguins, B.; Neto, N.; Morais, J. A new methodology for automatic cluster-based kriging using K-nearest neighbor and genetic algorithms. Information
**2019**, 10, 357. [Google Scholar] [CrossRef] [Green Version] - Hernández-Peñaloza, G.; Beferull-Lozano, B. Field estimation in wireless sensor networks using distributed kriging. In Proceedings of the IEEE International Conference on Communications, Ottawa, ON, Canada, 10–15 June 2012; pp. 724–729. [Google Scholar] [CrossRef]
- Chowdappa, V.P.; Botella, C.; Beferull-Lozano, B. Distributed clustering algorithm for spatial field reconstruction in wireless sensor networks. IEEE Veh. Technol. Conf.
**2015**, 2015. [Google Scholar] [CrossRef] - Park, J.; Wang, S.; Elgabli, A.; Oh, S.; Jeong, E.; Cha, H.; Kim, H.; Kim, S.L.; Bennis, M. Distilling on-device intelligence at the network edge. arXiv
**2019**, arXiv:1908.05895v1. [Google Scholar] - Deng, S.; Zhao, H.; Fang, W.; Yin, J.; Dustdar, S.; Zomaya, A.Y. Edge intelligence: The confluence of edge computing and artificial intelligence. IEEE Internet Things J.
**2020**, 7, 7457–7469. [Google Scholar] [CrossRef] [Green Version] - Xu, D.; Li, T.; Li, Y.; Su, X.; Tarkoma, S.; Hui, P. A survey on edge intelligence. arXiv
**2020**, arXiv:2003.12172. [Google Scholar] - Zhou, Z.; Chen, X.; Li, E.; Zeng, L.; Luo, K.; Zhang, J. Edge intelligence: Paving the last mile of artificial intelligence with edge computing. Proc. IEEE
**2019**, 107. [Google Scholar] [CrossRef] [Green Version] - Jeong, E.; Oh, S.; Kim, H.; Park, J.; Bennis, M.; Kim, S.L. Communication-efficient on-device machine learning: Federated distillation and augmentation under non-iid private data. arXiv
**2018**, arXiv:1811.11479. [Google Scholar] - Yang, Q.; Liu, Y.; Chen, T.; Tong, Y. Federated machine learning: Concept and applications. ACM Trans. Intell. Syst. Technol.
**2019**, 10, 1–19. [Google Scholar] [CrossRef] - Tobler, W.R. A Computer Movie Simulating Urban Growth in the Detroit Region. Econ. Geogr.
**1970**, 46, 234–240. [Google Scholar] [CrossRef] - Lähderanta, T.; Leppänen, T.; Ruha, L.; Lovén, L.; Harjula, E.; Ylianttila, M.; Riekki, J.; Sillanpää, M.J. Edge computing server placement with capacitated location allocation. J. Parallel Distrib. Comput.
**2021**, in press. [Google Scholar] - Ruha, L.; Lähderanta, T.; Lovén, L.; Kuismin, M.; Leppänen, T.; Riekki, J.; Sillanpää, M.J. Capacitated spatial clustering with multiple constraints and attributes. arXiv
**2020**, arXiv:2010.06333. [Google Scholar] - Lovén, L.; Lähderanta, T.; Ruha, L.; Leppänen, T.; Peltonen, E.; Riekki, J.; Sillanpää, M.J. Scaling up an Edge Server Deployment. In Proceedings of the 2020 IEEE International Conference on Pervasive Computing and Communications Workshops (PerCom Workshops), online, 23–27 March 2020; pp. 1–7. [Google Scholar]
- Fix, E.; Hodges, J.L. Discriminatory Analysis. Nonparametric Discrimination: Consistency Properties; Technical Report; USAF School of Aviation Medicine: Randolph Field, TX, USA, 1951. [Google Scholar]
- Nychka, D.; Furrer, R.; Paige, J.; Sain, S. Fields: Tools for Spatial Data. R Package Version 11.6; CRAN. 2017. Available online: https://cran.r-project.org/web/packages/fields/index.html (accessed on 23 March 2021).
- Dimoudi, A.; Kantzioura, A.; Zoras, S.; Pallas, C.; Kosmopoulos, P. Investigation of urban microclimate parameters in an urban center. Energy Build.
**2013**, 64, 1–9. [Google Scholar] [CrossRef] - McLean, D.J.; Volponi, M.A.S. trajr: An R package for characterisation of animal trajectories. Ethology
**2018**, 124. [Google Scholar] [CrossRef] [Green Version] - Pebesma, E.J. Multivariable geostatistics in S: The gstat package. Comput. Geosci.
**2004**, 30, 683–691. [Google Scholar] [CrossRef] - Gräler, B.; Pebesma, E.; Heuvelink, G. Spatio-Temporal Interpolation using gstat. RFID J.
**2016**, 8, 204–218. [Google Scholar] [CrossRef] - Pebesma, E. Spacetime: Spatio-Temporal Data in R. J. Stat. Softw.
**2012**, 51, 1–30. [Google Scholar] [CrossRef] [Green Version] - Bivand, R.S.; Pebesma, E.; Gomez-Rubio, V. Applied Spatial Data Analysis with R, 2nd ed.; Springer: New York, NY, USA, 2013. [Google Scholar]
- Ahmad, I.; Shahabuddin, S.; Malik, H.; Harjula, E.; Leppanen, T.; Lovén, L.; Anttonen, A.; Sodhro, A.H.; Mahtab Alam, M.; Juntti, M.; et al. Machine Learning Meets Communication Networks: Current Trends and Future Challenges. IEEE Access
**2020**, 8, 223418–223460. [Google Scholar] [CrossRef] - Karsisto, V.; Lovén, L. Verification of road surface temperature forecasts assimilating data from mobile sensors. Weather Forecast.
**2019**, 34, 539–558. [Google Scholar] [CrossRef] - Lovén, L.; Gilman, E.; Riekki, J.; Läärä, E.; Sukuvaara, T.; Mäenpää, K.; Sillanpää, M.J.; Pirttikangas, S. Pilot study: Road–tyre friction prediction by statistical methods and data fusion. In In Proceedings of the 2017 International Workshop on Smart Sensing System (IWSSS17), Oulu, Finland, 7–8 August 2017; University of Oulu: Oulu, Finland, 2017; pp. 1–2. [Google Scholar]

**Figure 1.**Overview of EDISON. The device layer comprises fixed sensors as well as mobile sensors mounted on vehicles. IoT gateways provide connectivity, store mobile sensor observations, and provide local computational capabilities. The edge layer enhances the fixed sensors with connectivity and further computational capacity. Cloud provides coordination and centralized processing.

**Figure 2.**EDISON operational states. Calibration and distributed learning are employed once, in the beginning of operation, after which distributed inference is the standard operative state.

**Figure 3.**EDISON distributed learning. Cloud partitions the training set, the partitioned data is transmitted to the edge layer, and edge servers train local interpolation models.

**Figure 4.**EDISON distributed inference. Edge servers partition newly-observed data, transmit the partitions to their designated edge servers, and use the local new data for interpolation.

**Figure 5.**Simulated sensor trajectories. We marked 250 mobile sensor trajectories in blue, with the shade implying the time step. Fixed sensors marked in dark red.

**Figure 6.**EDISON partitioning of data. The sill values of the pointwise variograms (

**left panel**) clearly identify the boundaries between the four data-generating processes. Subsequent clustering (

**middle panel**) finds those boundaries reasonably well. In the inference state, new observations can be partitioned among the clusters (

**right panel**).

**Figure 7.**Ground truth, observations, and interpolations. The interpolations are conducted with oracle clustering, EDISON clustering, as well as a global variogram with no clustering. First three time frames of test data set are shown, from left to right.

**Figure 8.**RMSE values. While interpolation with oracle clustering results in the lowest RMSE values, EDISON improves on both the baseline clustering as well as an unclustered, global interpolation.

**Figure 9.**EDISON with Multi-Access Edge Computing (MEC). Coverage of each base station shown in yellow. Adapting EDISON for MEC requires a rethinking of the cluster architecture, based now around the BS locations. Further, due to the near-constant connectivity offered by 5G, data flow in the distributed inference state must be carefully reconsidered.

Symbol | Range | Description | |
---|---|---|---|

N | $\in \mathbb{N}$ | the number of observations in the training set | |

L | $\in \mathbb{N}$ | the number of observations for inference | |

M | $\in \mathbb{N}$ | size of neighbourhood (i.e., n. of obs.) around each observation | |

K | $\in \mathbb{N}$ | number of fixed sensors/clusters | |

${\mathit{neigbourhood}}_{i}$ | observations in the neighbourhood around observation i | ||

O | $\in \mathbb{N}$ | the number of raster cells on the map | |

${x}_{o},\phantom{\rule{0.277778em}{0ex}}o\in [1,\cdots ,O]$ | $\in {\mathbb{R}}^{2}$ | coordinates of the center of raster cell o | |

${f}_{l},\phantom{\rule{0.277778em}{0ex}}l\in [1,\cdots ,K]$ | $\in {\mathbb{R}}^{2}$ | location of fixed sensor l | |

Q | $\in \mathbb{N}$; | the dimension of the interpolation model parameters | |

${\theta}_{i},\phantom{\rule{0.277778em}{0ex}}i\in [1,\cdots ,N]$ | $\in {\mathbb{R}}^{Q}$ | interpolation model parameters of the ngbh. around observation i | |

${\theta}_{o},\phantom{\rule{0.277778em}{0ex}}o\in [1,\cdots ,O]$ | $\in {\mathbb{R}}^{Q}$ | mean of the interpolation model parameters at raster cell o | |

${\theta}_{l},\phantom{\rule{0.277778em}{0ex}}l\in [1,\cdots ,K]$ | $\in {\mathbb{R}}^{Q}$ | mean of the interpolation model parameters at ${f}_{l}$ | |

${y}_{ij}$ | $\in [0,1]$ | membership of observation i to cluster j | |

$\lambda $ | $\in [0,1]$ | tradeoff between proximity and similarity in clustering | |

d | $\in \mathbb{N}$ | size of neighbourhood for knn | |

z | the interpolation by the cluster model | ||

$d(\xb7,\xb7)$ | $\in [0,\infty ]$ | distance between two locations | |

$\{\xb7\}$ | set |

Region | ${\mathit{a}}_{\mathit{p}}$ | Component | Cov. Funct. | Range | Smoothness | phi |
---|---|---|---|---|---|---|

1 | 12 | Spatial | Matern | 1 | 1.7 | 0.5 |

2 | 11 | Spatial | Matern | 9 | 0.7 | 2 |

3 | 15 | Spatial | Matern | 6 | 0.6 | 1.5 |

4 | 14 | Spatial | Matern | 0.5 | 1.7 | 0.1 |

Approach | Mobile Sensors | ||
---|---|---|---|

150 | 250 | 300 | |

Global | 1.30 | 1.19 | 1.14 |

Baseline | 1.24 | 1.14 | 1.11 |

E2 | 1.25 | 1.15 | 1.14 |

EDISON (this study) | 1.17 | 1.12 | 1.10 |

Improvement over global | 10% | 6% | 4% |

Improvement over baseline | 6% | 2% | 1% |

Improvement over E2 | 6% | 3% | 4% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lovén, L.; Lähderanta, T.; Ruha, L.; Peltonen, E.; Launonen, I.; Sillanpää, M.J.; Riekki, J.; Pirttikangas, S.
EDISON: An Edge-Native Method and Architecture for Distributed Interpolation. *Sensors* **2021**, *21*, 2279.
https://doi.org/10.3390/s21072279

**AMA Style**

Lovén L, Lähderanta T, Ruha L, Peltonen E, Launonen I, Sillanpää MJ, Riekki J, Pirttikangas S.
EDISON: An Edge-Native Method and Architecture for Distributed Interpolation. *Sensors*. 2021; 21(7):2279.
https://doi.org/10.3390/s21072279

**Chicago/Turabian Style**

Lovén, Lauri, Tero Lähderanta, Leena Ruha, Ella Peltonen, Ilkka Launonen, Mikko J. Sillanpää, Jukka Riekki, and Susanna Pirttikangas.
2021. "EDISON: An Edge-Native Method and Architecture for Distributed Interpolation" *Sensors* 21, no. 7: 2279.
https://doi.org/10.3390/s21072279