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

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## 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 |

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**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% |

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**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