# Improving Signal-Strength Aggregation for Mobile Crowdsourcing Scenarios

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## Abstract

**:**

## 1. Introduction

- Measurements are not uniformly distributed in the area, as they are defined by human mobility patterns [19].
- The number of measurements in small areas (e.g., coverage area of a single cell) could not be high enough to be considered representative enough [1].
- The measurements present accuracy errors in both signal-strength values and geographic coordinates [1].

## 2. Related Work

## 3. Common Pitfalls in Using Log-Scaled Signal Strength

**logarithms**are defined for numbers, but

**have no physical correspondence in operations involving actual physical quantities**”.

**Lemma**

**1.**

**do not**meet Bridgman’s principle of absolute significance of relative magnitude.

**Proof.**

#### 3.1. Averaging Signal Strength

- (2017) Sabu et al. [11] conducted a correlation study between signal strength and rainfall intensity in an area of interest, where logarithmic ASU values were aggregated by taking the arithmetic mean. As result, the authors concluded that the drop of signal strength during rainfall was not as significant as expected by the theoretical hypothesis.
- (2018) In the data exploration section provided by Sung et al. [8], an area of interest was divided in smaller square areas. For each square area, signal strength was reported by taking the mean of several logarithmic ASU measurements. As result, a weak geographical correlation between signal strength and throughput was found.
- (2015) In the research work of Marina et al. [13], signal-strength samples in dBm units were divided according to their context (indoor or outdoor), and then aggregated by taking the arithmetic mean. Then, the authors analyzed the great impact of user context (indoor or outdoor) on the received signal strength.
- (2013) Sonntag et al. [1] created signal-strength coverage maps by taking the arithmetic mean of values represented as percentages. The percentage values are calculated linearly from logarithmic signal strength, so they are also logarithmic values. The authors concluded that coverage maps created from crowdsourced signal strength were not very good at presenting the actual transport quality.

#### 3.2. Comparing Signal Strength

- (2019) Recently, Alimpertis et al. [45] proposed a new method based on machine learning to perform signal-strength prediction, i.e., given a set of signal-strength measurements in an area, estimate signal-strength values in other singular points. They claimed that their method consistently obtains lower prediction errors than related state-of-art algorithms. Nevertheless, it can be shown that the comparison of errors used by them leads to inconclusive results. Using the values shown in Table 4 of [45], for cell ID $x204$, their method obtains an average error of $2.3\phantom{\rule{0.277778em}{0ex}}\mathrm{dB}$, outperforming Ordinary Kriging (OK) and Ordinary Kriging Detrending method (OKD) which obtains average errors of $3.85\phantom{\rule{0.277778em}{0ex}}\mathrm{dB}$ and $2.99\phantom{\rule{0.277778em}{0ex}}\mathrm{dB}$ respectively. However, if we consider the case in which their method’s error is $+2.3\phantom{\rule{0.277778em}{0ex}}\mathrm{dB}$, OK’s error is $-3.85\phantom{\rule{0.277778em}{0ex}}\mathrm{dB}$, and OKD’s error is $-2.99\phantom{\rule{0.277778em}{0ex}}\mathrm{dB}$, all in relation to the expectation of the signal strength of cellID $x204$ ($-96\phantom{\rule{0.277778em}{0ex}}\mathrm{dBm}$), we have that OK method has an error $26\%$ lower than their method’s error, and OKD method has an error $30\%$ lower than their method’s error (using linear watt scale). In that case (a possible case given the prediction errors stated in the paper), their method actually gets worse results than related state-of-art algorithms.

## 4. Signal-Strength Aggregation

#### 4.1. Arithmetic Mean

**watts**) a

**large number**of individual RF measurements” [16]. Nevertheless, some works use the wrong methodology by explicitly applying the arithmetic mean over signal strength in logarithmic scale [1,8,11,13] as stated in Section 3.1.

#### 4.2. Median Value

#### 4.3. Our Proposal: Average Based on Interpolation (ABOI Method)

## 5. Mathematical Foundation for the Use of the ABOI Method

#### 5.1. ABOI Theorem

**Theorem**

**2.**

**Proof.**

#### 5.2. Improvement on Arithmetic Mean

**Preposition**

**2.1.**

**Preposition**

**2.2.**

**Proof**

**of**

**Preposition**

**2.1.**

**Proof**

**of**

**Preposition**

**2.2.**

## 6. Experimental Results

- We considered areas with signal strength coming from only one base transceiver station (BTS).

#### 6.1. Simulated Scenario

- Completely uniform distribution on the grid, which is commonly used, but not realistic for Mobile Crowdsourcing scenarios, as discussed in Section 1.
- Considering the mobility model based on social network theory proposed by Musolesi et al. [19]. This model is closer to the spatial distribution of Mobile Crowdsourcing measurements, as they are defined by human mobility.

#### 6.2. Real Data

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

QoS | Quality of Service |

ABOI | Average Based on Interpolation |

TCP | Transmission Control Protocol |

BTS | Base Transceiver Station |

ASU | Arbitrary Strength Unit |

OK | Ordinary Kriging |

OKD | Ordinary Kriging Detrending |

MAE | Mean Absolute Error |

MAPE | Mean Absolute Percentage Error |

MASE | Mean Absolute Scaled Error |

MSE | Mean Square Error |

RMSE | Root Mean Squared Error |

LTE | Long-Term Evolution |

## Appendix A. Proof of Mathematical Expression (16)

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**Figure 1.**Example of set N with $n=30$ positions of initial measurements (

**left**), and set M with $m=3481$ equispaced positions over A where to interpolate signal strength (

**right**).

**Figure 3.**Example of spatial distribution for 100 samples using uniform distribution (

**left**) and distribution based on social network theory (

**right**).

**Figure 4.**Simulated scenario. Boxplots for ${\overline{P}}_{A}$ prediction using the three aggregation methods and different sample sizes, selected by uniform distribution. Real ${\overline{P}}_{A}$ value in red line.

**Figure 5.**Simulated scenario. RMSE for ${\overline{P}}_{A}$ prediction for different sample sizes with uniform distribution.

**Figure 6.**Simulated scenario. Boxplots for ${\overline{P}}_{A}$ prediction using the three aggregation methods and different sample sizes, selected by distribution based on social network theory. Real ${\overline{P}}_{A}$ value in red line.

**Figure 7.**Simulated scenario. RMSE for ${\overline{P}}_{A}$ prediction for different sample sizes with distribution based on social network theory.

**Figure 8.**Real signal strength around the vicinity of a single LTE BTS. Color represents the dBm value of samples.

**Figure 9.**Real scenario. Boxplots for ${\overline{P}}_{A}$ prediction using the three aggregation methods and different sample sizes, selected by uniform distribution. Calculated ${\overline{P}}_{A}$ value in red line.

**Figure 10.**Real scenario. RMSE for ${\overline{P}}_{A}$ prediction for different sample sizes with uniform distribution.

**Figure 11.**Real scenario. Boxplots for ${\overline{P}}_{A}$ prediction using the three aggregation methods and different sample sizes, selected by distribution based on social network theory. Calculated ${\overline{P}}_{A}$ value in red line.

**Figure 12.**Real scenario. RMSE for ${\overline{P}}_{A}$ prediction for different sample sizes with distribution based on social network theory.

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**MDPI and ACS Style**

Madariaga, D.; Madariaga, J.; Bustos-Jiménez, J.; Bustos, B.
Improving Signal-Strength Aggregation for Mobile Crowdsourcing Scenarios. *Sensors* **2021**, *21*, 1084.
https://doi.org/10.3390/s21041084

**AMA Style**

Madariaga D, Madariaga J, Bustos-Jiménez J, Bustos B.
Improving Signal-Strength Aggregation for Mobile Crowdsourcing Scenarios. *Sensors*. 2021; 21(4):1084.
https://doi.org/10.3390/s21041084

**Chicago/Turabian Style**

Madariaga, Diego, Javier Madariaga, Javier Bustos-Jiménez, and Benjamin Bustos.
2021. "Improving Signal-Strength Aggregation for Mobile Crowdsourcing Scenarios" *Sensors* 21, no. 4: 1084.
https://doi.org/10.3390/s21041084