# Indoor Localization Based on Weighted Surfacing from Crowdsourced Samples

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## Abstract

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## 1. Introduction

## 2. Related Work and System Overview

#### 2.1. Related Work

#### 2.2. System Overview

`Weighting crowdsourced samples`assigns each crowdsourced sample a reliability weight based on our proposed cross-domain cluster intersection algorithm.

`Fitting radio surfaces`constructs a radio propagation surface for each AP based on the weighted samples.

`Weighting fitted surfaces`further assigns each fitted surface with two weights for discriminating their contributions for online localization.

`Constructing subarea fingerprints`creates an RSS fingerprint for each subarea from its fitted and weighted surfaces.

`Subarea determination`first locates an online test fingerprint into one subarea according to our proposed weighted signal distance.

`Location search`searches the coordinate for the test fingerprint based on the gradient search on the constructed surfaces. Figure 1 presents the main flowchart of the proposed system, and Table 1 lists the symbols used in this paper as well as their notations.

## 3. The Offline Weighted Surfacing Algorithm

#### 3.1. Weighting Crowdsourced Samples

#### 3.2. Fitting Radio Surfaces

#### 3.3. Weighting Fitted Surfaces

#### 3.4. Constructing Subarea Fingerprints

## 4. The Online Positioning Algorithm

**Subarea Determination:**Let ${\overrightarrow{f}}_{t}$ denote the RSS vector of a test sample, and ${\overrightarrow{f}}_{s}$ the sth subarea fingerprint. Let ${\mathcal{A}}_{int}$ denote the set of hearable APs by both ${\overrightarrow{f}}_{t}$ and ${\overrightarrow{f}}_{s}$. We compute the weighted signal distance between ${\overrightarrow{f}}_{t}$ and ${\overrightarrow{f}}_{s}$ as:

**Location Search:**Assume that the sth subarea is selected in the first phase. We next search a space point $(\widehat{x},\widehat{y})$ in this subarea to minimize the weighted signal difference between ${\overrightarrow{f}}_{t}$ and subarea surfaces:

## 5. Field Measurements and Experiments

#### 5.1. Experiment Settings

**Experiment Schemes**: According to their annotated locations, crowdsourced samples can be assigned into different grids to construct grid fingerprints. Similarly, they can also be grouped into different clusters in the signal space to obtain cluster fingerprints. We tested the following peer localization schemes to examine these typical approaches.

`FGrid`emulates the traditional site-survey fingerprinting based on grid fingerprints, which divides the subarea into several non-overlapping grid cell to contain samples, and assigns each new sample into its nearest grid cell. For each grid cell, a grid fingerprint is composed by averaging all samples located within the grid cell, and the location of the grid fingerprint is annotated as the grid center. In the online phase, we used the nearest neighbor algorithm.`SGrid`is similar to the`FGrid`to obtain grid fingerprints. We then constructed surfaces based on these fingerprints in the offline phase. In the online phase, we used the same surface search method as the one in our proposed`SWSample`.`SRaw`retains the original position of every crowdsourced sample and fits propagation surfaces based on them. In the online phase, we used the same surface search method as the one in our proposed`SWSample`.`SCluster`clusters the samples in signal domain only. For each cluster, we obtained a cluster fingerprint, which is the average of its cluster members’ RSS vectors. The location of a cluster fingerprint is the geometric center of the cluster members. We fitted the propagation surfaces for every AP based on these cluster fingerprints. In the online phase, we used the surface search method the same as the one in our proposed`SWSample`.`SWSample`is the proposed scheme.

#### 5.2. Surface Fitting Examples

`SWSample`fitted surface in Figure 7, we also color the sample weight as shown by the weight color spectrum alongside the graph. It can be seen that the proposed scheme could produce a smoother surface, compared with other schemes. If we assume that this AP is located at the coordinate around the highest RSS value, then we could observe that the surface in Figure 7 is more like an attenuated sphere centered at the AP. The Keenan–Motley path loss model has been widely adopted to characterize the radio propagation in mobile cellular networks. If such a model could still be applicable in a small and open space such as a room, then our fitted surface resembles the most to this model, which might also help to explain the effectiveness of our weighted surface fitting.

#### 5.3. Experiment Results

**Uniformly distributed samples:**We first considered the scenario that all crowdsourced samples are uniformly distributed in the experiment environment, that is, we used crowdsourced samples from ${\mathcal{S}}_{site}\bigcup {\mathcal{S}}_{walk}$. Figure 8 plots the average localization error (ALE) against the number of crowdsourced samples randomly drawn from ${\mathcal{S}}_{site}\bigcup {\mathcal{S}}_{walk}$. It was first observed that all the surfacing schemes outperform the grid fingerprinting

`FGrid`, which validates the effectiveness of using fitted radio propagation surfaces for localization. When the number of samples increases, from about $0.33{M}_{g}$ to $20{M}_{g}$ with ${M}_{g}$ the number of total grid cells, the ALE of the surfacing schemes first decreases and then increases. At first, the number of samples is not large enough to well fit actual surfaces. In this case, our scheme

`SWSample`has a slightly higher ALE than other surfacing schemes (see the first two points in Figure 8) due to its sample selection. On the other hand, if the noisy samples are too many, the surfaces may be overfitted for unreliable samples. However, ours presents a decent degradation and the ALE of using all 27,040 samples is 1.54 m, slightly higher than the best case of 1.45 m of using 3605 samples. The positioning accuracy improvement of our scheme are $36.71\%$ over

`FGrid`and $9.41\%$ over

`SRaw`, respectively. Compared with the

`SRaw`scheme, the improvement can be attributed to our sample weighting and selection algorithm, which only chooses those reliable samples for weighted surface fitting, leading to a more accurate radio map and better positioning results.

`SWSample`still performs the best. Figure 10 plots the cumulative distribution function (CDF) of localization error. It is worth noting that, besides a low median localization error of 1.51 m, our

`SWSample`has a low 90% percentile error of only 2.64 m. To provide the exact numbers, Table 2 summarizes the localization error results for three situations, namely, $\sigma =0.6$ m, $\sigma =0.9$ m, and $\sigma =1.2$ m, respectively.

**Non-uniformly distributed samples:**It is also often the case that crowdsourced samples are not uniformly distributed in the whole environment. To examine this nonuniform density issue, we only use the samples from ${\mathcal{S}}_{walk}$ to fit surfaces. That is, the subregion of chairs and desks in each room do not contain crowdsourced samples. However, as we intentionally include location annotation errors, some samples may still be annotated to locations within such a vacant subregion. As shown in Figure 11 and Figure 12, it is not unexpected to observe that all schemes suffer from such a nonuniform density situation, comparing with the results in Figure 8. However, our

`SWSample`scheme can still outperform other schemes in most of cases. The positioning accuracy improvements are $36.85\%$ over

`FGrid`and $18.79\%$ over

`SRaw`, respectively. Furthermore, the median and 90% localization errors in Figure 13 are 2.04 m and 3.24 m, respectively, which are comparable to the uniform density case. Table 2 summarizes the localization error results from three situations for non-uniformly distributed samples. It can be observed that our proposed scheme has great potential to obtain a better result in this non-uniformly distributed case, which illustrates its robustness for tackling the nonuniform density challenge.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The flowchart of the proposed system: In the offline phase, crowdsourced samples are each weighted according to our algorithm. For each access point and for one subarea, its radio propagation surface is firstly fitted and also weighted from those selected and weighted samples. Subarea fingerprints are then composed from fitted surfaces. In the online phase, a test sample is first compared with subarea fingerprints to determine its belonging subarea, and then a gradient search is used to estimate its exact location.

**Figure 2.**Illustration of the cross-domain cluster intersection algorithm: In the physical space, samples are clustered according to their annotated coordinates. In the signal space, samples are clustered according to the RSS distances. The weight of a sample is determined by the common samples between its belonged physical cluster and signal cluster.

**Figure 3.**The layout of the indoor environment. A grid lattice has been used to collect samples, with in total 1368 grid cells each with size $0.6\times 0.6$ m${}^{2}$. Besides, pedestrian trajectories have also been used to collect samples for the corridor and walkable pathways in each room.

**Figure 4.**Illustration of fitted surface by

`SGrid`. We choose one AP for Room A and fit its surface from 1800 samples randomly drawn from ${\mathcal{S}}_{site}\bigcup {\mathcal{S}}_{walk}$. Crowdsourced samples are assigned to grid cells. A

`grid fingerprint`is composed by averaging all samples in the grid cell, and its location is the grid center. The fitted surface is based on the grid fingerprints.

**Figure 5.**Illustration of fitted surface by

`SRaw`. We choose one AP for Room A and fit its surface from 1800 samples randomly drawn from ${\mathcal{S}}_{site}\bigcup {\mathcal{S}}_{walk}$. All crowdsourced samples are used for surface fitting, without sample weighting and selection.

**Figure 6.**Illustration of fitted surface by

`SCluster`. We choose one AP for Room A and fit its surface from 1800 samples randomly drawn from ${\mathcal{S}}_{site}\bigcup {\mathcal{S}}_{walk}$. All crowdsourced samples are first clustered in the signal space. For each cluster, a

`cluster fingerprint`is composed by averaging the RSS vectors of its cluster members, and its location is the geometric center of the cluster members. The fitted surface is based on the cluster fingerprints.

**Figure 7.**Illustration of fitted surface by our proposed

`SWSample`. We choose one AP for Room A and fit its surface from 1800 samples randomly drawn from ${\mathcal{S}}_{site}\bigcup {\mathcal{S}}_{walk}$. Crowdsourced samples are weighted and selected for surface construction. The sample weight is illustrated by the dot color in the figure.

**Figure 8.**Comparison of localization performance. The average localization error (ALE) vs. the number of crowdsourced samples ${M}_{all}$, when using crowdsourced samples from ${\mathcal{S}}_{site}\bigcup {\mathcal{S}}_{walk}$. The standard deviation of location offset $\sigma =1.2$ m.

**Figure 9.**Comparison of localization performance. The average localization error (ALE) vs. the standard deviation $\sigma $ of location offset, where ${M}_{all}=27,040$.

**Figure 10.**Comparison of cumulative distribution function (CDF) localization error, where ${M}_{all}$ = 27,040 and $\sigma =1.2$ m.

**Figure 11.**Comparison of localization performance, when using crowdourced samples only from ${\mathcal{S}}_{walk}$. The average localization error (ALE) vs. the number of crowdsourced samples ${M}_{all}$, where $\sigma =1.2$ m.

**Figure 12.**Comparison of localization performance, when using crowdsourced samples only from ${\mathcal{S}}_{walk}$. The average localization error (ALE) vs. the standard deviation $\sigma $ of location offset, where ${M}_{all}=4456$.

**Figure 13.**Comparison of cumulative distribution function (CDF) localization error with ${M}_{all}=4456$ and $\sigma =1.2$ m, when using crowdourced samples only from ${\mathcal{S}}_{walk}$.

Symbol | Definition |
---|---|

$\mathcal{S}$ | A set of crowdsourced samples in one subarea. |

M | The number of crowdsourced samples in $\mathcal{S}$, $M=\left|\mathcal{S}\right|$. |

${\mathit{s}}_{i}$ | The ith crowdsourced sample in $\mathcal{S}$. |

${\overrightarrow{l}}_{i}$ | The annotated location of the ith crowdsourced sample. |

${\overrightarrow{r}}_{i}$ | The RSS vector of the ith crowdsourced sample. |

N | The maximum number of hearable AP in $\mathcal{S}$. |

K | The number of clusters. |

${\mathcal{C}}^{p}$ | The set of clusters in the physical space. |

${\mathcal{C}}^{s}$ | The set of clusters in the signal space. |

${\gamma}_{i}$ | The cross-domain cluster coefficient of the ith sample. |

${\omega}_{i}$ | The reliability weight of the ith sample. |

$\varphi (x,y)$ | The RSS surface function. |

${p}_{th}$ | The percentile threshold in sample selection method. |

${\omega}_{th}$ | The weight threshold in sample selection method. |

$\overrightarrow{\omega}$ | The increasing order of sample reliability weight. |

${\omega}_{k}$ | The reliability weight at the ${p}_{th}$ percentile in $\overrightarrow{\omega}$. |

${\mathcal{S}}^{\prime}$ | The set of select samples. |

$\mathcal{A}$ | The set of hearable Aps by samples in ${\mathcal{S}}^{\prime}$. |

${\alpha}_{ij}$ | The surface coefficient of the RSS surface function. |

$\mathcal{R}$ | The set of RSS values from an AP in ${\mathcal{S}}^{\prime}$. |

${\overline{r}}_{i}$ | The normalized elements in $\mathcal{R}$. |

$\eta $ | The entropy-like quantity for each AP in $\mathcal{A}$. |

${\rho}_{n}^{sub}$ | The surface weight of nth AP in $\mathcal{A}$ for subarea determination. |

${\rho}_{n}^{loc}$ | The surface weight of nth AP in $\mathcal{A}$ for location search. |

$\overrightarrow{f}$ | Subarea fingerprint. |

$\mathcal{G}$ | The set of grid cells in one subarea. |

G | The number of grids in $\mathcal{G}$, $G=\left|\mathcal{G}\right|$. |

${\overrightarrow{f}}_{t}$ | The RSS vector of a test sample. |

${\overrightarrow{f}}_{s}$ | The sth subarea fingerprint. |

${\mathcal{A}}_{int}$ | The set of hearable APs by both ${\overrightarrow{f}}_{t}$ and ${\overrightarrow{f}}_{s}$. |

${D}_{s}$ | The weighted signal distance between the test sample and a subarea. |

${M}_{g}$ | The number of grid cells. |

$\sigma $ | The standard deviation of location offset. |

${\mathcal{S}}_{site}$ | The set of samples from site survey. |

${\mathcal{S}}_{walk}$ | The set of samples from pedestrian trajectories. |

Error (m) | $\mathit{\sigma}=0$ m | $\mathit{\sigma}=0.6$ m | $\mathit{\sigma}=1.2$ m | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean | 50% | 90% | Mean | 50% | 90% | Mean | 50% | 90% | ||

Uni. | FGrid | 2.479 | 2.448 | 3.672 | 2.284 | 2.086 | 3.744 | 2.421 | 2.217 | 3.868 |

SGrid | 1.571 | 1.353 | 2.595 | 1.726 | 1.630 | 2.884 | 1.898 | 1.757 | 3.048 | |

SRaw | 1.575 | 1.370 | 2.645 | 1.618 | 1.524 | 2.694 | 1.711 | 1.688 | 2.873 | |

SCluster | 1.552 | 1.364 | 2.550 | 1.708 | 1.657 | 2.875 | 1.916 | 1.879 | 3.111 | |

SWSample | 1.373 | 1.124 | 2.413 | 1.374 | 1.243 | 2.470 | 1.513 | 1.366 | 2.640 | |

Non-uni. | FGrid | 2.897 | 2.776 | 3.672 | 2.982 | 2.813 | 4.477 | 3.059 | 2.932 | 4.502 |

SGrid | 2.164 | 1.691 | 3.522 | 2.086 | 1.679 | 3.402 | 2.169 | 1.795 | 3.499 | |

SRaw | 2.155 | 1.713 | 3.459 | 2.221 | 1.732 | 3.594 | 2.322 | 1.898 | 3.647 | |

SCluster | 2.063 | 1.602 | 3.497 | 2.009 | 1.584 | 3.287 | 2.144 | 1.752 | 3.477 | |

SWSample | 1.854 | 1.497 | 3.172 | 1.951 | 1.472 | 3.217 | 2.043 | 1.625 | 3.242 |

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## Share and Cite

**MDPI and ACS Style**

Lin, J.; Wang, B.; Yang, G.; Zhou, M. Indoor Localization Based on Weighted Surfacing from Crowdsourced Samples. *Sensors* **2018**, *18*, 2990.
https://doi.org/10.3390/s18092990

**AMA Style**

Lin J, Wang B, Yang G, Zhou M. Indoor Localization Based on Weighted Surfacing from Crowdsourced Samples. *Sensors*. 2018; 18(9):2990.
https://doi.org/10.3390/s18092990

**Chicago/Turabian Style**

Lin, Junhong, Bang Wang, Guang Yang, and Mu Zhou. 2018. "Indoor Localization Based on Weighted Surfacing from Crowdsourced Samples" *Sensors* 18, no. 9: 2990.
https://doi.org/10.3390/s18092990