# A Mixed Approach to Similarity Metric Selection in Affinity Propagation-Based WiFi Fingerprinting Indoor Positioning

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## Abstract

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## 1. Introduction

## 2. System Model

#### 2.1. A General Model for WkNN Deterministic Algorithms

- Selection of k: Previous works showed the impact of the methodology of selecting the set of k RPs and the value of k itself on the positioning performance of deterministic WkNN algorithms [5,6,7], but no univocal and general way to select the value of k was provided. Section 2.2 presents the two selection schemes considered in this work.
- Clustering algorithm: Two-step algorithms considered in this work require the selection of a clustering algorithm to be used in the offline phase. Several works proposed the affinity propagation algorithm [13] as a viable solution to fulfill this task [12,14,15,16], and the algorithm was also adopted in this work. The affinity propagation algorithm and its application to RP clustering and coarse localization steps are discussed in Section 2.3.
- Similarity metrics: Different similarity metrics can be adopted in two-step algorithms for coarse and fine localization steps, respectively, leading to ${\mathit{sim}}_{i,n}^{C}\ne {\mathit{sim}}_{i,n}^{F}$, and thus, introducing an additional degree of freedom in the algorithm design. This possibility, not yet explored in the literature, will be thoroughly addressed in Section 3.

#### 2.2. k Selection Schemes

#### 2.3. Affinity Propagation Clustering for Indoor Positioning

#### 2.3.1. RP Clustering

- Responsibility $\mathit{resp}({\mathit{s}}_{{n}_{1}},{\mathit{s}}_{{n}_{2}})$: This reflects the accumulated evidence for how well-suited ${\mathrm{RP}}_{{n}_{2}}$ is to serve as the exemplar for ${\mathrm{RP}}_{{n}_{1}}$, taking into account other potential exemplars for ${\mathrm{RP}}_{{n}_{1}}$.
- Availability $\mathit{avail}({\mathit{s}}_{{n}_{1}},{\mathit{s}}_{{n}_{2}})$: This reflects the accumulated evidence for how appropriate it would be for ${\mathrm{RP}}_{{n}_{1}}$ to choose ${\mathrm{RP}}_{{n}_{2}}$ as its exemplar, taking into account the support from other RPs that ${\mathrm{RP}}_{{n}_{2}}$ should be an exemplar.

- Degeneracies: Degeneracies can arise, for example, if the similarity metric is commutative and two elements (RPs) are isolated from all of the others. In this case, oscillations in deciding which of the two elements should be the exemplar might appear. The solution proposed in [13] is to add a small amount of random noise to similarities values to avoid such a deadlock situation.
- Outliers: The algorithm might occasionally lead to an RP belonging to a cluster, but being physically far away from the cluster exemplar. In [12], taking advantage of the knowledge of the position of each RP, each outlier is forced to join the cluster characterized by the exemplar at the minimum distance from the outlier itself.

#### 2.3.2. Cluster Selection

- Similarity to the exemplar fingerprints (Criterion I): the similarity ${\mathit{sim}}_{i,{n}_{c}}^{C}$ between the i-th online reading and each ${n}_{c}$-th exemplar fingerprint (with ${n}_{c}=1,2,\cdots ,{N}_{c}$) is evaluated, and clusters corresponding to exemplars with similarity values above a predefined threshold α are selected.
- Similarity to the cluster average fingerprints (Criterion II): in this case, a cluster fingerprint is computed by averaging the RP fingerprints within the cluster. The similarity ${\mathit{sim}}_{i,{n}_{c}}^{C}$ between the i-th online reading and each ${n}_{c}$-th cluster fingerprint (with ${n}_{c}=1,2,\cdots ,{N}_{c}$) is then evaluated, and the clusters with similarity values above α are selected.

#### 2.3.3. Similarity Metric for RP Clustering and Cluster Selection Steps

## 3. Similarity in the Context of WiFi Fingerprinting Indoor Positioning

#### 3.1. Offline Phase Similarity Metrics

- RP positions ${\mathit{p}}_{n}$ (with $n=1,2,\cdots ,N$).
- RP RSS fingerprints ${\mathit{s}}_{n}$ (with $n=1,2,\cdots ,N$).

#### 3.1.1. A Spatial Distance-Based Similarity Metric

#### 3.2. Offline/Online Phase Similarity Metrics

#### 3.2.1. Minkowski Distance-Based Metrics: Manhattan and Euclidean

#### 3.2.2. Inner Product-Based Metrics: Cosine Similarity and Pearson Correlation

#### 3.2.3. A Frequentist Approach: p-values from the Pearson Correlation

_{0}, that the correlation $\rho \left({S}_{i},{S}_{n}\right)$ between two variables is zero, given the $R({\mathit{s}}_{i},{\mathit{s}}_{n})$ value computed on the sample variables. A p-value lower than a threshold significance level ${\alpha}_{SL}$ indicates that H

_{0}should be rejected and that the evaluated $R({\mathit{s}}_{i},{\mathit{s}}_{n})$ has indeed a statistical significance, numerically associated with the p-value.

#### 3.2.4. Exploring Interdisciplinary Metrics: Shepard Similarity

#### 3.3. A Comparative Framework for RSS-Based Similarity Metrics

**Figure 1.**Example of reference point (RP) masks obtained for a specific test point (TP). (

**a**) ${\text{M}}_{i,ideal}$, (

**b**) ${\text{M}}_{i,{\left({D}^{2}\right)}^{-1}}$.

## 4. Experimental Results and Discussion

#### 4.1. Testbed Implementation and Performance Indicators

**Figure 3.**Positions of the ${N}_{1}$ RPs on the Department of Information Engineering, Electronics and Telecommunications (DIET) first floor.

#### 4.2. Flat Algorithms

- Ideal sorting/ideal weighting (ISIW): This case corresponds to an ideal upper bound benchmark where the spatial distances between each TP and the RPs are assumed to be known and then used for both RP sorting and weighting.
- Ideal sorting/real weighting (ISRW): In this case, the spatial distances between each TP and the RPs are assumed to be known and used during the RP sorting phase. However, once the RPs are sorted, different RSS-based metrics are evaluated as RP weights, in order to isolate the impact of RSS-based metrics on the weighting phase.
- Real sorting/real weighting (RSRW): This case represents the only feasible use case, where the spatial distances between each TP and the RPs are unknown. In this case, RSS-based metrics are evaluated and then used in both RP sorting and weighting phases.

**Figure 4.**CDF of the 3D positioning error ϵ with $k=3$. ISIW, ideal sorting/ideal weighting; ISRW, ideal sorting/real weighting; RSRW, real sorting/real weighting. (

**a**) ISIW (${d}^{-1}$) vs. ISRW (${R}^{2}$, $(p$-value${)}^{-1}$, ${\left({D}^{2}\right)}^{-1}$ and ${S}^{2}$); (

**b**) ISIW (${d}^{-1}$) vs. RSRW (${R}^{2}$, $(p$-value${)}^{-1}$, ${\left({D}^{2}\right)}^{-1}$ and ${S}^{2}$).

**Figure 5.**Average 3D error $\overline{\u03f5}$ as a function of k. (

**a**) ISIW (${d}^{-1}$) vs. ISRW (${R}^{2}$, $(p$-value${)}^{-1}$, ${\left({D}^{2}\right)}^{-1}$ and ${S}^{2}$); (

**b**) ISIW (${d}^{-1}$) vs. RSRW (${R}^{2}$, $(p$-value${)}^{-1}$, ${\left({D}^{2}\right)}^{-1}$ and ${S}^{2}$).

#### 4.3. Affinity Propagation-Based Algorithms

#### 4.3.1. Topology

- Clustering was performed separately for the two floors composing the testbed, assuming the knowledge of the floor for each RP.
- With reference to the degeneracy issue identified in Section 2.3.1, the solution proposed in [13] of adding random noise to similarity values was not implemented, since different similarities are characterized by significantly different ranges of values, making it impossible to add noise with the same power to all similarity metrics.
- On the other hand, the outlier issue, also identified in Section 2.3.1, was addressed by actually implementing the solution proposed in [12] to eliminate such outliers. This leads in all cases to clusters occupying convex regions on each floor. Figure 6 shows as an example the clusters obtained on the DIET first floor, using the gen metric.

**Figure 6.**RP clusters and exemplars obtained on the DIET first floor with gen as the similarity metric (areas represented in different colors indicate different clusters; larger dots indicate exemplars).

Metric | ${\mathit{N}}_{\mathit{c}}$ | $\overline{\left|\mathit{C}\right|}$ | $\mathbf{max}\left\{\left|\mathit{C}\right|\right\}$ | $\mathbf{min}\left\{\left|\mathit{C}\right|\right\}$ | $\mathbf{var}\left\{\left|\mathit{C}\right|\right\}$ |
---|---|---|---|---|---|

${d}^{-1}$ | 37 | 3.62 | 6 | 2 | 1.07 |

gen | 13 | 10.31 | 15 | 5 | 9.06 |

${R}^{2}$ | 13 | 10.31 | 19 | 4 | 27.73 |

$(p$-value${)}^{-1}$ | 43 | 3.11 | 6 | 2 | 1.10 |

$CS$ | 13 | 10.31 | 15 | 5 | 7.90 |

${\left({D}^{1}\right)}^{-1}$ | 29 | 4.62 | 8 | 2 | 3.17 |

${\left({D}^{2}\right)}^{-1}$ | 27 | 4.96 | 8 | 2 | 4.19 |

${S}^{1}$ | 34 | 3.94 | 7 | 2 | 2.54 |

${S}^{2}$ | 20 | 67 | 69 | 2 | 216.01 |

- The first group including gen, ${R}^{2}$ and $CS$ metrics, leading to the creation of relatively few, large clusters, although with different variances.
- The second group, including ${d}^{-1}$, $(p$-value${)}^{-1}$, ${\left({D}^{1}\right)}^{-1}$, ${\left({D}^{2}\right)}^{-1}$ and ${S}^{1}$ metrics, leading to small clusters with low variances. $(p$-value${)}^{-1}$ in particular leads to the largest number of clusters with the lowest variance among all metrics.

#### 4.3.2. Positioning Accuracy: A Backward Approach

- Cluster matching criterion selection: The first step in the backward approach was the selection of one of the cluster matching criteria introduced in Section 2.3.2 in order to focus on a single criterion in the following steps of the analysis. In order to do so, both ${m}_{1}$ and ${m}_{2}$ were set equal to the gen metric, while the four metrics characterized by the smallest ${\Delta}^{k}$, already considered in the analysis of flat algorithms (see Section 3.3 and Section 4.2), were adopted as the ${m}_{3}$ metric, in order to identify which of the two criteria performs better under different conditions.
- ${m}_{2}/{m}_{3}$ selection: Having selected the best cluster matching criterion, the second step focused on the analysis and possibly the selection of the best metrics to be used during both coarse and fine localization steps. To do so, the ${m}_{1}$ metric was kept set to the gen metric, while ${m}_{2}$ and ${m}_{3}$ metrics were allowed to change. In particular, all of the RSS-based metrics introduced in Section 3.2 were considered as candidates for the role of the ${m}_{2}$ metric, while, based on the analysis already carried out in Section 3.3 and Section 4.2, candidates for the role of the ${m}_{3}$ metric were restricted to the four metrics with the smallest ${\Delta}^{k}$. Based on the positioning errors achieved by the different gen/${m}_{2}/{m}_{3}$ combinations, this phase concluded with a joint ${m}_{2}$ and ${m}_{3}$ selection.
- ${m}_{1}$ selection: The last step was the analysis of the impact of different metrics on the RP clustering step, aiming at the selection of the best one. In this case, all metrics defined in both Section 3.1 and Section 3.2 were eligible for the role of the ${m}_{1}$ metric, while keeping both ${m}_{2}$ and ${m}_{3}$ set to the metrics selected as a result of the ${m}_{2}/{m}_{3}$ selection step.

**Figure 7.**Impact of the matching cluster Criteria I/II on the average 3D error $\overline{\u03f5}$. (

**a**) flat vs. gen/gen/${R}^{2}$ combinations; (

**b**) flat vs. gen/gen/$(p$-value${)}^{-1}$ combinations; (

**c**) flat vs. gen/gen/${\left({D}^{2}\right)}^{-1}$ combinations; (

**d**) flat vs. gen/gen/${S}^{2}$ combinations.

**Figure 8.**Impact of the ${m}_{2}$ and ${m}_{3}$ metrics on the average 3D error $\overline{\u03f5}$. (

**a**) flat vs. gen/${m}_{2}$/${R}^{2}$ combinations; (

**b**) flat vs. gen/${m}_{2}$/$(p$-value${)}^{-1}$ combinations; (

**c**) flat vs. gen/${m}_{2}$/${\left({D}^{2}\right)}^{-1}$ combinations; (

**d**) flat vs. gen/${m}_{2}$/${S}^{2}$ combinations.

- Impact of the ${m}_{3}$ metric: The use of different ${m}_{3}$ metrics significantly affects the positioning accuracy. Among all metrics, $(p$-value${)}^{-1}$ emerged as the best candidate to play the role of ${m}_{3}$, since at the same time, it minimizes the impact of the RP selection threshold and the positioning error, with a value around $2.76$ m below any other metric.
- Impact of the ${m}_{2}$ metric: Different ${m}_{2}$ metrics have a negligible effect on the positioning accuracy, with a difference in the evaluated average errors in a range of about twenty centimeters. The substantial independence of the performance from the selected ${m}_{2}$ metric is confirmed by Figure 9, showing the impact of metrics on the value of ${\overline{N}}_{i}$ (defined in Section 4.1 as the average number of RPs within the clusters selected after the coarse localization step).

**Figure 10.**Impact of the ${m}_{1}$ metric on the average 3D error $\overline{\u03f5}$: flat vs. ${m}_{1}$/gen/$(p$-value${)}^{-1}$ combinations.

- Figure 11 shows that if a ${m}_{3}$ metric different from $(p$-value${)}^{-1}$, characterized by a lower selectivity, is selected, the impact of ${m}_{1}$ can be significantly larger, especially when many RPs are selected.

#### 4.3.3. Computational Complexity

- As expected, the number of computed similarity values is always equal to N for a flat algorithm.
- Two-step algorithms significantly reduce the average computational complexity of the online phase. Moreover, ${m}_{1}$ metrics can be divided into two different groups: (1) ${m}_{1}$ metrics that minimize the number of formed clusters and conversely maximize the number of RPs selected after the coarse localization step; and (2) ${m}_{1}$ metrics that maximize the number of formed clusters and conversely minimize the number of RPs remaining after the coarse localization step.
- The first group of metrics shows a ${\overline{N}}_{sim}$ value slightly lower than the second group. $CS$ in particular obtained the lowest computational complexity; at the same time, both groups show a ${\overline{N}}_{sim}$ value significantly lower than the flat algorithm.
- As a final note, it can be observed that the ${m}_{1}=(p$-value${)}^{-1}$ metric, which minimizes positioning error as found in Section 4.3.2, is not the metric minimizing ${\overline{N}}_{sim}$.

**Figure 11.**Impact of the ${m}_{1}$ metric on the average 3D error $\overline{\u03f5}$: flat vs. ${m}_{1}$/gen/${\left({D}^{2}\right)}^{-1}$ combinations.

**Figure 12.**${\overline{N}}_{sim}$ for both flat and two-step algorithms (with several ${m}_{1}$ metrics).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Caso, G.; De Nardis, L.; Di Benedetto, M.-G.
A Mixed Approach to Similarity Metric Selection in Affinity Propagation-Based WiFi Fingerprinting Indoor Positioning. *Sensors* **2015**, *15*, 27692-27720.
https://doi.org/10.3390/s151127692

**AMA Style**

Caso G, De Nardis L, Di Benedetto M-G.
A Mixed Approach to Similarity Metric Selection in Affinity Propagation-Based WiFi Fingerprinting Indoor Positioning. *Sensors*. 2015; 15(11):27692-27720.
https://doi.org/10.3390/s151127692

**Chicago/Turabian Style**

Caso, Giuseppe, Luca De Nardis, and Maria-Gabriella Di Benedetto.
2015. "A Mixed Approach to Similarity Metric Selection in Affinity Propagation-Based WiFi Fingerprinting Indoor Positioning" *Sensors* 15, no. 11: 27692-27720.
https://doi.org/10.3390/s151127692