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Continuous Space Estimation: Increasing WiFi-Based Indoor Localization Resolution without Increasing the Site-Survey Effort^{ †}

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## Abstract

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## 1. Introduction

## 2. Related Work

- Deterministic:
- -
- Propagation model based: Localization is carried out estimating the distance to nearby APs by means of a WiFi signal propagation model [27,28,29,30,31,32]. Propagation models describe how the signal is propagated in the environment, and they are used to translate the RSS into a distance. The APs’ location in the environment is normally known a priori. Using the distance to the APs and their positions in the environment, the typical choice is to use lateration algorithms to perform the localization.
- -
- Fingerprint based: These systems use a fingerprint database stored in a training stage to obtain the estimated position of the device by means of different classification algorithms [14,15,16,33]. The fingerprint database stores information, typically the RSS, at certain locations of the environment, modeling the characteristics of the signal using either discrete (fingerprints) or continuous (surfaces) representations.

## 3. Continuous Space Estimator

#### 3.1. Training Stage

- Creation of an RSS map for each AP using discrete information:First, the environment is site-surveyed as in the usual fingerprint-based method: RSS is measured at some positions of the environment and stored in the fingerprint database FP (Equation (1)).$$\mathrm{FP}=\left(\begin{array}{cccc}RS{S}_{A{P}_{1}}\left({P}_{1}\right)& RS{S}_{A{P}_{2}}\left({P}_{1}\right)& \dots & RS{S}_{A{P}_{n}}\left({P}_{1}\right)\\ RS{S}_{A{P}_{1}}\left({P}_{2}\right)& RS{S}_{A{P}_{2}}\left({P}_{2}\right)& \dots & RS{S}_{A{P}_{n}}\left({P}_{2}\right)\\ \vdots & \vdots & \ddots & \vdots \\ RS{S}_{A{P}_{1}}\left({P}_{p}\right)& RS{S}_{A{P}_{2}}\left({P}_{p}\right)& \dots & RS{S}_{A{P}_{n}}\left({P}_{p}\right)\end{array}\right)$$Then, n matrices $RS{S}_{A{P}_{i}}$ (one per AP) are created using the information contained in the fingerprint database (Equation (2)).$$RS{S}_{A{P}_{i}}=\begin{array}{cc}& \begin{array}{cccc}{y}_{1}& {y}_{2}& \dots & {y}_{{p}_{2}}\end{array}\\ \begin{array}{c}{x}_{1}\\ {x}_{2}\\ \vdots \\ {x}_{{p}_{1}}\end{array}& \left(\begin{array}{cccc}RSS({x}_{1},{y}_{1})& RSS({x}_{1},{y}_{2})& \dots & RSS({x}_{1},{y}_{{p}_{2}})\\ RSS({x}_{2},{y}_{1})& RSS({x}_{2},{y}_{2})& \dots & RSS({x}_{2},{y}_{{p}_{2}})\\ \vdots & \vdots & \ddots & \vdots \\ RSS({x}_{{p}_{1}},{y}_{1})& RSS({x}_{{p}_{1}},{y}_{2})& \dots & RSS({x}_{{p}_{1}},{y}_{{p}_{2}}\end{array}\right)\end{array}$$An example of this matrix is represented in Figure 2a, where the colored dots represent the RSS at the site-surveyed coordinates, and the white background covers the coordinates with unknown RSS.
- Estimation of the continuous reference surfaces:The continuous reference surfaces are created using the information contained in the discrete matrices defined in the previous step. To do so, an ϵ-SVR algorithm [46] is used to infer the RSS values for the coordinates with unknown values.This process is performed in a two-step approach:First, ϵ-SVR is used to create a function ${f}_{A{P}_{i}}\left(\mathit{v}\right)$ (Equation (3)) for each $A{P}_{i},\text{}\mathrm{with},\phantom{\rule{4pt}{0ex}}i=1,\cdots ,n$, that is able to estimate the target outputs for the input training data $\mathit{v}$ with at most ϵ deviation.For a given training dataset $\mathit{v}={\mathit{v}}_{A{P}_{i}}=\{({\mathit{v}}_{1},{z}_{1}),\phantom{\rule{4pt}{0ex}}\cdots ,\phantom{\rule{4pt}{0ex}}({\mathit{v}}_{p},{z}_{p})\}$ for each AP, the function for linear SVR is defined as:$${f}_{A{P}_{i}}\left(\mathit{v}\right)=\langle \mathit{\omega},\mathit{v}\rangle +b$$Then, the RSS expected to be measured in the coordinates with unknown values are estimated using this function. The resulting continuous surfaces can be seen as maps of estimated coverage for each AP.Under the selection of the parameters $\u03f5>0$ and $C>0$ (C is the regularization parameter, determining the trade-off between training error and model complexity), the function is obtained by:$$\begin{array}{cc}\underset{\mathit{\omega},\phantom{\rule{4pt}{0ex}}b,\phantom{\rule{4pt}{0ex}}{\xi}_{j},\phantom{\rule{4pt}{0ex}}{\xi}_{j}^{*}}{\mathrm{minimize}}\hfill & \frac{1}{2}{\u2225\mathit{\omega}\u2225}^{2}+C\sum _{j=1}^{p}({\xi}_{j}+{\xi}_{j}^{*})\hfill \\ \mathrm{subject}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\hfill & {z}_{j}-\langle \mathit{\omega},\mathit{v}\rangle -b\phantom{\rule{4pt}{0ex}}\le \phantom{\rule{4pt}{0ex}}\u03f5+{\xi}_{j}\hfill \\ & \langle \mathit{\omega},\mathit{v}\rangle +b-{z}_{j}\phantom{\rule{4pt}{0ex}}\le \phantom{\rule{4pt}{0ex}}\u03f5+{\xi}_{j}^{*}\hfill \\ & {\xi}_{j},\phantom{\rule{4pt}{0ex}}{\xi}_{j}^{*}\phantom{\rule{4pt}{0ex}}\ge \phantom{\rule{4pt}{0ex}}0\hfill \end{array}$$$${\left|\xi \right|}_{\u03f5}:=\left\{\begin{array}{ccc}0\hfill & & if\phantom{\rule{4pt}{0ex}}\left|z-f\left(\mathit{v}\right)\right|\le \u03f5\hfill \\ \left|\phantom{\rule{4pt}{0ex}}z-f\left(\mathit{v}\right)\right|-\u03f5\hfill & & otherwise.\hfill \end{array}\right.$$This optimization problem is computationally simpler to solve when transformed into its Lagrange dual formulation (called the dual problem). The dual formula is obtained introducing the ${\alpha}_{j}$ and ${\alpha}_{j}^{*}$ multipliers by:$$\begin{array}{cccc}& \mathrm{minimize}\hfill & & L\left(\alpha \right)=\frac{1}{2}\sum _{j=1}^{p}\sum _{k=1}^{p}({\alpha}_{j}-{\alpha}_{j}^{*})({\alpha}_{k}-{\alpha}_{k}^{*})\langle {\mathit{v}}_{j},{\mathit{v}}_{k}\rangle +\u03f5\sum _{j=1}^{p}({\alpha}_{j}+{\alpha}_{j}^{*})-\sum _{j=1}^{p}{z}_{j}({\alpha}_{j}^{*}-{\alpha}_{j})\hfill \\ & \mathrm{subject}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\hfill & & \sum _{j=1}^{p}({\alpha}_{j}-{\alpha}_{j}^{*})=0\hfill \\ & & & 0\le {\alpha}_{j}\le C\hfill \\ & & & 0\le {\alpha}_{j}^{*}\le C\hfill \end{array}$$The
**ω**parameter can be described as a linear combination of the training observations using the equation:$$\mathit{\omega}=\sum _{j=1}^{p}({\alpha}_{j}-{\alpha}_{j}^{*}){z}_{j}$$The function ${f}_{A{P}_{i}}\left(\mathit{v}\right)$ for the dual problem is therefore:$${f}_{A{P}_{i}}\left(\mathit{v}\right)=\sum _{j=1}^{p}({\alpha}_{j}-{\alpha}_{j}^{*})\langle {z}_{j},z\rangle +b$$The bias term can be computed by using the Karush–Kuhn–Tucker (KKT) [47,48] conditions. These conditions state that the product between dual variables and constraints has to disappear at the optimal solution, obtaining the following conditions:$$\begin{array}{c}{\alpha}_{j}(\u03f5+{\xi}_{j}-{z}_{j}+\langle \mathit{\omega},{\mathit{v}}_{j}\rangle +b)=0\hfill \\ {\alpha}_{j}^{*}(\u03f5+{\xi}_{j}^{*}+{z}_{j}-\langle \mathit{\omega},{\mathit{v}}_{j}\rangle -b)=0\hfill \\ (C-{\alpha}_{j}){\xi}_{j}=0\hfill \\ (C-{\alpha}_{j}^{*}){\xi}_{j}^{*}=0\hfill \end{array}$$According to these conditions, b can be computed as follows:$$\begin{array}{ccc}b={z}_{j}-\langle \mathit{\omega},{\mathit{v}}_{j}\rangle -\u03f5\hfill & & \mathrm{for}\phantom{\rule{4pt}{0ex}}{\alpha}_{i}\in (0,C)\hfill \\ b={z}_{j}-\langle \mathit{\omega},{\mathit{v}}_{j}\rangle +\u03f5\hfill & & \mathrm{for}\phantom{\rule{4pt}{0ex}}{\alpha}_{i}^{*}\in (0,C)\hfill \end{array}$$Finally, the dual formula is extended to support nonlinear functions by replacing the dot product $\langle \mathit{v},{\mathit{v}}^{\prime}\rangle $ with a nonlinear kernel function $k(\mathit{v},{\mathit{v}}^{\prime})=\langle \mathsf{\Phi}\left(\mathit{v}\right),\mathsf{\Phi}\left({\mathit{v}}^{\prime}\right)\rangle $, where $\mathsf{\Phi}\left(\mathit{v}\right)$ is used to transform $\mathit{v}$ to a high-dimensional space. This way, the original problem comes to:$$\begin{array}{cccc}& \mathrm{minimize}\hfill & & L\left(\alpha \right)=\frac{1}{2}\sum _{j=1}^{p}\sum _{k=1}^{p}({\alpha}_{j}-{\alpha}_{j}^{*})({\alpha}_{k}-{\alpha}_{k}^{*})\mathit{k}(\mathit{v},{\mathit{v}}^{\prime})+\u03f5\sum _{j=1}^{p}({\alpha}_{j}+{\alpha}_{j}^{*})-\sum _{j=1}^{p}{z}_{j}({\alpha}_{j}^{*}-{\alpha}_{j})\hfill \\ & \mathrm{subject}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\hfill & & \sum _{j=1}^{p}({\alpha}_{j}-{\alpha}_{j}^{*})=0\hfill \\ & & & 0\le {\alpha}_{j}\le C\hfill \\ & & & 0\le {\alpha}_{j}^{*}\le C\hfill \end{array}$$$${f}_{A{P}_{i}}\left(\mathit{v}\right)=\sum _{j=1}^{p}({\alpha}_{j}-{\alpha}_{j}^{*})k({\mathit{v}}_{\mathit{j}},\mathit{v})+b$$In this work, we have used the Radial Basis Function kernel (RBF kernel), which is defined as:$$k(\mathit{v},{\mathit{v}}^{\prime})=exp\left(-\frac{{\u2225\mathit{v}-{\mathit{v}}^{\prime}\u2225}^{2}}{2{\sigma}^{2}}\right)$$

#### 3.2. Localization Stage

- Measurement of the RSS:An RSS sample is collected from every AP using the WiFi device to be located.
- Search for the RSSs in the continuous surfaces:The RSS from each AP is searched in the reference surface corresponding to that AP. This search is performed adding a noise margin to the RSS to deal with the noise characteristic of WiFi technology. An analysis of how this margin has been selected will be exposed in Section 4.3.1. The coordinates where the RSS is exactly the same as the RSS sample will obtain the maximum score, then the score will decrease as the RSS differs from the RSS sample until the difference between both values is higher than the margin. The rest of coordinates will get no score. This way, a new localization subsurface for each AP is created containing the scores for the coordinates that can be the location of the device (represented from light blue to red depending on their score in Figure 3). Figure 3 shows the subsurface obtained for the estimated reference surface shown in Figure 2b and an RSS of −97 dBm.
- Addition of the localization subsurfaces:The localization subsurfaces generated for each AP are now summed to obtain a new surface with higher values in the coordinates with higher probabilities of being the location of the device. An example of the results surface can be seen in Figure 4. In this surface, the red coordinates are more likely to be the location of the device.
- Application of an environment mask:The surface generated in the previous step can contain inaccessible areas outside the building. In addition, in certain applications, only certain areas of the environment are reachable for users. For instance, a museum visitor cannot access the storage rooms. To remove these unreachable areas, a mask is applied to the surface obtaining a new surface where only the reachable coordinates have a score (Figure 5a).
- Estimation of the device location:Finally, the location of the device will be estimated as the coordinate with the highest score in the resulting masked surface (Figure 5b).

## 4. Experimental Analysis

#### 4.1. Experimental Setup

#### 4.2. Experimental Results

#### 4.2.1. Topological Positions

#### 4.2.2. Trajectories

#### 4.3. Parameters and Resolution Evaluation

#### 4.3.1. Noise Margin Evaluation

#### 4.3.2. Training Resolution

#### 4.3.3. Reference Surfaces Resolution

#### 4.4. Statistical Validation of the Reported Results

- Ranking: The means of the results achieved by two or more methods under consideration (CSE, RF, SVM and RADAR) are the same.
- Post-hoc with control method: The mean of the results of the control method and against each other group is equal (compared in pairs).

- CSE vs. RF: ${H}_{0}$ is rejected with the p-value equal to 0.00001.
- CSE vs. SVM: ${H}_{0}$ is rejected with the p-value equal to 0.00114.
- CSE vs. RADAR: ${H}_{0}$ is rejected with the p-value equal to 0.00061.

## 5. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Discrete information and continuous reference surface estimated for an AP. (

**a**) Discrete information matrix; (

**b**) estimated RSS reference surface.

**Figure 3.**Localization subsurface for an AP: possible coordinates having a certain RSS collected at an unknown coordinate.

**Figure 4.**Resulting surface: sum of all of the APs’ localization subsurfaces for a test sample collected at an unknown coordinate.

**Figure 5.**Resulting surface applying a mask of the environment to allow only reachable coordinates. (

**a**) Masked surface to the reduce allowed locations; (

**b**) location of the device.

**Figure 11.**Mean error evolution changing the noise margin used to create the localization subsurfaces.

**Figure 12.**Localization subsurfaces created using different noise margins. (

**a**) Margin 1; (

**b**) Margin 5; (

**c**) Margin 10; (

**d**) Margin 15.

**Figure 13.**Manually selected configurations for training resolution experimentation. (

**a**) Configuration 1: 18 positions; (

**b**) Configuration 2: 12 positions; (

**c**) Configuration 3: 10 positions; (

**d**) Configuration 4: eight positions.

**Figure 14.**Mean error evolution changing the number of randomly-selected training positions. (

**a**) Trajectory 1: mean error; (

**b**) Trajectory 1: CSE boxplot; (

**c**) Trajectory 2: mean error; (

**d**) Trajectory 2: CSE boxplot.

**Table 1.**Experimental results: test using the topological positions dataset. CSE, Continuous Space Estimator.

Mean Error | ||||
---|---|---|---|---|

CSE | RF [53] | SVM | RADAR [14] | |

Topological positions | 1.53 m | 0.83 m | 1.53 m | 1.56 m |

Mean Error | ||||
---|---|---|---|---|

CSE | RF [53] | SVM | RADAR [14] | |

Trajectory 1 | 1.52 m | 4.04 m | 4.08 m | 4.14 m |

Trajectory 2 | 1.76 m | 5.47 m | 5.35 m | 5.57 m |

All Positions | Config. 1 | Config. 2 | Config. 3 | Config. 4 | |
---|---|---|---|---|---|

Number of positions | 30 | 18 | 12 | 10 | 8 |

Minimum distance between positions | 2.23 m | 5.50 m | 6.80 m | 8.00 m | 14.53 m |

Mean distance between positions | 26.58 m | 27.46 m | 29.63 m | 29.75 m | 31.35 m |

Mean Error | ||||||
---|---|---|---|---|---|---|

All Pos. | Config. 1 | Config. 2 | Config. 3 | Config. 4 | ||

Trajectory 1 | CSE | 1.52 m | 1.71 m | 3.38 m | 3.39 m | 4.02 m |

RF [53] | 4.04 m | 4.31 m | 8.79 m | 6.56 m | 6.10 m | |

SVM | 4.08 m | 4.93 m | 5.16 m | 5.97 m | 5.63 m | |

RADAR [14] | 4.14 m | 4.20 m | 5.69 m | 6.06 m | 6.16 m | |

Trajectory 2 | CSE | 1.76 m | 2.01 m | 2.80 m | 3.67 m | 3.57 m |

RF [53] | 5.47 m | 5.67 m | 9.18 m | 8.43 m | 5.87 m | |

SVM | 5.35 m | 5.65 m | 6.24 m | 6.06 m | 6.06 m | |

RADAR [14] | 5.57 m | 5.90 m | 6.40 m | 6.45 m | 5.94 m |

© 2017 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**

Hernández, N.; Ocaña, M.; Alonso, J.M.; Kim, E. Continuous Space Estimation: Increasing WiFi-Based Indoor Localization Resolution without Increasing the Site-Survey Effort. *Sensors* **2017**, *17*, 147.
https://doi.org/10.3390/s17010147

**AMA Style**

Hernández N, Ocaña M, Alonso JM, Kim E. Continuous Space Estimation: Increasing WiFi-Based Indoor Localization Resolution without Increasing the Site-Survey Effort. *Sensors*. 2017; 17(1):147.
https://doi.org/10.3390/s17010147

**Chicago/Turabian Style**

Hernández, Noelia, Manuel Ocaña, Jose M. Alonso, and Euntai Kim. 2017. "Continuous Space Estimation: Increasing WiFi-Based Indoor Localization Resolution without Increasing the Site-Survey Effort" *Sensors* 17, no. 1: 147.
https://doi.org/10.3390/s17010147