4. The Kullback-Leibler multivariate Gaussian (KLMvG) model

Another approach, specifically the KLMvG model, has recently been used in fingerprintingbased methods to estimate the position of the objects. This model exploits the interdependencies between the RPs, such as the signal model and the geometry that can be quantified to find the correlations among the RPs. Milioris [29] proposed a KLMvG model to measure the similarity between the RSS measurements of test points and the RPs, defined as

$$\text{KL}\_{\text{MVG}}\left(p\|q\_{j}^{(\boldsymbol{\upgamma})}\right) = \frac{1}{2}\left(\left(\boldsymbol{\upmu}\_{q,j}^{\boldsymbol{\upgamma}} - \boldsymbol{\upmu}\_{p}^{\boldsymbol{\upbeta}}\right)^{\mathrm{T}}\left(\boldsymbol{\upSigma}\_{j,q}^{\boldsymbol{\upgamma}}\right)^{-1}\left(\boldsymbol{\upmu}\_{q,j}^{\boldsymbol{\upgamma}} - \boldsymbol{\upmu}\_{p}^{\boldsymbol{\upbeta}}\right) + tr\left(\boldsymbol{\upSigma}\_{\mathbb{R}}^{\boldsymbol{\upgamma}}\left(\boldsymbol{\upSigma}\_{j,q}^{\boldsymbol{\upgamma}}\right)^{-1} - I\right) - \ln\left|\boldsymbol{\upSigma}\_{p}^{\boldsymbol{\upgamma}}\left(\boldsymbol{\upSigma}\_{j,q}^{\boldsymbol{\upgamma}}\right)^{-1}\right|\right) \tag{6}$$

where S represents the matrix of RSS values from the different APs at specific locations and j represents the cell of the fingerprint location where

$$\mathcal{S}\_{\dot{\gamma}}^{(\circ)} = \left\{ \mu\_{\dot{\gamma}}^{(\circ)}, \Sigma\_{\dot{\gamma}}^{(\circ)} \right\} \tag{7}$$

μj �ð Þ is the mean of Jth column of the RSS measurement and Σ<sup>j</sup> �ð Þ represents the covariance matrix, where j j Σ is the determinant of Σ. Now, using a KLMvG model, we can formulate a probability kernel-based approach. The kernel regression scheme allows us to estimate the PDF of the training datasets and the true positives (TPs) from the online phase that are used to estimate the location of the object. The KLMVG model is used to measure the distance between the likelihood of the input sample and the RPs in order to determine which class it belongs to. The RSS distribution can be defined as

$$D(p, q\_{\ell}) = \exp\left(-\frac{\text{KL}\_{\text{MVG}}\left(p \| q\_{j}^{(\circ)}\right)}{2\sigma^{2}}\right) \tag{8}$$

where σ is the kernel smoothing factor. The probability will be equal to 1 if p = q, and the output will decrease when the difference between p and q becomes larger.

#### Algorithm 1. The Kullback-Leibler multivariate Gaussian positioning method.


Δ �ð Þ <sup>j</sup> <sup>¼</sup> <sup>Δ</sup> �ð Þ 1,j ;Δ �ð Þ 2,j ; Δ �ð Þ 3,j

146 Machine Learning - Advanced Techniques and Emerging Applications

Δ �ð Þ i,j <sup>¼</sup> <sup>1</sup> t � 1

<sup>j</sup> ) with q

q �ð Þ <sup>j</sup> ¼ q �ð Þ 1,j ; q �ð Þ 2,j ; q �ð Þ 3,j

During the online phase, the RSS measurement is denoted as

i,j is the variance for AP i at RP j with orientation � ��

�ð Þ

where

where Δ �ð Þ

Radio Map is (xj, yj

KLMVG pkqj

μj

�ð Þ � �

¼ 1 <sup>2</sup> <sup>μ</sup><sup>S</sup>

The RSS distribution can be defined as

q,j � <sup>μ</sup><sup>S</sup> p � �<sup>T</sup>

represents the cell of the fingerprint location where

, q �ð Þ <sup>j</sup> ,<sup>Δ</sup> �ð Þ ; ::…Δ �ð Þ L,j

� �<sup>2</sup>

; ::………; q

<sup>þ</sup> tr <sup>Σ</sup><sup>s</sup>

� �

<sup>R</sup> Σ<sup>s</sup> j, q � ��<sup>1</sup>

� �

� I

� ln <sup>Σ</sup><sup>s</sup> <sup>p</sup> Σ<sup>s</sup> j, q � ��<sup>1</sup> �

� � �

�ð Þ represents the covariance

A (8)

� � � �

(6)

(7)

h i

pr ¼ p1,r; p2,r; :……; pL,r h i

Another approach, specifically the KLMvG model, has recently been used in fingerprintingbased methods to estimate the position of the objects. This model exploits the interdependencies between the RPs, such as the signal model and the geometry that can be quantified to find the correlations among the RPs. Milioris [29] proposed a KLMvG model to measure the

�ð Þ i,j

> �ð Þ L,j

(2)

(3)

(4)

(5)

; thus, the database table of the

h i

X<sup>t</sup> <sup>τ</sup>¼<sup>1</sup> <sup>q</sup> �ð Þ i,j ð Þ� τ q

<sup>j</sup> defined as

4. The Kullback-Leibler multivariate Gaussian (KLMvG) model

similarity between the RSS measurements of test points and the RPs, defined as

Sj

�ð Þ is the mean of Jth column of the RSS measurement and Σ<sup>j</sup>

D p; q<sup>ℓ</sup>

� � <sup>¼</sup> exp �

μS q,j � <sup>μ</sup><sup>S</sup> p � �

�ð Þ <sup>¼</sup> <sup>μ</sup><sup>j</sup>

where S represents the matrix of RSS values from the different APs at specific locations and j

matrix, where j j Σ is the determinant of Σ. Now, using a KLMvG model, we can formulate a probability kernel-based approach. The kernel regression scheme allows us to estimate the PDF of the training datasets and the true positives (TPs) from the online phase that are used to estimate the location of the object. The KLMVG model is used to measure the distance between the likelihood of the input sample and the RPs in order to determine which class it belongs to.

> 0 @

�ð Þ;Σ<sup>j</sup> �ð Þ n o

KLMVG pkq

2σ<sup>2</sup>

�ð Þ j � �

1

Σs j, q � ��<sup>1</sup>

