Data-Efficient Training of Gaussian Process Regression Models for Indoor Visible Light Positioning

. 2024 Dec 16;24(24):8027. doi: 10.3390/s24248027

Algorithm 1 GPR-VLP with Data-Efficient Training Based on Active Learning (AL-GPR)

Input:

training dataset D

, the numbers of the initial and final effective training data (m and E

, respectively), kernel function k

, measurement s_{T}

at an unknown location.
Output:

estimated position {\hat{p}}_{T}

and its confidence

σ_{T}^{2} .

* Active Learning:
Initialization:

Divide D

into an initial effective training dataset D_{e}^{0}

(|D_{e}^{0}| = m

) with m data randomly drawn from D

and the corresponding complement dataset D_{*}^{0} (|D_{e}^{0}| = |D| - m) .

While

the number of data in D_{e}^{i}

after i th iteration is no larger than E

(i.e.,

|D_{e}^{i}| <

E

) do
1. evaluate the similarity of data in

D_{*}^{i}

D_{e}^{i}

using (4) with D_{e}^{i}

as the training dataset . To be specific, for each data d_{x} = (s_{x}, p_{x}) \in D_{*}^{i}

, calculate σ_{x}^{2} = K (s_{x}, s_{x}) - K (s_{x}, S_{D_{e}^{i}}) {K (S_{D_{e}^{i}}, S_{D_{e}^{i}})}^{- 1} K (S_{D_{e}^{i}}, s_{x})

2 . update D_{e}^{i}

and D_{*}^{i}

by moving k data with k lowest similarity levels from D_{*}^{i}

to D_{e}^{i}

. Mathematically, the step is conducted as follows : D_{e}^{(i + 1)} = D_{e}^{i} \cup {d_{x} \in D_{*}^{i} w i t h {k l o w e s t σ}_{x}^{2}}

, D_{*}^{(i + 1)} = {d_{x} \in D_{*}^{i} w i t h (|D_{*}^{i}| - {k) h i g h e s t σ}_{x}^{2}}

. The size of the two datasets after the update are |D_{e}^{(i + 1)}| = |D_{e}^{i}| + k a n d |D_{*}^{(i + 1)}| = |D_{*}^{i}| - k

.
3. i = i + 1.
End
AL Output:

The final effective training dataset D_{e}

from the final iteration . Note |D_{e}| =

E

.
Positioning Stage: Calculate the estimated position

{\hat{p}}_{T}

and the confidence of estimation

σ_{T}^{2}

for the measured RSS s_{T}

at an unknown location using GPR based on (3) and (4) with

D_{e}

as the training dataset.