Infinite Mixtures of Gaussian Process Experts with Latent Variables and its Application to Terminal Location Estimation from Multiple-Sensor Values

Abstract

This study proposes a probabilistic method that estimates the locations of sensor nodes (terminals) in a wireless sensor network using the multi-sensor data from sensors located on terminals and the hop counts between terminals. The proposed method involves the use of a novel probabilistic generative model, called the infinite mixture model of Gaussian process experts with latent input variables (imGPE-LVM) that enables us to inversely infer the value of an explanatory variable from that of a response variable for a piece-wise continuous function. Based on an imGPE-LVM, where the sensor data measured by sensors on a terminal are represented by observed variables and the location of the terminal is represented by a latent variable, the proposed method maximizes the posterior probability of the latent variable given sensor values with assuming the terminal location estimated by the DV-Hop algorithm as a prior. This method yields more precise estimates of terminal locations compared with the localization techniques utilizing a Gaussian process latent variable model and those using the DV-Hop algorithm, which is solely based on the hop counts between two terminals.

Appendices

Appendix

A Infinite Mixture of Gaussian Process Experts

In the infinite mixture of Gaussian process experts [13], the input space is (probabilistically) divided by a gating function into regions within which specific separate experts make predictions. Using Gaussian process (GP) models as experts, the model has an additional advantage that the computation for each expert is cubic only in the number of data points in its region, rather than in the entire dataset. Each GP-expert learns different characteristics of the function (e.g., lengths scales, noise variances, etc.).

Let $y \in \mathbb {R}$ be an output variable, let $x \in \mathbb {R}$ be an input variable, and let c be the discrete indicator variable assigning a data point to an expert. The joint probability of the output data $\mathbf{y} = \{ y^{(i)} \}_{i=1}^{N}$ and the corresponding indicators $\mathbf{c} = \{ c^{(i)} \}_{i=1}^{N}$ given the input data $\mathbf{x} = \{ x^{(i)} \}_{i=1}^{N}$ is expressed by

$$\begin{aligned} \begin{aligned}&p ( \mathbf { y }, \mathbf{c} | \mathbf { x } , {\varvec{\theta }} ) \\&= \left[ \prod _ { j } p \left( \left\{ y ^ { (i) }: c ^ { (i) } = j \right\} | \left\{ x ^ { (i) } : c ^ { (i) } = j \right\} , \theta ^ { (j) } \right) \right] p ( \mathbf { c } | \mathbf { x } , \phi ), \end{aligned} \end{aligned}$$

(13)

where ${\varvec{\theta }} = \{ \theta ^{(j)} \}$ denotes the set of the parameters in the Gaussian process regressor for each of the experts.

In inference, with the posterior of $\mathbf{c}$ given, the input data $\mathbf{x}$ and the output data $\mathbf{y}$ is calculated by Gibbs sampling. Once the expert each data point is assigned to using the maximum posterior estimation for the posterior of $\mathbf{c}$ is determined, we obtain the Gaussian process regression by class. For a new input, we output the expectation of the predictions produced by the experts with respect to the posterior distributions of the classes.

B Distance Vector Hop

The distance vector hop (DV-Hop) algorithm [12, 17] estimates terminal locations using the hop count between a terminal and an anchor node with known location in a wireless sensor network. It consists of the following stages:

• 1. Initially, all anchors transmit their locations to the other terminals in the network. The messages are propagated hop by hop where a hop counter is included in the message. Each node maintains an information table on the anchor nodes and counts the minimum number of hops that separates it from each anchor.

• 2. When an anchor node receives a message from another, it estimates the average distance in terms of hops using the locations of two anchors and the hop counter, and returns it to the network as a correction factor. An anchor estimates the average distance of a hop accompanied by reception of hop counts from anchor nodes of all unknown terminals, this computes the distance to the anchor node based on hop counts and the minimum hop count.

• 3. The DV-Hop uses a multilateration method to calculate the unknown terminal’s location according to the distance to each anchor node obtained in the stage 2.

C Details on Experiment

1.1 C.1 Experimental Conditions

• 1. The scale of the environment is assumed to be $10.00\$m. That is, the environment is an area of 10.00 m $\times 10.00$ m.

• 2. The wireless communicable range is assumed to be $10.00 \times \sqrt{2}/3\$m. This setting enables us to prevent a terminal from communicating with all other terminals directly and to preventing generation of a terminal incapable of communicating with others for a small number of terminals.

• 3. The three terminals used as anchor nodes are randomly selected from the terminals in the environment that are at least $10.00 \times \sqrt{2}/2 \$m away from each other. This constraint on the selection of the three anchors causes the DV-Hop to use the locations of the three and the sensor values on them.

• 4. We used three kinds of sensors, a microphone, a thermometer, and an illuminometer. Referring to the specification of ADMP504 (microphone), that of LM35D (thermometer), that of PICMD01-1 (illuminometer), we assumed their dynamic ranges are: 20–120 (dB) for the microphone, 0–100 ($^\circ$C) for the thermometer, and 0–1500 (lux) for the illuminometer, with errors of $\pm 1$ (dB) for the microphone, $\pm 1$ ($^\circ$C) for the thermometer, and $\pm 5$ (lux) for the illuminometer.

• 5. As sensor values, we used the values sampled from the distributions of physical quantities at the true terminal locations added with the Gaussian noises with means of 0 and standard deviations of one third of the sensor errors.

1.2 C.2 Parameter Settings in Inference in imGPE-LVM

In Sect. 3.3, we described the inference in imGPE-LVM. This appendix describes the parameter settings in the experiment.

In sampling, the sample size is 20, 000 and the burn-in size is 2, 000. The initial locations of the terminals $\mathbf{x}^{(i)}$, ${i=1}$, $\ldots$, N, are the locations estimated by DV-Hop, that are denoted by $\mathbf{x}_\mathrm{dvh}^{(i)}$, ${i=1}$, $\ldots$, N. The initial classes of the experts $c^{(i)}$, ${i=1}$, $\ldots$, N, are set to 0. Those of the parameters of the Gaussian process regression for class 0 are $\theta _0^{(0)} = 1.00$, $\theta _1^{(0)} = 1.00$, and $\theta _2^{(0)} = 1.00 \times 10^{2}$. Those of the parameters, $\alpha$ and $\phi$, of the hyper-parameters for the Chinese restaurant process are $\alpha = 1.00$ and $\phi = 2.50 \times 10^{-3}$. We set the kernel parameters so that the class 0 is removed during sampling.

In sampling by the Hamiltonian Monte Carlo, (1) for the terminal locations $\mathbf{x}^{(i)}$, ${i=1}$, $\ldots$, N, the step size is set to $1.00 \times 10^{-2}$ and the number of the leap frog steps is set to 10, (2) for the kernel parameters ${\varvec{\theta }}_0$, ${\varvec{\theta }}_1$, and ${\varvec{\theta }}_2$ in the Gaussian process regression, the step size is set to $1.00 \times 10^{-2}$ and the number of the leap frog steps is set to 2, (3) for the hyper-parameters $\alpha$ in the Chinese restaurant process, the step size is set to $1.00 \times 10^{-3}$ and the number of the leap frog steps is set to 2, and (4) for the hyper-parameters $\phi$ in the Chinese restaurant process, the step size is set to $1.00 \times 10^{-2}$ and the number of the leap frog steps is set to 2. The parameters of the hyper-parameters $a_{\alpha }$, $b_{\alpha }$, $a_{\phi }$, and $b_{\phi }$ are fixed to $a_{\alpha } = 1.00 \times 10^{-2}$, $b_{\alpha } = 1.00 \times 10^{2}$, $a_{\phi } = 1.00 \times 10^{-2}$, and $b_{\phi } = 1.00 \times 10^{2}$, because the hyper-parameters $\alpha$ and $\phi$ obey the noninformative Gamma distribution.

The parameters, $b_{\theta _0}$, $b_{\theta _1}$, and $b_{\theta _2}$, of the hyper-parameters are fixed to 1.00. The initial values of the parameters, $a_{\theta _0}$, $a_{\theta _1}$, and $a_{\theta _2}$, the hyper-parameters are set to $a_{\theta _0} = 1.00 \times 10^{1}$, $a_{\theta _1} = 1.00 \times 10^{1}$, and $a_{\theta _2} = 1.00$.