Accurate estimation of the radio frequency maps is key in practical indoor localization, but this requires dense sampling of the electromagnetic eld, which is also known as ngerprinting. To decrease the time-consuming ngerprinting process, we both aim to estimate probabilistic radio frequency maps (RM) using arti cial neural networks (ANN), and automatically pick the most informative ngerprint positions following an active learning strategy based on uncertainty sampling, aided by Gaussian Processes (GP). Once we have an estimate of the RM of the environment, the problem can be formulated as a tracking problem with a Hidden Markov Model (HMM), and the RM can be used as the observation densities of the HMM. The trajectories of sequential positions are approximated with a sequential Monte Carlo lter. The results indicate that ngerprint measurements can be reduced by 30% in return of 8.1% decrease in the localization performance.
distributions for each position and stored them in a probabilistic RM. They used both maximum
likelihood estimation (MLE) and neural networks in localization stage and achieved 0:27-meter
and 1-meter positioning error respectively. In [
In [
In this work, we hypothetize that both neighboring ngerprints and their coordinates are related to each other. Modeling these relations with Gaussian Processes (GP) and Arti cial Neural Networks (ANN), we propose a novel reduction strategy to be applied to any ngerprint-based localization problem. Our reduction strategy combines two methods: ANNs are used to estimate the RM of ngerprints in the positions where no ngerprint data is originally available, and GPs are used to model the uncertainty of ngerprints. For evaluation, we compare the localization error with and without these methods. We approach our problem as a tracking problem based on HMM and use SMC to approximate the original trajectory. We calculate the localization error by comparing the estimated trajectory with the ground truth. 2 2.1
The relationship between an RSSI distribution at a position and its 2D position coordinates is a complex relationship, as it bases on the degenerative e ects on signals, such as path loss, shadowing, or even the architecture of the environment. ANNs are good at modeling complex relationships and multi-class classi cation problems where the output layer represents a probability distribution.
Five di erent sets of input features, Si, are constructed with di erent combinations of the following features: the distances between a BLE transmitter (i.e. beacon) and a bluetooth receiver (i.e., dongle), ngerprints in the form of RSSI histograms on the neighboring positions and their distances to the beacon. In Table 1, P T stands for the distance of the position to the beacon, P Fi for the distance of the position to the ith closest ngerprint, T Fi for the distance between the ith closest ngerprint and the beacon and Hi for the histogram on the ith closest ngerprint, with indices for the rank of the proximity.
S1 S2 S3 S4 S5
P T
P F1; T F1; H1
P F1;2T F1;2; H1;2
P F1;2;3; T F1;2;3; H1;2;3
P T ; P F1;2;3;4; T F1;2;3;4; H1;2;3;4
We train a Single Layer Neural Network (SNN) using these ve feature sets. For every set we select di erent number of units in the hidden layer ranging from 20 to 520. As an SNN may not be successful enough to approximate the complex function between our input features and output, we also train a Deep Neural Network (DNN) with two hidden layers. We feed the features to the input layers and we estimate the discrete probability distribution of RSSI values ranging from 100 to 20.
We compare two distributions using a cross-bin comparison metric that is also called the squared
Earth Mover's Distance (EMD2) [
EMD2(p; q) / jjFP (p)
FQ(q)jj22 where p and q are original and predicted distributions respectively, and FP (p) and FQ(q) are their cumulative distribution functions.
For intermediate and top layers of ANNs, we select the sigmoid function as the activation
function to model complex relations of features. The weights and biases are initialized with the
Glorot Initializer and as zero respectively. Weights and bias values are optimized via the RMSProp
algorithm [
We use GP to model the mapping functions between the positions and the ngerprints together with the uncertainty in these positions. GP is composed of random variables, f , any nite set of which have a joint Gaussian distribution. The positions and the RSSI values are the observations, and the functions are the latent variables (see Fig. 1). When the functions for the given observations are known, the function value, f , for a new position, x , can be calculated.
r1 f1 x1 r2 f2 x2 r::: f::: x::: r f x
We assume that the distances between the positions makes a di erence in RSSI distributions.
We use a squared exponential function as the covariance function with the kernel k(xi; xj ). The
hyper-parameters of the kernel, f , n, and l are optimized with gradient ascent by maximizing
the log marginal likelihood as in [
N ( ;
) = m(x) + K(x ; x)[K(x; x) + n2I] 1 (r
m(x)) = K(x ; x )
K(x ; x)[K(x; x) + n2I] 1K(x; x ) where r and x are the RSSI observations and positions respectively. The function value can be predicted only by calculating the correlation between existing positions and a new coordinate. The output value r di ers from f only in the sense that it has an additional noise term in its covariance, + n2I.
Predictive variance, , is used for iterative active ngerprint selection. In the initial step, a random ngerprint is selected for training. At each step, the ngerprint that has the greatest variance is added to the training set. Retraining with the newly added ngerprint, predictive variance for each of the remaining positions is recalculated and this procedure continues until we reach a target number of ngerprints. (1) (2) 2.3
We model the tracking problem as HMM, as in Fig. 2. Measurements, rt, are the RSSI values and the latent variables, xt, are the positions. We are to estimate the current position, xt, given the observations so far. Filtering, p(xt; r1:t), is a recursive process: x0 x1 r1 :::
::: xt rt xT rT
For the transition density, p(xt i+1jxt i), a di usion-based motion model is used as explained
in [
We conducted our experiments on two di erent datasets, collected from two di erent areas, the
rst of which, A1, is a living room (5:28 6:35 m2) containing six BLE beacons, logged at 50
positions each for 24 hours [
We used 70% of these ngerprints for training, and the remaining for validation and test purposes. The training data were used for optimizing the weights and biases, and the validation data for selecting the best model. The test data were used only for evaluating the localization performance. All computations were performed on an Intel Core i7 based computer and 8GB of RAM. The models were constructed on Keras 2.1.5 with Tensor ow as backend.
For the test area of A1, we trained the SNN with ve feature sets (see Table 1) and 25 di erent hidden unit sizes. We repeated the training process ten times, constructing ten di erent randomly selected ngerprint con gurations which are then used to compare the results of these feature sets. We evaluate the performance by measuring the average validation loss over all beacons. Fig. 3 shows that S1 yields much greater error than the remaining feature sets, meaning the estimations that depend solely on the distance to the beacons perform very poorly. S4 has the best performance with minimum EMD2 loss with 60 units in the hidden layer. It shows that the input feature with the four nearest ngerprint data, their distances to the beacons, and their histograms performs better than the other feature sets. We trained the DNN with hidden unit sizes ranging from 10 to 100. Finally, the con guration of S4 and 40 units is the best architecture for each layer, as seen in Fig. 3. Repeating the same experiments for the ngerprints collected in the A2, we found that S3 and 40 hidden units in the hidden layers performs better for both SNN and DNN (see Fig. 3).
To evaluate our strategy, we trained ANNs using (i) all collected ngerprints, (ii) randomly selected 70% of the ngerprints and (iii) actively selected 70% of the ngerprints. With the trained networks, we made estimations of dense RMs with 0.1-meter intervals for ten times.
We used the constructed maps as the observation distribution in the particle lter to estimate the trajectory points. We applied the SMC with 2000 particles and at 400 trajectory points in A1 and A2. In each run, the rst 32 estimated points were omitted as burn-in period. We run the ltering algorithm ten times for each radio frequency map and each time particles were initialized randomly. This prevented us from getting poor performance because of the random initialization. We reported the performance of our methodology as median tracking error instead of the mean absolute error, because errors form skewed distributions, which are clearly non-Gaussian. Tracking accuracies in A1 and A2 are given in Fig. 5a.
We have three observations. The rst one is that deep neural networks cause higher median error in both areas. Second one is when the training size is decreased from 100% to 70% of data, median error increases. Finally, the median error for the actively selected data in each test area is lower than the one for the randomly selected data regardless of the neural network type.
As seen in Table 2, we can reduce the number of training data by 30% while increasing the tracking accuracy by at most 8:1% in return. For A1, the resulting increase are 1:9% and 2:3% for SNN and DNN, respectively. For A2, the resulting increase are 8:1% and 3:96%.
We propose a novel approach to reduce the training size required to estimate probabilistic radio maps by picking training positions adaptively. We have two conclusions showing the success of our approach. First, we pick the ngerprint positions in an adaptive manner, we estimate ngerprints at remaining positions better than we do with the randomly selected ngerprints in both test areas. Second, we reduced the training size by 30% with an increase in the median error by a maximum of 8:1%. Since ngerprinting is a time-consuming procedure in indoor localization eld, this study would be used to reduce the ngerprinting process of any other study in the indoor localization research area.