Unsupervised Learning for Solving RSS Hardware Variance Problem in WiFi Localization

Abstract

Hardware variance can significantly degrade the positional accuracy of RSS-based WiFi localization systems. Although manual adjustment can reduce positional error, this solution is not scalable as the number of new WiFi devices increases. We propose an unsupervised learning method to automatically solve the hardware variance problem in WiFi localization. This method was designed and implemented in a working WiFi positioning system and evaluated using different WiFi devices with diverse RSS signal patterns. Experimental results demonstrate that the proposed learning method improves positional accuracy within 100 s of learning time.

APPENDIX: Analytical model for the balanced AP distribution

A typical location in an RSS fingerprint localization system is characterized by several (RSS, AP) pairs, where RSS is not a single value but rather a distribution of signals collected from the training phase and often modeled by a Gaussian distribution. While tracking a device, the probability of a set of observed (RSS, AP) pairs against a certain location is then computed by multiplying all the probabilities acquired from the previously modeled Gaussian probability distribution function. The location with the highest joint probability is the output result.

The above localization system is assumed here. Further, without loss of generality, the following assumptions are made:

• 1. RSS decays linearly;

• 2. The variances of all pairs are identical.

Consider the one-dimensional example in Fig. 12. Two access points AP _a and AP _b are at either side of a tracking device. Suppose the tracking device is at an arbitrary position u ₀ on the line from 0 (the leftmost position) to z (the rightmost position).

Figure 12

Highest probability distribution at location u ₀

According to the first assumption above, if the RSS directly beneath an access point is s, the distributions of (RSS=s/u ₀ , AP=AP _a ) and (RSS=s/(z-u ₀ ), AP=AP _b ) at position u ₀ are identical, and u _a and u _b are located at u ₀ . If the above two RSS signal patterns are entered into an RSS-based positioning engine, the estimated location will be u ₀ .

As Fig. 13 shows, if the tracking device differs from the training device with a linear RSS mapping function with slope = 1 as in the first assumption, the RSS distribution is simply shifted.

Figure 13

Highest probability distribution at location u ₀ shifted by hardware difference with linearity of y = x + b

Although u ₀ is no longer the most probable location for both APs, the multiplied probability is still the highest. This outcome is demonstrated by comparing the multiplied probability at each position. Since the RSS variances are assumed identical, in the p.d.f. of Gaussian distribution $\frac{1}{{\sigma \sqrt {2\pi } }}e^{ - \frac{{\left( {x - \mu } \right)^2 }}{{2\sigma ^2 }}}$, we need only compare the (x - u) ² part. Restated, the smaller the value, the higher the probability.

At u ₀ , after multiplication, the next procedure would be

$$\left[ {u_0 - \left( {u_0 - b} \right)} \right]^2 + \left[ {u_0 - \left( {u_0 + b} \right)} \right]^2 = 2b^2 .$$

Assume an arbitrary position denoted as (u ₀ +d) in Fig. 13. After substitution, (x - u) ² becomes

$$\left[ {\left( {u_0 + d} \right) - \left( {u_0 - b} \right)} \right]^2 + \left[ {\left( {u_0 + d} \right) - \left( {u_0 + b} \right)} \right]^2 = 2\left( {d^2 + b^2 } \right).$$

Since 2(d ₂ + b ₂ ) > 2b ₂ for any nonzero d, u ₀ is the location with maximum probability.