WiFi case study
Measured indoor signal strength follows the exponentiated Ornstein–Uhlenbeck model. On held-out traces, one well-placed measurement raises the deployed payoff by about half.
The data
The Feng, Nguyen and Luo (2024) dataset covers a 92 by 15 metre floor on a 0.6 metre grid. It has 642 reference points, 120 samples each, and 13 access points.
The model holds
After detrending by a path-loss model, the residual signal strength along traces is close to Gaussian. Its spatial correlation is exponential with a length of about 11 grid units, 6.6 to 7.1 metres, plus a short-scale component carrying roughly half the variance.
Two independent implementations, with different detrending, agree. This matches the classical shadowing model of Gudmundson (1991).
Replaying the policies
The 50 traces were split into 25 training and 25 held-out traces before anything was fitted. The path-loss trend was fitted on the observations outside the held-out traces, then frozen. The covariance model was fitted on the training traces alone.
Fitted model: variance 1.139, correlation length 9.43 grid units, OU share 0.49.
Each of the 125 episodes on held-out traces starts from a random incumbent point. The searcher makes one more measurement and then deploys at one point of the trace. The payoff is the exponentiated signal at the deployed point, the objective the policies optimize.
Model deployment picks the point with the highest expected payoff under the fitted model. An observed point is worth its known value; an unobserved point is worth its conditional log-normal mean.
| trial | deployment | mean payoff | gain over the incumbent |
|---|---|---|---|
| none | incumbent | 2.899 | |
| two-shot step | better observed point | 3.875 | +33.7% |
| two-shot step | model | 4.235 | +46.1% |
| far end of the trace | better observed point | 4.300 | +48.3% |
| three-shot rule | model | 4.380 | +51.1% |
The two-shot step moves to correlation $\operatorname{clip}(b,0,1)$ from the incumbent, where $b$ is the incumbent's standardized value. It is a heuristic trial here. The paper's two-shot game pays the second point, not the better of two.
The three-shot rule chooses the trial that maximizes the expected payoff of the model deployment that follows it. It is computed for the fitted model on the finite trace, nugget included, by searching every grid point and integrating over the trial's predicted value.
The three-shot rule leads the far probe by 0.079, with a paired standard error of 0.260. The two are not distinguished by these 125 episodes.
Fast reversion and far probes
The correlation length of 9 to 11 grid units is short against traces of 15 to 92, so far probes are nearly fresh draws. For small bracket correlation $\rho$ the exact formulas give a revisiting window
$$\big(b_-,\ e^{2\Theta}\big) \;\to\; \big(0.586\sqrt\rho,\ 1 + 2\rho\big),$$and an interior premium of order $\rho(1 - b^2)$. Placing the final point between two nearly independent observations gains little.
Placing it outside keeps its value. A final point stepped out from an observed value $y$ by the two-shot rule has log expected payoff
$$\zeta(y) \;=\; \tfrac{1}{2}\left(\max(y,0)^{2}+1\right) \;>\; y \qquad (y < 1),$$so the exterior option is worth order one even when the interior premium vanishes. Where the trial goes also matters. Small bracket correlation does not make probing far and taking the better observation optimal.
Refinements
- Report gains conditional on the incumbent's standardized value and position, where the phase diagram predicts the three-shot rule beats far probing.
- Add a per-trace random intercept for the elevated mean of detected traces.
- Replay more held-out episodes, or several splits, to separate the leading policies.
Code: casestudy/wifi/.