Copula in statistics
Updated: Apr 2, 2021
Recently I came across this method commonly applied in financial statistics for computing joint distributions from marginal distributions and a dependency metric. A recent paper that I submitted to Globecom tries to leverage this technique to create a cost function for localizing a wireless transmitter by measuring its signal strength by multiple wireless receivers placed at known locations in an indoor environment.
Signal strength measured by receivers without processing is not a good indicator of the position of the transmitter due to fading. Fading can be either fast changing multipath fading or a slow varying shadow fading.
Multipath fading occurs when duplicate copies of the original transmitter signal reach the receiver at the same time but varying in signal phase. The duplicate copies are created due to reflection, refraction and diffraction of wireless signals by objects within the indoor environment.
Shadow fading is a slow changing phenomenon caused by relevant obstructions between the transmitter and receiver. Movement of people or machinery, doors etc might obstruct the wireless signal path between the transmitter and receiver. This obstructions causes the measured signal strength to go down.
In my localization method, I remove the fast fading using a Ornstein-Uhlenbeck mean reverting filter followed by GARCH filtration. This end result is a zero mean, unit variance stochastic variable that corresponds to the shadow fading measured at each receiver. Now I construct a student-t copula from these shadow fading residuals along with shadow fading dependency between receivers. Shadow fading dependency between receivers is a measure of how similar the shadow fading noise are between a receiver pair. Higher the similarity the receiver/transmitter are close to each other and vice versa. Finally, the joint distribution of shadow fading residuals acts as my cost function for the Maximum Likelihood Estimator (MLE) for the transmitter position.
I am hoping this paper gets published. I have extensive simulation and real test results of my algorithm. For my real test, I collected localization accuracy from the food court of a local mall. Though this method resulted in a mean accuracy of around 2m, it helps to show that even under heavy people traffic, there is a way to infer the location of the transmitter