Distributed linear regression under privacy constraints
Regression analysis is used to estimate the unknown effect of changing one variable over another. While running a linear regression, it is assumed that there is a linear relationship between regressand and regressor variables and this relationship is additive. Various regression algorithms are used for modeling a data set using linear functions as well as estimating the unknown model parameters from data. The Recursive Least Square(RLS) algorithm is one of the most popular regression analysis method with extremely fast convergence rate. But a regression analysis method does not perform well if the analysis is performed based on a set of observations representing only a small portion of independent input variables of the system. If a set of agents collaboratively read the observations of a system, then agents can also per- form regression analysis collaboratively, i.e. by sharing all their readings and then forming an estimate. But in some systems, the agents are unwilling to share their observations due to privacy reasons, but do not mind sharing estimates with others for getting similar estimates in return. To this end, a RLS based distributed regression algorithm is developed for wireless sensor networks (WSNs) whereby sensors exchange local estimates with one-hop neighbors to consent on the network-wide estimates adaptively and thereby converging to the estimates formed by a regression analysis based on all ob- servations. The proposed algorithm has been obtained through recasting of the weighted linear least-squares cost into a separable form. Numerical sim- ulations demonstrate that for higher number of 1 hop neighbors, this RLS based distributed regression algorithm implementation by an agent performs similar to the RLS algorithm based on full information.