Multiple Composite Hypothesis Testing: A Competitive Approach

There are many problems in science and engineering where the signals of interest depend simultaneously on continuous and q-ary parameters, i.e. parameters which can take only one out of q possible values. This problem is generally known as multiple composite hypothesis testing. The probability function of the observed data for a given hypothesis is uncertain, as it depends on the parameters of the system. When there is no statistical model for the unknown continuous parameters, the GLRT is the usual criterion for the binary case. Although the GLRT philosophy can be extended to accommodate multiple composite hypotheses, unfortunately the solution is not satisfactory in the general case. In this paper, we restrict the general scenario and consider problems with q-ary input vectors and linear dependence on a unique set of continuous parameters; i.e. all the hypotheses depend on the same set of parameters. Direct application of the GLRT is feasible in this case, but it suffers from an exponential increase in complexity with data length. In this paper, we derive a low-complexity stochastic gradient procedure for this problem. The resulting algorithm, which resembles the LMS, updates the unknown parameters only along the direction of the winning hypothesis. This approach also presents similarities with competitive learning techniques, in the sense that at each iteration the different hypotheses compete to train the parameters. The validity of the proposed approach is shown by applying it to blind system identification/equalization, and chaotic AR(1) model estimation.