Bootstrap multiple tests applied to sensor location

A method for finding optimal locations of vibration sensors within a group of sensors for detecting knock in combustion engines is proposed. It differs from other techniques in that only signal processing and statistical tests are used. The method is based on linearly predicting the in-cylinder pressure in a combustion chamber from the output signals of a group of vibration sensors being distributed on the engine wall. The irrelevancy of a sensor in the group is characterized by the closeness to zero of a “coherence gain” explained by this sensor at some frequencies of interest. We formulate multiple hypotheses, define suitable statistics, approximate their sampling distributions by the nonparametric bootstrap method, and construct the generalized sequentially rejective Bonferroni multiple test. The level of accuracy of bootstrap tests is superior to that of asymptotic tests, provided the test statistic used is asymptotically pivotal. To achieve pivoting, we transform the test statistic into a variance stable scale where we use a bootstrap technique to approximate the variance stabilizing transformation. Simulations results as well as results of an experiment performed on a test bed with a four-cylinder engine emphasize the applicability of the method