Detection of multiple changes in mean by sparse parameter estimation - Neubauer, Jirí; Veselý, Vítezslav (Nonlinear Analysis: Modelling and Control, 2013)
The contribution is focused on detection of multiple changes in the mean in a onedimensional stochastic process by sparse parameter estimation from an overparametrized model. The authors’ approach to change point detection differs entirely from standard statistical techniques. A stochastic process residing in a bounded interval with changes in the mean is estimated using dictionary (a family of functions, the so-called atoms, which are overcomplete in the sense of being nearly linearly dependent) and consisting of Heaviside functions. Among all possible representations of the process we want to find a sparse one utilizing a significantly reduced number of atoms. This problem can be solved by ℓ1-minimization. The basis pursuit algorithm is used to get sparse parameter estimates. In this contribution the authors calculate empirical probability of successful change point detection as a function depending on the number of change points and the level of standard deviation of additive white noise of the stochastic process. The empirical probability was computed by simulations where locations of change points were chosen randomly from uniform distribution. The authors’ approach is compared with LASSO algorithm, ℓ1 trend filtering and selected statistical methods. Such probability decreases with increasing number of change points and/or standard deviation of white noise. The proposed method was applied on the time series of nuclear magnetic response during the drilling of a well.