An empirical evidence of inconsistency of the l1 trend filtering in change point detection

The fused LASSO signal approximator (FLSA) can be applied to find change points from the data having piecewise constant mean structure. It is well-known that the FLSA is inconsistent in change points detection. This inconsistency is due to a total-variation denoising penalty of the FLSA. l(1) trend filter, one of the popular tools for finding an underlying trend from data, can be used to identify change points of piecewise linear trends. Since the l(1) trend filter applies the sum of absolute values of slope differences, it can be inconsistent for change points recovery as the FLSA. However, there are few studies on the inconsistency of the l(1) trend filtering. In this paper, we demonstrate the inconsistency of the l(1) trend filtering with a numerical study.