A trend filtering method closely related to $$\ell _{1}$$ ℓ 1 trend filtering

The filtering method developed by Kim et al. (SIAM Rev 51:339–360, 2009), $$\ell _{1}$$ ℓ 1 trend filtering, is attractive because it enables us to estimate a continuous piecewise linear trend. This paper introduces a new filtering method closely related to $$\ell _{1}$$ ℓ 1 trend filtering in order to contribute to the accumulation of knowledge on $$\ell _{1}$$ ℓ 1 trend filtering. We show that the piecewise linearity, which is the key feature of $$\ell _{1}$$ ℓ 1 trend filtering, is derived from the new filtering. For this reason, we refer to the filtering as ‘pure’ $$\ell _{1}$$ ℓ 1 trend filtering. We also demonstrate some other miscellaneous results concerning the new filtering.