A linear transformation and its properties with special applications in time series filtering

✍ Scribed by Estela Bee Dagum; Alessandra Luati

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 220 KB
• Volume: 388
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00397-5

✦ Synopsis

The main purpose of this paper is to introduce a linear transformation, called t, and to derive its algebraic properties by means of permutation matrices that represent it.

To demonstrate the importance of the t-transformation for the estimation of latent variables in time series decomposition, we obtain a general expression for smoothing matrices characterized by symmetric and asymmetric weighting systems.

We show that the submatrix of the symmetric weights (to be applied to central observations) is t-invariant whereas the submatrices of the asymmetric weights (to be applied to initial and final observations) are the t-transform of each other. By virtue of this relation, the properties of the t-transformation provide useful information on the smoothing of time series data.

Finally, we illustrate the role of the t-transformation on the weighting systems of several smoothers often applied for trend-cycle estimation, such as the locally weighted regression smoother (loess), the cubic smoothing spline, the Gaussian kernel and the 13-term trend-cycle Henderson filter.