Linear Transformation and its Properties with Application in Time Series Filtering

Abstract

The primary objective of this article is to justify the concept of a linear transformation that derives its algebraic properties by means of variation matrices represented so far. The mathematical formulas will be used to elaborate and justify the argument of linear transformation. By displaying the significance of the t-transformation for estimation of latent variables in the time series decomposition, the paper obtains a general expression for smoothing matrices characterized by symmetric and asymmetric weighting system. The article further elaborates the concept of sub-matrix of the symmetric weights that is t-invariant, whereas the sub matrices of the asymmetric weights are the t-transformation of each other. By virtue of this relation, the properties of the t-transformation project useful imperative information on the smoothing of time series data, which may be required to clarify the concept of the topic raised so far. Eventually, the paper illustrates 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, the cube smoothing spine, the Gaussian Kernel and 13-term trend cycle Henderson filter. By so doing, the article will pose the interrelated facts of linear transformation, from which the prospective researchers can benefit.