A linear transformation and its properties with special applications in time series filtering
A linear transformation and its properties with special applications in time series filtering
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线性变换及其性质在时间序列滤波中的特殊应用
DOI：
10.1016/s0024-3795(03)00397-5
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发表时间：
2004
影响因子：
1.1
通讯作者：
A. Luati
中科院分区：
文献类型：
--
作者：
E. Dagum;A. Luati
在时间序列分析中，通常假设数据生成过程可以分解为代表趋势，周期性波动，季节性影响和不规则性的各种不可观察成分。这些潜在变量通过以移动的方式对观测值应用线性滤波器或权重系统来估计。滤波器可以以矩阵形式布置，使得应用于观测向量，产生相应的估计值。如果线性滤波器是对称的，比如长度为2 m + 1，其中m是正整数，并且应用于长度N> 2 m + 1的序列，则很明显，对于第一个和最后m个观测值，分量不能被估计。然而，由于对政策和决策制定来说，对潜变量的估计直到并包括最近的观测值非常重要，因此必须对序列的开始和结束m值应用非对称滤波器。整个预测矩阵在这里示出相对于称为t的线性变换是不变的，并且该线性变换是由给定矩阵的两个适当维度的置换矩阵的前乘和后乘产生的。特别地，我们证明了预测矩阵是中心对称的，并且它是由对称权重的子矩阵形成的（应用于中心观测），它是t不变的，或者等价地，矩形中心对称的，并且通过非对称权重的子矩阵（应用于初始和最终观测），它们是彼此的t变换。我们要注意的是，t变换被不适当地（例如，参见Farebrother，[7]）被Weaver [23]称为“反射”，或被Krafft和Schaever [11]称为“旋转”。在本文中，我们定义和研究的t-变换的性质，并突出其在时间序列滤波的作用。
In time series analysis, it is often assumed that the data generating process can be decomposed into various unobservable components representing the trend, cyclical fluctuations, seasonal effects and irregulars. These latent variables are estimated by applying linear filters or systems of weights to the observations, in a moving manner. The filters can be arranged in matrix form such that, applied to the vector of observations, produce the corresponding estimated values. If the linear filters are symmetric, say of length 2m+ 1, with m positive integer, and applied to a series of length N> 2m+ 1, then it is evident that the components cannot be estimated for the first and last m observations. How-ever, since for policy and decision making is of great importance to have estimates of the latent variables up to and including the most recent observations, asymmetric filters must be applied to the beginning and ending m values of the series. The entire predictor matrix is here shown to be invariant with respect to a linear transformation called t and which results from pre-and postmultiplication of a given matrix by two permutation matrices of suitable dimensions. In particular, we show that the predictor matrix is centrosymmetric and that it is forned by a submatrix of symmetric weights (to be applied to central observations) which is t-invariant or, equivalently, rectangular centrosymmetric, and by submatrices of asymmetric weights (to be applied to initial and final observations) which are the t-transform of each other.We would like to remark that the t-transformation has been improperly (see, for example, Farebrother,[7]) referred to either as a “reflection'by Weaver [23] or as a “rotation'by Krafft and Schaever [11]. In this paper, we define and study the properties of the t-transformation and highlight its role in time series filtering.