A Fast Algorithm for the Repeated Evaluation of the Likelihood of a General Linear Process for Long Series
长序列一般线性过程似然重复计算的快速算法AI 解读 ↓
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长序列一般线性过程似然重复计算的快速算法
- 研究问题
- 对同一长时间序列反复评估候选线性过程的似然时，如何利用协方差结构减少运算并支持更一般的模型？
- 方法与思路
- 作者将似然中的二次型变换为具有对称 Toeplitz 结构的形式，利用固定滞后的乘积和与预计算信息加速评估。
- 材料陈述的贡献
- 摘要称，完成所需自协方差计算后，近似似然评估的运算量不随序列长度增长，并给出纯自回归过程的简洁精确似然。
- 阅读关注点
- 应核对预计算的时间与存储成本、近似精度及其控制方式，以及一般线性过程、季节模型和分数差分模型的适用条件。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；运算量表述有预计算前提，不能理解为总成本与长度无关；差分阶数选择和均值估计不在讨论范围。
原文摘要
The likelihood function of a stationary and invertible autoregressive moving average (ARMA) process for a given series is a function of the determinant of the covariance matrix of the series and a quadratic form involving the inverse of this matrix. In this article the quadratic form is transformed into a new quadratic form, whose matrix is evaluated readily and has constant elements along its diagonals; that is, it is a Symmetrie Toeplitz matrix. It allows the computation of the quadratic form to be performed conveniently, by multiplying these constant diagonals by the sums of the products of the elements placed at constant lags in the vector appearing in the new quadratic form. This approach leads to an algorithm with the following main properties: 1. Once the autocovariances of the series have been computed, the number of arithmetic operations required to obtain a good approximation to the likelihood function is independent of the size of the observed series.2. The algorithm can cope with any general linear process, in addition to those included in the dass of ARMA(p,q) models.3. It gives a very concise expression for the exact likelihood of purely autoregressive processes.4. It enables easy use of the pattem of zeros appearing in the inverse autocovariances of seasonal ARMA(p,q) models.5. It enables efficient use of the fact that the likelihood function of each tentative model has to be evaluated several times for the same observed series, to obtain the maximum likelihood estimates of the parameters of each tentative model. Neither the problem of how to determine the degree of differencing needed to render the process stationary nor that of estimating the mean function of the process efficiently is the concern of this article. Consequently, the method is applied first to the detrended ARMA(p,q) model, discussing explicitly the important AR (p), MA(1), ARMA(1,1) and seasonal processes, and second to the stationary and invertible ARIMA (p,d,q) model, in which the degree of differencing is allowed to take fractional values. The case of a model, π(B)yt = at in which the function π(B) is the product of the π(B) functions of any two given linear models, is also considered.