The common methods of spectral analysis for multivariate ($n$-dimensional) time series, like discrete Frourier transform (FT) or Wavelet transform, are based on Fourier series to decompose discrete data into a set of trigonometric model components, e. g. amplitude and phase. Applied to discrete data with a finite range several limitations of (time discrete) FT can be observed which are caused by the orthogonality mismatch of the trigonometric basis functions on a finite interval. However, in the general situation of non-equidistant or fragmented sampling FT based methods will cause significant errors in the parameter estimation. Therefore, the classical Lomb-Scargle method (LSM), which is not based on Fourier series, was developed as a statistical tool for one dimensional data to circumvent the inconsistent and erroneous parameter estimation of FT. The present work deduces LSM for $n$-dimensional data sets by a redefinition of the shifting parameter $\tau$, to maintain orthogonality of the trigonometric basis. An analytical derivation shows, that $n$-D LSM extents the traditional 1D case preserving all the statistical benefits, such as the improved noise rejection. Here, we derive the parameter confidence intervals for LSM and compare it with FT. Applications with ideal test data and experimental data will illustrate and support the proposed method.
翻译:多变量(美元-维)时间序列的光谱分析常见方法,如离散Frourier变换(FT)或Wavelet变换(Wavelet)等,以Fourier序列为基础,将离散数据分解成一组三角测量模型元件,例如振幅和相位。将(时间离位)FT(时间离位)的几种限制应用到具有有限范围的离散数据,这些限制是由于三角测基函数在一定间隔下对美元-维基数据集的异位错配造成的。然而,在非等离异或零散抽样FT基方法的一般情况下,在参数估计方面将造成重大错误。因此,古典Lomb-Scargle(LSM)方法(LSM)不是基于Fourier序列的,是作为一种统计工具,用来绕过FTFT的不一致和错误的参数估计。目前的工作通过重新界定变换参数($-tau)的参数来推断LSM(LSM),以保持三角测算基础的或测深度为基础。分析推算方法将显示SM(SM-SM)和Smal-Cxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx的计算。