Univariate Spectral Density with Daniell Method

Tutorial

  1. Open the sample project file in Origin, go to Folder Spectral Analysis using the Project Explorer. Activate the workbook Univariate spectral data.
    Uni dan pe data.png
  2. Highlight column B in worksheet. Click the Time Series Analysis App icon Time Series Analysis icon.png in the Apps Gallery window.
  3. Choose Spectral Analysis tab. Click Univariate Spectral Density with Daniell Method icon to open the dialog.
    Uni dan toolbar.png
  4. In the Setting branch, choose No correction. Enter 0.2,10 and 0.5 in Tapering Proportion, Smoothing Window Width and Shape Ratio of Trapezium Window respectively.
    Uni dan dialog.png
  5. Click Preview button to display smoothed spectrum.
  6. Click OK button to output the report.
    Uni dan report.png

Algorithm

\[\left\{\begin{array}{ll}\frac{1}{2}(1-cos(\frac{\pi(t-\frac{1}{2})}{T}))&1\leqslant t\leqslant T \cr\frac{1}{2}(1-cos(\frac{\pi(n-t+\frac{1}{2})}{T}))&n+1-T\leqslant t\leqslant n \cr1&Otherwise\end{array}\right.\]
where \(T=[\frac{np}{2}]\) and \(p\) is the tapering proportion.
The unsmoothed sample spectrum
\[f^*(\omega) = \frac{1}{2}\pi|\sum_{t=1}^{n}x_texp(i\omega t)|^2\]
is calculated for frequency values
\[\omega_k = \frac{2\pi k}{K},k=0,1...,[K/2]\]
where [ ] denotes the integer part.
The smoothed spectrum is returned as a subset of these frequencies for which \(K\) is a multiple of a chosen value \(r\), i.e.,
\[\omega_{rl}=v_l=\frac{2\pi l}{L},l=0,1,...,[L/2]\]
where \(K=r\times L\)

Reference

  1. nag_tsa_spectrum_univar (g13cbc)