2.2.6.2 Univariate Spectral Density with Lag Window


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 Lag Window icon to open the dialog.
    Uni lag toolbar.png
  4. In the Setting branch, choose Mean correction. Enter 0.2 in Tapering Proportion. Choose Parzen window type. Enter 50 in Cut-off of Lag Window.
    Uni lag dialog.png
  5. Click Preview button to display smoothed spectrum.
  6. Click OK button to output the report.
    Uni lag report.png

Algorithm

  • Tapering factors
\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.
  • Smoothed sample spectrum
\hat{f}(\omega) = \frac{1}{2\pi}(C_0+2\sum_{k=1}^{M-1}\omega_kC_kcos(\omega_k))
where M is the window width, and is calculated for frequency values
\omega_i = \frac{2\pi i}{L},i=0,1...,[L/2]
where [ ] denotes the integer part.
  • Smoothing window
\omega_k = W\frac{k}{M}, k\leqslant M-1
which for the various windows is defined over 0< \alpha < 1 by
rectangular: W(\alpha)=1
Bartlett: W(\alpha)=1-\alpha
Tukey: W(\alpha)=\frac{1}{2}(1+cos(\pi \alpha))
Parzen:W(\alpha)=\left\{\begin{array}{ll}1-6\alpha^2+6\alpha^3&0\leqslant \alpha\leqslant \frac{1}{2} \cr2(1-\alpha)^3&\frac{1}{2}< \alpha <1\end{array}\right.

Reference

  1. nag_tsa_spectrum_univar_cov (g13cac)
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