Time Series Analysis Package

The SPECTRA object uses the finite Fourier transform to decompose data series into a sum of sine and cosine terms of different amplitudes and wavelengths. The finite Fourier transform decomposition of the series is

where

Functions of the Fourier coefficients and can be plotted against frequency or against wavelength to form periodograms. The amplitude periodogram is defined as

The periodogram is an estimate at the discrete frequencies of the spectral density function that characterizes the series . Smoothing the periodogram enables an improved spectral density estimate with reduced variance and bias to be achieved at these points. Smoothing can be accomplished either by using a spectral window smoothing function or by applying a lag window filter to the series autocovariance function.

When the Value of the DOMAIN argument is FREQUENCY, the spectral density estimates are computed by using the following equation to smooth the periodogram ordinates:

where is the spectral window function whose form is specified by either the SetKernel method or the SetWeights method; M is the kernel scale parameter, which acts as a frequency scaling factor in the spectral window smoothing function; and values of that fall outside are mapped to values inside this range by the equations presented previously.

When the Value of the DOMAIN argument is TIME, the spectral density values are estimated by applying a lag window filter, , to the series autocovariance function. Then, the spectral density estimate can be computed from the filtered autocovariance function by using the following equation:

In this case, the kernel scale parameter, M, serves as a scale factor for the lag length, h, in the time domain. In the lag window formulation, the spectral density estimate is a consistent estimator as under the conditions for and . These conditions lead to the following parameterization of M:

where the values and satisfy the consistency conditions. To specify the kernel scale parameter explicitly, set c to the desired scale factor and set .

For uniformity and computational efficiency, all spectral density estimates are calculated using a spectral window weighting function, , which is applied to the periodogram ordinates. In the case where the Value of the DOMAIN argument is TIME, the effective spectral window weighting function is computed by the following equation:

Because the kernel scale parameter, M, serves as a lag scale factor in the time domain and as a bandwidth scale factor in the frequency domain, the impact of M on spectral density estimates depends on the Value of the DOMAIN argument. When the Value of the DOMAIN argument is specified to be FREQUENCY, increasing values of M decrease variance and increase bias in the spectral density estimates; when the Value of the DOMAIN argument is TIME, increasing values of M increase variance and decrease bias.

Using Kernel Specifications

You can specify one of 10 different kernel smoothing functions in the SPECTRA object. Five smoothing functions are available in the SetKernel method, and five complementary smoothing functions that correspond to lag window filters are available when the SetKernel method is used and the Value of the DOMAIN argument is TIME.

For example, a Parzen kernel with a support of 11 periodogram ordinates in the frequency domain can be specified using the SetKernel method:

rc = SetKernel('PARZEN', 5, 0);

The SPECTRA class of the TSA package supports the following spectral window kernel functions in the frequency domain, where :

BARTLETT (Bartlett kernel):

PARZEN (Parzen kernel):

QS (quadratic spectral kernel):

TUKEY (Tukey-Hanning kernel):

TRUNCAT (truncated kernel):

When the Value of the DOMAIN argument is TIME, the five preceding kernel functions are interpreted as lag window filters on the autocovariance function. The lag window kernel functions correspond to the following spectral window smoothing functions, where :

BARTLETT (Bartlett equivalent lag window filter):

PARZEN (Parzen equivalent lag window filter):

QS (quadratic spectral equivalent lag window filter):

TUKEY (Tukey-Hanning equivalent lag window filter):

TRUNC (truncated equivalent lag window filter):

array group1[5]/nosymbols;group1[1]=1;group1[2]=2;group1[3]=3;
      group1[4]=2;group1[5]=1;
rc = spectra.SetWeights(group1);