Blackman-Tukey Spectrum

The Blackman-Tukey spectrum (or correlogram) is defined as follows

where $w_k$ are window weights and $\hat r_k$ are standard sample covariance (or correlation) estimates

Implementation

The Blackman-Tukey periodogram is implemented in the demetra.data.SmoothedPeriodogram class.

Details

The estimation of the spectrum is defined as follows:

• The series is corrected for mean effect ($y_t^c=y_t-\overline y$)
• Optionally, tapering is applied on the series
• The $M-1$ first sample correlations are computed
• The spectrum is computed, using the provided smoothing window (Tukey window by default)
• it is evaluated at frequencies defined by $2 \pi k / \left(r\left(2M-1\right)\right)$, where $r$ is the relative (in $]0,1]$) resolution defined by the user (.5 by default)

Example

        SmoothedPeriodogram periodogram = SmoothedPeriodogram.builder()
                .data(DoubleSequence.of(data))
                .taper(new TukeyHanningTaper(.1))
                .windowLength(85)
                .relativeResolution(1)
                .windowFunction(DiscreteWindowFunction.Tukey)
                .build();