Optimal Data-based Kernel Estimation of Evolutionary Spectra

Complex demodulation of evolutionary spectra is formulated as a two-dimensional kernel smoother in the time-frequency domain. In the first stage, a tapered Fourier transform, $y_{nu}(f,t)$, is calculated. Second, the log-spectral estimate, $\hat{\theta}_{\nu}(f,t) \equiv \ln(|y_{nu}(f,t)|^2$, is smoothed. As the characteristic widths of the kernel smoother increase, the bias from temporal and frequency averaging increases while the variance decreases. The demodulation parameters, such as the order, length, and bandwidth of spectral taper and the kernel smoother, are determined by minimizing the expected error. For well-resolved evolutionary spectra, the optimal taper length is a small fraction of the optimal kernel half-width. The optimal frequency bandwidth, $w$, for the spectral window scales as $w^2 \approx \lambda_F/ \tau$, where $\tau$ is the characteristic time, and $\lambda_F$ is the characteristic frequency scale-length. In contrast, the optimal half-widths for the second stage kernel smoother scales as $h \approx 1/(\tau \lambda_F)^{1 \over ( p+2) }$, where $p$ is the order of the kernel smoother. The ratio of the optimal frequency half-width to the optimal time half-width satisfies $h_F / h_T ~ (|\partial_t ^p \theta | / |\partial_f^p \theta|)$. Since the expected loss depends on the unknown evolutionary spectra, we initially estimate $|\partial_t^p \theta|^2$ and $|\partial_f^p \theta|^2$ using a higher order kernel smoothers, and then substitute the estimated derivatives into the expected loss criteria.