Notes 2023.09.29

advanced spectral estimate techniques:

What to do when your process is nonstationary?

Climate science:

• good data coverage has only existed since the '70s. Quite short relative to climatological time scales
• Quasi-periodicity
• externally forced cycle
  • e.g. seasonal cycle is HUGE peak, and small leakage can obliterate fidelity of nearby frequencies
• Trends (potentially nonlinear) are not periodic

Multiple taper method:

• What can you play with if the data length is short?
  • Taper function is main lever we can pull

Suppose $a_t$ such that $a^2$ are a partition of unity.

$A(f)$ is the sum of fourier transforms of each taper. Suppose we wanna ecrease spectral leakage outtside of a bandwith $2W$, me want to maximize $$\lambda(N, W) = \frac{\int_{-W}^W |A(f)|^2 \intd{f}}{\int_{0.5}^{0.5} |A(f)|^2 \intd{f}}$$ and inverting the transform, $$\lambda(N, W) = \frac{a_tC_{tt'}a_{t'}}{a_t\cdot a_{t'}}, \qquad C_{tt'} = \frac{\sin[2\pi W(t-T')]}{\pi (t-t')}$$ gives eigenspectrum problem $$C a_t - \lambda(N, W) a_t = 0$$

Adaptive leakage resistance estimate

• useful for quasi-periodicity

Useful link

Maximum entropy method: