Notes 2023.09.25

Last time: basic fourier transform theory

Assumes stationarity (related to periodicity).

• E.g. trend vs annual cycle. In this case we have scale separation in spectral space.

Data in practice are always multiplied by a window function. I.e. you have an observation window. Therefore, choose a window function which decays in frequency domain as quickly as possible.

Autoregressive process:

• White noise has no autocorrelation.
• Random walk: expectation zero, but variance increases.

How to compute power spectrum of random walk? $$x(t) = ax(t-\Delta t) + (1-a^2)^{1/2}\varepsilon(t), \quad 0 < a < 1$$ which has mean zero and $\overline{x^2} = 1$. Then find autocorrelation $r(\Delta t) = \overline{x(t)x(t-\Delta t)} = a$ using the fact that $\varepsilon(t) \bot \varepsilon(t+\Delta t)$

Then define $T = \frac{-\Delta t}{\log(a)}$ and we get $$r(n\Delta t) = r^n(\Delta t) = a^n = \exp\left(-\frac{n\Delta t}{T}\right)$$. If $T=n\Delta t$ gives $r(n\Delta t) = e^{-1}$ so this is effectively the $e$-folding time.

Therefore the power spectrum is $$\Phi(\omega) = \int r(\tau) e^{-i\omega t} \intd{\tau} = \frac{2T}{1+\omega^2T^2}$$

statistical significance and time series

$\chi^2$ test: known sigma, or $F$ test: do they come from the same process? What does $n$ mean for a spectral test

$N$: total number of samples, $M_{sp}$ number of spectral estimates, $f_\omega$ spectral correction for windowing function. $n = \frac{N}{M_{sp}}f_\omega$ for e.g. $\chi^2$ or $F$ test.

In practice: how to increase statistical significance? If $N$ fixed, then make $M_{sp}$ smaller.

• Welch's method