Datenanalyse und Stochastische Modellierung
7. Spectra

### The Autocorrelation Function and the MSD

As shown in previous chapters, the MSD is directly connected to the velocity autocovariance

$\langle (x(t-t_0)-x(t_0))^2 \rangle = 2 \int_{0}^{t} \mathrm{d}t_1 \int_{t_1}^t \mathrm{d}t_2 \langle \dot{x}(t_0+t_1) \dot{x}(t_0+t_2) \rangle = 2 \int_{0}^{t} \mathrm{d}t_1 \int_{t_1}^t \mathrm{d}t_2 C(t_2-t_1) \sigma^2(t_0+t_1)$

The ensemble-averaged MSD with a fixed initial time, e.g. x(t0)=0, depends on non-stationarity

By calculating the time average, the integral over the total time eliminates this time dependence

$\langle \delta^2(\Delta) \rangle = \frac{2}{t-\Delta} \int_0^{t-\Delta} \mathrm{d}t_0 \int_{0}^{\Delta} \mathrm{d}t_1 \int_{t_1}^\Delta \mathrm{d}t_2 \langle \dot{x}(t_0+t_1) \dot{x}(t_0+t_2) \rangle$ $\;\;\;\;\;\;\;\;\;\;\; = \frac{2}{t-\Delta} \int_0^{t-\Delta} \mathrm{d}t_0 \int_{0}^{\Delta} \mathrm{d}t_1 \int_{t_1}^\Delta \mathrm{d}t_2 C(t_2-t_1) \sigma^2(t_0+t_1)$

Example: power-laws: Using the exponents defined in the previous chapters J, M, and L, it can be shown $\langle x^2(t) \rangle \sim t^{2J+2M+2L-1} \;\;\;\;\;\; \langle \delta^2(\Delta) \rangle \sim \Delta^{2J}$

### Sinusoidal signals

Autocorrelation function and time-averaged MSD both oscillate - no nice measures for the frequency of oscillation

### Problems of the autocorrelation function

Very noisy for long lag times Timescales mix

### The Power Spectrum

The Power Spectrum is the square of the Fourier transform of the signal; it therefore nicely resolves oscillations

$X(\nu) = \int \mathrm{d}t\; x(t) e^{-\frac{2\pi i}{N}t\nu}$ $x(t) = \int \mathrm{d}\nu\; X(\nu) e^{\frac{2\pi i}{N}t\nu}$ $|X(\nu)|^2 = X(\nu)\overline{X}(\nu) = \int \mathrm{d}t^\prime \int \mathrm{d}t\; x(t) x(t^\prime) e^{-\frac{2\pi i}{N}(t-t^\prime)\nu}$

### The Wiener-Khinchin Theorem

The connection between the autocorrelation function and the power spectral density can be calculated straightforwardly. The autocorrelation function is $C(\Delta)=\int_{-\infty}^\infty \overline{x}(t) x(t+\Delta) \mathrm{d}t$ The Fourier transform of X and its complex conjugate is $x(t)=\int_{-\infty}^\infty X(\nu) e^{-2\pi i \nu t} \mathrm{d}\nu \mbox{ } \mbox{ } \mbox{ and } \mbox{ } \mbox{ } \overline{x}(t)=\int_{-\infty}^\infty \overline{X}(\nu) e^{2\pi i \nu t} \mathrm{d}\nu$ If we now plug in the Fourier transform into the definition of the autocorrelation function, we get the Fourier transform of the power spectrum $C(\Delta) = \int_{-\infty}^\infty \left[ \int_{-\infty}^\infty X(\nu) e^{-2\pi i \nu (t+\Delta)} \mathrm{d}\nu \right] \left[ \int_{-\infty}^\infty \overline{X}({\nu'}) e^{2\pi i \nu' t} \mathrm{d}\nu' \right] \mathrm{d}t \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\mbox{ } = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \overline{X}(\nu) X({\nu'}) e^{-2\pi i (\nu'-\nu) t} e^{-2\pi i \nu \Delta} \mathrm{d}t \mathrm{d}\nu \mathrm{d}\nu = \int_{-\infty}^\infty \int_{-\infty}^\infty \overline{X}(\nu) X({\nu'}) \delta(\nu'-\nu) e^{-2\pi i \nu' \Delta} \mathrm{d}\nu \mathrm{d}\nu^\prime$ $= \int_{-\infty}^\infty \overline{X}(\nu) X({\nu}) e^{-2\pi i \nu \Delta}\mathrm{d}\nu = \int_{-\infty}^\infty |X({\nu})|^2 e^{-2\pi i \nu \Delta}\mathrm{d}\nu = \mathcal{F}_\nu \left[ |X(\nu) |^2 \right](\Delta) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

### Examples

Autocorrelation function and power spectrum for

• a signal with exponentially decaying autocorrelations
• a signal with oscillatory behavior
• a signal with power-law decay of correlations

### Wavelets

$d_{j,k} = \int_{-\infty}^\infty x(t) \Psi_{j,k}^* \mathrm{d}t$

Where the wavelet Psi is a function

• which is localized in the time-domain
• the integral over the function is zero
• wavelets with different j are orthogonal to each other

### The Wavelet Transform

$d_{j,k} = \int_{-\infty}^\infty x(t) \Psi_{j,k}^* \mathrm{d}t$ $\Psi_{j,k}(t) = \frac{1}{\sqrt{2^j}} \Psi(2^{-j}t-k)$ $x(t) = \frac{1}{C} \int_0^\infty \sum_{j=0}^\infty d_{j,k} \Psi_{j,k} \frac{\mathrm{d}j\;\mathrm{d}k}{2^j}$ Wavelet transform of a the increments of a trajectory of a tracked animal

The parameter k can resolve non-stationarities in the signal

### Averaging over the the k-parameter

$F_j = \frac{1}{K}\sum_{k=0}^K |d_{j,k}|^2$

Is a scaling function similar to the time-averaged mean squared displacement (TAMSD)

### The Poor Man's Wavelet and the TAMSD

$\Psi(t) = \delta(t) - \delta(t-\tau)$

satisfies the conditions of a wavelet

The wavelet transform with above expression yields $d_{j,k} = \int_{-\infty}^\infty x(t) \frac{1}{\sqrt{2^j}} (\delta(2^{-j}t-k) - \delta(2^{-j}t-k-\tau)) \mathrm{d}t$ $=\frac{1}{\sqrt{2^j}} [x(2^{j}k) - x(2^{j}(k+\tau))]$ The corresponding scaling function is $F_j = \frac{1}{2^jK}\sum_{k=0}^K [x(2^{j}k) - x(2^{j}(k+\tau))]^2$ which is the same as the TAMSD

### Smoothening via the Wavelet transform

The wavelet transform can be used as a filter

• Set all elements for small j to zero and back-transform to physical space: small timescales (noise) are removed
• Set all elements for large j to zero and back-transform to physical space: long timescales (trends) are removed

### DFA to the Autocorrelation Function

Detrended fluctuation analysis [Peng et al. (1994)]:

$F_q^2(s)=\left\langle\frac{1}{s}\sum_{t=1}^{s} \Big(y(t+(n -1) s)-p_{n,s}^{(q)}(t)\Big)^2\right\rangle_n$

where p is a polynomial of order q

DFA0 corresponds to the TA MSD with a specific averaging $\overline{\delta^2}(\Delta) = \frac{1}{t-\Delta} \sum_{t_0=0}^{t-\Delta} \left[ y(t_0+\Delta) - y(t_0)\right]^2$ $F^2(s)=\frac{1}{N}\sum_{n=1}^{N} \mbox{var}\Big(y(t)|_{(n-1)s\leq t < ns}\Big)$

Larger q lead to filtering out slow trends in the signal

### Summary

The following measures can express equivalent information as the autocorrelation function, however, they emphasis on different properties of the signal

• Autocorrelation function $C(\Delta )=\frac{1}{T\sigma^2}\sum_{t=1}^{T} x(t)x(t+\Delta)$
• Time average mean squared displacement $\overline{\delta^2}(t,\Delta) = \frac{1}{t-\Delta} \sum_{t_0=0}^{t-\Delta} \left[ y(t_0+\Delta) - y(t_0)\right]^2$
• Detrended fluctuation analysis $F_q^2(s)=\left\langle\frac{1}{s}\sum_{t=1}^{s} \Big(y(t+(n -1) s)-p_{n,s}^{(q)}(t)\Big)^2\right\rangle_n$
• Power Spectral Density $S(f)=|\mathcal{F}x(t)|^2$
• Wavelet analysis