Datenanalyse und Stochastische Modellierung
4. Non-Gaussian Prozesses

### The Gaussian Normal Distribution

$\rho(x)=\frac{1}{\sqrt{2\pi} \sigma} e^-\frac{(x-\mu)^2}{2\sigma^2}$

### Sum of two Random Numbers

• Look at histogram $P(x+y=z) = \sum_{x\leq z} P(x) P(y=z-x)$
• Continuous limit: convolution $\rho_{x+y}(z) = \int \rho_x(x)\rho_y(z-x)\mathrm{d}x$

### Sum of two Gaussian Random Numbers

$\rho_{x+y}(z) = \int_{-\infty}^\infty \rho_x(x) \rho_y(z-x) \mathrm{d}x = \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi} \sigma_x} e^-\frac{(x-\mu_x)^2}{2\sigma_x^2} \frac{1}{\sqrt{2\pi} \sigma_y} e^-\frac{(z-x-\mu_y)^2}{2\sigma_y^2} \mathrm{d}x$ $... = \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}\sqrt{2\pi}\sigma_x\sigma_y} e^{-\frac{x^2(\sigma_x^2\sigma_y^2)-2x(\sigma_x^2(z-\mu_y)+\sigma_y^2\mu_x) + \sigma_x^2(z^2+\mu_y^2-2z\mu_y)+\sigma_y^2\mu_x^2}{2\sigma_y^2\sigma_x^2}} \mathrm{d}x$ $\mbox{with } \sigma_{x+y}^2=\sigma_x^2 + \sigma_y^2 \; \; \mbox{ and } \mu_{x+y}=\mu_x + \mu_y$ $... = \frac{1}{\sqrt{2\pi}\sigma_{x+y}} e^{-\frac{(z-\mu_{x+y})^2}{2\sigma_{x+y}^2}} \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}\frac{\sigma_x\sigma_y}{\sigma_{x+y}}} e^{-\frac{\left(x-\frac{\sigma_x^2(z-\mu_y)+\sigma_y^2\mu_x}{\sigma_{x+y}^2}\right)^2}{2\left(\frac{\sigma_x\sigma_y}{\sigma_{x+y}}\right)^2}} \mathrm{d}x = \frac{1}{\sqrt{2\pi}\sigma_{x+y}} e^{-\frac{(z-\mu_{x+y})^2}{2\sigma_{x+y}^2}}$

### Moments

A probability distribution can be defined by its moments

$M_i = \int x^i \rho(x) \mathrm{d}x$

In the case of the Gaussian distribution, the moments are

$M_1 = \frac{1}{2\pi \sigma^2} \int x e^-\frac{(x-\mu)^2}{2\sigma^2} \mathrm{d}x = \mu. \mbox{ In the following we set }\mu=0,$ i.e. we calculate the central moments

$M_2 = \frac{1}{2\pi \sigma^2} \int x^2 e^-\frac{x^2}{2\sigma^2} \mathrm{d}x = \sigma^2, \mbox{ } M_3 = \frac{1}{2\pi \sigma^2} \int x^3 e^-\frac{x^2}{2\sigma^2} \mathrm{d}x = 0, and$ $M_4 = \frac{1}{2\pi \sigma^2} \int x^4 e^-\frac{x^2}{2\sigma^2} \mathrm{d}x = 3\sigma^4, \mbox{ i.e. } M_{i} = \left\lbrace\begin{array}{2} 0 & \mbox{i odd}\\ (i-1)!! \sigma^i & \mbox{i even}\end{array}\right.$
• The distribution is characterized by two parameters: mean and variance

### The Central Limit Theorem

Given random variables x

• x_t independent
• x_t identically distributed (with zero mean for simplicity)
• the distribution has a finite variance

Then the sum of these random variables $X_n=\sum_{t=1}^n x_t/\sqrt{n}$ for large n goes to a Gaussian normal distribution $\rho(X)=\frac{1}{2\pi \sigma^2} e^-\frac{X^2}{2\sigma^2}$

### Moments of the sum of independent random variables

In order to proof the Central Limit Theorem, one can directly calculate the moments of $X_n=\sum_{t=1}^n x_t/\sqrt{n}$

$\mbox{Calculation rules: } \langle x_t\rangle = 0 = \langle X_n\rangle, \langle x_t^2\rangle = \sigma^2, \mbox{ and } \langle x_t x_s\rangle = \langle x_t\rangle \langle x_s\rangle = 0$

$\langle X_n^2\rangle = \frac{\sum_{t} \langle x_t^2\rangle}{n} + \frac{\sum_{t\neq s} \langle x_t x_s\rangle}{n} = \sigma^2 , \mbox{ } \langle X_n^3\rangle = \frac{\sum_{t} \langle x_t^3\rangle}{n^{3/2}} + 3\frac{\sum_{t\neq s} \langle x_t^2 x_s\rangle}{n^{3/2}} + \frac{\sum_{t\neq s \neq q} \langle x_t x_s x_q\rangle}{n^{3/2}} \propto \frac{n}{n^{3/2}} \rightarrow 0$ $\langle X_n^4\rangle = \frac{\sum_{t} \langle x_t^4\rangle}{n^{2}} + 4\frac{\sum_{t\neq s} \langle x_t^3 x_s\rangle}{n^{2}} + 3\frac{\sum_{t\neq s} \langle x_t^2 x_s^2\rangle}{n^{2}} + 6\frac{\sum_{t\neq s \neq q} \langle x_t^2 x_s x_q\rangle}{n^{2}} + \frac{\sum_{t\neq s \neq q\neq p} \langle x_t x_s x_q x_p\rangle}{n^{2}}$ $\rightarrow 3\frac{\sum_{t\neq s}\langle x_t^2\rangle \langle x_s^2\rangle}{n^{2}} = 3\sigma^4\frac{n(n-1)}{n^2}\rightarrow 3\sigma^4$

So only for even moments, the combinations of squared variables survive. The number of possible combinations defines the pre-factor

$\langle X_n^i \rangle = \left\lbrace\begin{array}{2} 0 & \mbox{i odd}\\ (i-1)!! \sigma^i & \mbox{i even}\end{array}\right.$

### The Mean Squared Displacement (MSD)

As we can see from $\langle X_n^2\rangle = \frac{\sum_{t} \langle x_t^2\rangle}{n} + \frac{\sum_{t\neq s} \langle x_t x_s\rangle}{n} = const,$ the MSD of the sum of independent and identically distributed random variables with finite variance scales linearly $\langle y_t \rangle = 2Dt \mbox{ with }y_t=\sum_{n=1}^t x_n.$

• The linear scaling and convergence to the Gaussian distribution holds in the long t limit if the random variables are correlated (not independent) with a finite correlation time $\tau = \sum_{t=1}^\infty C(t) < \infty.$
• This can be seen by looking at the coarse grained time series with uncorrelated elements

### Beyond the central limit theorem

Processes with increments x that

• do not have a finite variance (extreme events)
• are not identically distributed (changes over time)
• are not independent - event in the long time limit (in a later lecture)

### Levy-stable Distributions

$\mathcal{F}[\rho_{\gamma,\beta}(x,\mu,\sigma)]=\rho_{\gamma,\beta}(k,\mu,\sigma) =\exp\left[ i\mu k - \sigma^\gamma |k|^\gamma \left( 1 - i\beta \frac{k}{|k|}w(k,\gamma)\right)\right]$ $\rho_{\gamma,\beta}\left(\frac{\sum_{n=1}^t v(n)}{t^{1/\gamma}}\right)=\rho_{\gamma,\beta}\left(\frac{x(t)}{t^{1/\gamma}}\right)$

### The Noah-Effect

(Mandelbrot and Wallis) $\rho(x(t)) = t^L \rho\left(x(t)/t^L\right)$ $y(t)=\sum_{s=1}^t x(s), \; \mbox{ with } \; \langle x(s)x(s+\Delta)\rangle=\delta(\Delta), \; \mbox{ and } \; \lim_{x\rightarrow\infty}\rho(|x|)\propto |x|^{-3+2L}.$
• Power law tails lead to anomalous scaling (non-linear) $\langle y^2(t) \rangle \propto t^{2L}$
• Slower decay in tails of the distribution has no effect on the MSD and the density approaches a Gaussian

### Non-stationary Processes

• Idealized case: Increment variance grows or decays with power law over time (Scaled Brownian Motion) $x_t=t^{M-1/2}\xi_t \;\; \Rightarrow \;\; y(t)=\sum_{s=1}^t x(s) = t^{M} \sum_{s=1}^t \xi_t/\sqrt{t}$
• Power law growth/ decay lead to anomalous scaling $\langle x^2(t) \rangle \propto t^{2M}$

Non-stationary increments: What is shape of distribution and MSD scaling?

variance grows/decaysvariance random/periodic
Gaussian incrementsGaussian/anomalousGaussian/normal
Non-Gaussian incrementsNon-Gaussian/anomalousNon-Gaussian/normal

### Air Pressure: Non-stationarity

• Air pressure data from Potsdam
• Density is Non-Gaussian
• Trajectory shows: Variance changes with the Seasons

### Correlated Volatility

• Look at S & P 500 index
• Financial assets are characterized by log-return
• Variance (volatility) is not constant, but correlated
• GARCH(1,1) Model $x_t = \sigma_t\xi_t$ $\sigma_t^2 = c + ax_{t-1}^2 + b \sigma_{t-1}^2$

### Exterimental Observation: Non-Gausian with normal scaling of MSD

Experimental setup for non-Gaussian diffusion with linear scaling of MSD (see Pastore et al. 2022)

### Unicellular Dictyostelium discoideum: Anomalous scaling and non-Gaussian

Non-Gaussian diffusion and anomalous scaling in the diffusion of amoeboid cells (see Cherstvy et al. 2018)

### Ergodicity

Remember: a requirement for ergodicity is, that the dynamics is measure preserving

• Non-stationary increments and increments with diverging variance both lead to linear scaling in the time-averaged MSD, while the ensemble averaged MSD depends on the exact definition of the dynamics

### Telomeres Diffusion: Non-ergodicity

In the diffusion of Telomeres, the time average of the MSD exhibits a different scaling from the ensemble average (see Bronstein et al. 2009)

### Random Events

Examples:

• Volcanic erruptions

A random event might occur at any moment in time with probability lambda

• What is the probability distribution of the event happening at time t $p(t=0) = \lambda$ $p(t=1) = (1-\lambda) \lambda$ $p(t) = (1-\lambda)^t\lambda \approx e^{-\lambda t} \lambda \; \; \mbox{with} \; \lambda=\frac{1}{\tau}$

### The Poisson Prozess

A process with interevent times

$W(\tau) = {\lambda} \exp(-\lambda \tau)$

Generally, the probability of N events in the time interval t is given by the binomial distribution

$\Lambda = t\lambda$ $\frac{t!}{N!(t-N)!} \lambda^N(1-\lambda)^{t-N} \approx \frac{\sqrt{2\pi t}(t/e)^t}{\sqrt{2\pi(t-N)}((t-N)/e)^{t-N}}\lambda^N (1-\lambda)^{t-N}$ $\approx \frac{t^t \lambda^N(1-\lambda)^{t-N}e^{-N}}{(t-N)^{t-N} N!} \approx \frac{t^t (\Lambda/t)^N(1-\Lambda/t)^{t-N}e^{-N}}{t^{t-N}(1-N/t)^{t-N} N!} \approx \frac{\Lambda^N (1-\Lambda/t)^{t} e^{-N}}{(1-N/t)^{t} N!} \approx \frac{\Lambda^N e^{-\Lambda}}{N!}$

### Continuous time random walks

• There is a general class of processes with interevent duration distribution W(t)
• In addition, we can define a random walk y(t), where the time between two steps is defined by W(t)
• The distribution of jumplengths can then be drown from a second probability distribution
• Or both distribution can be coupled, i.e. the joint probability distribution reads
• $\Psi(\chi,\tau) = W(\tau) \frac{1}{2} [ \delta(\chi-f(\tau)) + \delta(\chi+f(\tau)) ]$

### Levy walks

Instead of jumps after each waiting period, the velocity can also be constant or grow/decay during the waiting period

More on these processes later, when we talk about Long Range Correlations

### Aging

What happens if the distribution of waiting times W(t) has no finite mean?

$W(\tau) = \frac{u^\alpha}{\tau^{1+\alpha}} \;\; \mbox{ with } \;\; 1 > \alpha > 0$
• The probability of an observation x(t) happening during a waiting period increases over time

### Anomalous scaling: How to distinguish extreme events and non-stationarity

[Chen et al. 2017]

Look at first and second moments of the increments

• Non-stationary increments: $\langle \sum_t |x_t| \rangle = \sum_t t^{M-1/2} \langle |\xi_t| \rangle = t^{M+1/2} \langle |\xi_t| \rangle$ $\langle \sum_t x_t^2 \rangle = \sum_t t^{2M-1} \langle \xi_t^2 \rangle = t^{2M} \langle \xi_t^2 \rangle$
• Infinit variance: $\rho(|x|) \rightarrow |x|^{-3+2L} \mbox{ for } |x|\rightarrow\infty \mbox{ with } 1>L>1/2$ $\langle |x_t| \rangle \mbox{integrable, constant}$ $\langle x_t^2 \rangle \mbox{not integrable} \Rightarrow \mbox{grows with time}$

Aging systems: L=3/4-M/2, both moments have the same time-dependent

### Time-Reversal Symmetry

Gaussian Prozesses are time-reversal processes

When is time-reversal symmetry violated