Scaling of shot noise processes

TL;DR

This paper studies how the distribution, power spectral density, and time above threshold of shot noise processes change under scaling transformations.

Contribution

It introduces a detailed analysis of the scaling behavior of shot noise processes and their statistical properties.

Findings

Derived scaling laws for shot noise distribution

Analyzed spectral density scaling behavior

Examined the impact on time above threshold

Abstract

In this contribution, we investigate the scaling of the distribution of the shot noise process, its power spectral density and its time above threshold.

Equations

$$\Phi(t)=\sum_{k=0}^{\infty}A_{k}\phi\mathopen{}\mathclose{{}\left(\frac{t-t_{k}}{\tau_{k}}}\right).$$

$$\ln C_{\Phi}(u)=\gamma\sum_{n=1}^{\infty}\frac{(\text{i}u)^{n}}{n!}\mathopen{}\mathclose{{}\left<A^{n}}\right>I_{n}$$

$${\Omega }_{\tilde{\Phi }}(\omega )=\frac{1}{{\tau }_{\text{d}}}\underset{0}{\overset{\infty }{\int }}\text{d}\tau {\tau }^2{P}_{\tau }(\tau )\phi (\omega \tau ).$$

$$p_{S}(S)=\frac{1}{\tau_{\text{d}}}p_{A}\mathopen{}\mathclose{{}\left(\frac{S}{\tau_{\text{d}}}+L|\frac{S}{\tau_{\text{d}}}+L>L}\right)=\frac{1}{\tau_{\text{d}}\mathopen{}\mathclose{{}\left<A}\right>}\exp\mathopen{}\mathclose{{}\left(-\frac{S}{\tau_{\text{d}}\mathopen{}\mathclose{{}\left<A}\right>}}\right),\,S>0.$$

$$p_{△T}(△T)$$

$$p_{△T}(\lambda △T)$$

$$=\underset{0}{\overset{\infty }{\int }}\text{d}\frac{\tau }{\lambda }{p}_{\tau }(\tau )p_{△T}(△T|\tau /\lambda )$$

$$={\lambda }^{-\alpha }\underset{0}{\overset{\infty }{\int }}\text{d}\frac{\tau }{\lambda }{p}_{\tau }(\tau /\lambda )p_{△T}(△T|\tau /\lambda )$$

$$={\lambda }^{-\alpha }{p}_{△T}(△T).$$

$$p_S(\lambda S)={\lambda }^{-\alpha }{p}_S(S),$$

$$\varphi (\theta )=c(\theta +m{)}^{-\alpha },0\le \theta \le \Delta ,$$

$$p_{A}(A|L/\phi(0)<A<L/\phi(\Delta))=\frac{1}{\mathopen{}\mathclose{{}\left<A}\right>}\frac{\exp(-A/\mathopen{}\mathclose{{}\left<A}\right>)}{\exp\mathopen{}\mathclose{{}\left(-L/\mathopen{}\mathclose{{}\left<A}\right>\phi(0)}\right)-\exp\mathopen{}\mathclose{{}\left(-L/\mathopen{}\mathclose{{}\left<A}\right>\phi(\Delta)}\right)}.$$

$$p_{{\triangle T}}({\triangle T}|0<{\triangle T}<\tau_{\text{d}}\Delta)=\frac{\alpha L}{c}\frac{{\triangle T}^{\alpha-1}}{\tau_{\text{d}}^{\alpha}}p_{A}\mathopen{}\mathclose{{}\left[\frac{L}{c}\mathopen{}\mathclose{{}\left(\frac{{\triangle T}}{\tau_{\text{d}}}}\right)^{\alpha}|L/\phi(0)<A<L/\phi(\Delta))}\right]$$

$$p_{{\triangle T}}({\triangle T}|0<{\triangle T}<\tau_{\text{d}}\Delta)\propto{\triangle T}^{\alpha-1}\exp\mathopen{}\mathclose{{}\left(-\frac{L}{c\mathopen{}\mathclose{{}\left<A}\right>}\frac{{\triangle T}^{\alpha}}{\tau_{\text{d}}^{\alpha}}}\right).$$

$$S=\frac{c\tau_{\text{d}}}{1-\alpha}A\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{Ac}{L}}\right)^{-1+1/\alpha}-m^{1-\alpha}}\right].$$

$$\displaystyle\frac{1-\alpha}{c\tau_{\text{d}}\mathopen{}\mathclose{{}\left<A}\right>}\mathopen{}\mathclose{{}\left(\frac{L}{c\mathopen{}\mathclose{{}\left<A}\right>}}\right)^{-1+1/\alpha}S$$

$$\hat{S}$$

$$p_A(A)=\frac{1-\alpha }{M^{1-\alpha }-m^{1-\alpha }}{A}^{-\alpha },m<A<M,$$

$$p_A(A|A>L)=c{A}^{-\alpha },m<A<M,$$

$$c=\{\begin{array}{ll}\frac{1-\alpha }{M^{1-\alpha }-m^{1-\alpha }},& L<m,\\ \frac{1-\alpha }{M^{1-\alpha }-L^{1-\alpha }},& m<L<M,\\ 0,& L>M.\end{array}$$

$$p_{{\triangle T}}({\triangle T})=\frac{L}{\tau_{\text{d}}}\exp\mathopen{}\mathclose{{}\left(\frac{{\triangle T}}{\tau_{\text{d}}}}\right)p_{A}(L\exp({\triangle T}/\tau_{\text{d}})|A>L)=c\frac{L^{1-\alpha}}{\tau_{\text{d}}}\exp\mathopen{}\mathclose{{}\left[(1-\alpha)\frac{{\triangle T}}{\tau_{\text{d}}}}\right]$$

Full text

The shot noise process is given by

[TABLE]

We assume exponential waiting times and uncorrelated random variables. We have that the characteristic function of $\Phi$ is given by theodorsen-ppcf

[TABLE]

where $\gamma=\langle\tau\rangle/\tau_{\text{w}}$ and $I_{n}=\int_{-\infty}^{\infty}\phi(\theta)^{n}\text{d}\theta$.

The power spectral density of $\widetilde{\Phi}=(\Phi-\langle\Phi\rangle)/\Phi_{\text{rms}}$ is given by garcia-pop-1

[TABLE]

The general method for deriving the time above threshold in the limit of no pulse overlap is given in theodorsen-pre . In the normal limit, the shot noise process $\widetilde{\Phi}$ is a process with standard normal distribution and power spectrum given by Eq. (3). If the power spectral density scales as a power law, $\Omega\sim\omega^{-\beta}$, the shot noise process in the normal limit will follow a fractional Brownian motion (fBM) with Hurst parameter given by $\beta=2H+1$, $0<H<1$. It is known that the first return time for fBM (which is equivalent to time above threshold) scales as ${\triangle T}^{H-2}$ rypdal-pre . Thus, in the normal limit we straightforwardly have $\nu=(5-\beta)/2$. In the same manner, it can be shown that the mass above threshold $S=A\int_{0}^{{\triangle T}}\phi(t/\tau_{\text{d}})\text{d}t$ scales as $p_{S}(S)\sim S^{-\chi}$ with $\chi=2/(1+H)=4/(1+\beta)$.

Unless indicated, duration times are assumed to be degenerately distributed, pulses are exponential functions and the amplitudes are exponentially distributed.

[TABLE]

I Standard shot noise

Probability distribution

In this case, the probability distribution is known to be a Gamma distribution with no power-law scaling.

Power spectral density

This has been treated in previous publications. The scaling is 0 for low values and 2 for large values, but this is the power spectrum of exponential decay.

Duration above threshold - intermittent limit

In theodorsen-pre , this is shown to be a Gumbel distribution which lacks power law scaling.

Duration above threshold - normal limit

This is a known result for the Ornstein-Uhlenbeck process.

Mass above threshold - intermittent limit

With the exponential pulse, ${\triangle T}=\tau_{\text{d}}\ln(A/L)$, giving $S=\tau_{\text{d}}(A-L)$. This means that $A>L\rightarrow S>0$. We then have

[TABLE]

There is no scaling here.

Mass above threshold- normal limit

This is a known result for the Ornstein-Uhlenbeck process kearney-2005 .

II Power law pulse duration times

Probability distribution

The probability density function of $\Phi$ only depends on $\langle\tau\rangle$, and so does not scale with any scaling parameter of $\tau$.

Power spectral density

From Eq. (3), the scaling of the power spectrum is straightforward. It has been seen that this only holds for $1<\alpha<3$.

Duration above threshold - intermittent limit

Assuming exponential pulses and exponentially distributed amplitudes, $p_{{\triangle T}}({\triangle T}|\tau)$ is given in theodorsen-pre in the strongly intermittent limit. By inspection, $p_{{\triangle T}}(\lambda{\triangle T}|\tau)=\frac{1}{\lambda}p_{{\triangle T}}({\triangle T}|\tau/\lambda)$. If $p_{\tau}\sim\tau^{-\alpha}$, we then have that $p_{\tau}(\tau)=\lambda^{-\alpha}p_{\tau}(\tau/\lambda)$, and

[TABLE]

Duration above threshold - normal limit

Calculated from $\nu=(5-\beta)/2$.

Mass above threshold - intermittent limit

Since $p_{S}(S|\tau)$ is an exponential distribution with mean value \tau\mathopen{}\mathclose{{}\left<A}\right>, we have that $p_{S}(\lambda S|\tau)=\frac{1}{\lambda}p_{S}(S|\tau/\lambda)$, the calculation follows analogously to the one above and we have

[TABLE]

giving the scaling $p_{S}(S)\sim S^{-\alpha}$.

Mass above threshold- normal limit

The scaling is derived from the expression for known $\beta$.

II.1 The rate parameter $\lambda$

Note that several authors have used the rate parameter $1/\tau$ instead of $\tau$. For $p_{\tau}\sim\tau^{-\alpha}$, $p_{\lambda}\sim\lambda^{\alpha-2}$. Thus at $\alpha=1$, these are equal, but in general there is a shift in the distribution. Thus, uniform $\lambda$, which is known to give $\Omega\sim\omega^{-1}$ requires $\tau^{-2}$. Several authors have wrongly assumed $\tau^{-1}$ leads to $\Omega\sim\omega^{-1}$ .

III Power law pulses

Probability distribution

Possible presence by inspection of probability distribution.

Power spectral density

The scaling of $\Omega$ is given by lowen-teich . Note that this requires $0<\alpha<1$.

Duration above threshold - intermittent limit

We consider the pulse shape

[TABLE]

where $c=[1-\alpha]/[(\Delta+m)^{1-\alpha}-m^{1-\alpha}]$ is a normalization constant. For the pulse to trigger, we require $A\phi(0)>L$. If $A\phi(\Delta)>L$, the duration is the full pulse duration, ${\triangle T}=\tau_{\text{d}}\Delta$. The possible interesting scaling therefore happens for $L/\phi(0)<A<L/\phi(\Delta)$. The truncated exponential distribution for $A$ is therefore

[TABLE]

We then have that (RECHECK, POSSIBLY ${\triangle T}/\tau_{\text{d}}\rightarrow{\triangle T}/\tau_{\text{d}}+m$!)

[TABLE]

giving

[TABLE]

We note that the scaling $\alpha=0\rightarrow p_{{\triangle T}}\sim{\triangle T}^{-1}$ is not possible, as in this case $\phi(0)=\phi(\Delta)$ and there are no possible events. For small values of the exponent (or in the limit ${\triangle T}\rightarrow 0$), the exponential function approaches 1 and $p_{{\triangle T}}\sim{\triangle T}^{\alpha-1}$.

Duration above threshold - normal limit

Calculated from $\nu=(5-\beta)/2$.

Mass above threshold - intermittent limit

The pulse is the same as above. For $L/\phi(0)<A<L/\phi(\Delta)$, we have ${\triangle T}=\tau_{\text{d}}(Ac/L)^{1/\alpha}-\tau_{\text{d}}m$, and

[TABLE]

We can rewrite this equation as (using that X=A/\mathopen{}\mathclose{{}\left<A}\right> is a standard exponential distribution):

[TABLE]

Note that $L/\phi(0)<A\rightarrow c_{1}^{\alpha/(1-\alpha)}<X$ is equivalent to $S>0$. On the other side, we require $X<c_{1}^{\alpha/(1-\alpha)}(1+\Delta/m)^{\alpha}$. Above this, $S=A\tau_{\text{d}}\Delta$. By visual inspection of numerically generated PDFs, there is no power law scaling in this function.

Mass above threshold- normal limit

The scaling is derived from the expression for known $\beta$.

IV Power law amplitudes

Probability distribution

Present by inspection of probability distribution

Power spectral density

The pulse amplitude distribution does not affect the power spectral density.

Duration above threshold - intermittent limit

Following theodorsen-pre , we find that for a truncated Pareto distribution,

[TABLE]

the conditional amplitude distribution for amplitudes above threshold are

[TABLE]

where

[TABLE]

With the exponential pulse, we get that

[TABLE]

which has no power law scaling in $\alpha$.

Duration above threshold - normal limit

In the normal limit, we have a process with normal probability distribution and exponential correlation (OU-process). This gives a scaling as 1/2.

Mass above threshold - intermittent limit

In this case, we still have $S=\tau_{\text{d}}(A-L)$ from the calculation preceding Eq. (4). It follows straightforwardly that $p_{A}\sim A^{-\alpha}\rightarrow p_{S}\sim S^{-\alpha}$.

Mass above threshold- normal limit

This is a known result for the Ornstein-Uhlenbeck process kearney-2005 .