Statistical properties of a filtered Poisson process with additive random noise: Distributions, correlations and moment estimation
Filtered Poisson processes are often used as reference models for intermittent fluctuations in physical systems. Such a process is here extended by adding a noise term, either as a purely additive term to the process or as a dynamical term in a stochastic differential equation. The lowest order moments, probability density function, auto-correlation function and power spectral density are derived and used to identify and compare the effects of the two different noise terms. Monte-Carlo studies of synthetic time series are used to investigate the accuracy of model parameter estimation and to identify methods for distinguishing the noise types. It is shown that the probability density function and the three lowest order moments provide accurate estimations of the parameters, but are unable to separate the noise types. The auto-correlation function and the power spectral density also provide methods for estimating the model parameters, as well as being capable of identifying the noise type. The number of times the signal crosses a prescribed threshold level in the positive direction also promises to be able to differentiate the noise type.
Intermittent fluctuations are found in a variety of physical systems such as atmospheric winds Narashima et al. (2007), astrophysical plasmas Robinson (1995); Kristensen et al. (1991), fission chambers Elter et al. (2015), diodes and electric circuits Rice (1944, 1945) and magnetic confinement experiments Endler et al. (1995); Carreras (2005); Liewer (1985); D’Ippolito et al. (2011); Zweben et al. (2007), as well as in fields such as finance Jang (2004) and physiology Fesce et al. (1986). In several such systems, treatments of intermittent effects as a superposition of random variables has been fruitful, see for instance Refs. Hole et al., 2004, 2006. In this contribution, we will focus on a particular reference model for intermittent fluctuations, the filtered Poisson process (FPP) (also called a generalized shot noise process) Elter et al. (2015); Rice (1944); Jang (2004); Fesce et al. (1986). This model consists of a super-position of uncorrelated pulses with a uniform pulse shape and randomly distributed pulse amplitudes, arriving according to a Poisson process Garcia (2012); Garcia et al. (2016a). The FPP has been considered by e. g. Refs. Bondesson, 1982; Pécseli, 2000; Parzen, 1999; Kotler et al., 1999; Lowen and Teich, 2005.
This contribution is primarily motivated by turbulent flows in the far scrape-off layer of magnetically confined plasmas. Evidence points towards these fluctuations being caused by filamentary structures moving radially outwards, transporting particles and heat through the scrape-off layer towards main chamber walls Carreras (2005); D’Ippolito et al. (2011); Zweben et al. (2007). Time series obtained from probe measurements and gas puff imaging diagnostics exhibit similar behavior for a wide range of machine parameters, having skewed and flattened probability distribution functions (PDFs) resembling Gamma distributions and large amplitude fluctuations with exponential pulse shapes and exponentially distributed amplitudes, arriving according to a Poisson process Garcia et al. (2013a, b); Graves et al. (2005); Kube et al. (2016); Theodorsen et al. (2016); Garcia et al. (2007). The FPP with exponentially distributed pulse amplitudes and a pulse shape consisting of a rapid rise and exponential decay can be shown to be Gamma distributed Garcia (2012); Garcia et al. (2016a); Bondesson (1982). By adding an independent, normally distributed variable to the process, the PDF of the resulting process is a convolution of a Gamma PDF and a normal PDF. This result has been shown to be in very good agreement with probe measurements from the Alcator C-Mod and KSTAR tokamaks Kube et al. (2016); Garcia et al. (2016b).
In this contribution, we will extend the reference FPP model by adding normally distributed noise in two different ways, either as a purely additive term to the process, modeling measurement noise or other processes unconnected to the dynamics of the FPP, or as a dynamical noise term in the stochastic differential equation for the reference model, resembling an Ornstein-Uhlenbeck process. We will mainly consider how these different noise terms affect the PDF of the resulting signal, its auto-correlation function and its power spectral density. Additionally, we will consider the rate at which the processes cross a certain threshold level in the positive direction. The goal of this contribution is to find methods for discriminating the two types of noise and to identify reliable methods for estimating the model parameters in a given realization of the process.
This contribution is organized as follows: In Sec. II, the pure FPP is considered. In Sec. III, the two types of noise are considered. The lowest order moments and the PDF of the FPP with additional noise are discussed in Sec. IV, and the power spectral density and auto-correlation function of this process are discussed in Sec. V. In order to differentiate the types of noise and to compare different methods of parameter estimation, Monte-Carlo studies of synthetic data are presented in Sec. VI. Sec. VII concludes the contribution. In Appendix A, a list of symbols and the most important analytical results of this contribution are collected. Appendix B contains derivations relating to the power spectral densities and auto-correlation functions discussed.
Ii Filtered Poisson process
In this section, we present the FPP to be analyzed in this contribution. This process is constructed as a super-position of pulses arriving in a time interval :
where the pulse duration time is taken to be the same for all pulses. The pulse amplitudes are taken to be exponentially distributed with mean value ,
where and here and in the following indicates the average over all random variables.
As an idealization of a pulse with a fast rise and an exponential decay, we use the one-sided pulse form
where is the Heaviside step function and is a dimensionless variable. Compared to a two-sided exponential pulse function, this simplification does not affect the moments or PDF of the process, simplifies the auto-correlation and power spectra (see LABEL:theodorsen-ppcf-16 for the auto-correlation of this process with finite growth) and allows the formulation of the process by a simple stochastic differential equation as described below.
The pulses are assumed to arrive according to a Poisson process with constant rate ( see for example Refs. Pécseli, 2000, Parzen, 1999, Ch. 4.1 or Stark and Woods, 2012, p. 562). Thus the (non-negative) number of arrivals is Poisson distributed,
where we have taken the mean value to be . It can be shown that the waiting time between consecutive pulses is exponentially distributed with mean value (Parzen, 1999, p. 135), while the arrival times are independent and uniformly distributed on the interval (Parzen, 1999, p. 140). The ratio between pulse duration time and average time between pulses,
is in the following referred to as the intermittency parameter.
While the FPP is a continuous process, any experimental data or synthetic realization of the process is discrete. As the time resolution of numerical data may be important for the noise processes, we also introduce the normalized time step,
where is the time step for synthetically generated signals. Details on the synthetically generated signals will be discussed in Sec. VI. Some realizations of for various values of are presented in Fig. 1, where we have used the normalization
Here and in the following, we will use a tilde to denote a normalized variable with zero mean and unit standard deviation. For , pulses arrive rarely and the signal spends large amounts of time close to zero value, resulting in a strongly intermittent signal. For , pulses overlap and the signal begins to resemble random and symmetric fluctuations around a mean value.
where due to the non-negative pulse amplitudes and pulse functions. The lowest order moments of are the mean , the standard deviation , the skewness and the flatness , giving a parabolic relationship between the skewness and flatness moments of Garcia (2012); Garcia et al. (2016a),
In Sec. IV it will be shown how additive noise alters this relationship.
It is possible to write the FPP as a stochastic differential equation. In the case of a one-sided exponential pulse function, it takes the form Kontorovich and Lyandres (1996)
where is the Dirac delta function. The model described by Eq. (10) can be seen as a train of -pulses, arriving at times with amplitudes . These pulses pass through a filter causing an exponential decay with characteristic decay time , giving the process its name.
We remark that Eq. (10) can be written as
where is the linear operator and the forcing is given by
A Green’s function for the operator , that is any function fulfilling , is given by
The solution of Eq. (11) is then the convolution
where can be interpreted as the filter which passes through. Since both and are non-zero only for positive arguments, the integration in Eq. (14) could be taken over the interval , and this is done in Sec. VI. Here, however, we take the integration limits to infinity in order to remain consistent with the Fourier transforms, discussed below and in the appendices.
In Appendix B, the power spectral density (PSD) of the FPP is shown to be
The mean value of the signal gives a zero frequency contribution to the PSD, while the fluctuations around the mean value give rise to a Lorenzian power spectrum. Since the pulses are uncorrelated, there is no explicit dependence on the average waiting time , apart from contributing to the value of and . Moreover, we see from Eqs. (66) and (72) that the Poisson point process provides the zero frequency contribution as well as a flat contribution independent from the frequency due to the lack of correlation between the pulses, while the Lorenzian spectrum comes entirely from the filter .
The auto-correlation and power spectral density are Fourier transform pairs under the Fourier transform over the entire real line. Thus we readily obtain the auto-correlation function
where denotes the inverse Fourier transform. We see that the time dependence of the auto-correlation function comes entirely from the Green’s function , as is expected, since the pulses are uncorrelated and thus the average time between pulses does not appear explicitly in the auto-correlation function.
It can be shown that the PSD and auto-correlation function of the normalized variable are given by
These expressions are presented in Figs. 2 and 3, respectively. The solid line in Fig. 2 represents the PSD of , while the solid line in Fig. 3 represents the auto-correlation function of . The other elements in these figures will be discussed further in Sec. V. From Eq. (18), it is evident that the e-folding time of is the pulse duration time . This corresponds to in Fig. 2, giving the approximate frequency where the PSD changes from a flat spectrum to power law behavior.
Iii Additive noise
In this section, we will expand the FPP with two forms of additional noise terms, which will be referred to as either observational or dynamical noise. In the following, the specific realization of the FPP (that is, the pulse amplitudes, arrival times and number of pulses for a particular realization) plays no role. Thus, for simplicity of notation, the subscript will be suppressed in the following.
Observational noise denotes noise unconnected to the FPP. Thus, a noise term is simply added to a realization of the process ;
where is a normally distributed process with vanishing mean and unit standard deviation and is the noise intensity parameter, effectively describing the standard deviation of the noise process.
In the case of dynamical noise, the noise term is added as random forcing in the stochastic differential equation (10), and is therefore connected to the pulses:
is an Ornstein-Uhlenbeck process, and the random variable is normally distributed with zero mean and unit standard deviation.
We also introduce a parameter , describing the relative noise level. Using as a collective symbol for both and , we define as
Although and have the same probability distributions, they exhibit very different dynamical behavior, as illustrated by realizations of the processes presented in Fig. 4. While fluctuates rapidly on the sampling time scale around the zero value, wanders around the zero value, with finite temporal correlations. This is quantified in Appendix B.2, where the auto-correlation functions and power spectral densities of and are derived. Note that both and are independent of the process .
Realizations of the processes , and for different parameters , and are presented in Figs. 1, 5 and 6. Realizations of the pure FPP for various are presented in Fig. 1. For , the signal is very intermittent and spends large amounts of time close to zero. For , pulse overlap is much more significant, washing out the intermittent features. As becomes very large, the signal resembles symmetric fluctuations around the mean value, and it can be shown that in the limit , the filtered Poisson process has a probability distribution resembling a normal distribution Garcia (2012); Garcia et al. (2016a); Rice (1944); Pécseli (2000).
Realizations of the FPP with dynamical noise for various are presented in Fig. 5. For very small , there is very little difference between the pure process and the process with dynamical noise. For larger , the noise process plays a larger role, concealing all but the largest pulses.
The effect of changing is clearest for the FPP with observational noise. Realizations of this process for various values of are presented in Fig. 6. Here we see that for large , the process resembles a pure FPP. For small the noise process dominates, even though its rms-value is th the rms-value of the FPP in this case. A smaller time step means more data points in a given time interval and thus more chances for large values of the noise process. In contrast, the pure FPP itself is much less sensitive to changes in time resolution, the primary effect being that separate pulses may be counted as one in the computation, if they are close enough. The FPP with dynamical noise is also less sensitive to due to the exponential damping the noise.
Iv Moments and distributions
In this section, we present the lowest order moments and the PDF of the FPP with additive noise and describe how the model parameters can be estimated from these for a realization of the process. The probability density function of the stationary process is given in Eq. (8), and both and are normally distributed with vanishing mean and unit standard deviation. Thus, we write
where the random variable has the probability density function
Note that has the same distribution as both and , while has the same distribution as both and . In other words, the PDF of the FPP with additional noise is the same for both types of noise considered here. In this section, we assume continuous random variables and . It can be shown that the convergence of the moments of to their true values depends on . A discussion on this and the estimation of the moments of in general is given in Refs. Garcia et al., 2016a; Kube and Garcia, 2015. Under the assumption that and are independent, and using that , the lowest order moments of are readily calculated as
Using Eq. (23), we have the moments
Also in this case, we can find a parabolic relation between the skewness and flatness moments,
The effect of additional noise is to increase the pre-factor in the parabolic relationship.
The model parameters and can be estimated from the moments in a several different ways. Using Eqs. (27a) and (27b) to eliminate . The relative fluctuation level is then related to the model parameters by
clearly showing how the additional noise amplifies the fluctuation level. As the estimators for lower order moments are more accurate than those for higher order moments, it is reasonable to assume the most accurate estimators for the model parameters come from using the lowest order moments, and :
In experimental fluctuation data time series there can sometimes be reasons not to trust the mean value of a signal, for example due to externally imposed low frequency noise or trends Garcia et al. (2013a, b). Such problems typically do not affect the large-amplitude fluctuations, leaving the higher order moments trustworthy. In this case, using the normalization in Eq. (7) and observing that and , we have
The probability density function of a sum of two independent random variables and is a convolution of their respective probability density functions Stark and Woods (2012):
As a consistency check, it should be noted that in the limit , should be the probability density function of a pure FPP. This is indeed the case, as in this limit , and by definition,
and the Gamma distribution of the FPP without noise is recovered. Thus, in the following, we take the case to signify a FPP without additive noise.
The expression for the probability density function of is given in the appendix, Eq. (50). In order to illustrate the effect of pulse overlap and additional noise, the PDF for is shown for various values of and , and in Figs. 7, 8 and 9, respectively. Clearly, as increases beyond unity, the probability distribution changes towards a normal distribution. For , the random variable is non-negative, causing an abrupt halt in the distribution for . This jump does not exist for . Thus negative values for , or equivalently, , signifies a deviation from a pure FPP.
V Spectra and correlations
The PSD of the sum of two independent random variables is the sum of their respective PSDs. The power spectra for the FPP with additional noise are derived in Appendix B. With Eqs. (15) and (76), the power spectral density of the FPP with dynamical noise is therefore
where and are given by Eqs. (27a) and (27b), respectively. We have also used the relation . Note that the spectrum in Eq. (33) is identical to that for the pure FPP. Thus, we conclude that the auto-correlation functions of and have the same functional shape as well, up to the difference in the first two moments.
This function is qualitatively different from the power spectral densities of and , although it converges to both in the limit . These differences are now explored in detail.
The power spectral density of is presented in Figs. 2 and 10. In Fig. 2, the difference between or and is presented. Higher ratio of noise signal to FPP decreases the value of the power spectral density for low frequencies and causes a transition from a power law spectrum to a constant spectrum at higher frequencies. The Nyquist frequency for is indicated by the vertical line,
and shows that for small , this transition happens at too high frequencies to be reliablely observed. In Fig. 10, the power spectral density of is presented for and various values of . For low frequencies, the difference between and is too small to be of practical use, while the effect of noise for high frequencies can only be observed without aliasing for very low .
The auto-correlation function of is given in Eq. (16). Since the power spectral density of the rescaled processes and are identical, we conclude that their auto-correlation functions are as well,
where the auto-correlation of a rescaled random process is
With the auto-correlation function of from Eq. (79), we have that the auto-correlation function of is
Note that as required by the normalization, while for correlation times longer than the sampling time, the correlation function of the normalized variable has a value of times the correlation function of . Thus, Eq. (38) can also be written as
If there is no appreciable drop from to , and the correlation functions overlap for , the observational noise is negligible.
As expected, in the limit , approaches . In practice, the auto-correlation function should be better at revealing the presence of observational noise than the power spectral density, as this difference is largest for small time lags, where the auto-correlation function is the most accurate. Still, and , a difference which may be difficult to verify in practice for small, but appreciable (say, ). An example of this is seen in Fig. 3. In this figure, the auto-correlation function of is presented as a function of for and various values of . For large , both the initial drop and the reduced value of compared to is clear, but this is not the case for . The behavior of the auto-correlation of for and varying is shown in Fig. 11. All functions fulfill , and for , all have the same value.
Vi Parameter estimation
In this section, we present results from numerical analysis using synthetically generated time series. The time series are generated following the convolutions in Eqs. (14) and (22), with integration limits . The convolutions are performed by a fast Fourier transform numerical convolution. All random numbers are generated using a Mersenne Twister. The time array is constructed as , with and . We have set , so we vary by varying . The pulse amplitudes are drawn from an exponential distribution with , the arrival times correspond to integers uniformly distributed on , giving
where is the Kronecker delta function for integers and . is an array of independent and identically distributed normal variables with vanishing mean and unit standard deviation, while the discrete version of is an equally shaped array of independent and identically distributed normal variables with vanishing mean and standard deviation .
vi.1 Reliability of parameter estimation
The aim of this section is to numerically test and verify results from the prior sections. We present analysis of parameter estimation for synthetic time series with the goal of separating the two types of noise, where the analysis is performed on 1000 time series of each type, each of length data points, with parameters close to experimental values (see Refs. Kube et al., 2016; Theodorsen et al., 2016; Garcia et al., 2007); , and . This gives time series of duration .
Given a time series, we can estimate and by comparing the estimated PDF for the time series to the PDF in Eq. (50), or by comparing the sample estimate of the moments to the moments in Eqs. (30) or (31). These two methods do not discriminate the nature of the noise. However, comparing Eqs. (37) and (39), we see that Eq. (39) gives the auto-correlation function of a filtered Poisson process with observational noise for and the auto-correlation function of a filtered Poisson process with dynamical noise or without noise for . Thus Eq. (39) can potentially separate observational from dynamical noise. In addition, it gives us an estimate of . The same considerations hold for the power spectral densities in Eq. (35a) as compared to Eq. (35b). In all the following cases where we fit synthetic data to an analytical function, the initial values are given as the true parameter values of the process.
The estimated PDFs of the synthetic signals are presented in Fig. 12. The thick lines give the average PDF for all synthetic signals, and the thin lines indicate the maximal deviation from this mean value. Visually, the only reliable difference between the signals is the elevated tail for negative values of and compared to . In Fig. 13, we present PDFs for the estimated parameters and from the moments and the PDF of the processes. Here and in the following, the hat symbol indicates an estimated value. The values in Figs. 12(a) and 12(b) were obtained by estimating the relative fluctuation level and skewness of the synthetic data and using Eq. (30). The values in Figs. 12(c) and 12(d) were obtained by estimating the skewness and flatness and using Eq. (31). In Figs. 12(e) and 12(f), the values were obtained by fitting the function in Eq. (50) to the estimated PDF of the synthetic data with a least squares routine under the constraints and to ensure convergence. The rms-value of the distributions of and are presented in Tables 1 and 2, respectively.
As is evident from Fig. 13, using the three lowest order moments or the PDF of the signal to estimate the parameters is far preferable to using the estimated skewness and flatness moments of the signal. Figs. 12(c) and 12(d) have far broader distributions than the other methods of parameter estimation. Due to the significant overlap between the distributions in Fig. 12(d), the presence or absence of noise can be difficult to determine reliably. For the underlying parameters used here, Table 1 shows that the three lowest order moments give a better estimate for than the PDF, while Table 2 shows the reverse for . This result is likely highly dependent on the estimation methods employed, as well as the properties of the time series in question. Thus, the authors recommend a full Monte Carlo analysis, as presented here, in order to determine errors in parameter estimation. As expected, the type of noise cannot be determined from the moments or PDF of the signal.
The auto-correlation functions of the synthetic signals are presented in Fig. 13(a), while the PSDs are presented in Fig. 13(b). Again the thick lines give the average result from all signals, while the thin lines give the maximal deviation from the average. The PSD was computed using Welch’s method with data points window size, overlap and a Hanning window. In both cases, the observational noise gives a clear visual difference compared to the pure FPP and the same process with dynamical noise, having an elevated tail as predicted in Eq. (35b). Note also the slight lifting of the tail in Fig. 13(b) for and . This is most likely caused by roundoff errors near the Nyquist frequency. Changing the window size or the windowing function does not correct the problem. Direct computation of the periodogram using a fast Fourier transform does not have this problem, although it presents other problems for parameter estimation. The effect of this lifting of the tail on parameter estimation is discussed below.
In order to estimate and , we have fitted the auto-correlation function of the synthetic signals to the discrete version of Eq. (39):
where we ignore the contribution, since this equals unity for all processes discussed. Due to the uncertainties in the auto-correlation function for large time lags, only the first 50 time steps are used. The PSD of the synthetic signals is fitted to the function in Eq. (35b). In both cases, a non-linear least-squares fit routine with the true values as initial values was used.
In Figs. 13(c) and 13(e) the PDFs of and from the auto-correlation are presented, respectively. As is the same for all three classes of signal, the over-lapping PDFs are no surprise, and we note the small variation around the true value ( for all processes). Concerning , the auto-correlation function does indeed separate the observational noise from the dynamical noise. For the observational noise, the rms-value is low, , and the auto-correlation function can be used to estimate the noise level. The PDFs of and from the PSD are presented in Figs. 13(d) and 13(f), respectively. Again the rms-values of all parameters is small, of the original parameters, with the exception of the –parameter in Fig. 13(f) for and , with . In these figures, we have a clear bias in both and . This is most likely created by the lifting of the tail of the power spectral density estimate, as discussed above. An elevated tail signifies observational noise, so an artificial elevated tail leads to an over-estimation of , as seen in Fig. 13(f). This leads to a bias in the estimation of , as seen in Fig. 13(d). Eliminating this bias by restricting the fit range is not recommended, as this will compromise the accuracy in separating the PSDs. As this bias is reproducible, it does not present a significant problem for parameter estimation, and the power spectral density could be used as a sanity check for the auto-correlation function.
vi.2 Level crossing statistics
Another measure which would intuitively separate the two types of noise is the number of upwards crossings above a certain threshold level per unit time, or the rate of level crossings. This quantity has been explored in Refs. Rice, 1945; Theodorsen and Garcia, 2016; Biermé and Desolneux, 2012; Bar-David and Nemirovsky, 1972; Garcia et al., 2016a for the pure FPP, in Refs. Barakat, 1988; Hopcraft et al., 2007 for a gamma distributed random process and in Refs. Sato et al., 2012; Fattorini et al., 2012 for atmospheric plasma. The rate of level crossings above a threshold as a function of the threshold is presented in Fig. 15 for the synthetic data discussed in the previous section. The thick lines give the mean values for the given threshold, while the thin lines represent the minimal and maximal value for all synthetic time series generated. The threshold is in units of signal rms-value above signal mean value. In agreement with intuition, the FPP with observational noise crosses the threshold much more frequently than the two others due to the rapid fluctuations around the mean value of the pure FPP at any amplitude triggering spurious crossings. The difference between the pure FPP and the FPP with dynamical noise is largest for small threshold values, where the number of threshold crossings is largest. Note that while has its maximum number of level crossings for , this is down-shifted for the processes with noise, since the noise does not contribute to the mean value of the process. While we know of no theoretical estimate for the rate of level crossings for a FPP with noise, this value can still be found from synthetic signals, generated by estimating and from the PDF of a measurement signal and from its auto-correlation function. Comparing the true rate of level crossings to the rate of level crossings for synthetic signals with different types of noise could separate the noise types.
Motivated by previous analysis of measurement data from magnetically confined plasmas, we have here investigated a FPP with normally distributed noise included as either observational or dynamical noise. The PDF, PSD and auto-correlation function of the pure FPP have been presented. These have also been extended to include noise, showing how the gamma probability distribution of the FPP moves towards a normal distribution as the influence of noise increases and how dynamical noise gives the same auto-correlation and PSD as the pure FPP. The model including noise has a PDF in accordance with recent measurements Kube et al. (2016); Garcia et al. (2016b) and a parabolic relation between the skewness and flatness moments, as found in a variety of experiments on magnetized plasmas Kube et al. (2016); Theodorsen et al. (2016); Garcia et al. (2007); Sattin et al. (2009); Garcia et al. (2013a, b). The model also has an approximately exponentially decaying auto-correlation function and a power law PSD, in accordance with measurements from the scrape-off layer of a range of fusion plasmas Theodorsen et al. (2016); Garcia et al. (2007); Kube et al. (2016); Cziegler et al. (2010); Dewhurst et al. (2008); Tanaka et al. (2009). The previous analysis of experimental data demonstrate that significant noise levels can be present, and the extension of the FPP presented here is indeed necessary in order to reliably estimate the underlying parameters of the process.
Using synthetically generated time series with experimentally relevant parameters, we have shown that the PDF of the FPP with additional noise is indeed capable of separating a process with noise from a noise-less process, and the sample variance of the estimated parameters provide an indication of the error of estimation. In addition, it was shown that estimating the intermittency parameter and the noise parameter from the estimated relative fluctuation level and skewness of the process is comparable in accuracy to estimating the parameters from the estimated PDF of the process, while using the estimated skewness and flatness gives a much worse estimate. It was furthermore shown that the auto-correlation function can provide an estimate of the characteristic decay time of the FPP, and is also capable of separating the types of noise. In the case of observational noise, the auto-correlation function can also provide an estimate of the noise level, but can not do this in the case of dynamical noise. The PSD has comparable capabilities to the auto-correlation function, but care must be taken in its use, in particular in the case of marginal sampling frequency. Lastly, the rate of level crossings was proposed as another way to differentiate the types of noise. This method takes advantage of the visual differences in the noise demonstrated in Fig. 4, and was shown to be capable of detecting both the presence of noise and differentiating the types of noise.
It should be noted that the results presented here were not a parameter scan, and some of the conclusions may depend on the model parameters and especially the duration of the signal. For instance, from Fig. 6, it is evident that the rate of level crossings for is dependent on . In applying the methods from this contribution to experimental data, one should carry out a full Monte Carlo analysis as presented here, using parameters estimated from the PDF and auto-correlation function of the data set.
In this contribution, we have presented an extension of a reference model for intermittent fluctuations in physical systems. Its main application will be for analysis of fluctuation data time series from probe measurements in the scrape-off layer of magnetically confined plasmas, although it is sufficiently general to be applicable elsewhere. This model has no built-in long range correlations, interaction between pulses or other non-stationary effects, meaning it could serve as a useful null hypothesis for investigations of such effects.
This work was supported with financial subvention from the Research Council of Norway under grant 240510/F20. Discussions with H. L. Pécseli and F. C. Geisler are gratefully acknowledged.
Appendix A List of symbols and results
a.1 Time series
|The filtered Poisson process.|
|Normally distributed, uncorrelated noise with vanishing mean and unit standard deviation.|
|An Ornstein-Uhlenbeck process with vanishing mean and unit standard deviation.|
|A collective term for either or , where is a noise intensity parameter.|
|Denotes , it is a collective term for both and .|
We use the normalization
a.2 Moments and probability density functions
The PDF of is a gamma distribution with shape parameter and scale parameter , given by Eq. (8). Denoting the mean of the process by , its standard deviation by , its skewness by and its flatness by , we have
where . Setting in the above equation gives the moments of .
We then have
This expression is independent of due to the normalization of . Comparing this distribution to a realization of the process gives and as fit parameters. In the limit of vanishing , this expression reduces as expected to a Gamma distribution for .
a.3 Power spectral densities
The PSD of a random process is defined as
is the Fourier transform of the random variable over the domain . Analytical functions which fall rapidly enough to zero [such as the Greens function given in Eq. (13)] have the Fourier transform
and the inverse transform
We thus have the relevant PSDs,
a.4 Auto-correlation functions
The auto-correlation of a random variable is the inverse Fourier transform of its PSD, , where the inverse Fourier transform is given by Eq. (54). We thus have
Appendix B Derivation of the power spectral density and auto-correlation function
Results for the autocorrelation function and PSD of a white noise process or Ornstein-Uhlenbeck process are numerous in the literature, see for instance Refs. Uhlenbeck and Ornstein, 1930; Stark and Woods, 2012; Schuss, 2010. The same results for filtered Poisson processes are also readily available, see e. g. Refs. Rice, 1944; Parzen, 1999; Lowen and Teich, 2005; Garcia et al., 2016a. For completeness, we present full derivations in this appendix.
b.1 Power spectral density of the filtered Poisson process
To find the PSD of the FPP, we start from Eq. (14), and take the Fourier transform
where we have exchanged the functions in the convolution given by Eq. (14). A change of variables gives
Note that is only non-zero for positive and is negligible after a few . Moreover, since no pulses arrive for negative times, for . Assuming , we can therefore approximate the limits of the second integral in Eq. (62) as , and the two integrals become independent. This gives
The power spectral density (PSD) of the stationary process is thus
where is independent of , since the average is over all random variables. The Fourier transform of the Green’s functions is easily computed as , giving
We also readily find the Fourier transform of the forcing,
Multiplying this expression with its complex conjugate and averaging over all random variables gives
In this equation, there are terms where and terms where , for which all events are independent. Summing over all these terms, we have
where we have used that for an exponentially distributed variable, . Thus, averaging over all and using and gives
The second term in this equation resembles a Dirac delta function in the limit . With the appropriate normalization this gives
which together with Eq. (66) gives the PSD of the FPP as
b.2 Power spectral density and auto-correlation function of the noise processes
Since is constructed as a convolution in the same way as the FPP, we have an analogue of Eq. (65) for this process:
With Eq. (66) and the relation
we have the power spectral density for the Ornstein-Uhlenbeck process,
The power spectral density of is therefore Lorentzian with the same parameter as for the PSD of , implying that the auto-correlation function of is also an exponentially decaying function with the same rate as the auto-correlation function of :
For observational noise we require that is a unit-less variable. One way of realizing such a process is by using integrated increments of the Wiener process:
where is the sampling time (in this case, each sample , is normally and independently, identically distributed with zero mean and unit standard deviation). In this case, the most direct route to the power spectral density is via the auto-correlation function. We find that
Using the normalized sampling time from Eq. (6), this expression can be written as
For small , the cosine function can be expanded around 0, and we have
In this limit, approaches white noise, which has a flat power spectrum. Since we have demanded , the higher resolution granted by means the total power of is divided among a greater number of frequencies, reducing the power per frequency.
Since is the smallest time we will observe, it makes more sense to use the i.i.d. random sequence and to translate the auto-correlation to discrete time:
The discrete Fourier transform corresponding to is then
Intuitively, this corresponds to the case where we don’t see the effects of , that is the regime . Since this is also the spectral density we will observe for a realization of the process , we will use Eq. (84) when discussing the power spectral density of .
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