Complexity measures and occurrence of the breaking point in the neutron and gamma-rays time series measured with organic scintillators

This paper deals with the first analysis of the neutron and gamma time series measured with organic scintillators from plutonium samples by using information measures. Fast neutron detection with organic scintillators has been widely used for various nuclear applications and homeland security. One of the significant attributes of special nuclear materials (SNM) is the high multiplicity events in a short period of time. The time distributions of neutron and gamma-rays events for the plutonium metal plates designed as fuel plates for the Idaho National Laboratory (INL) Zero Power were measured with the Fast Neutron Multiplicity Counter (FNMC) consisting of 8 EJ-309 liquid scintillators and 8 stilbene detectors. Since the neutron correlated counts within the coincidence window of 40 ns are related to 240Pu effective mass of plutonium metal plates we investigated the randomness of the measured neutron and gamma-rays events. To access such information, we resort to complexity measures in the hope of being able to connect complexity values with the reliability of detection. That was done through (i) application of Kolmogorov complexity (KC) and its derivatives [Kolmogorov spectrum and its highest value (KCM) and running complexity (RKC)] and (ii) establishing the breaking point after which there exists a sharp drop in the running Kolmogorov complexity of the neutron and gamma-rays time series. It was found that the complexity of all the time series detected from the sample with 5, 9, 11, 13, and 15 plutonium plates had the high almost identical values of KC while the sample with 3 plates had by one-third smaller KC values than all the others. These calculations were supplemented by the Lypaunov exponents for a time series and the NIST tests. The low KC values can be addressed to the different sources of uncertainties in measuring procedure with the sample consisting of three plates.


Detection of special nuclear materials
Detection of special nuclear materials (SNM) such as plutonium, uranium-233, or uranium-235 (undeclared material and the missing of declared material) is of high importance in the field of nuclear safeguards and homeland security (Robinson et al., 2011). Passive neutron coincidence counting is the frequent technique for the non-destructive assay (NDA) of SNM. Detection systems based on organic scintillators have been applied successfully in recent nuclear safeguards applications for measurements on the nanosecond time-scales (Knoll, 2010). Taking into account that the neutron lifetime within organic scintillators is several orders of magnitude shorter compared to the neutron lifetime in thermal neutron counters it is possible to reduce the coincidence window significantly and achieve lower statistical measurement uncertainty compared to thermal neutron systems (Chichester, 2015). SNM emits multiple neutrons and gamma-rays simultaneously, through spontaneous fission (SF) or neutron-induced fission (IF) reactions. Since signatures for fissile materials are the high multiplicity events in thenanosecondtime-scale, detection systems based on organic scintillators are suitable for the detection of spontaneous and neutron-induced fission events. Neutrons from the spontaneous fission,(alpha, n) reactions and background, induce fissions, and generate some very short-lived chains in most cases (Pazsit et al., 2015).
For this study we used the measured neutron and gamma-rays time series for 3,5,7,9,11,13, and 15 PAHN Pu metal plates. The measurements were performed by using the Fast Neutron Multiplicity Counter (FNMC) consisting of 8 EJ-309 liquid organic scintillators and 8 stilbene detectors at Idaho National Laboratory (INL) (Di Fulvio, et al., 2017). Both types of organic scintillators are sensitive to fast neutrons and gamma rays. The pulse-shape discrimination (PSD) between fast neutrons and gamma rays was carried out using a standard charge integration method (Polack et al., 2015). The neutron time interval distributions depend on the source activities providing a possibility to evaluate 240 Pu effective mass of the samples with high accuracy (Di Fulvio et al., 2017). Since the neutron time distribution of events detected with the FNMC in a coincidence acquisition mode can be useful for extracting some fissile material properties such as the fissile material mass, it is of interest to investigate the randomness of neutron and gamma-rays events in the process of their detection.

Studying the complexity of neutron and gamma-rays time series
Understanding the dynamics of nuclear fission remains a challenge in nuclear physics.
Nuclear fission is a stochastic process and does not always take place in the same way, which means that two identical fission fragments are not obtained with a high probability or that the same number of neutrons and gamma-rays are always emitted. Spontaneous fission within the coincidence window of 40 ns is related to the fissile material massand one of the ways to better understand the dynamics of SF process is to measure the complexity of their time series. The assumption is that complexity represents an important variable that comes from the intuitive understanding of the unpredictability of chaos in radioactive decay (SF can be treated as a form of radioactive decay emitting multiple neutrons and gamma-rays in a short time interval) and the high complexity and low entropy of the measured time series. To access such information, we resort to complexity measures in the hope of being able to connect complexity values with various external causes and reliability of detection.
The structure of the paper is as follows. Section 2 presents the clear mathematical consideration and description of Kolmogorov complexity and its derivatives (subsection 2.1) that includes Kolmogorov complexity (2.1.1), Kolmogorov spectrum and its highest value (2.1.2) and running complexity (RKC), and "breaking point" of the running complexity time series (2.1.3), in which we defined this term by means of numerical experiments with a logistic equation (LE).
Subsections 2.4 and 2.5 are devoted to description of statistical randomness and NIST tests and calculating the Lyapunov exponent from a time series.
In Section 3 we describe the experimental set up that includes (i) technical details and (ii) acquisition and processing of Pu time series (subsections 3.1 and 3.2, respectively). The results are discussed and conclusions are given in Section 4.

Methods
Experiment is one of the crucial methods in physics. Paraphrasing Duhem (1954), it could be said that an experiment can always remove a theory from power, but never prove it definitively. In this paper, we deal with occurrence of the "breaking point" in the neutron and gamma-rays time series detected by organic scintillators. On that occasion, we use the term randomness, which have different interpretations. Therefore, we will consider its possible interpretations and explain why we decided on the one that will be described below in details.
If we ask ourselves a question: what is randomness and where it comes from, then we must be aware that "this is one scary place to venture in" (Perez, 2017). We take for approved the randomness in our reality (consciously or unconsciously; with or without intent). To avoid this we use the probability theory to compensate for the randomness. In a really enlightening paper "Introduction to foundations of probability and randomness" about the randomness, Andrei Khrennikov explored the different interpretations of randomness. He also elaborated in more detail Kolmogorov's proposal on this subject (Khrennikov, 2014). He summarizes the three different interpretations of randomness: (1) Randomness as unpredictability; (2) randomness as typicality and (3) randomness as complexity. He observed that: "As we have seen, none of the three basic mathematical approaches to the notion of randomness (based on unpredictability, typicality, and algorithmic complexity) led to the consistent and commonly accepted theory of randomness." Among these interpretations, only Kolmogorov's interpretation is of an objective nature. To our surprise, this view point is not actually the predominant viewpoint in our understanding of probability. Moreover, Kolmogorov complexity is not even computable. Therefore, randomness is either a subjective measure or a non-computable objective measure. According to his discussion it might be that randomness is not mathematical, but physical notion. Namely, it is the physical procedure where the true randomness is hidden. It is plausibly possible that strictly mathematical approach to randomness and formalization of its definition cannot be reached, since probably it is insufficient to use only the mathematical methods for the theoretical construction of concept of the randomness itself. Let it note that in Kolmogorov's interpretation there is "there is no room high enough" for level of randomness.
Many scientific analyzes use the term "algorithmic" randomness, which is directly related with the definition of complexity developed by Kolmogorov (Kolmogorov, 1965). It is quantified by the algorithmic complexity, which is a measure of how long an algorithm would take to complete given an input of size n. If an algorithm has to scale, it should compute the result within a finite and practical time bound even for large values of n. For this reason, complexity is calculated asymptotically as n approaches infinity as a measure of randomness. Note that a mistreatment of language allows us to attribute greater randomness to a sequence with a higher complexity value. However, we maintain the term level in sense of the above approach. This is a typical situation in science (in particular, in physics and engineering sciences). Namely, in choosing the model or approach, scientists often apply a heuristic technique that could be defined as any approach to problem solving that makes use of a practical method not guaranteed to be optimal or perfect but sufficient either for the immediate goals or until a better approach is reached.

Kolmogorov complexity
The Kolmogorov complexity (KC) can be defined as the complexity of an object, such as a piece of text, to be a measure of the computability resources needed to specify the object. This complexity in existing terms is the minimum piece of code/program that one can write to generate a particular string. This measure is non-computable. Therefore, for a given time series, , = 1,2,3,4, … , it is calculated approximately using the Lempel-Ziv algorithm (LZA, Lempel and Ziv, 1976), or some of its variants. This algorithm includes the following steps: (1) Encode the time series by constructing a sequence of the characters0 and 1 written as ( ), = 1,2, … , according to the rule ( ) = 0, < or 1 if > , where is the mean value of the time series samples, selected as the threshold (Mihailović et al., 2015). Other encoding schemes are also used, depending on the application (Radhakrishnan et al., 2000). although the values of KC can exceed value 1.

Kolmogorov complexity spectrum and its highest value
The Kolmogorov complexity has two weaknesses: (i) It cannot distinguish between time series with different amplitude variations and that with similar random components; and (ii) in the conversion of a time series into a binary string, its complexity is unseen in the rules of the applied procedure. Therefore, in defining a threshold for a criterion for coding, some information about the composition of time series could be lost. In our analysis, two measures were used: (i) Kolmogorov complexity spectrum (KC spectrum) and (ii) the highest value of the KC spectrum, introduced by Mihailović et al. (2015) who described the procedure for calculating the KC spectrum. Figure 1schematically shows how to calculate the KC spectrum ( 1 , 2 , 3 , … , ) for plutonium (neutron and gamma-rays) time series ( 1 , 2 , 3 , … , ). This spectrum allows investigating therange of amplitudes in a time series that represents a complex system with highly enhanced stochastic components. The binarization of time series reflects an "average" since it depends on what value is used to binarize the series. This problem is also present in statistical randomness tests. However, the KC spectrum takes this problem into account, since it calculates the complexity taking each element of a time series as a threshold. Ultimately, this relates probability distribution with compressibility.
It may be noted that for a large number of samples of a time series, the computation of the KC spectrum can be computationally consumable. Therefore, it is reasonable to divide the domain, including all values between the minimum ( ) and maximum ( ), in the time series into subintervals, which then are used as thresholds. The highest value as in this series, i.e., = { }, is the highest value of the Kolmogorov complexity spectrum (KCM), as seen in Figure 1.

Running Kolmogorov complexity and "breaking point"
In nuclear statistics, a running mean is often calculated to analyze data by creating a time series of averages of different subsets of the full data set that is a type of finite impulse response filter. This procedure was used to calculate the Running Kolmogorov complexity (RKC). For a given time series of plutonium samples, we extracted a fixed window as in running mean procedure, and then the procedure for calculating the KC ( Fig. 1) is applied. After that, the window is moved forward for a step, and the KC algorithm is applied, until the end of Pu, the time series is reached. Note that the running temporal means offers a consistent and easily interpretable summary of the time series not only for KC but also for other measures.
On epistemological grounds, it is quite disturbing that there exists randomness to nature.
We mentioned the word randomness. In everyday language but also in some scientific branches, people tend to use the words "random" and "chaotic" synonymously. Here, we make a clear Here, we will show computationally that in the condition of high chaos and randomness, the KC complexity can unexpectedly go down or even more to enter an increasing mode. For this purpose, we will use the logistic equation (LE) that deterministically generates chaotic sequences (logistic parameter greater than 3.57). One of the indicators of chaotic behavior is the Lyapunov exponent whose positive value indicates the presence of chaos. We generated six time series, each of length 3000, covering an interval of from 3.948 to 3.953 with an increment of 0.001.
Then we calculated the running mean of Lyapunov exponent for a fixed window (of sizes 300, 400, 500, and 2800) for each ( Figure 2a). From this figure is seen that all values of are positive (most of them indicate a high chaotic regime).
The same experiments were performed for KC complexity. For the shortest window, a size of 200 steps was selected according to numerical experiments, which indicate that it is the minimum length at which calculating the KC complexity, by the LZA, is applicable (Mihailović et al., 2014). The results of the experiments are given in Fig. 2b. For some values of the logistic parameter the randomness of the sequence, quantified by KC complexity, at some point starts to go down. We will define that point as a "breaking point" of the running KC complexity (or shortly "breaking point") for randomness. Red circle indicates position of the "breaking point".

Statistical randomness
A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal die roll, or the digits of Pi or some of other the best known transcendental numbers exhibit "statistical randomness. It is one contrast to the algorithmic randomness which per definitionem does not use probabilities. Technology (NIST). However, The NIST tests (Shen et al., 2016) are quite expensive in computational time requiring very long series. Therefore we used a version that is less time consumable (Sys et al., 2015). The complete battery includes 15 tests but we used the simplest 6 following Kovalsky et al (2018

Determining Lyapunov exponents from a time series
Since a random process is non-deterministic, numerical computation of a "Lyapunov exponent" is not well defined but. Wolf's algorithm (applied to a sequence) looks at the closest neighbor 0 = of a point and uses it as the initial value of another sequence. The average value of the rate of separation ln( ‖ +1 − +1 ‖)/‖ − ‖)‖ − ‖) is used as a measure of the Lyapunov exponent. Calculations were performed following Kodba et. al. (2005) with software which is available on http://www.matjazperc.com/time/ which is based on Wolf's algorithm that is broadly used although it is also designed on a heuristic way.

Experimental setup
3.1 Technical details

Experimental configuration
Experimental validation of the FNMC measurement system consisting of sixteen organic scintillation detectors was performed at Idaho National Laboratory (INL) (Di Fulvio et al., 2017).The 3D model of the FNCM is given in Figure 3. A passive assay in a coincidence acquisition mode of plutonium metal plates (3,5,7,9,11,13, and 15 PAHN plates) was carried out in the mass range between 75.14 g and 375.69 g. The experiments were done with lead shielding to reduce the gamma-ray background for a symmetric position of the plutonium samples with respect to the detectors. for charge-integration, pulse-shape discrimination. and estimation ofneutron/photon misclassification in organic scintillators was applied in the measurements (Polack et al., 2015).
Since stilbene detectors have better PSD properties, a wider dynamic range of 336-6500 keV neutron energy deposited for stilbene detectors was used in the experiment compared to a range of 520-5530 keV neutron energy deposited for EJ-309 detectors. The digital acquisition was carried out by using a 500-MS/s, 14-bit, 16-channel digitizer (V1730 by CAEN Technologies, Viareggio, Italy (S.P.A., 2016).

Acquisition and processing of Pu (neutron and gamma-rays) time series
The time   In that regard, we relied on our experience in working on (i) logistic equation , (ii) time series of velocity data of turbulent flow in laboratory channel (Mihailović et al., 2017), and river streamflow . were calculated using the windows with length of 200 time units. As in the running mean procedure, the procedure for calculating the KC (Fig. 1) was applied. After that, the window is moved forward for a step with a length of one-time unit and the KC algorithm is applied, until the end of the Pu time series is achieved. In this study, 500 thresholds were used to obtain all spectra.
Results reported here were computed in FORTRAN90 while graphics was done in MATLAB.

Results and discussion
In a nonlinear dynamic analysis of neutron and gamma-rays time series from the metal Pu samples measured with organic scintillators, we encounter two sources of randomness that come (i) from nature these time series and (ii) background of scintillators. All the measured time series have high randomness and also elements of chaotic behavior, thus their analysis, like astrophysical and cosmic rays time series is concerned with time-varying signals that contain a non-trivial element of randomness (Vaughan, 2013). The changes in that randomness can occur because of uncertainties and characteristic limits of organic scintillators.

1 Measuring evidence
Plutonium as a SNM emits multiple neutrons and gamma rays simultaneously and therefore they are correlated in time.  Cosmic rays can create a neutron background that can also initiate fission chains. A significant contribution of cosmic ray events to the interpretation of the signal was determined in the case of passive neutron counting of high enriched uranium (HEU) samples of low multiplication (Nakae et al., 2011). The contribution of accidental coincidences due to the source and random background events is not negligible in the measurements with fast organic scintillators. Taking into accout that fission emits multiple neutrons simultaneouly, the most of (neutron, neutron) coincidences can be attributed to fission reaction with a low but measurable contribution of accidentals.
The number of neutrons emitted in spontaneous fission of 240 Pu varies from zero to six or more whereas the number of neutrons emitted in induced fission (IF) of 239 Pu can vary from zero to eight. Neutron multiplicity distributions for 240 Pu and 239 Pu are given in Figure 4. The average SF neutron and gamma-rays multiplicity in 240 Pu is about 2.156 (Ensslin et al., 1998) and 8.2 (Oberstedt et al., 2016) respectively, while the average IF neutron multiplicity in 239 Pu is about 3.16 induced by neutrons with energy of 2 MeV (Zucker and Holden, 1986).   (Ensslin et al., 1998).

The randomness of time series quantified by KC complexity and its derivatives
Complexity measures are used in a number of applications including extraction of information from data such as environmental time series (cosmic rays, solar and UV radiation, river streamflow), biomedical signals (brain and heart activity, epilepsy, schizophrenia), testing of random number generators, etc. Measuring the complexity of a time series can give crucial insights into the functioning of the system under investigation (here, a focus will be set on KC complexity and its derivatives calculated by the LZA algorithm). Let us count some examples: (i) The high value of KC in measured solar radiation time series points out the presence of very changeable cloudiness (Mihailović et al., 2018). (ii) Low values of KC for streamflow time series of turbulent rivers are often a result of human activity (Mihailović et al., 2014). (iii) In the biomedical signal analysis, a high value of KC of measurements (heart-rate, respiration rate, and blood oxygen concentration) might indicate a particular stage of sleep-apnea (Nagaraj et al., 2013). (iv) If the KC is higher the neuron activity will be more disordered; thus, the decreased KC complexity in the tasking state might be related to the increase in neuronal activity synchronization (Thilakvathi et al., 2017). However, different complexity measures proposed in the literature are ineffective in analyzing short time series that are further corrupted with noise.
Note, that in our case the KC complexity is the right choice since we analyze the level of randomness for time series of very large lengths. We have computed Kolmogorov complexity (KC) and its derivatives (Kolmogorov spectrum and KC spectrum highest value -KCM) for 8 and 16 detectors, which are given in Tables 3 and 4     detectors.

The sharp drop of complexity ("breaking point") in highly random Pu time series
The problem of defining and studying the complexity of a time series has the scientific community interested for years. Mostly, it was in the context of getting predictive information through information measures. Usually, KC complexity and Shannon has been used. Thus, Bialek et al. (2001) pointed out that our intuitive notion of complexity corresponds to statements about the underlying process, and not directly to Kolmogorov complexity. A dynamic process with an unpredictable and random output (large KC) may be as small as the dynamics producing the predictable constant outputs (small KC) -although "really" complex processes lie somewhere in between (Prokopenko et al., 2009). This fact is visualized by Crutchfield and Young (1990) via the complexity spectrum in Figure1 where is drown complexity C as a function of the diversity of patterns against the normalized Shannon entropy (Fig. 10). Regular data have low entropy; very random data have maximal entropy. However, their complexities are both low.

Figure 10
The complexity spectrum: complexity C as a function of the diversity of patterns.
The latter is measured with the (normalized) Shannon entropy H (Crutchfield and Young, 1990).
The different areas of ellipses symbolically show a transition of the Pu time series complexity level from high complexity (yellow) and low Shannon entropy (and also low KC) towards the "breaking point" and approaching to zero (red ellipses).
The comments by Crutchfield and Young (1990) and Figure 10 (Tables 3 and 4). From Figure 11 is seen that the running complexity of PAHN_7 is nearly constant (it is the same for time series of higher number with Pu plates, which are not drawn), while for PAHN_3 time series a "breaking point" is occurred. This situation we can attribute to the neutron and gamma-rays interaction with detector rather than to the nuclear process itself. Namely, the architecture of detectors configuration and influence of surrounding environment on the detector can introduce some uncertainties in the measuring procedure. Thus, all the Pu time series (except PAHN_3) are measured reliably.
Complexities of these time series lie in the (C, H) space symbolically covered by the yellow ellipse ( Figure 11). In contrast to that, the PAHN_3 time series have high KC complexities and then reaches a point ("breaking point") after which they quickly go down.

Figure 11
The running complexity of the neutron and gamma time series with 3 and 7 PAHN plates. The "breaking points" are rounded by ellipses. The position of sample on the xaxis is normalized by its the length of a time series. BP is notation for "breaking point".
We have shown that for 3 PAHN plates there is a drop of KC, same as in the logistic map, All Pu time series have high KC complexity with an exception for PAHN_3 plate, whose time series (gamma and neutron) have Lyapunov exponent in the region of the high chaos and lower value of the KC (isolated island in the left side of Fig. 12). This is not contradictory since the complexity is collective dynamical behavior from simple interactions between large numbers of subunits while chaos is the generation of complicated, aperiodic, apparently random behavior from the iteration of a simple rule. "Stand alone chaos and complexity have absolutely nothing to do with generating formal function. Chaotic systems are not necessarily complex, and complex systems are not necessarily chaotic" (Rickles et al., 2007;Bertuglia, and Vaio F, 2005).

Figure 12
The Lyapunov exponent of the neutron and gamma time series for all Pu plates as a function of KC complexity.
We have performed NIST tests for 28 time series [7 (gamma rays) and 7 (neutrons) for 8 and 16 detectors]. The test having p> 0.01 has been noted as passed; otherwise notation was no- and 16 detectors). It means that these time series are no-random. Having in mind these facts, we can say that the PAHN_3 and PAHN_5 time series are chaotic because of the nature of neutron and gamma radiation. Lower KC complexity and no-random nature (relative to other time series) of PAHN_3 time series can be attributed to the different sources of uncertainties in detection procedure with the sample consisting of three plates.

Conclusions
One of the measures to investigate the reliability of the Pu time series measurement was used to obtain a general estimate of the complexity of Pu time series. To access this information, After the "breaking point" they have a very pronounced drop in running complexity (i) for neutron and gamma rays and (ii) for both clusters with 8 and 16 detectors. (4) From this statistics, it could be concluded that neutron and gamma-rays from Pu samples measured either by 8 or 16 detectors show nearly the same results for their very high randomness, which is calculated using KC complexity and the highest value of in the KC spectra (KCM). It is a sign that the detectors reliably measured the neutron and gamma events on the nanosecond time-scale, i.e. without any drop in their complexity. (5) The running complexity of the PAHN_3 time series has the "breaking point". This situation can attribute to the sensitivity and stability of the measurement equipment during the measurement procedure as well as to the contribution of radiation background which was not constant during the acquisition time rather than to the nuclear process itself.