Dynamic Complexity Measures Research Articles

COMPLEXITY ANALYSIS OF TIME SERIES BASED ON GENERALIZED FRACTIONAL ORDER DUAL-EMBEDDED DIMENSIONAL MULTIVARIATE MULTISCALE DISPERSION ENTROPY

Based on the theories of multivariate multiscale dispersion entropy, we propose an improved new model — dual-embedded dimensional multivariate multiscale dispersion entropy (mvMDE), and generalize the new model to fractional order (GmvMDE). The mvMDE and GmvMDE simultaneously consider the cross-correlation between multiple channels, and they provide a dynamic complexity measure for measuring the multivariable data observed in one system. We introduce dual embedded dimensions to construct subsequences, which makes the model more flexible and applicable to a variety of application scenarios. Through empirical analysis of simulated data and stock market data, this paper verifies that both mvMDE and GmvMDE can effectively measure and distinguish the sequence complexity, and GmvMDE can better capture the small evolution of sequences, with higher stability and accuracy. On the basis of mvMDE research, we also propose the multiscale system dispersion entropy (MSDE) and its fractional order form (GMSDE). Using the vector-to-vector distance quantization method, the relationship between individuals in the system can be better extracted, and the systemic complexity of financial portfolio can be measured from the time dimension and the space dimension simultaneously. The empirical analysis based on the data of major industries in Chinese stock market shows that MSDE and GMSDE can accurately capture the structural information in the system and effectively distinguish the complexity of financial portfolios.

Complexity Measure for Engineering Systems Incorporating System States and Behavior

Complex engineering systems have structures characterized by a large number of components, interconnections, and elaborated configurations. Time-dependent functionality and multiple system states defy accurate predictions of complex systems behavior. Although many measures exist to quantify complexity, most of them consider either structure or function only. States and behavior are critical aspects to consider simultaneously with structure and function when assessing the complexity of modern systems as all of them have a profound impact on system design and future system performance. A more complete understanding of system complexity could improve the accuracy of scheduling and cost estimates during system development. This article proposes an objective, dynamic, and comprehensive complexity measure derived from molecular quantum mechanics and signal processing principles to quantify engineering system complexity. The proposed measure formalizes complexity as a single quantifiable system property that incorporates structure, functionality, and behavior, enabling unbiased comparisons of different conceptual designs, identification of complex elements, and analysis of complexity management. It provides systems architects and systems engineers with a strong basis for evaluating and communicating complexity during system design and architectural modeling. The formulation of the measure and methodology to calculate system complexity is illustrated through test cases using model-based systems engineering.

Changes in nonlinear dynamic complexity measures of blood pressure during anesthesia for cardiac surgeries using cardio pulmonary bypass.

Nonlinear complexity measures computed from beat-to-beat arterial BP dynamics have shown associations with standard cardiac surgical risk indices. They reflect the physiological adaptability of a system and has been proposed as dynamical biomarkers of overall health status. We sought to determine the impact of anesthetic induction and cardiopulmonary bypass (CPB) upon the complexity measures computed from perioperative BP time series. In this prospective, observational study, 300 adult patients undergoing cardiac surgery were included. Perioperative period was divided as: (1) Preoperative (PreOp); (2) ORIS-induction to sternotomy; (3) ORSB- sternotomy to CPB; (4) ORposB-post CPB and within 30min before leaving OR and (5) postoperative phase (PostOp)-initial 30min in the cardiac surgical intensive care unit. BP waveforms for systolic (SAP), diastolic (DAP), mean arterial pressure (MAP) and pulse pressure (PP) were recorded, and their corresponding complexity index (MSE∑) was calculated. Significant decrease in MSE∑ from Preop to PostOp phases was observed for all BP time series. Maximum fall was seen during post anesthetic induction (ORIS) phase. Mild recovery during the subsequent phases was observed but they never reached the baseline values. In an exploratory analysis, preoperative MSE∑ showed a significant correlation with postoperative length of ICU stay. Blood pressure complexity varies at different time points and is not fixed for a given individual. Preoperative BP Complexity decreased significantly following anesthetic induction and did not recover to baseline until 30min after surgery. Prevention of this significant fall may offer restoration of MSE∑ throughout surgery. Furthermore, preoperative BP complexity needs to be explored as a predictor of major postoperative adverse events by itself or in addition with the current risk indices.

Dynamic Complexity Measures for Use in Complexity-Based System Design

Difficulty predicting system behaviors introduces a certain level of complexity to a system design. The INCOSE systems engineering handbook indicates that system complexity is one of the seven key challenges influencing development when engineering a system of systems. The scope of this paper is first to survey systems engineering relevant definitions of complexity for latter application to the complexity evaluation framework. The literature search also includes state-of-the-art works on system complexity measurement. Before proposing new techniques, the current complexity-based system, and interface measurement and design techniques are explored. As the state-of-the-art only includes static/structural complexity quantification, entropy-based measures for dynamic complexity quantification are proposed. A sample system is evaluated using the proposed dynamic complexity measures and the results are discussed. The methods proposed herein provide a first step in the path to an enhanced system/interface complexity evaluation framework using dynamic complexity measures.

Object-oriented dynamic complexity measures for software understandability

Complexity is an inherent property when developing software. Software measures are a way to quantify the structural complexity of software. A rich body of literature exists for object-oriented (OO)...

Global and local complexity of intracranial EEG decreases during NREM sleep.

Key to understanding the neuronal basis of consciousness is the characterization of the neural signatures of changes in level of consciousness during sleep. Here we analysed three measures of dynamical complexity on spontaneous depth electrode recordings from 10 epilepsy patients during wakeful rest (WR) and different stages of sleep: (i) Lempel–Ziv complexity, which is derived from how compressible the data are; (ii) amplitude coalition entropy, which measures the variability over time of the set of channels active above a threshold; (iii) synchrony coalition entropy, which measures the variability over time of the set of synchronous channels. When computed across sets of channels that are broadly distributed across multiple brain regions, all three measures decreased substantially in all participants during early-night non-rapid eye movement (NREM) sleep. This decrease was partially reversed during late-night NREM sleep, while the measures scored similar to WR during rapid eye movement (REM) sleep. This global pattern was in almost all cases mirrored at the local level by groups of channels located in a single region. In testing for differences between regions, we found elevated signal complexity in the frontal lobe. These differences could not be attributed solely to changes in spectral power between conditions. Our results provide further evidence that the level of consciousness correlates with neural dynamical complexity.

Dynamical complexity in the C.elegans neural network

We model the neuronal circuit of the C.elegans soil worm in terms of Hindmarsh-Rose systems of ordinary differential equations, dividing its circuit into six communities pointed out by the walktrap and Louvain methods. Using the numerical solution of these equations, we analyze important measures of dynamical complexity, namely synchronicity, the largest Lyapunov exponent, and the $\Phi_{\mbox{AR}}$ auto-regressive integrated information theory measure, which has been suggested to reflect different levels of consciousness. We show that $\Phi_{\mbox{AR}}$ provides a useful measure of the information contained in the C.elegans brain dynamic network. Our analysis reveals that the C.elegans brain dynamic network generates more information than the sum of its constituent parts, and that attains higher levels of integrated information for couplings for which either all its communities are highly synchronized, or there is a mixed state of highly synchronized and desynchronized communities. Both situations are characterized by relatively low chaotic behavior.

Discontinuous Patterns of Brain Activation in the Psychotherapy Process of Obsessive-Compulsive Disorder: Converging Results from Repeated fMRI and Daily Self-Reports

This study investigates neuronal activation patterns during the psychotherapeutic process, assuming that change dynamics undergo critical instabilities and discontinuous transitions. An internet-based system was used to collect daily self-assessments during inpatient therapies. A dynamic complexity measure was applied to the resulting time series. Critical phases of the change process were indicated by the maxima of the varying complexity. Repeated functional magnetic resonance imaging (fMRI) measurements were conducted over the course of the therapy. The study was realized with 9 patients suffering from obsessive-compulsive disorder (subtype: washing/contamination fear) and 9 matched healthy controls. For symptom-provocative stimulation individualized pictures from patients’ personal environments were used. The neuronal responses to these disease-specific pictures were compared to the responses during standardized disgust-provoking and neutral pictures. Considerably larger neuronal changes in therapy-relevant brain areas (cingulate cortex/supplementary motor cortex, bilateral dorsolateral prefrontal cortex, bilateral insula, bilateral parietal cortex, cuneus) were observed during critical phases (order transitions), as compared to non-critical phases, and also compared to healthy controls. The data indicate that non-stationary changes play a crucial role in the psychotherapeutic process supporting self-organization and complexity models of therapeutic change.

SU-E-J-135: Measurements of Non-Linearity Features of Breathing Patterns Using Recurrence Quantification Analysis (RQA) and Dynamic Complexity (DC)

Purpose: To investigate if there exists a difference in breathing patterns between patients treated with SBRT and IMRT using RQA and DC (using K2: correlation entropy and D2: correlation dimension) measures. Methods: 9 patients treated with SBRT and 8 treated with IMRT were scanned with 4D CT and the breathing patterns were acquired. One of the SBRT patients was scanned with and without meditation. Each breathing signal consisted of a scalar time series and recurrence quantification analysis (RQA) was utilized to determine the following measures: Periodicity of the system as percentage of recurrence points (%RR), determinism (DET), maximum diagonal line length (L_max) whose inverse the divergence (DIV) is measure for how fast trajectories diverge from each other, the average diagonal line length (L) that can be interpreted as the mean prediction time of the signal, and the entropy (ENTR) a measure of information complexity. In addition the invariant measures of K2 and D2 were also estimated. A locally nonlinear forecast was applied to predict future breathing signals of N time step ahead for the patient with and without meditation. Results: Our results showed %RR has significant correlation with L_max and has inverse correlation with DIV. DET has significant correlation with Lmax, L, ENTR and DIV. Independent t test suggests there is no difference between the SBRT and IMRT groups in terms of the RQA measures and K2. Patient that had undergone meditation showed improvement in %RR, L_max, DIV, K2 and had an estimated correlation dimension of 1.7. Prediction showed similar results for one and three time step ahead but meditation one had better prediction horizon when time step was higher. Conclusion: RQA is a powerful tool that allows one to analyze the dynamic nature of breathing pattern. No significant difference was found in the dynamical complexity of SBRT and IMRT patients.

Effects of Clozapine on Heart Rate Dynamics and Their Relationship With Therapeutic Response in Treatment-Resistant Schizophrenia

Previous studies have suggested the utility of nonlinear complexity measures of heart rate variability (HRV) in evaluating the regulatory capacity of the neuroautonomic system. The purpose of the present study was to investigate the effects of clozapine on the nonlinear complexity measures of HRV in patients with treatment-resistant schizophrenia to find novel electrophysiological markers that indicate response to clozapine treatment. Forty patients with treatment-resistant schizophrenia were evaluated during 8 weeks of clozapine monotherapy. For nonlinear complexity measures of HRV, the approximate entropy (ApEn) and sample entropy (SampEn) values were obtained. The response rate to clozapine was 37.5%. The results of multivariate analysis of covariance revealed that the ApEn and the SampEn values of HRV at week 8 were significantly higher in the responders than in the nonresponders. Repeated-measures analysis of covariance showed a significant group by time interaction effect in the ApEn and SampEn indices. The responder group showed an increasing pattern of change in these complexity measures after administration of clozapine, whereas the nonresponder group showed a decreasing pattern of change. These results suggest that the nonlinear dynamic complexity measures of HRV, which indicate the irregularity and complexity of the biosystem, may be useful in evaluating the therapeutic changes of neuroautonomic function in schizophrenia. The response to clozapine treatment is expected to be more favorable when the plasticity of the neuroautonomic system reflected in the nonlinear complexity measures is high.

Extending the information-theoretic measures of the dynamic complexity of manufacturing systems

This paper revisits and extends the structural and dynamic complexity measures of manufacturing systems, which are based on an information-theoretic interpretation of the amount of information that is needed to describe the state of a manufacturing system. In this paper, a generic manufacturing system is modelled as a set of interacting resources and queues. At any moment in time, the combined specific states of the resources and queues define the overall state of the manufacturing system. The main contribution of this paper is a set of dynamic complexity measures that quantify the rate at which information is generated by the facility, and which may be applied with or without reference to the facility's schedule. A worked example demonstrates the step-by-step calculations to obtain values of the complexity measures, and guidance is provided on how to obtain the data needed to carry out the calculations. The links between these complexity measures and the behaviour of a facility are identified and described.

Causal density and integrated information as measures of conscious level

An outstanding challenge in neuroscience is to develop theoretically grounded and practically applicable quantitative measures that are sensitive to conscious level. Such measures should be high for vivid alert conscious wakefulness, and low for unconscious states such as dreamless sleep, coma and general anaesthesia. Here, we describe recent progress in the development of measures of dynamical complexity, in particular causal density and integrated information. These and similar measures capture in different ways the extent to which a system's dynamics are simultaneously differentiated and integrated. Because conscious scenes are distinguished by the same dynamical features, these measures are therefore good candidates for reflecting conscious level. After reviewing the theoretical background, we present new simulation results demonstrating similarities and differences between the measures, and we discuss remaining challenges in the practical application of the measures to empirically obtained data.

The π measure

A novel software measure, called the π measure, used for evaluating the fault-detection effectiveness of test sets, for measuring test-case independence and for measuring code complexity is proposed. The π measure is interpreted as a degree of run-time control and data difference at the code level, resulting from executing a program on a set of test cases. Unlike other well-known static and dynamic complexity measures, the π measure is an execution measure, computed using only run-time information. The Diversity Analyzer computes the π measure for programs written in C, C++, C# and VB in .NET. The experimental data presented here show a correlation between the π measure, test case independence and fault-detection rates.

Restarting automata with restricted utilization of auxiliary symbols

The restarting automaton is a restricted model of computation that was introduced by Jančar et al. to model the so-called analysis by reduction, which is a technique used in linguistics to analyse sentences of natural languages. The most general models of restarting automata make use of auxiliary symbols in their rewrite operations, although this ability does not directly correspond to any aspect of the analysis by reduction. Here we put restrictions on the way in which restarting automata use auxiliary symbols, and we investigate the influence of these restrictions on their expressive power. In fact, we consider two types of restrictions. First, we consider the number of auxiliary symbols in the tape alphabet of a restarting automaton as a measure of its descriptional complexity. Secondly, we consider the number of occurrences of auxiliary symbols on the tape as a dynamic complexity measure. We establish some lower and upper bounds with respect to these complexity measures concerning the ability of restarting automata to recognize the (deterministic) context-free languages and some of their subclasses.

Theories and measures of consciousness: an extended framework.

A recent theoretical emphasis on complex interactions within neural systems underlying consciousness has been accompanied by proposals for the quantitative characterization of these interactions. In this article, we distinguish key aspects of consciousness that are amenable to quantitative measurement from those that are not. We carry out a formal analysis of the strengths and limitations of three quantitative measures of dynamical complexity in the neural systems underlying consciousness: neural complexity, information integration, and causal density. We find that no single measure fully captures the multidimensional complexity of these systems, and all of these measures have practical limitations. Our analysis suggests guidelines for the specification of alternative measures which, in combination, may improve the quantitative characterization of conscious neural systems. Given that some aspects of consciousness are likely to resist quantification altogether, we conclude that a satisfactory theory is likely to be one that combines both qualitative and quantitative elements.

Quantitative measure of complexity of the dynamic event-related EEG data

Currently, the quantification of event-related EEG is usually based on power feature with the classical band power method. In this paper, the method quantifying the complexity and irregularity of event-related EEG data in relation to hand motor imagery is presented. Two groups of the complexity indexes: Kolmogorov complexity ( Kc) and Fourier spectral entropy ( FSE) are discussed. The event-related desynchronization/synchronization (ERD/ERS) time course is analyzed and characterized by two parameters Kc and FSE, respectively. The percentage of EEG complexity during imagination of the unilateral hand movement relative to that during reference period is calculated for quantifying the complexity measure of ERD/ERS time course. The method is applied to two sets of movement-related EEG data recorded over the primary sensorimotor area from two subjects. In addition, the validity of the quantitative measure of complexity of the event-related EEG is testified by evaluating the performance of feature extraction and classification. The results show that both Kc and FSE can effectively describe the dynamic complexity of event-related EEG and also display the consistent and similar behaviors. The relative increase and decrease of event-related EEG complexity could be an indicator of ERD/ERS, which is also independent of the power changes. Thus, the dynamic complexity measure of event-related EEG quantified by Kc and FSE provides another evidence for ERD/ERS and can be meaningful for analyzing the event-related EEG.

Risk assessment of software-system specifications

This paper presents a methodology and an example of risk assessment of functional-requirement specifications for complex real-time software systems. A heuristic risk-assessment technique based on CPN (colored Petri-net) models is presented. This technique is used to classify software functional-requirement specification components according to their relative importance in terms of such factors as severity and complexity. A dynamic complexity measure, based on concurrence in the functional requirements, is introduced. This technique is applied on the Earth Operation Commanding Center (EOC COMMANDING), a large component of the NASA Earth Observing System (EOS) project. Two specification models of the system are considered. Results of applying this technique to both CPN models are presented. The risk assessment methodology in this paper suggests the following conclusions: (i) risk assessment at the functional-requirement specification phase can be used to classify functional requirements in terms of their complexity and severity; (ii) dynamic complexity metrics and the concurrence metric can be important in assessing the risk factors based on the complexity of functional specifications; (iii) the concurrence complexity metric is an important aspect of dynamic complexity; and (iv) CPN models can be used to build an executable specification of the system, which helps the analyst not only to acquire deep understanding of the system but also to study the dynamic behavior of the system by simulating the model.

PROBLEMS WITH COMPLEXITY IN GOLD'S PARADIGM OF INDUCTION Part I: Dynamic Complexity

In 1967, the computer scientist E. M. Gold published an article on inductive inference, which founded a new paradigm for the study of induction. This approach to induction has since become known as “Gold's Paradigm”. The purpose of this paper is to introduce Gold's paradigm to a broad audience, and to raise a problem with the paradigm, a problem which has largely been ignored by computer scientists. In particular, which complexity measures are suitable for inductive inference in Gold'ls paradigm? This paper is the first of two papers on this topic. The paper begins with an introduction to Gold'ls paradigm. It then introduces Blum's axioms for dynamic complexity measures. It is argued that further progress in Gold's paradigm requires strengthening Blum's axioms. As they stand, Blum's axioms allow measures which are not suitable for inductive inference. The second paper deals with static complexity measures.

PROBLEMS WITH COMPLEXITY IN GOLD'S PARADIGM OF INDUCTION Part II: Static Complexity

This is the second of two papers discussing Gold's paradigm. The first paper introduced Gold's paradigm and critically examined the use of dynamic complexity measures in the paradigm. This second paper looks at the use of static complexity measures. Both papers explore the question of which complexity measures are suitable for inductive inference. Blum's axioms for static complexity measures are examined, and they are found to be too weak for inductive inference. That is, the class of complexity measures which the axioms define is too broad. The Kolmogorov-Chaitin approach to defining static complexity is also examined, and it is also found to be too weak for inductive inference. It is argued that further progress in Gold's paradigm requires a strong definition of static complexity.