Proposed Complexity Measure Research Articles

Measuring the complexity of micro and nanostructured surfaces

Nanostructured surfaces usually exhibit complicated morphologies that cannot be described in terms of Euclidean geometry. Simultaneously, they do not constitute fully random noise fields to be characterized by simple stochastics and probability theory. In most cases, nanomorphologies consist of complicated mixtures of order and randomness, which should be described quantitatively if one aims to control their fabrication and properties. In this work, inspired by recent developments in complexity theory, we propose a method to measure nanomorphology complexity that is based on the deviation from the average symmetry of surfaces. We present the methodology for its calculation and the validation of its performance, using a series of synthetic surfaces where the proposed complexity measure obtains a maximum value at the most heterogeneous morphologies between the fully ordered and fully random cases. Additionally, we measure the complexity of experimental micro and nanostructured surfaces (polymeric and metallic), and demonstrate the usefulness of the proposed method in quantifying the impact of processing conditions on their morphologies. Finally, we hint at the relationship between the complexity measure and the functional properties of surfaces.

Complexity Measures of Voice Recordings as a Discriminative Tool for Parkinson's Disease.

In this paper, we have investigated the differences in the voices of Parkinson’s disease (PD) and age-matched control (CO) subjects when uttering three phonemes using two complexity measures: fractal dimension (FD) and normalised mutual information (NMI). Three sustained phonetic voice recordings, /a/, /u/ and /m/, from 22 CO (mean age = 66.91) and 24 PD (mean age = 71.83) participants were analysed. FD was first computed for PD and CO voice recordings, followed by the computation of NMI between the test groups: PD–CO, PD–PD and CO–CO. Four features reported in the literature—normalised pitch period entropy (Norm. PPE), glottal-to-noise excitation ratio (GNE), detrended fluctuation analysis (DFA) and glottal closing quotient (ClQ)—were also computed for comparison with the proposed complexity measures. The statistical significance of the features was tested using a one-way ANOVA test. Support vector machine (SVM) with a linear kernel was used to classify the test groups, using a leave-one-out validation method. The results showed that PD voice recordings had lower FD compared to CO (p < 0.008). It was also observed that the average NMI between CO voice recordings was significantly lower compared with the CO–PD and PD–PD groups (p < 0.036) for the three phonetic sounds. The average NMI and FD demonstrated higher accuracy (>80%) in differentiating the test groups compared with other speech feature-based classifications. This study has demonstrated that the voices of PD patients has reduced FD, and NMI between voice recordings of PD–CO and PD–PD is higher compared with CO–CO. This suggests that the use of NMI obtained from the sample voice, when paired with known groups of CO and PD, can be used to identify PD voices. These findings could have applications for population screening.

Multiscale analysis of financial time series by Rényi distribution entropy

It is of great interests to analyze the complexity or dependence of time series of the daily stock closing price in different regions. Entropy is typically a fundamental technique to explore the complexity of time series. The well−studied irregularity measure, sample entropy (SampEn), and a more recently proposed complexity measure, distribution entropy (DistEn) are treated as classification features. In our study, based on Rényi entropy, which has been recently suggested as a measure of complexity in nonlinear systems, we propose multiscale Rényi distribution entropy (MRDE) to research the complexity of nonlinear time series over multiple time scales. Considering the validity and accuracy, Rényi entropy obtains more information than the Shannon entropy for time series mingled with much noise like financial time series, which is mainly reflected in the choice of parameters for frequent events and rare events. Both the simulated signals (logistic map and Gaussian noise) and the financial time series with three different stock markets are calculated and compared using an innovative method in different situations. For different stock markets, the entropy decreases as the scale factor increases, except when the parameter is the maximum in the special case. Meanwhile, We illustrate the necessity and advantage of Rényi distribution entropy (RDE) method by comparing RDE results with the SampEn results about stability and consistency on synthetic data and financial time series. It is applied to measure the dependence by detecting the length of the time series and special algorithm parameters.

Partially ordered permutation complexity of coupled time series

A new information-theoretic complexity measure of coupling among multiple time series is proposed based on ordinal patterns. We regard ordinal patterns extracted from time series as total orders and exploit their algebraic properties to introduce a multiplication among them. The result of the multiplication is in general a partial order. A Kullback–Leibler divergence for the partial orders gives rise to a complexity measure of coupling in the sense that it vanishes in two extreme cases: when time series are uncoupled and when they are identical. The performance of the proposed complexity measure is tested on two model multivariate dynamical systems and its advantage over the existing complexity measures is discussed.

Quantification of Graph Complexity Based on the Edge Weight Distribution Balance: Application to Brain Networks

The aim of this study was to introduce a novel global measure of graph complexity: Shannon graph complexity (SGC). This measure was specifically developed for weighted graphs, but it can also be applied to binary graphs. The proposed complexity measure was designed to capture the interplay between two properties of a system: the 'information' (calculated by means of Shannon entropy) and the 'order' of the system (estimated by means of a disequilibrium measure). SGC is based on the concept that complex graphs should maintain an equilibrium between the aforementioned two properties, which can be measured by means of the edge weight distribution. In this study, SGC was assessed using four synthetic graph datasets and a real dataset, formed by electroencephalographic (EEG) recordings from controls and schizophrenia patients. SGC was compared with graph density (GD), a classical measure used to evaluate graph complexity. Our results showed that SGC is invariant with respect to GD and independent of node degree distribution. Furthermore, its variation with graph size [Formula: see text] is close to zero for [Formula: see text]. Results from the real dataset showed an increment in the weight distribution balance during the cognitive processing for both controls and schizophrenia patients, although these changes are more relevant for controls. Our findings revealed that SGC does not need a comparison with null-hypothesis networks constructed by a surrogate process. In addition, SGC results on the real dataset suggest that schizophrenia is associated with a deficit in the brain dynamic reorganization related to secondary pathways of the brain network.

Interest Rate Swap Market Complexity and Its Volatility Implication

Many regulators and researchers have notionally attributed the recent financial crisis to the complexity of the financial system. However, very few direct measures of such complexity exist. The primary objective of this paper is to study the network properties of the interest rate swap market and propose an effective complexity measure to capture the dynamic behaviors of the swap contract flows with respect to interlinkages and interdependencies. We use a unique swap dataset extracted from the Swap Data Repositories (SDRs) to illustrate the effectiveness of our proposed complexity measure. We find a) the interest rate swap market follows a scale-free network where the power law exponent is less than 2, which indicates that few of its important entities have significant number of contracts within their subsidiaries; b) market volatility Granger-causes swap users to increase their risk sharing diversification at both system and local level simultaneously; c) sufficient saturation of the swap market tends to ease the market volatility toward an equilibrium regime not causing a positive feedback. These findings provide the market regulators and swap users a better understanding of interest rate swap market behaviors and provide a tool for them to monitor the swap market risks and control major contributing risk factors as a result.

Operation-based Configuration Complexity Measurement for Manufacturing System

Configuration not only determines the material flow pattern but also influences the production cost in a manufacturing system. The diversity of the products and growing demands for system flexibility increases the complexity of the configuration. In this paper, an operation-based approach is proposed to measure the configuration complexity of a manufacturing system. The configuration complexity models of the stations are built with single operation, several operations and their parallel types. Subsequently, an operation-based configuration complexity model of the overall system is used to measure the configuration complexity of a manufacturing system using the information entropy. Then, the relationship of the complexity between operations and stations is quantitatively described. An assembly line example is presented to validate the model. The results show that the proposed complexity measurement may evaluate the configuration complexity of a manufacturing system.

SIG-Blocks: Tangible game technology for automated cognitive assessment

This paper presents the SIG-Blocks system developed for automated cognitive assessment via tangible geometric games (TAG-Games). Computerized game administration and real-time cognitive and behavior assessments were realized by wireless self-synchronization in communication, decentralized hybrid-sensing, assembly and motion detection, and graphical visualization. The measurable performance data included time and accuracy at each manipulation step, overall speed of manipulative motions, and the total number of rotational motions. For preliminary evaluation, three types of TAG-Games were designed: TAG-GameA for assembly, TAG-GameS for shape matching, and TAG-GameM for memory. As a part of the game design, a computational measure of play complexity was defined for each TAG-Game based on the geometric properties and the number of blocks in the item. An evaluation with 86 participants assessed both reliability of the TAG-Game items using split-half and test-retest reliability tests and validity of the proposed complexity measures by comparing the results with three subtests of the Wechsler Adults Intelligence Scale 4th Edition (WAIS-IV), i.e. Block Design (BD), Matrix Reasoning (MR), and Digit Span (DS). The high reliability coefficients showed that TAG-Games were reliable. Regarding validity, correlations were found between TAG-GameA and BD and between TAG-GameS and MR. Behavioral analysis also showed that the TAG-Game performance was positively correlated with the manipulation speed, but not correlated with the total number of rotations applied to the blocks.

Genetic programming approach to evaluate complexity of texture images

We adopt genetic programming (GP) to define a measure that can predict complexity perception of texture images. We perform psychophysical experiments on three different datasets to collect data on the perceived complexity. The subjective data are used for training, validation, and test of the proposed measure. These data are also used to evaluate several possible candidate measures of texture complexity related to both low level and high level image features. We select four of them (namely roughness, number of regions, chroma variance, and memorability) to be combined in a GP framework. This approach allows a nonlinear combination of the measures and could give hints on how the related image features interact in complexity perception. The proposed complexity measure MGP exhibits Pearson correlation coefficients of 0.890 on the training set, 0.728 on the validation set, and 0.724 on the test set. MGP outperforms each of all the single measures considered. From the statistical analysis of different GP candidate solutions, we found that the roughness measure evaluated on the gray level image is the most dominant one, followed by the memorability, the number of regions, and finally the chroma variance.

Predicting Complexity Perception of Real World Images.

The aim of this work is to predict the complexity perception of real world images. We propose a new complexity measure where different image features, based on spatial, frequency and color properties are linearly combined. In order to find the optimal set of weighting coefficients we have applied a Particle Swarm Optimization. The optimal linear combination is the one that best fits the subjective data obtained in an experiment where observers evaluate the complexity of real world scenes on a web-based interface. To test the proposed complexity measure we have performed a second experiment on a different database of real world scenes, where the linear combination previously obtained is correlated with the new subjective data. Our complexity measure outperforms not only each single visual feature but also two visual clutter measures frequently used in the literature to predict image complexity. To analyze the usefulness of our proposal, we have also considered two different sets of stimuli composed of real texture images. Tuning the parameters of our measure for this kind of stimuli, we have obtained a linear combination that still outperforms the single measures. In conclusion our measure, properly tuned, can predict complexity perception of different kind of images.

Classification of 5-S Epileptic EEG Recordings Using Distribution Entropy and Sample Entropy.

Epilepsy is an electrophysiological disorder of the brain, the hallmark of which is recurrent and unprovoked seizures. Electroencephalogram (EEG) measures electrical activity of the brain that is commonly applied as a non-invasive technique for seizure detection. Although a vast number of publications have been published on intelligent algorithms to classify interictal and ictal EEG, it remains an open question whether they can be detected using short-length EEG recordings. In this study, we proposed three protocols to select 5 s EEG segment for classifying interictal and ictal EEG from normal. We used the publicly-accessible Bonn database, which consists of normal, interical, and ictal EEG signals with a length of 4097 sampling points (23.6 s) per record. In this study, we selected three segments of 868 points (5 s) length from each recordings and evaluated results for each of them separately. The well-studied irregularity measure—sample entropy (SampEn)—and a more recently proposed complexity measure—distribution entropy (DistEn)—were used as classification features. A total of 20 combinations of input parameters m and τ for the calculation of SampEn and DistEn were selected for compatibility. Results showed that SampEn was undefined for half of the used combinations of input parameters and indicated a large intra-class variance. Moreover, DistEn performed robustly for short-length EEG data indicating relative independence from input parameters and small intra-class fluctuations. In addition, it showed acceptable performance for all three classification problems (interictal EEG from normal, ictal EEG from normal, and ictal EEG from interictal) compared to SampEn, which showed better results only for distinguishing normal EEG from interictal and ictal. Both SampEn and DistEn showed good reproducibility and consistency, as evidenced by the independence of results on analysing protocol.

Entropic Movement Complexity Reflects Subjective Creativity Rankings of Visualized Hand Motion Trajectories.

In a previous study we have shown that human motion trajectories can be characterized by translating continuous trajectories into symbol sequences with well-defined complexity measures. Here we test the hypothesis that the motion complexity individuals generate in their movements might be correlated to the degree of creativity assigned by a human observer to the visualized motion trajectories. We asked participants to generate 55 novel hand movement patterns in virtual reality, where each pattern had to be repeated 10 times in a row to ensure reproducibility. This allowed us to estimate a probability distribution over trajectories for each pattern. We assessed motion complexity not only by the previously proposed complexity measures on symbolic sequences, but we also propose two novel complexity measures that can be directly applied to the distributions over trajectories based on the frameworks of Gaussian Processes and Probabilistic Movement Primitives. In contrast to previous studies, these new methods allow computing complexities of individual motion patterns from very few sample trajectories. We compared the different complexity measures to how a group of independent jurors rank ordered the recorded motion trajectories according to their personal creativity judgment. We found three entropic complexity measures that correlate significantly with human creativity judgment and discuss differences between the measures. We also test whether these complexity measures correlate with individual creativity in divergent thinking tasks, but do not find any consistent correlation. Our results suggest that entropic complexity measures of hand motion may reveal domain-specific individual differences in kinesthetic creativity.

Structural complexity of minerals: information storage and processing in the mineral world

AbstractStructural complexity of minerals is characterized using information contents of their crystal structures calculated according to the modified Shannon formula. The crystal structure is considered as a message consisting of atoms classified into equivalence classes according to their distribution over crystallographic orbits (Wyckoff sites). The proposed complexity measures combine both size- and symmetry-sensitive aspects of crystal structures. Information-based complexity parameters have been calculated for 3949 structure reports on minerals extracted from the Inorganic Crystal Structure Database. According to the total structural information content, IG, total, mineral structures can be classified into very simple (0–20 bits), simple (20–100 bits), intermediate (100–500 bits), complex (500–1000 bits), and very complex (&gt; 1000 bits). The average information content for mineral structures is calculated as 228(6) bits per structure and 3.23(2) bits per atom. Twenty most complex mineral structures are (IG, total in bits): paulingite (6766.998), fantappieite (5948.330), sacrofanite (5317.353), mendeleevite-(Ce) (3398.878), bouazzerite (3035.201), megacyclite (2950.928), vandendriesscheite (2835.307), giuseppetite (2723.097), stilpnomelane (2483.819), stavelotite-(La) (2411.498), rogermitchellite (2320.653), parsettensite (2309.820), apjohnite (2305.361), antigorite (m = 17 polysome) (2250.397), tounkite (2187.799), tschoertnerite (2132.228), farneseite (2094.012), kircherite (2052.539), bannisterite (2031.017), and mutinaite (2025.067). The following complexity-generating mechanisms have been recognized: modularity, misfit relationships between structure elements, and presence of nanoscale units (clusters or tubules). Structural complexity should be distiguished from topological complexity. Structural complexity increases with decreasing temperature and increasing pressure, though at ultra-high pressures, the situation may be different. Quantitative complexity measures can be used to investigate evolution of information in the course of global and local geological processes involving formation and transformation of crystalline phases. The information-based complexity measures can also be used to estimate the 'ease of crystallization' from the viewpoint of simplexity principle proposed by J.R. Goldsmith (1953) for understanding of formation of simple and complex mineral phases under both natural and laboratory conditions. According to the proposed quantitative approach, the crystal structure can be viewed as a reservoir of information encoded in its complexity. Complex structures store more information than simple ones. As erasure of information is always associated with dissipation of energy, information stored in crystal structures of minerals must have an important influence upon natural processes. As every process can be viewed as a communication channel, the mineralogical history of our planet on any scale is a story of accumulation, storage, transmission and processing of structural information.

Structural and topological complexity of zeolites: An information-theoretic analysis

Shannon information theory is applied to the quantitative evaluation of topological and structural complexity of zeolite frameworks. Within this approach, content of the reduced unit cell volume is considered as a message with atoms as symbols. Two atoms are considered equivalent, if they belong to the same crystallographic orbit. The framework complexity is characterized by its total topological information content, IG,total. Topological complexity parameters are calculated for the 201 zeolite structure types listed in the International Zeolite Association Zeolite Structure Database. On the basis of the proposed complexity measures, zeolite frameworks can be classified into very simple, simple, intermediate, complex, and very complex. The six most complex zeolite frameworks are [IG,total in bits]: SFV [19557.629], ITV [2825.097], IMF [2614.111], CLO [2413.449], and LTN [2285.985]. The six simplest zeolite frameworks are: SOD [16.529], BCT [19.020], ABW [23.020], NPO [28.529], EDI and NAB [32.603]. The sodalite–cancrinite group of zeolite frameworks is used to investigate evolution of complexity within modular series of structurally related zeolites. The topological complexity measures can be used to investigate evolution of complexity during crystallization and transformation of zeolites.

로봇 운동솜씨의 작업 복잡도

Measuring task complexity of movement skill is an important factor to evaluate a difficulty of learning and/or imitating a task for autonomous robots. Although many complexity-measures are proposed in research areas such as neuroscience, physics, computer science, and biology, there have been little attention on the robotic tasks. To cope with measuring complexity of robotic task, we propose an information-theoretic measure for task complexity of movement skills. By modeling proprioceptive as well as exteroceptive sensor data as multivariate Gaussian distribution, movements of a task can be modeled as probabilistic model. Additionally, complexity of temporal variations is modeled by sampling in time and modeling as individual random variables. To evaluate our proposed complexity measure, several experiments are performed on the real robotic movement tasks.

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A geometric approach to complexity

We develop a geometric approach to complexity based on the principle that complexity requires interactions at different scales of description. Complex systems are more than the sum of their parts of any size and not just more than the sum of their elements. Using information geometry, we therefore analyze the decomposition of a system in terms of an interaction hierarchy. In mathematical terms, we present a theory of complexity measures for finite random fields using the geometric framework of hierarchies of exponential families. Within our framework, previously proposed complexity measures find their natural place and gain a new interpretation.

A Novel Method for Detection of Epilepsy in Short and Noisy EEG Signals Using Ordinal Pattern Analysis

Introduction: In this paper, a novel complexity measure is proposed to detect dynamical changes in nonlinear systems using ordinal pattern analysis of time series data taken from the system. Epilepsy is considered as a dynamical change in nonlinear and complex brain system. The ability of the proposed measure for characterizing the normal and epileptic EEG signals when the signal is short or is contaminated with noise is investigated and compared with some traditional chaos-based measures. Materials and Methods: In the proposed method, the phase space of the time series is reconstructed and then partitioned using ordinal patterns. The partitions can be labeled using a set of symbols. Therefore, the state trajectory is converted to a symbol sequence. A finite state machine is then constructed to model the sequence. A new complexity measure is proposed to detect dynamical changes using the state transition matrix of the state machine. The proposed complexity measure was applied to detect epilepsy in short and noisy EEG signals and the results were compared with some chaotic measures. Results: The results indicate that this complexity measure can distinguish normal and epileptic EEG signals with an accuracy of more than 97% for clean EEG and more than 75% for highly noised EEG signals. Discussion and Conclusion: The complexity measure can be computed in a very fast and easy way and, unlike traditional chaotic measures, is robust with respect to noise corrupting the data. This measure is also capable of dynamical change detection in short time series data.

Design of irregular LDPC codes with optimized performance-complexity tradeoff

The optimal performance-complexity tradeoff for error-correcting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the minimization of decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. A sufficient condition is presented for convexity of the complexity measure in the variable edge-degree distribution; when it is not satisfied, numerical experiments nevertheless suggest that the local minimum is unique. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform threshold-optimized codes at long block lengths, within the ensemble of irregular codes.

A Fuzzy Logic Approach to Measure Complexity of Generic Aspect-Oriented Systems.

Aspect-oriented programming (AOP) is an emerging technique that provides a mechanism to clearly encapsulate and implement concerns that crosscut other modules. It is claimed that this technique improves code modularization and therefore reduces complexity of object-oriented programs (OOP). Most of the proposed complexity measurement frameworks for AOP are for AspectJ programming language. In this paper, a generalized framework for assessment of complexity of aspect-oriented (AO) systems, has been defined that takes into account three, the most well known families of available AOP languages, AspectJ, CaesarJ and Hyper/J. In order to automate complexity measurement, a tool has been developed using fuzzy logic, in which some set of rules have been defined as rule base. Using this tool, complexity of majority of AOP languages can be measured, which will further help in the measurement of external software qualities, such as maintainability, reusability, adaptability and understandability.