Measures Of Spatial Complexity Research Articles

Spatial complexity method for tracking brain development and degeneration using functional near-infrared spectroscopy.

Brain complexity analysis using functional near-infrared spectroscopy (fNIRS) has attracted attention as a biomarker for evaluating brain development and degeneration processes. However, most methods have focused on the temporal scale without capturing the spatial complexity. In this study, we propose a spatial time-delay entropy (STDE) method as the spatial complexity measure based on the time-delay measure between two oxy-hemoglobin (Δ[HbO]) or two deoxy-hemoglobin (Δ[Hb]) oscillations within the 0.01-0.1 Hz frequency band. To do this, we analyze fNIRS signals recorded from infants in their sleeping state, children, adults, and healthy seniors in their resting states. We also evaluate the effects of various noise to STDE calculations and STDE's performance in distinguishing various developmental age groups. Lastly, we compare the results with the normalized global spatial complexity (NGSC) and sample entropy (SampEn) measures. Among these measures, STDEHbO (STDE based on Δ[HbO] oscillations) performs best. The STDE value increases with age throughout childhood (p < 0.001), and then decreases in adults and healthy seniors in the 0.01-0.1 Hz frequency band. This trajectory correlates with cerebrovascular development and degeneration. These findings demonstrate that STDE can be used as a new tool for tracking cerebrovascular development and degeneration across a lifespan based on the fNIRS resting-state measurements.

Unravelling Spatial Privacy Risks of Mobile Mixed Reality Data

Previously, 3D data---particularly, spatial data---have primarily been utilized in the field of geo-spatial analyses, or robot navigation (e.g. self-automated cars) as 3D representations of geographical or terrain data (usually extracted from lidar). Now, with the increasing user adoption of augmented, mixed, and virtual reality (AR/MR/VR; we collectively refer to as MR) technology on user mobile devices, spatial data has become more ubiquitous. However, this ubiquity also opens up a new threat vector for adversaries: aside from the traditional forms of mobile media such as images and video, spatial data poses additional and, potentially, latent risks to users of AR/MR/VR. Thus, in this work, we analyse MR spatial data using various spatial complexity metrics---including a cosine similarity-based, and a Euclidean distance-based metric---as heuristic or empirical measures that can signify the inference risk a captured space has. To demonstrate the risk, we utilise 3D shape recognition and classification algorithms for spatial inference attacks over various 3D spatial data captured using mobile MR platforms: i.e. Microsoft HoloLens, and Android with Google ARCore. Our experimental evaluation and investigation shows that the cosine similarity-based metric is a good spatial complexity measure of captured 3D spatial maps and can be utilised as an indicator of spatial inference risk.

Information Dynamics at a Phase Transition

We propose a new way of investigating phase transitions in the context of information theory. We use an information-entropic measure of spatial complexity known as configurational entropy (CE) to quantify both the storage and exchange of information in a lattice simulation of a Ginzburg-Landau model with a scalar order parameter coupled to a heat bath. The CE is built from the Fourier spectrum of fluctuations around the mean-field and reaches a minimum at criticality. In particular, we investigate the behavior of CE near and at criticality, exploring the relation between information and the emergence of ordered domains. We show that as the temperature is increased from below, the CE displays three essential scaling regimes at different spatial scales: scale free, turbulent, and critical. Together, they offer an information-entropic characterization of critical behavior where the storage and processing of information is maximized at criticality.

An approach to optimize sample preparation for MALDI imaging MS of FFPE sections using fractional factorial design of experiments.

A standardized workflow for matrix-assisted laser desorption/ionization imaging mass spectrometry (MALDI imaging MS) is a prerequisite for the routine use of this promising technology in clinical applications. We present an approach to develop standard operating procedures for MALDI imaging MS sample preparation of formalin-fixed and paraffin-embedded (FFPE) tissue sections based on a novel quantitative measure of dataset quality. To cover many parts of the complex workflow and simultaneously test several parameters, experiments were planned according to a fractional factorial design of experiments (DoE). The effect of ten different experiment parameters was investigated in two distinct DoE sets, each consisting of eight experiments. FFPE rat brain sections were used as standard material because of low biological variance. The mean peak intensity and a recently proposed spatial complexity measure were calculated for a list of 26 predefined peptides obtained by in silico digestion of five different proteins and served as quality criteria. A five-way analysis of variance (ANOVA) was applied on the final scores to retrieve a ranking of experiment parameters with increasing impact on data variance. Graphical abstract MALDI imaging experiments were planned according to fractional factorial design of experiments for the parameters under study. Selected peptide images were evaluated by the chosen quality metric (structure and intensity for a given peak list), and the calculated values were used as an input for the ANOVA. The parameters with the highest impact on the quality were deduced and SOPs recommended.

An index of floodplain surface complexity

Abstract. Floodplain surface topography is an important component of floodplain ecosystems. It is the primary physical template upon which ecosystem processes are acted out, and complexity in this template can contribute to the high biodiversity and productivity of floodplain ecosystems. There has been a limited appreciation of floodplain surface complexity because of the traditional focus on temporal variability in floodplains as well as limitations to quantifying spatial complexity. An index of floodplain surface complexity (FSC) is developed in this paper and applied to eight floodplains from different geographic settings. The index is based on two key indicators of complexity, variability in surface geometry (VSG) and the spatial organisation of surface conditions (SPO), and was determined at three sampling scales. FSC, VSG, and SPO varied between the eight floodplains and these differences depended upon sampling scale. Relationships between these measures of spatial complexity and seven geomorphological and hydrological drivers were investigated. There was a significant decline in all complexity measures with increasing floodplain width, which was explained by either a power, logarithmic, or exponential function. There was an initial rapid decline in surface complexity as floodplain width increased from 1.5 to 5 km, followed by little change in floodplains wider than 10 km. VSG also increased significantly with increasing sediment yield. No significant relationships were determined between any of the four hydrological variables and floodplain surface complexity.

Quantifying spatiotemporal complexity of cardiac dynamics using ordinal patterns.

Analyzing the dynamics of complex excitation wave patterns in cardiac tissue plays a key role for understanding the origin of life-threatening arrhythmias and for devising novel approaches to control them. The quantification of spatiotemporal complexity, however, remains a challenging task. This holds in particular for the analysis of data from fluorescence imaging (optical mapping), which allows for the measurement of membrane potential and intracellular calcium at high spatial and temporal resolution. Hitherto methods, like dominant frequency maps and the analysis of phase singularities, address important aspects of cardiac dynamics, but they consider very specific properties of excitable media, only. This article focuses on the benchmark of spatial complexity measures over time in the context of cardiac cell cultures. Standard Shannon Entropy and Spatial Permutation Entropy, an adaption of [1], have been implemented and applied to optical mapping data from embryonic chicken cell culture experiments. We introduce spatial separation of samples when generating ordinal patterns and show its importance for Spatial Permutation Entropy. Results suggest that Spatial Permutation Entropies provide a robust and interpretable measure for detecting qualitative changes in the dynamics of this excitable medium.

Tortuosity entropy: A measure of spatial complexity of behavioral changes in animal movement

The goal of animal movement analysis is to understand how organisms explore and exploit complex and varying environments. Animals usually exhibit varied and complicated movements, from apparently deterministic behaviours to highly random behaviours. It has been a common method to assess movement efficiency and foraging strategies by means of quantifying and analyzing movement trajectories. Here we introduce a tortuosity entropy (TorEn), a simple measure for quantifying the behavioral change in animal movement data. In our approach, the differences between pairwise successive track points are transformed into symbolic sequences, then we map these symbols into a group of pattern vectors and calculate the information entropy of pattern vectors. We test the algorithm on both simulated trajectories and real trajectories to show that it can accurately identify not only the mixed segments in simulated data, but also the different phases in real movement data. Tortuosity entropy can be easily applied to arbitrary real-world data, whether deterministic or stochastic, stationary or non-stationary. It could be a promising tool to reveal behavioral mechanism in movement data.

Information-entropic measure of energy-degenerate kinks in two-field models

We investigate the existence and properties of kink-like solitons in a class of models with two interacting scalar fields. In particular, we focus on models that display both double and single-kink solutions, treatable analytically using the Bogomol'nyi–Prasad–Sommerfield bound (BPS). Such models are of interest in applications that include Skyrmions and various superstring-motivated theories. Exploring a region of parameter space where the energy for very different spatially-bound configurations is degenerate, we show that a newly-proposed momentum–space entropic measure called Configurational Entropy (CE) can distinguish between such energy-degenerate spatial profiles. This information-theoretic measure of spatial complexity provides a complementary perspective to situations where strictly energy-based arguments are inconclusive.

Effects of CPAP-Therapy on Brain Electrical Activity in Obstructive Sleep Apneic Patients: A Combined EEG Study Using LORETA and Omega Complexity

Effects of initiation of continuous positive airway pressure (CPAP) therapy on EEG background activity were investigated in patients with obstructive sleep apnea syndrome (OSAS, N=25) to test possible reversibility of alterations of brain electrical activity caused by chronic hypoxia. Normal control group (N=14) was also examined. Two EEG examinations were done in each groups: at night and in the next morning. Global and regional (left vs. right, anterior vs. posterior) measures of spatial complexity (Omega complexity) were used to characterize the degree of spatial synchrony of EEG. Low resolution electromagnetic tomography (LORETA) was used to localize generators of EEG activity in separate frequency bands. Before CPAP-treatment, a significantly lower Omega complexity was found globally and over the right hemisphere. Due to CPAP-treatment, these significant differences vanished. Significantly decreased Omega complexity was found in the anterior region after treatment. LORETA showed a decreased activity in all of the beta bands after therapy in the right hippocampus, premotor and temporo-parietal cortex, and bilaterally in the precuneus, paracentral and posterior cingulate cortex. No significant changes were seen in control group. Comparing controls and patients before sleep, an increased alpha2 band activity was seen bilaterally in the precuneus, paracentral and posterior cingulate cortex, while in the morning an increased beta3 band activity in the left precentral and bilateral premotor cortex and a decreased delta band activity in the right temporo-parietal cortex and insula were observed. These findings indicate that effect of sleep on EEG background activity is different in OSAS patients and normal controls. In OSAS patients, significant changes lead to a more normal EEG after a night under CPAP-treatment. Compensatory alterations of brain electrical activity in regions associated with influencing sympathetic outflow, visuospatial abilities, long-term memory and motor performances caused by chronic hypoxia could be reversed by CPAP-therapy.

High-order Stochastic Simulation of Complex Spatially Distributed Natural Phenomena

Spatially distributed and varying natural phenomena encountered in geoscience and engineering problem solving are typically incompatible with Gaussian models, exhibiting nonlinear spatial patterns and complex, multiple-point connectivity of extreme values. Stochastic simulation of such phenomena is historically founded on second-order spatial statistical approaches, which are limited in their capacity to model complex spatial uncertainty. The newer multiple-point (MP) simulation framework addresses past limits by establishing the concept of a training image, and, arguably, has its own drawbacks. An alternative to current MP approaches is founded upon new high-order measures of spatial complexity, termed “high-order spatial cumulants.” These are combinations of moments of statistical parameters that characterize non-Gaussian random fields and can describe complex spatial information. Stochastic simulation of complex spatial processes is developed based on high-order spatial cumulants in the high-dimensional space of Legendre polynomials. Starting with discrete Legendre polynomials, a set of discrete orthogonal cumulants is introduced as a tool to characterize spatial shapes. Weighted orthonormal Legendre polynomials define the so-called Legendre cumulants that are high-order conditional spatial cumulants inferred from training images and are combined with available sparse data sets. Advantages of the high-order sequential simulation approach developed herein include the absence of any distribution-related assumptions and pre- or post-processing steps. The method is shown to generate realizations of complex spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data. In addition, it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated realizations, whereas the training images are only used to fill in high-order relations that cannot be inferred from data. Compared to the MP framework, the proposed approach is data-driven and consistently reconstructs the lower-order spatial complexity in the data used, in addition to high order.

Characteristic Changes in Brain Electrical Activity Due to Chronic Hypoxia in Patients with Obstructive Sleep Apnea Syndrome (OSAS): A Combined EEG Study Using LORETA and Omega Complexity

EEG background activity of patients with obstructive sleep apnea syndrome (OSAS, N = 25) was compared to that of normal controls (N = 14) to reflect alterations of brain electrical activity caused by chronic intermittent hypoxia in OSAS. Global and regional (left vs. right, anterior vs. posterior) measures of spatial complexity (Omega) were used to characterize the degree of spatial synchrony of EEG. Low resolution electromagnetic tomography (LORETA) was used to localize generators of EEG activity in separate frequency bands. Comparing patients to controls, lower Omega complexity was found globally and in the right hemisphere. Using LORETA, an increased medium frequency activity was seen bilaterally in the precuneus, paracentral and posterior cingulate cortex. These findings indicate that alterations caused by chronic hypoxia in brain electrical activity in regions associated with influencing emotional regulation, long-term memory and the default mode network. Global synchronization (lower Omega complexity) may indicate a significantly reduced number of relatively independent, parallel neural processes due to chronic global hypoxic state in apneic patients as well as over the right hemisphere.

Evaluation of object-oriented spatial complexity measures

Spatial complexity measures help in the estimation of the effort re-quired in the process of program comprehension. These spatial com-plexity measures have been proposed for procedure-oriented software as well as object-oriented software. In this paper, an at-tempt has been made to evaluate object-oriented spatial complexity measures using formal evaluation frameworks proposed by Weyu-ker and Briand et al. A practical and useful complexity measure must satisfy most of the properties given in these frameworks. The results of this study show that object-oriented spatial metrics satisfy all properties and parameters required by these two evaluation frameworks and thus these spatial measures are robust and useful.

Object-Oriented Cognitive-Spatial Complexity Measures

Software maintenance and mainly software comprehension pose the largest costs in the software lifecycle. In order to assess the cost of software comprehension, various complexity measures have been proposed in the literature. This paper proposes new cognitive-spatial complexity measures, which combine the impact of spatial as well as architectural aspect of the software to compute the software complexity. The spatial aspect of the software complexity is taken into account using the lexical distances (in number of lines of code) between different program elements and the architectural aspect of the software complexity is taken into consideration using the cognitive weights of control structures present in control flow of the program. The proposed measures are evaluated using standard axiomatic frameworks and then, the proposed measures are compared with the corresponding existing cognitive complexity measures as well as the spatial complexity measures for object-oriented software. This study establishes that the proposed measures are better indicators of the cognitive effort required for software comprehension than the other existing complexity measures for object-oriented software. Keywords—cognitive complexity, software comprehension, software metrics, spatial complexity, Object-oriented software

On the meaning and interpretation of global descriptors of brain electrical activity. Including a reply to X. Pei et al.

Global descriptors of the brain's electrical activity, Σ, Φ, and Ω, provide a comprehensive characterisation of brain functional states. Recently, Pei et al. [Pei, X., Zheng, C., Zhang, A., Duan, F., Bin, G., 2005. Discussion on “Towards a quantitative characterisation of functional states of the brain: from the nonlinear methodology to the global linear description” by J. Wackermann. Int. J. Psychophysiol. 56, 201–207] discussed the effects of signal power on the global measure of spatial complexity, Ω, and suggested a modification consisting in epoch-wise and channel-wise normalisation of input data to unit power. In the present paper, the basic principles of the global approach are reviewed, and the issues of Pei et al.'s approach are assessed. The original and the modified measures of spatial complexity are compared in two case studies. Numerical simulation shows that both methods veridically estimate small numbers of signal sources, but systematically underestimate as the number increases; the modified method yields a minor relative improvement. A study on real EEG data shows that the two measures sensibly differ only where artefactual inhomogeneities in channel variances affect the data; a combined procedure, consisting in record-wise equalisation of channel variances before Ω calculations, is suggested as the optimal strategy. Differences between the original objectives of the global methodology and the proposed modifications are pointed out and critically discussed.

Functionally relevant measures of spatial complexity in neuronal dendritic arbors

We introduce a set of scaling exponents for characterizing global 3D morphologic properties of mass distribution, branching and taper in neuronal dendritic arbors, capable of distinguishing functionally relevant changes in dendritic complexity that standard Sholl analysis and fractal analysis cannot. We demonstrate that the scaling exponent for mass distribution, d M , comprises a sum of independent scaling exponents for branching, d N , and taper, d T . The accuracy of experimental measurements of the scaling exponents was verified using computer generated self-similar binary trees of known fractal dimension, and with prescribed amounts of branching and taper. The theory was applied to measuring 3D spatial complexity in the apical and basal dendritic trees of two functionally distinct types of macaque monkey neocortical pyramidal neurons: long corticocortical projection neurons from superior temporal cortex to area 46 of the prefrontal cortex (PFC), and local projection neurons within area 46 of the PFC. Two distinct scaling subregions (proximal and medial) were identified in both apical and basal trees of the two neuron types, and scaling exponents were fitted. A small but significant difference in mass scaling in the proximal region distinguished long from local projection neurons. Interestingly, both classes of neuron exhibited a homeostatic pattern of mass distribution across the two regions: despite large differences between proximal and medial regions in branching and tapering exponents, these effects were compensatory, resulting in a uniform, slow reduction of mass with distance from the soma, over both scaling regions of the apical and basal trees. Given a uniformly excitable membrane, the electrotonic properties of dendritic arbors depend entirely upon mass distribution, and its relative contributions from dendritic branching and taper. By capturing each of these complex morphologic properties in a single, globally descriptive parameter, the new 3D scaling exponents introduced in this study permit efficient morphometric characterization of complex dendritic arbors in the fewest possible parameters, that can be directly related to their electrotonic properties, and hence to neuronal function.

Avian community response to a golf-course landscape unit gradient

Although golf-course construction significantly alters natural wildlife habitat, the resulting remnant, disturbed, and introduced landscape patches may provide valuable habitat for birds. Considering the current rate of new course construction, the effects of land consumption and habitat alteration on breeding bird communities in coastal South Carolina are of immediate concern. During summer 2000 and 2001, we sampled 24 golf-course landscape units (GCLU) to assess their value to the breeding bird community. We defined a GCLU as the legally owned parcel of land where a golf course was sited, including the course and all associated development (e.g., residential housing). Sample units (n = 24) were selected to represent a gradient of GCLUs ranging from low to high landscape alteration and were subjectively classified a priori into 1 of 3 alteration groups (G1 [low], G2 [medium] or G3 [high]). We conducted Analysis of Variance procedures to determine whether estimations of species richness, Neotropical migrant richness, and degree of conservation concern differed across the gradient. We explored relative strengths of associations between landscape structure (landscape composition and spatial configuration) and avian community parameters at 2 spatial scales using stepwise multiple regression techniques. We used simple linear regression to assess the relationship between percent forested area of the GCLU and avian community parameters. Total number of species and number of Neotropical migrant species, as well as degree of conservation concern of the species present, were higher in less-altered GCLUs (F2,21 <14, P < 0.05), and were significantly influenced by percent forested area (adj. R2=0.39–0.57). The majority of birds associated with less-developed landscapes were woodland and scrub-shrub breeding species, while urban-breeding species were found primarily in the more-altered landscapes (χ2=440.3, df=6, n=4757, P < 0.001). The area of forest and disturbance patches, size variability in managed turfgrass patches, and measures of spatial complexity proved most useful in explaining variability of response variables due to landscape structure (adj. R2=0.57–0.90).

Measurement of object-oriented software spatial complexity

One of the important activities of the maintenance phase is to understand the source-code first, and then change it. Understandability of the software gets affected by psychological complexity of the source-code and cognitive abilities are needed to understand it. The correlation between the orientation and location of various entities with their processing needs to be established by the programmers, which requires spatial abilities. These spatial abilities play an important role in object-oriented software, in which the use of data as well as the methods of the class needs to be understood in a combined way. This paper presents two measures of spatial complexity of object-oriented software, which are based on definition and usage of classes and objects. The values of proposed measures get affected suitably because of inheritance and polymorphism as well, due to change in the distances. The significance of object-oriented spatial complexity has been demonstrated with the help of 15 object-oriented projects of varied length and results have been validated with the help of reverse engineering data and perfective maintenance data.

Code and data spatial complexity: two important software understandability measures

In order to maintain the software, the programmers need to understand the source code. The understandability of the source code depends upon the psychological complexity of the software, and it requires cognitive abilities to understand the source code. The individual needs to correlate the orientation and location of various entities with their processing, which requires spatial abilities. This paper presents two measures of spatial complexity, which are based on two important aspects of the program—code as well as data. The measures have been applied to 15 different software projects and results have been used to draw many conclusions. The validation of the results has been done with help of perfective maintenance data. Lower values of code as well as data spatial complexity denote better understandability of the source code.

Towards modeling the query processing relevant shape complexity of 2D polygonal spatial objects

The shape complexity of two-dimensional (2D) polygonal spatial objects has implications for how the object can be best represented in a spatial database, and for the query-processing performance of that object. Nevertheless few useful definitions of query-processing relevant spatial complexity are available. A query-processing oriented shape complexity measure is likely to be different from a fractal measure of shape complexity that focused on compression/decompression or a shape complexity measure that would be used for pattern recognition, and should give better performance for the analysis of query processing. It could be used to classify spatial objects, cluster spatial objects in multiprocessor database systems. In a recent paper Brinkhoff et al. (T. Brinkhoff, H-P. Kriegel, R. Shneider, A. Braun, Measuring the complexity of spatial objects, Proceedings of the 3rd ACM International Workshop on Advances in Geographic Information Systems, Baltimore, MD, 1995, pp. 109–117) demonstrated the usefulness of a spatial complexity measure. They did not, however, offer much theoretical justification for their choice of parameters nor for the functional form that they used. In this paper we present a conceptual framework for discussing the query processing oriented shape complexity measures for spatial objects. It is hoped that this will lead to the development of improved measures of spatial complexity.

Temporal and spatial complexity measures for electroencephalogram based brain-computer interfacing

There has been much interest recently in the concept of using information from the motor cortex region of the brain, recorded using non-invasive scalp electrodes, to construct a crude interface with a computer. It is known that movements of the limbs, for example, are accompanied by desynchronisations and synchronisations within the scalp-recorded electroencephalogram (EEG). These event-related desynchronisations and synchronisations (ERD and ERS), however, appear to be present when volition to move a limb occurs, even when actual movement of the limb does not in fact take place. The determination and classification of the ERD/S offers many exciting possibilities for the control of peripheral devices via computer analysis. To date most effort has concentrated on the analysis of the changes in absolute frequency content of signals recorded from the motor cortex. The authors present results which tackle the issues of both the interpretation of changes in signals with time and across channels with simple methods which monitor the temporal and spatial 'complexity' of the data. Results are shown on synthetic and real data sets.