Book•
Wavelet Methods for Time Series Analysis
01 Feb 2006-
TL;DR: Wavelet analysis of finite energy signals and random variables and stochastic processes, analysis and synthesis of long memory processes, and the wavelet variance.
Abstract: 1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.
Citations
More filters
TL;DR: It is concluded that correlated, low-frequency oscillations in human fMRI data have a small-world architecture that probably reflects underlying anatomical connectivity of the cortex, and could provide a physiological substrate for segregated and distributed information processing.
Abstract: Small-world properties have been demonstrated for many complex networks. Here, we applied the discrete wavelet transform to functional magnetic resonance imaging (fMRI) time series, acquired from healthy volunteers in the resting state, to estimate frequency-dependent correlation matrices characterizing functional connectivity between 90 cortical and subcortical regions. After thresholding the wavelet correlation matrices to create undirected graphs of brain functional networks, we found a small-world topology of sparse connections most salient in the low-frequency interval 0.03–0.06 Hz. Global mean path length (2.49) was approximately equivalent to a comparable random network, whereas clustering (0.53) was two times greater; similar parameters have been reported for the network of anatomical connections in the macaque cortex. The human functional network was dominated by a neocortical core of highly connected hubs and had an exponentially truncated power law degree distribution. Hubs included recently evolved regions of the heteromodal association cortex, with long-distance connections to other regions, and more cliquishly connected regions of the unimodal association and primary cortices; paralimbic and limbic regions were topologically more peripheral. The network was more resilient to targeted attack on its hubs than a comparable scale-free network, but about equally resilient to random error. We conclude that correlated, low-frequency oscillations in human fMRI data have a small-world architecture that probably reflects underlying anatomical connectivity of the cortex. Because the major hubs of this network are critical for cognition, its slow dynamics could provide a physiological substrate for segregated and distributed information processing.
2,345 citations
TL;DR: Efficiency was reduced disproportionately to cost in older people, and the detrimental effects of age on efficiency were localised to frontal and temporal cortical and subcortical regions.
Abstract: Brain anatomical networks are sparse, complex, and have economical small-world properties. We investigated the efficiency and cost of human brain functional networks measured using functional magnetic resonance imaging (fMRI) in a factorial design: two groups of healthy old (N = 11; mean age = 66.5 years) and healthy young (N = 15; mean age = 24.7 years) volunteers were each scanned twice in a no-task or “resting” state following placebo or a single dose of a dopamine receptor antagonist (sulpiride 400 mg). Functional connectivity between 90 cortical and subcortical regions was estimated by wavelet correlation analysis, in the frequency interval 0.06–0.11 Hz, and thresholded to construct undirected graphs. These brain functional networks were small-world and economical in the sense of providing high global and local efficiency of parallel information processing for low connection cost. Efficiency was reduced disproportionately to cost in older people, and the detrimental effects of age on efficiency were localised to frontal and temporal cortical and subcortical regions. Dopamine antagonism also impaired global and local efficiency of the network, but this effect was differentially localised and did not interact with the effect of age. Brain functional networks have economical small-world properties—supporting efficient parallel information transfer at relatively low cost—which are differently impaired by normal aging and pharmacological blockade of dopamine transmission.
2,208 citations
TL;DR: This work investigates the role of modularity in human learning by identifying dynamic changes of modular organization spanning multiple temporal scales and develops a general statistical framework for the identification of modular architectures in evolving systems.
Abstract: Human learning is a complex phenomenon requiring flexibility to adapt existing brain function and precision in selecting new neurophysiological activities to drive desired behavior. These two attributes—flexibility and selection—must operate over multiple temporal scales as performance of a skill changes from being slow and challenging to being fast and automatic. Such selective adaptability is naturally provided by modular structure, which plays a critical role in evolution, development, and optimal network function. Using functional connectivity measurements of brain activity acquired from initial training through mastery of a simple motor skill, we investigate the role of modularity in human learning by identifying dynamic changes of modular organization spanning multiple temporal scales. Our results indicate that flexibility, which we measure by the allegiance of nodes to modules, in one experimental session predicts the relative amount of learning in a future session. We also develop a general statistical framework for the identification of modular architectures in evolving systems, which is broadly applicable to disciplines where network adaptability is crucial to the understanding of system performance.
1,529 citations
Cites methods from "Wavelet Methods for Time Series Ana..."
...Accordingly, we used MODWT to decompose each regional time series into wavelet scales corresponding to specific frequency bands [19]....
[...]
TL;DR: It is concluded that people with schizophrenia tend to have a less strongly integrated, more diverse profile of brain functional connectivity, associated with a less hub-dominated configuration of complex brain functional networks.
Abstract: Schizophrenia has often been conceived as a disorder of connectivity between components of large-scale brain networks. We tested this hypothesis by measuring aspects of both functional connectivity and functional network topology derived from resting-state fMRI time series acquired at 72 cerebral regions over 17 min from 15 healthy volunteers (14 male, 1 female) and 12 people diagnosed with schizophrenia (10 male, 2 female). We investigated between-group differences in strength and diversity of functional connectivity in the 0.06-0.125 Hz frequency interval, and some topological properties of undirected graphs constructed from thresholded interregional correlation matrices. In people with schizophrenia, strength of functional connectivity was significantly decreased, whereas diversity of functional connections was increased. Topologically, functional brain networks had reduced clustering and small-worldness, reduced probability of high-degree hubs, and increased robustness in the schizophrenic group. Reduced degree and clustering were locally significant in medial parietal, premotor and cingulate, and right orbitofrontal cortical nodes of functional networks in schizophrenia. Functional connectivity and topological metrics were correlated with each other and with behavioral performance on a verbal fluency task. We conclude that people with schizophrenia tend to have a less strongly integrated, more diverse profile of brain functional connectivity, associated with a less hub-dominated configuration of complex brain functional networks. Alongside these behaviorally disadvantageous differences, however, brain networks in the schizophrenic group also showed a greater robustness to random attack, pointing to a possible benefit of the schizophrenia connectome, if less extremely expressed.
1,264 citations
TL;DR: This study provides new evidence that there is disrupted organization of functional brain networks in AD, and suggests that these network measures may be useful as an imaging-based biomarker to distinguish AD from healthy aging.
Abstract: Functional brain networks detected in task-free (“resting-state”) functional magnetic resonance imaging (fMRI) have a small-world architecture that reflects a robust functional organization of the brain. Here, we examined whether this functional organization is disrupted in Alzheimer's disease (AD). Task-free fMRI data from 21 AD subjects and 18 age-matched controls were obtained. Wavelet analysis was applied to the fMRI data to compute frequency-dependent correlation matrices. Correlation matrices were thresholded to create 90-node undirected-graphs of functional brain networks. Small-world metrics (characteristic path length and clustering coefficient) were computed using graph analytical methods. In the low frequency interval 0.01 to 0.05 Hz, functional brain networks in controls showed small-world organization of brain activity, characterized by a high clustering coefficient and a low characteristic path length. In contrast, functional brain networks in AD showed loss of small-world properties, characterized by a significantly lower clustering coefficient (p<0.01), indicative of disrupted local connectivity. Clustering coefficients for the left and right hippocampus were significantly lower (p<0.01) in the AD group compared to the control group. Furthermore, the clustering coefficient distinguished AD participants from the controls with a sensitivity of 72% and specificity of 78%. Our study provides new evidence that there is disrupted organization of functional brain networks in AD. Small-world metrics can characterize the functional organization of the brain in AD, and our findings further suggest that these network measures may be useful as an imaging-based biomarker to distinguish AD from healthy aging.
1,086 citations
References
More filters
01 Feb 1989
TL;DR: In this paper, the authors provide an overview of the basic theory of hidden Markov models (HMMs) as originated by L.E. Baum and T. Petrie (1966) and give practical details on methods of implementation of the theory along with a description of selected applications of HMMs to distinct problems in speech recognition.
Abstract: This tutorial provides an overview of the basic theory of hidden Markov models (HMMs) as originated by L.E. Baum and T. Petrie (1966) and gives practical details on methods of implementation of the theory along with a description of selected applications of the theory to distinct problems in speech recognition. Results from a number of original sources are combined to provide a single source of acquiring the background required to pursue further this area of research. The author first reviews the theory of discrete Markov chains and shows how the concept of hidden states, where the observation is a probabilistic function of the state, can be used effectively. The theory is illustrated with two simple examples, namely coin-tossing, and the classic balls-in-urns system. Three fundamental problems of HMMs are noted and several practical techniques for solving these problems are given. The various types of HMMs that have been studied, including ergodic as well as left-right models, are described. >
21,819 citations
TL;DR: In this paper, it is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2 /sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.
Abstract: Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions. In L/sup 2/(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. >
20,028 citations
Book•
01 Jan 1995TL;DR: Detailed notes on Bayesian Computation Basics of Markov Chain Simulation, Regression Models, and Asymptotic Theorems are provided.
Abstract: FUNDAMENTALS OF BAYESIAN INFERENCE Probability and Inference Single-Parameter Models Introduction to Multiparameter Models Asymptotics and Connections to Non-Bayesian Approaches Hierarchical Models FUNDAMENTALS OF BAYESIAN DATA ANALYSIS Model Checking Evaluating, Comparing, and Expanding Models Modeling Accounting for Data Collection Decision Analysis ADVANCED COMPUTATION Introduction to Bayesian Computation Basics of Markov Chain Simulation Computationally Efficient Markov Chain Simulation Modal and Distributional Approximations REGRESSION MODELS Introduction to Regression Models Hierarchical Linear Models Generalized Linear Models Models for Robust Inference Models for Missing Data NONLINEAR AND NONPARAMETRIC MODELS Parametric Nonlinear Models Basic Function Models Gaussian Process Models Finite Mixture Models Dirichlet Process Models APPENDICES A: Standard Probability Distributions B: Outline of Proofs of Asymptotic Theorems C: Computation in R and Stan Bibliographic Notes and Exercises appear at the end of each chapter.
16,079 citations
Book•
01 Jan 1965
TL;DR: This chapter discusses the concept of a Random Variable, the meaning of Probability, and the axioms of probability in terms of Markov Chains and Queueing Theory.
Abstract: Part 1 Probability and Random Variables 1 The Meaning of Probability 2 The Axioms of Probability 3 Repeated Trials 4 The Concept of a Random Variable 5 Functions of One Random Variable 6 Two Random Variables 7 Sequences of Random Variables 8 Statistics Part 2 Stochastic Processes 9 General Concepts 10 Random Walk and Other Applications 11 Spectral Representation 12 Spectral Estimation 13 Mean Square Estimation 14 Entropy 15 Markov Chains 16 Markov Processes and Queueing Theory
13,886 citations
Book•
01 Jan 1968
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Abstract: Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.
13,153 citations