Canonical information flow decomposition among neural structure subsets

The Partial Directed Coherence (PDC) and its generalized formulation (gPDC) are popular tools for investigating, in the frequency domain, the concept of Granger causality among multivariate (MV) time series. PDC and gPDC are formalized in terms of the coefficients of an MV autoregressive (MVAR) model which describes only the lagged effects among the time series and forsakes instantaneous effects. However, instantaneous effects are known to affect linear parametric modeling, and are likely to occur in experimental time series. In this study, we investigate the impact on the assessment of frequency domain causality of excluding instantaneous effects from the model underlying PDC evaluation. Moreover, we propose the utilization of an extended MVAR model including both instantaneous and lagged effects. This model is used to assess PDC either in accordance with the definition of Granger causality when considering only lagged effects (iPDC), or with an extended form of causality, when we consider both instantaneous and lagged effects (ePDC). The approach is first evaluated on three theoretical examples of MVAR processes, which show that the presence of instantaneous correlations may produce misleading profiles of PDC and gPDC, while ePDC and iPDC derived from the extended model provide here a correct interpretation of extended and lagged causality. It is then applied to representative examples of cardiorespiratory and EEG MV time series. They suggest that ePDC and iPDC are better interpretable than PDC and gPDC in terms of the known cardiovascular and neural physiologies.

Electroencephalographic (EEG) signals recorded from the scalp surface are generally highly correlated. Each channel is a linear mixture of concurrently active brain and non-brain electrical sources whose activities are volume conducted to the scalp electrodes with broadly overlapping patterns (Nunez et al., 1997). This property is particularly relevant to connectivity analyses, which seek to detect and characterize active interactions between brain regions. Therefore, meaningful connectivity patterns can be derived only from measures of cortical source activities and not directly from EEG channel activities (Michel and Murray, 2012). However, this poses a serious problem, since estimating the nature, number, brain (or non-brain) locations, and time courses of the active sources contributing to the scalp EEG is not straightforward (Baillet et al., 2001). Several methods have been proposed to estimate source activities from multi-channel EEG recordings, thereby removing the confounding effects of volume conduction. These methods can be grouped into three categories: (a) simple spatial filters that seek to reduce correlations between scalp channels based on idealized assumptions; (b) more complex spatial filters that seek to estimate net activities within ROIs based on detailed neurophysiological head models; and (c) blind spatial source separation methods that seek to separately identify source signals by exploiting source signal information differences. Whereas simple spatial filters such as bipolar derivations and Laplacian filters can reduce, to some extent, correlations among scalp-recorded channels (Fisch, 2012), more sophisticated spatial filtering methods use inverse imaging methods to estimate the time courses of cortical sources in given or estimated region of interest (ROI) (Baillet et al., 2001). Blind source separation techniques, in particular independent component analysis, by contrast, learn spatial filters from the EEG time courses that separate the data into constituent independent source activities. Their corresponding brain (or non-brain) locations can then be estimated using neurophysiological inverse imaging methods (Makeig et al., 1996; Jung et al., 2001; Delorme et al., 2012). Vector autoregressive (VAR) models are versatile tools for analyzing multivariate time series, including multi-channel EEG or multivariate source activities. VAR models predict current values of time series from their recent past (Lutkepohl, 2005). Importantly, they can be used to derive various electrophysiological connectivity measures (Schlogl and Supp, 2006). Volume conduction in biological tissue can be modeled as instantaneous propagation of activity from sources to recording channels. The resulting zero-phase connectivity may be treated as noise added to lagged connectivity patterns of interest. Although some measures, including the imaginary part of the coherency (Nolte et al., 2004), are insensitive to zero-phase connectivity, measures derived from VAR model coefficients do not include such zero-phase terms. Thus, volume conduction effects are not accounted for by the model and affect the correlation structure of the model residuals, which are normally assumed to be uncorrelated. Popular connectivity measures derived from VAR models include the Directed Transfer Function (DTF) (Kaminski and Blinowska, 1991) and the Partial Directed Coherence (PDC) (Baccala and Sameshima, 2001). Whereas the PDC is defined in terms of the system matrix (a frequency domain representation of the VAR model), the DTF is based on the inverse of the system matrix. Viewing lagged dependencies between source signals as information flow, the DTF may be said to be normalized by the inflow of information to some sink, while the PDC is normalized by the outflow of information from some source. As we will demonstrate below, both the DTF and the PDC are indeed adversely affected by volume conduction from multiple sources to the scalp electrodes, in contrast to the claim of Kaminski and Blinowska in their recent opinion article (Kaminski and Blinowska, 2014). Thus, in general direct application of connectivity measures to scalp EEG signals produces less than accurate results and also does not allow their clear interpretation in terms of underlying source dynamics.

In this work, we will consider the dimension reduction of the set of time series, such as economic data, to find the meaningful basis vector for the set of data, and indicate which data use which basis vector. Usually each of the time series is analyzed independently in economics but here we will analyze the set of time series simultaneously. Since some of the economic data are measured as positive values and we want to decompose them as a mixture of the parts, we will apply non-negative matrix factorization to the economic data. Non-negative matrix factorization can compress dimensions by approximating a non-negative matrix with the product of two non-negative matrices. The two non-negative matrices are called the coefficient matrix and the basis matrix, and the basis matrix can be considered as a dimensionally compressed matrix. If the standard non-negative matrix factorization is used for economic data, the basis matrix may not be smooth. We think that the basis vectors should be smooth except a few special economical incidents. In the proposed method, a Graph-based non-negative matrix factorization is introduced to regularize the basis matrix of the time series. A path graph for representing the time series of economic data is incorporated into the non-negative matrix factorization as regularization. As a result, basis vectors that maintains the time series of economic data are decomposed. Furthermore, we propose to introduce a sparsity in the non-negative matrix factorization. Traditionally, the sparsity incorporated into non-negative matrix factorization has been used for basis vectors. However, the proposed method introduces the sparsity for coefficient vectors. Thus the proposed method, which simultaneously incorporates the sparsity for the coefficient vectors and the smoothness for the basis vectors, can extract the smooth basis vectors and the original economical data are approximated as the weighted sum of the few bases vectors. This allows us to discover economic trends and the best-fit trends for each data at the same time.

In seasonal adjustment a time series is considered as a juxtaposition of several components, the trend-cycle, and the seasonal and irregular components. The Bureau of the Census X-11 method, based on moving averages, correction of large errors and trading day adjustments, has long dominated. With the success of ARIMA modelling at the end of the 20th century, methods with better outlier detection and trading day corrections by regression with ARIMA errors have appeared, with the regARIMA module of Census X-12-ARIMA or Bank of Spain TRAMO-SEATS. SEATS consists of extracting the components by an ARIMA-model-based unobserved components approach. This means that models are used for each component such that the sum of the components is compatible with the ARIMA model for the corrected time series. The underlying theory of the SEATS program is studied in many papers but there is no complete and systematic description of its output. Our purpose is to examine SEATS text output and to explain the results in simple words and formulas. This is done on a simple example, a time series with a non-seasonal model so that the computations can be verified step by step. The principles behind SEATS are first described, including the admissible decompositions and the canonical decomposition, and the derivation of the Wiener-Kolmogorov filter. Then the example is introduced: the interest rates of US certificates of deposits. The text output from SEATS is presented in edited form in several tables. Finally, the main results are checked on the example by means of a Microsoft Excel workbook and direct computations. In particular, the forecasts and backcasts are obtained; the admissible and canonical decompositions with two components are discussed; the filters are first derived using autocorrelations of two auxiliary ARMA processes, then applied on the prolonged time series; and the characteristics of the estimates, the revisions and the growth rates are analyzed.

BackgroundMotor imagery based brain-computer interfaces (MI-BCIs) are systems that detect the mental rehearsal of movement from brain activity signals (EEG) for controlling devices that can potentiate motor neurorehabilitation. Considering the problem that MI proficiency requires training and it is not always achieved, EEG desirable features should be investigated to propose indicators of successful MI training.MethodsNine healthy right-handed subjects trained with a MI-BCI for four sessions. In each session, EEG was recorded for 30 trials that consisted of a rest and a dominant-hand MI sequence, which were used for calibrating the system. Then, the subject participated in 160 trials in which a cursor was displaced on a screen by performing MI or relaxing to hit a target. The session’s accuracy was calculated. For each trial from the calibration phase of the first session, the power spectral density (PSD) and the partial directed coherence (PDC) of the rest and MI EEG segments were obtained to estimate the event-related synchronization changes (ERS) and the connectivity patterns of the θ, α, β and γ bands that are associated with high BCI control (accuracy above 70% in at least one session). Finally, t-tests and rank-sum tests (p<0.05, with Benjamini-Hochberg correction) were used to compare the ERS/ERD and PDC values of subjects with high and low accuracy, respectively.ResultsProficient users showed greater α ERD on the right-hand motor cortex (left hemisphere). Furthermore, the β PDC related to the ipsilateral motor cortex is commonly weakened during motor imagery, while the contralateral motor cortex γ PDC is enhanced.ConclusionsMotor imagery proficiency is related to the focused and lateralized event-related α desynchronization patterns and the lateralization of β and γ PDC. Future analysis of these features could allow complimenting the information for assessment of subject-specific BCI control and the prediction of the effectiveness of motor-imagery training.

Background High temporal resolution and relatively low cost make electroencephalography (EEG) the most suitable noninvasive tool for studying brain dynamics. One of the major challenges is the determination of effective connectivity, i.e., directed (causal) information flow between brain areas. The common definition of effective connectivity is based on Granger’s argument, that “the cause must precede the effect” [1], which is implemented by the original Granger Causality (GC) score [1], the Directed Transfer Function (DTF) [3], Partial Directed Coherence (PDC) [2] and the Phase-slope Index (PSI) [4], all of which have been applied to EEG data previously. However, due to volume conduction in the head, the original causallyrelated sources are mixed into EEG channels. We conducted a realistic simulation study to investigate how volume conduction affects EEG sensor-space effective connectivity estimation.

The aim of this study was to investigate the temporal dynamic behavior of cardiovascular interactions between heart period and systolic blood pressure during a 20-min head-up tilt test at 70° in young women with orthostatic intolerance compared to healthy women. Methods included the lagged and extended partial directed coherence applied to short-term windows shifted by 5 seconds, extracted from a multivariate set of cardiovascular and respiratory time series. Findings revealed significantly increased information flow (p <; 0.01) in patients from: a) heart period to blood pressure during supine position which subsequently decreased and b) blood pressure to heart period during the progression of orthostatic phase. Controls developed balanced cardiovascular interactions with smaller information flows than patients.

The problem of determining directional coupling between neuronal oscillators from their time series is addressed. We compare performance of the two well-established approaches: partial directed coherence and phase dynamics modeling. They represent linear and nonlinear time series analysis techniques, respectively. In numerical experiments, we found each of them to be applicable and superior under appropriate conditions: The latter technique is superior if the observed behavior is "closer" to limit-cycle dynamics, the former is better in cases that are closer to linear stochastic processes.

In order to preserve the EEG time-frequency domain features while fully uncovering the information flow and spatial information in the causal connectivity of relevant brain regions, this paper proposes a multichannel EEG signal emotion recognition method based on partial directed coherence dense graph propagation. The proposed method mainly uses partial directed coherence (PDC) to construct causal brain networks, which is used to describe the interactions and information flow relationships between different nodes or brain regions in the brain, and uses a modified model of dense graph propagation (DGP) in graph convolutional neural networks to train the graph data modeled by multichannel causal connectivity information of EEG signals and single-channel time-frequency domain features. Unlike traditional graph convolutional neural networks, the PDC-DGP model focuses on the directedness of the graph and does not dilute the feature information of the graph data by extensive smoothing. We investigate on two publicly available datasets, DEAP and SEED, and the results show that the method has good recognition performance on both datasets, and the average accurate recognition rate can be as high as 85.25% on SEED and 77.37% and 76.92% on the DEAP dataset for valence and arousal, respectively.

Forecasting time series data is an integral component for management, planning and decision making. Following the Big Data trend, large amounts of time series data are available from many heterogeneous data sources in more and more applications domains. The highly dynamic and often fluctuating character of these domains in combination with the logistic problems of collecting such data from a variety of sources, imposes new challenges to forecasting. Traditional approaches heavily rely on extensive and complete historical data to build time series models and are thus no longer applicable if time series are short or, even more important, intermittent. In addition, large numbers of time series have to be forecasted on different aggregation levels with preferably low latency, while forecast accuracy should remain high. This is almost impossible, when keeping the traditional focus on creating one forecast model for each individual time series. In this paper we tackle these challenges by presenting a novel forecasting approach called cross-sectional forecasting. This method is especially designed for Big Data sets with a multitude of time series. Our approach breaks with existing concepts by creating only one model for a whole set of time series and requiring only a fraction of the available data to provide accurate forecasts. By utilizing available data from all time series of a data set, missing values can be compensated and accurate forecasting results can be calculated quickly on arbitrary aggregation levels.

The purpose of the article is to present the new method for electronic circuits simulation in frequency domain. The paper consists of an introduction, five sections, a conclusion and list of references. &#x0D; The first section is considered with the current state of the spectral balance method, which is a special versions of traditional harmonic balance method. In this method the linear and nonlinear parts of circuits are modeled in frequency domain only.&#x0D; The second section is devoted to the development of a spectra calculation method in nonlinear elements. The nonlinear elements into circuits simulation software usually described in time domain, so we need to develop the algorithm for calculation the spectra in nonlinear elements in frequency domain.&#x0D; In the third section the review of methods for tensor usage at simulation tasks into electronic CAD systems are discussed.&#x0D; Into the section four the methods of tensor decomposition are described. The canonical decomposition, Tacker decomposition and tensor train decomposition are discussed. &#x0D; The possible version of tensors usage in the tasks of spectral balance method for electronic CAD systems is presented in the section five.&#x0D; The conclusion contains the results of the work done, practical value and prospects are indicated.

A time series forecasting model—which is typically built on a single time series—is known as a local time series model (tsLM). In contrast, a forecasting model trained on multiple time series is referred to as a global time series model (tsGM). tsGMs can enhance forecasting accuracy and improve generalisation by learning cross-series information. As such, developing tsGMs has become a prominent research focus within the time series forecasting community. However, the benefits of tsGMs may not always be realised if the given set of time series is heterogeneous. While increasing model complexity can help tsGMs adapt to such a set of data, it can also increase the risk of overfitting and forecasting error. Additionally, the definition of homogeneity remains ambiguous in the literature. To address these challenges, this paper explores how to define data heterogeneity and proposes a two-stage modelling framework: At stage one, a tsGM is learnt to identify homogeneous patterns; and at stage two, tsLMs (e.g., ARIMA) or sub-tsGMs tailored to different groups are learnt to capture the heterogeneity. Numerical experiments on four open datasets demonstrate that the proposed approach significantly outperforms six state-of-the-art models. These results highlight its effectiveness in unlocking the full potential of global forecasting models for heterogeneous datasets.

While much is known about motor control during simple movements, corticomuscular communication profiles during compound movement control remain largely unexplored. Here, we aimed at examining frequency band related interactions between brain and muscles during different movement periods of a bipedal squat (BpS) task utilizing regression corticomuscular coherence (rCMC), as well as partial directed coherence (PDC) analyses. Participants performed 40 squats, divided into three successive movement periods (Eccentric (ECC), Isometric (ISO) and Concentric (CON)) in a standardized manner. EEG was recorded from 32 channels specifically-tailored to cover bilateral sensorimotor areas while bilateral EMG was recorded from four main muscles of BpS. We found both significant CMC and PDC (in beta and gamma bands) during BpS execution, where CMC was significantly elevated during ECC and CON when compared to ISO. Further, the dominant direction of information flow (DIF) was most prominent in EEG-EMG direction for CON and EMG-EEG direction for ECC. Collectively, we provide novel evidence that motor control during BpS is potentially achieved through central motor commands driven by a combination of directed inputs spanning across multiple frequency bands. These results serve as an important step toward a better understanding of brain-muscle relationships during multi joint compound movements.

This paper presents a unified mathematical derivation of the asymptotic behaviour of the three main forms of partial directed coherence (PDC). Numerical examples are used to contrast PDC, gPDC (generalized PDC) and iPDC (information PDC) as to meaning and applicability and, more importantly, to show their essential statistical equivalence insofar as connectivity inference is concerned.

Time series are an important type of data with applications in virtually every aspect of the real world. Often a large number of time series have to be monitored and analyzed in parallel. Sets of time series may show intrinsic hierarchical relationships and varying degrees of importance among the individual time series. Effective techniques for visually analyzing large sets of time series should encode the relative importance and hierarchical ordering of the time series data by size and position, and should also provide a high degree of regularity in order to support comparability by the analyst. In this paper, we present a framework for visualizing large sets of time series. Based on the notion of inter time series importance relationships, we define a set of objective functions that space-filling layout schemes for time series data should obey. We develop an efficient algorithm addressing the identified problems by generating layouts that reflect hierarchy and importance based relationships in a regular layout with favorable aspect ratios. We apply our technique to a number of real world data sets including sales and stock data, and we compare our technique with an aspect ratio aware variant of the well known TreeMap algorithm. The examples show the advantages and practical usefulness of our layout algorithm.