Granger causality for compressively sensed sparse signals.

Compressed sensing is a scheme that allows for sparse signals to be acquired, transmitted, and stored using far fewer measurements than done by conventional means employing the Nyquist sampling theorem. Since many naturally occurring signals are sparse (in some domain), compressed sensing has rapidly seen popularity in a number of applied physics and engineering applications, particularly in designing signal and image acquisition strategies, e.g., magnetic resonance imaging, quantum state tomography, scanning tunneling microscopy, and analog to digital conversion technologies. Contemporaneously, causal inference has become an important tool for the analysis and understanding of processes and their interactions in many disciplines of science, especially those dealing with complex systems. Direct causal analysis for compressively sensed data is required to avoid the task of reconstructing the compressed data. Also, for some sparse signals, such as for sparse temporal data, it may be difficult to discover causal relations directly using available data-driven or model-free causality estimation techniques. In this work, we provide a mathematical proof that structured compressed sensing matrices, specifically circulant and Toeplitz, preserve causal relationships in the compressed signal domain, as measured by Granger causality (GC). We then verify this theorem on a number of bivariate and multivariate coupled sparse signal simulations which are compressed using these matrices. We also demonstrate a real world application of network causal connectivity estimation from sparse neural spike train recordings from rat prefrontal cortex. In addition to demonstrating the effectiveness of structured matrices for GC estimation from sparse signals, we also show a computational time advantage of the proposed strategy for causal inference from compressed signals of both sparse and regular autoregressive processes as compared to standard GC estimation from original signals.

Compression complexity with ordinal patterns for robust causal inference in irregularly sampled time series.

Distinguishing cause from effect is a scientific challenge resisting solutions from mathematics, statistics, information theory and computer science. Compression-Complexity Causality (CCC) is a recently proposed interventional measure of causality, inspired by Wiener-Granger's idea. It estimates causality based on change in dynamical compression-complexity (or compressibility) of the effect variable, given the cause variable. CCC works with minimal assumptions on given data and is robust to irregular-sampling, missing-data and finite-length effects. However, it only works for one-dimensional time series. We propose an ordinal pattern symbolization scheme to encode multidimensional patterns into one-dimensional symbolic sequences, and thus introduce the Permutation CCC (PCCC). We demonstrate that PCCC retains all advantages of the original CCC and can be applied to data from multidimensional systems with potentially unobserved variables which can be reconstructed using the embedding theorem. PCCC is tested on numerical simulations and applied to paleoclimate data characterized by irregular and uncertain sampling and limited numbers of samples.

Time-Reversibility, Causality and Compression-Complexity.

Detection of the temporal reversibility of a given process is an interesting time series analysis scheme that enables the useful characterisation of processes and offers an insight into the underlying processes generating the time series. Reversibility detection measures have been widely employed in the study of ecological, epidemiological and physiological time series. Further, the time reversal of given data provides a promising tool for analysis of causality measures as well as studying the causal properties of processes. In this work, the recently proposed Compression-Complexity Causality (CCC) measure (by the authors) is shown to be free of the assumption that the "cause precedes the effect", making it a promising tool for causal analysis of reversible processes. CCC is a data-driven interventional measure of causality (second rung on the Ladder of Causation) that is based on Effort-to-Compress (ETC), a well-established robust method to characterize the complexity of time series for analysis and classification. For the detection of the temporal reversibility of processes, we propose a novel measure called the Compressive Potential based Asymmetry Measure. This asymmetry measure compares the probability of the occurrence of patterns at different scales between the forward-time and time-reversed process using ETC. We test the performance of the measure on a number of simulated processes and demonstrate its effectiveness in determining the asymmetry of real-world time series of sunspot numbers, digits of the transcedental number π and heart interbeat interval variability.

ChaosNet: A chaos based artificial neural network architecture for classification.

Inspired by chaotic firing of neurons in the brain, we propose ChaosNet-a novel chaos based artificial neural network architecture for classification tasks. ChaosNet is built using layers of neurons, each of which is a 1D chaotic map known as the Generalized Luröth Series (GLS) that has been shown in earlier works to possess very useful properties for compression, cryptography, and for computing XOR and other logical operations. In this work, we design a novel learning algorithm on ChaosNet that exploits the topological transitivity property of the chaotic GLS neurons. The proposed learning algorithm gives consistently good performance accuracy in a number of classification tasks on well known publicly available datasets with very limited training samples. Even with as low as seven (or fewer) training samples/class (which accounts for less than 0.05% of the total available data), ChaosNet yields performance accuracies in the range of 73.89%-98.33%. We demonstrate the robustness of ChaosNet to additive parameter noise and also provide an example implementation of a two layer ChaosNet for enhancing classification accuracy. We envisage the development of several other novel learning algorithms on ChaosNet in the near future.

Causal stability and synchronization.

Synchronization of chaos arises between coupled dynamical systems and is very well understood as a temporal phenomenon, which leads the coupled systems to converge or develop a dependence with time. In this work, we provide a complementary spatial perspective to this phenomenon by introducing the novel idea of causal stability. We then propose and prove a causal stability synchronization theorem as a necessary and sufficient condition for complete synchronization. We also provide an empirical criterion to identify synchronizing variables in coupled identical chaotic dynamical systems based on intrasystem causal influences estimated using time series data of the driving system alone. For this, a recently proposed measure, Compression-Complexity Causality (CCC), is used. The sign and magnitude of the estimated CCC value capture the nature of dynamical influences from each variable to rest of the subsystem and are thus able to determine whether or not the variable, when used to couple another system, will drive that system to synchronization.

Data-based intervention approach for Complexity-Causality measure.

Causality testing methods are being widely used in various disciplines of science. Model-free methods for causality estimation are very useful, as the underlying model generating the data is often unknown. However, existing model-free/data-driven measures assume separability of cause and effect at the level of individual samples of measurements and unlike model-based methods do not perform any intervention to learn causal relationships. These measures can thus only capture causality which is by the associational occurrence of 'cause' and 'effect' between well separated samples. In real-world processes, often 'cause' and 'effect' are inherently inseparable or become inseparable in the acquired measurements. We propose a novel measure that uses an adaptive interventional scheme to capture causality which is not merely associational. The scheme is based on characterizing complexities associated with the dynamical evolution of processes on short windows of measurements. The formulated measure, Compression-Complexity Causality is rigorously tested on simulated and real datasets and its performance is compared with that of existing measures such as Granger Causality and Transfer Entropy. The proposed measure is robust to the presence of noise, long-term memory, filtering and decimation, low temporal resolution (including aliasing), non-uniform sampling, finite length signals and presence of common driving variables. Our measure outperforms existing state-of-the-art measures, establishing itself as an effective tool for causality testing in real world applications.

Adaptive Spectral Envelope Estimation for Doppler Ultrasound.

Estimation of accurate maximum velocities and spectral envelope in ultrasound Doppler blood flow spectrograms are both essential for clinical diagnostic purposes. However, obtaining accurate maximum velocity is not straightforward due to intrinsic spectral broadening and variance in the power spectrum estimate. The method proposed in this paper for maximum velocity point detection has been developed by modifying an existing method-signal noise slope intersection, incorporating in it steps from an altered version of another method called geometric method. Adaptive noise estimation from the spectrogram ensures that a smooth spectral envelope is obtained postdetection of these maximum velocity points. The method has been tested on simulated Doppler signal with scatterers possessing a parabolic flow velocity profile constant in time, steady and pulsatile string phantom recordings, as well as in vivo recordings from uterine, umbilical, carotid, and subclavian arteries. The results from simulation experiments indicate a bias of less than 2.5% in maximum velocities when estimated for a range of peak velocities, Doppler angles, and SNR levels. Standard deviation in the envelope is low-less than 2% in the case of experiments done by varying the peak velocity and Doppler angle for steady phantom and simulated flow, and also less than 2% in the case of experiments done by varying SNR but keeping constant flow conditions for in vivo and simulated flow. Low variability in the envelope makes the prospect of using the envelope for automated blood flow measurements possible and is illustrated for the case of pulsatility index estimation in uterine and umbilical arteries.