Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis

Permutation Entropy (PE) is a powerful tool for quantifying the predictability of a sequence which includes measuring the regularity of a time series. Despite its successful application in a variety of scientific domains, PE requires a judicious choice of the delay parameter $\tau$. While another parameter of interest in PE is the motif dimension $n$, Typically $n$ is selected between $4$ and $8$ with $5$ or $6$ giving optimal results for the majority of systems. Therefore, in this work we focus solely on choosing the delay parameter. Selecting $\tau$ is often accomplished using trial and error guided by the expertise of domain scientists. However, in this paper, we show that persistent homology, the flag ship tool from Topological Data Analysis (TDA) toolset, provides an approach for the automatic selection of $\tau$. We evaluate the successful identification of a suitable $\tau$ from our TDA-based approach by comparing our results to a variety of examples in published literature

Delay Differential Analysis of Seizures in Multichannel Electrocorticography Data

High-density electrocorticogram (ECoG) electrodes are capable of recording neurophysiological data with high temporal resolution with wide spatial coverage. These recordings are a window to understanding how the human brain processes information and subsequently behaves in healthy and pathologic states. Here, we describe and implement delay differential analysis (DDA) for the characterization of ECoG data obtained from human patients with intractable epilepsy. DDA is a time-domain analysis framework based on embedding theory in nonlinear dynamics that reveals the nonlinear invariant properties of an unknown dynamical system. The DDA embedding serves as a low-dimensional nonlinear dynamical basis onto which the data are mapped. This greatly reduces the risk of overfitting and improves the method's ability to fit classes of data. Since the basis is built on the dynamical structure of the data, preprocessing of the data (e.g., filtering) is not necessary. We performed a large-scale search for a DDA model that best fit ECoG recordings using a genetic algorithm to qualitatively discriminate between different cortical states and epileptic events for a set of 13 patients. A single DDA model with only three polynomial terms was identified. Singular value decomposition across the feature space of the model revealed both global and local dynamics that could differentiate electrographic and electroclinical seizures and provided insights into highly localized seizure onsets and diffuse seizure terminations. Other common ECoG features such as interictal periods, artifacts, and exogenous stimuli were also analyzed with DDA. This novel framework for signal processing of seizure information demonstrates an ability to reveal unique characteristics of the underlying dynamics of the seizure and may be useful in better understanding, detecting, and maybe even predicting seizures

How to avoid potential pitfalls in recurrence plot based data analysis

Recurrence plots and recurrence quantification analysis have become popular in the last two decades. Recurrence based methods have on the one hand a deep foundation in the theory of dynamical systems and are on the other hand powerful tools for the investigation of a variety of problems. The increasing interest encompasses the growing risk of misuse and uncritical application of these methods. Therefore, we point out potential problems and pitfalls related to different aspects of the application of recurrence plots and recurrence quantification analysis

Automated Fourier space region-recognition filtering for off-axis digital holographic microscopy

Automated label-free quantitative imaging of biological samples can greatly benefit high throughput diseases diagnosis. Digital holographic microscopy (DHM) is a powerful quantitative label-free imaging tool that retrieves structural details of cellular samples non-invasively. In off-axis DHM, a proper spatial filtering window in Fourier space is crucial to the quality of reconstructed phase image. Here we describe a region-recognition approach that combines shape recognition with an iterative thresholding to extracts the optimal shape of frequency components. The region recognition technique offers fully automated adaptive filtering that can operate with a variety of samples and imaging conditions. When imaging through optically scattering biological hydrogel matrix, the technique surpasses previous histogram thresholding techniques without requiring any manual intervention. Finally, we automate the extraction of the statistical difference of optical height between malaria parasite infected and uninfected red blood cells. The method described here pave way to greater autonomy in automated DHM imaging for imaging live cell in thick cell cultures

Refining a Phase Vocoder for Vocal Modulation

Vocal harmonies are a highly sought-after effect in the music industry, as they allow singers to portray more emotion and meaning through their voices. The chords one hears when listening to nearly any modern song are constructed through common ratios of frequencies (e.g., the recipe for a major triad is 4:5:6). Currently, vocal melodies are only readily obtainable through a few methods, including backup singers, looper-effects systems, and post-process overdubbing. The issue with these is that there is currently no publicly-available code that allows solo-artists to modulate input audio to whatever chord structure is desired while maintaining the same duration and timbre in the successive layers. This thesis plans to address this issue using the phase vocoder method. If this modulation technique is successful, this could revolutionize the way vocalists perform. The introduction of real-time self harmonization would allow artists to have access to emphasized lyrical phrases and vocals without needing to hire and train backup vocalists. This phase vocoder would also allow for more vocal improvisation, as the individual would only need to know how to harmonize with themselves and would thus not be relying on interpreting how backup vocalists plan on moving the melody when creating more spontaneously

Dynamical system analysis and forecasting of deformation produced by an earthquake fault

We present a method of constructing low-dimensional nonlinear models describing the main dynamical features of a discrete 2D cellular fault zone, with many degrees of freedom, embedded in a 3D elastic solid. A given fault system is characterized by a set of parameters that describe the dynamics, rheology, property disorder, and fault geometry. Depending on the location in the system parameter space we show that the coarse dynamics of the fault can be confined to an attractor whose dimension is significantly smaller than the space in which the dynamics takes place. Our strategy of system reduction is to search for a few coherent structures that dominate the dynamics and to capture the interaction between these coherent structures. The identification of the basic interacting structures is obtained by applying the Proper Orthogonal Decomposition (POD) to the surface deformations fields that accompany strike-slip faulting accumulated over equal time intervals. We use a feed-forward artificial neural network (ANN) architecture for the identification of the system dynamics projected onto the subspace (model space) spanned by the most energetic coherent structures. The ANN is trained using a standard back-propagation algorithm to predict (map) the values of the observed model state at a future time given the observed model state at the present time. This ANN provides an approximate, large scale, dynamical model for the fault.Comment: 30 pages, 12 figure