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Hyde 1 , Jurriaan M. Peters 1 , Frank H.

Analysis, Manifolds and Physics, Part II Revised and Enlarged Edition Pt 2

Duffy 1 , Petr Stovicek 3 , Simon K. Warfield 1 , Rob S. Reconstruction of classical Lorenz system using Laplacian eigenmaps applied to delay coordinates of x component, demonstrated with increasing levels of noise added to observations.

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The Lorenz system [ a , top row] was synthesized using classical parameter choices see text , and increasing levels of zero-mean noise standard deviations of 0, 1, and 5 were added for each analysis [see a , b , and c , respectively, in the top row]. The noisy x component time series was taken as the only observation in each case [see a , b , and c in the middle row], and reconstructions were performed using delay embedding taking consecutive samples and Laplacian eigenmaps [taking the first three LE coordinates; see a , b , and c in the bottom row].


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Panels a and b show two different views of the same Laplacian eigenmaps LE visualization, highlighting collections of trajectories according to their pacing locations. We retain the convention often used in medicine of presenting results in the coordinate system facing the subject, so that left from the viewer's perspective corresponds to the subject's right.

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Thus a shows RV-paced trajectories highlighted with yellow, green, red, blue, and purple points, and all trajectories for contralaterally paced beats shown with copper points. Panel b shows LV-paced trajectories highlighted with red, green, blue, yellow, and purple points, and all trajectories for contralaterally paced beats shown with silver points. Colors used to group location indices correspond to those used to highlight LE trajectories. Panel e shows a visualization of the LV blood volume with spheres indicating LV pacing locations.

Analysis, manifolds, and physics /

Indices correspond to those shown on a map of LV segments. Omitted indices no. Electrograms from canine ventricular surface epicardial during QRS: three orthogonal views [ a , b , and c ] of points in LE coordinates. Each sphere represents a single time instant of data. Changes in activation times on epicardial surface of ventricles [ d , f , and h , showing front and back views of isochronal maps] are compared to changes in trajectories in LE coordinates [ e , g , and i ] during three stages of the ischemia experiment ordered, with time increasing left to right.

Interventions were performed to restrict blood supply and induce downstream ischemia. Normal beats are shown in d and e , beats from the fifth intervention are shown in f and g , and beats from the eighth intervention are shown in h and i. Relevant LE trajectories for each intervention are colored yellow, with the rest of the data shown as semitransparent red points.

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An analysis using the differential is shown in the fourth row [ j , k , and l ]. The differential was evaluated at a point during a normal beat [corresponds to potentials measured on heart surface, shown as isopotential j , and black point in LE coordinates k ].

In LE coordinates, a change vector was chosen to represent the direction of trajectory changes [green arrow in k ] during the experimental stages [yellow trajectories from e to g and then i ]. A change map corresponding to this change vector was obtained using the proposed vector mapping approach, and visualized as an isopotential map l. The change map shows one dominant region near the apex that changes the most. This region is the same one that suffers from late activations after eight ischemia interventions h. Scalp EEGs recorded during interictal epileptic spikes: two different spike types, slow and sharp waves, were identified and labeled in the data set.


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Panel a shows mean signals of time-aligned spikes from a single electrode with standard deviation bounds. Panels b and d show two visualizations of LE coordinates resulting from delay-embedding all spikes and all electrodes. In b color shows the progression of time, and in d color shows the type of spike. The voltage map in c of the example point [shown as black with green border in LE coordinates in d ] shows both spike types.

Voltage maps in e and f were synthesized by adding voltage maps corresponding to LE coordinate vectors [green arrows in d ] to voltage map in c. The vector in d pointing towards the slow waves was used to create the voltage map with a pronounced slow wave in e.

Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition - 1st Edition

The other vector, pointing towards sharp waves, resulted in the pronounced sharp wave in f. Extensions to a manifold learning framework for time-series analysis on dynamic manifolds in bioelectric signals Burak Erem, Ramon Martinez Orellana, Damon E. Analysis, manifolds and physics Home Analysis, manifolds and physics.

Analysis, manifolds and physics. Read more. Analysis, Manifolds and Physics. Differential manifolds and mathematical physics. Manifolds, tensor analysis, and applications. Manifolds Tensor Analysis and Applications. Manifolds Tensor analysis and Applications. Manifolds, Tensor Analysis and Applications. Geometry and Analysis on Manifolds. Revised Edition Part I.

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