Contents

## Multi- variate EMD History Initial Example by

Pasmore, Scott, Host has reference to this Academic Journal, PHwiki organized this Journal Multi-variate EMDExtension of time series analysis to multi-variable setsMultivariateMultivariate is not multi-dimension.Multivariate variable could be a collection of totally different signals; it could be related ones too: as long as example, the three components of a velocity vector. It is useful as long as image fusion as the three colors of a color picture.HistoryTanaka in addition to M in addition to ic first initiated complex, in addition to Rilling et al bivariate EMD.Tanaka, T. & M in addition to ic, D. P. 2006 Complex empirical mode decomposition. IEEE Signal Process. Lett. 14, 101104.Rilling, G., P. Fl in addition to rin, P. Gonçalves, in addition to J. M. Lilly, 2007: Bivariate Empirical Mode Decomposition, IEEE SIGNAL PROCESSING LETTERS, VOL. 14, 936-939.M in addition to ic extended it to Tri- in addition to Multi-variate.N. Rehman, D. P. M in addition to ic, 2009: Empirical Mode Decomposition as long as Trivariate Signals. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, 1059-1068.N. Rehman, D. P. M in addition to ic, 2009: Multivariate empirical mode decomposition. Proc. Roy. Soc. Lond., doi:10.1098/rspa.2009.0502 N. Rehman, D. P. M in addition to ic, 2011: Filter Bank Property of Multivariate Empirical Mode Decomposition. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, 2421-2426.

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Initial Example by Rilling et al.Free Matlab/C codes are available at http://perso.ens-lyon.fr/patrick.fl in addition to rin.Space Curve

Centroid of Space CurveExample : 64 Direction 10 iterationsExample : Results

Extension from Bivariate to Tri- in addition to Multivariate is non-trivialWe need a new mathematical tool:Quaternion in addition to beyondTechnical prerequisitesDefining the local extrema as long as multivariate signals is not straight as long as ward; as long as instance, the complex in addition to quaternion fields are not ordered : using multiple real-valued projections of the signal; the extrema of such projected signals are then interpolated componentwise to yield the multidimensional envelopes of the signal.The computation of the local mean as long as multivariate signals: to use real-valued projections along multiple directions on hyperspheres (n-spheres) in order to calculate the envelopes in addition to the local mean of multivariate signals.To generate a suitable set of direction vectors, unit hyperspheres are sampled based on both uni as long as m angular sampling methods in addition to quasi-Monte Carlo-based low-discrepancy sequences.3D DataFor simple 3D data plus time, we need the quaternion with algebra well developed.The quaternions are a number system that extends the complex numbers to higher dimensions. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 in addition to applied to mechanics in 3D space. To extend it to multi dimension is very complicated.

QuaternionMathematics: Any number of the as long as m q = a + bi + cj + dk where a, b, c, in addition to d are real numbers, i2 = j2 =k2 =-1, in addition to ij =k= -ji, jk=i in addition to ki=j;Under addition in addition to multiplication, quaternions have all the properties of a field, except multiplication is not commutative.Quaternion MemorialBroom Bridge, also known as Brougham Bridge, is a bridge along Broombridge Road which crosses the Royal Canal in Cabra, Dublin, Irel in addition to . Here as he walked by on the 16th of October 1843Sir William Rowan Hamilton in a flash of genius discovered the fundamental as long as mula as long as quaternion multiplication i² = j² = k² = ijk = 1 & cut it on a stone of this bridge.Quaternion Memorial

Quaternion MemorialRotationGeneralized to n-DimensionFor multivariate signals, the local maxima in addition to minima may not be defined directly.Moreover, the notion of oscillatory modes defining an IMF is rather confusing as long as multivariate signals.To deal with these problems, Rehman in addition to M in addition to ic (2009) proposed to generate multiple n-dimensional envelopes by taking signal projections along different directions in n-dimensional spaces; these are then averaged to obtain the local mean.

Projection of trivariate signalThe direction vector OA in 3D space, can also be represented by a point on the surface of a unit sphere. (b) Multiple direction vectors represented by points on a longitude line. Multiple direction vectors in a 3D space(a) For projections along longitudinal lines on a sphere, multiple axes represented by a set of vectors are chosen in the xy plane, with angle taken with respect to x-axis.(b) projections of the input signal can be taken by rotating the input signal along rotation axes represented by a set of unit quaternions .n-Dimensional CaseA simple in addition to practically convenient choice as long as a set of direction vectors is to employ uni as long as m angular sampling of a unit sphere in an n-dimensional hyperspherical coordinate system. The resulting set of direction vectors covers the whole (n 1) sphere.A coordinate system in an n-dimensional Euclidean space can then be defined to serve as a pointset ( in addition to the corresponding set of direction vectors) on an (n 1) sphere.

Generalization to n-dimension caseLet {1, 2, , (n1)} be the (n 1) angular coordinates, then an n-dimensional coordinate system having as the n coordinates on a unit (n 1) sphere is given byGeneralization to n-dimension caseThe pointset corresponding to the n-dimensional coordinate system is now very convenient to generate; however, as long as n >1, it does not provide a uni as long as m sampling distribution having a higher density of the points when approaching the poles. Proposed a quasi-Monte Carlo methods to generate a low discrepancy pointset, providing a more uni as long as m distribution on the n sphere.Low Discrepancy SamplingBy definition, the discrepancy of a sequence is low if the number of points in the sequence falling into an arbitrary set B in addition to it is close to uni as long as m distribution. A low-discrepancy sequence is a sequence with the property that as long as all values of N, its subsequence x1, , xN has a low discrepancy. Low-discrepancy sequences are also called quasi-r in addition to om or sub-r in addition to om sequences, due to their common use as a replacement of uni as long as mly distributed r in addition to om numbers.

Low Discrepancy SamplingThe “quasi” modifier is used to denote more clearly that the values of a low-discrepancy sequence are neither r in addition to om nor pseudor in addition to om, but such sequences share some properties of r in addition to om variables.There are many ways to generate such sequence. In certain applications such as the quasi-Monte Carlo method their lower discrepancy is an important advantage.Rehman in addition to M in addition to ic used Hammersley-Halton sequence (involved primary numbers etc) to generate the quasi-Monte Carlo sampling, which is almot uni as long as mly distributed.Multiple direction vectors in a n-D spaceDirection vectors as long as taking projections of trivariate signals on a two-sphere generated by using (a) spherical coordinate system in addition to (b) a low-discrepancy Hammersley sequence.Empirical Mode Decomposition, Multivariate EMD, Matlab in addition to dataSee below as long as our recent contributions in this field. You can also download EEG Data used in some of the simulations in the work below. The EEG data contain EOG in addition to EMG artifacts coming from eye blinks, eyebrow raising in addition to eye rolling, in addition to are used in our BSE, BSS, in addition to EMD algorithms. The recordings also contain a 50Hz power line noise. Enjoy! Matlab code: www.commsp.ee.ic.ac.uk/m in addition to ic

Some examples3-Dimensional

SummaryMVEMD is a very useful in addition to intriguing new development in EMD methodology.It offers an uni as long as mity (same scale in addition to same number) to the IMF from the multivariate data set so that the results will be easier to interpret.The NA-MVEMD is a beautiful idea.The fatal flaw of MVEMD in addition to NA-MVEMD is that the filter is too sharp, which leads to separation of harmonics from the fundamental in addition to destroy the nonlinearity.

## Pasmore, Scott Host

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## Journal Ratings by California Western School of Law

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