The-local-mean-decomposition-and-its-application-_decomposition_local mean_生物技术

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/625aa3d7be9ad24cfaec4b49/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625aa3d7be9ad24cfaec4b49/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">The local mean decomposition and its</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws0">application to EEG perception data</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls1 ws0">Jonathan S. Smith</div><div class="t m1 x4 h4 y4 ff1 fs2 fc0 sc0 ls1 ws0">†</div><div class="t m0 x5 h5 y5 ff2 fs3 fc0 sc0 ls1 ws0">Neurotechno Ltd, Marlow, Buckinghamshire SL7 1SJ, UK</div><div class="t m0 x6 h6 y6 ff3 fs3 fc0 sc0 ls1 ws0">This paper describes the local mean decomposition (LMD), a new iterative approach to</div><div class="t m0 x6 h6 y7 ff3 fs3 fc0 sc0 ls1 ws0">demodulating amplitude and frequency modulated signals. The new method decomposes</div><div class="t m0 x6 h6 y8 ff3 fs3 fc0 sc0 ls1 ws0">such signals into a set of functions, each of which is the product of an envelope signal and a</div><div class="t m0 x6 h6 y9 ff3 fs3 fc0 sc0 ls1 ws0">frequency modulated signal from which a time-varying instantaneous frequency can be</div><div class="t m0 x6 h6 ya ff3 fs3 fc0 sc0 ls1 ws0">derived. The LMD method can be used to analyse a wide variety of natural signals such as</div><div class="t m0 x6 h6 yb ff3 fs3 fc0 sc0 ls1 ws0">electrocardiograms, function al magnetic resonan ce imaging data, and earthquake data. The</div><div class="t m0 x6 h6 yc ff3 fs3 fc0 sc0 ls1 ws0">paper presents the results of applying LMD to a set of scalp electroencephalogram (EEG)</div><div class="t m0 x6 h6 yd ff3 fs3 fc0 sc0 ls1 ws0">visual perception data. The LMD instantaneous frequency and energy structure of the EEG</div><div class="t m0 x6 h6 ye ff3 fs3 fc0 sc0 ls1 ws0">is examined, and compared with results obtained using the spectrogram. The nature of visual</div><div class="t m0 x6 h6 yf ff3 fs3 fc0 sc0 ls2 ws0">perception is investigated by measuring the degree of EEG instantaneous phase</div><div class="t m0 x6 h6 y10 ff3 fs3 fc0 sc0 ls1 ws0">concentration that occurs following stimulus onset over multiple trials. The analysis</div><div class="t m0 x6 h6 y11 ff3 fs3 fc0 sc0 ls1 ws0">suggests that there is a statistically signiﬁcant difference between the theta phase</div><div class="t m0 x6 h6 y12 ff3 fs3 fc0 sc0 ls1 ws0">concentrations of the perception and no perception EEG data.</div><div class="t m0 x7 h7 y13 ff1 fs3 fc0 sc0 ls1 ws0">Keywords: demodulated signal time–frequency representation; electroencephalogram;</div><div class="t m0 x6 h7 y14 ff1 fs3 fc0 sc0 ls1 ws0">instantaneous frequency; local mean decomposition; phase concentration; visual perception</div><div class="t m0 x8 h7 y15 ff1 fs3 fc0 sc0 ls1 ws0">1. INTRODUCTION</div><div class="t m0 x8 h6 y16 ff3 fs3 fc0 sc0 ls1 ws0">The nature of information is one of the central issues in</div><div class="t m0 x8 h6 y17 ff3 fs3 fc0 sc0 ls1 ws0">modern science. Given a signal, or a set of data, how do</div><div class="t m0 x8 h6 y18 ff3 fs3 fc0 sc0 ls1 ws0">we extract information from it, or give meaning to it?</div><div class="t m0 x8 h6 y19 ff3 fs3 fc0 sc0 ls1 ws0">Fourier analysis is a particularly powerful method of</div><div class="t m0 x8 h6 y1a ff3 fs3 fc0 sc0 ls3 ws0">describing signals in terms of their energy and</div><div class="t m0 x8 h6 y1b ff3 fs3 fc0 sc0 ls1 ws0">frequency. Representing a signal as being the sum of a</div><div class="t m0 x8 h6 y1c ff3 fs3 fc0 sc0 ls1 ws0">set of plane waves, each of which has a constant phase,</div><div class="t m0 x8 h6 y1d ff3 fs3 fc0 sc0 ls1 ws0">frequency, and envelope, can often be very revealing.</div><div class="t m0 x8 h6 y1e ff3 fs3 fc0 sc0 ls4 ws0">For example, Fourier analysis can show that an</div><div class="t m0 x8 h6 y1f ff3 fs3 fc0 sc0 ls1 ws0">apparently very complicated quasiperiodic signal con-</div><div class="t m0 x8 h6 y20 ff3 fs3 fc0 sc0 ls1 ws0">sists merely of the sum of two tones with incomm ensu-</div><div class="t m0 x8 h6 y21 ff3 fs3 fc0 sc0 ls1 ws0">rate frequencies. However, it is generally the case tha t</div><div class="t m0 x8 h6 y22 ff3 fs3 fc0 sc0 ls1 ws0">complicated amplitude and frequency modulated sig-</div><div class="t m0 x8 h6 y23 ff3 fs3 fc0 sc0 ls1 ws0">nals, such as most biological signals, have correspond-</div><div class="t m0 x8 h6 y24 ff3 fs3 fc0 sc0 ls1 ws0">ingly complicated Fourier spectra. Such signals can also</div><div class="t m0 x8 h6 y25 ff3 fs3 fc0 sc0 ls1 ws0">be represented as being the product of an envelope</div><div class="t m0 x8 h6 y26 ff3 fs3 fc0 sc0 ls1 ws0">signal and a frequency-modulated signal, or as the sum</div><div class="t m0 x8 h6 y27 ff3 fs3 fc0 sc0 ls1 ws0">of a ﬁnite set of such product functions. A well-behaved</div><div class="t m0 x8 h6 y28 ff3 fs3 fc0 sc0 ls1 ws0">and physically meaningful time-varying instantaneous</div><div class="t m0 x8 h6 y29 ff3 fs3 fc0 sc0 ls1 ws0">frequency can then be calculated from the frequency</div><div class="t m0 x8 h6 y2a ff3 fs3 fc0 sc0 ls1 ws0">modulated signal.</div><div class="t m0 x9 h6 y2b ff3 fs3 fc0 sc0 ls5 ws0">The local mean decomposition (LMD) was</div><div class="t m0 x8 h6 y2c ff3 fs3 fc0 sc0 ls6 ws0">developed recently by the author to decompose</div><div class="t m0 x8 h6 y2d ff3 fs3 fc0 sc0 ls1 ws0">amplitude and frequency modulated signals into a</div><div class="t m0 x8 h6 y2e ff3 fs3 fc0 sc0 ls1 ws0">small set of product functions, each of which is the</div><div class="t m0 x8 h6 y2f ff3 fs3 fc0 sc0 ls7 ws0">product of an envelope signal and a frequency</div><div class="t m0 x8 h6 y30 ff3 fs3 fc0 sc0 ls1 ws0">modulated signal from which a time-varying instan-</div><div class="t m0 x8 h6 y31 ff3 fs3 fc0 sc0 ls1 ws0">taneous phase and instantaneous frequency can be</div><div class="t m0 xa h6 y32 ff3 fs3 fc0 sc0 ls1 ws0">derived. The instantaneous frequency and envelope</div><div class="t m0 xa h6 y33 ff3 fs3 fc0 sc0 ls1 ws0">values can then be plotted together in the form of a</div><div class="t m0 xa h6 y34 ff3 fs3 fc0 sc0 ls8 ws0">demodulated signal time–frequency representation</div><div class="t m0 xa h6 y35 ff3 fs3 fc0 sc0 ls1 ws0">(tfr). Although a form of instantaneous frequency can</div><div class="t m0 xa h6 y36 ff3 fs3 fc0 sc0 ls1 ws0">be calculated from the spectrogram (the conditional</div><div class="t m0 xa h6 y37 ff3 fs3 fc0 sc0 ls1 ws0">mean frequency, investigated by Loughlin 1999), it is</div><div class="t m0 xa h6 y38 ff3 fs3 fc0 sc0 ls1 ws0">window-dependent. The conventional method of deriv-</div><div class="t m0 xa h6 y39 ff3 fs3 fc0 sc0 ls1 ws0">ing a time-varying frequency from a signal is to use the</div><div class="t m0 xa h6 y3a ff3 fs3 fc0 sc0 ls1 ws0">so-called analytic signal developed by Gabor (1946).</div><div class="t m0 xa h6 y3b ff3 fs3 fc0 sc0 ls1 ws0">The instantaneous frequency is deﬁned as being the</div><div class="t m0 xa h6 y3c ff3 fs3 fc0 sc0 ls1 ws0">derivative of the phase of the analytic signal. The</div><div class="t m0 xa h6 y3d ff3 fs3 fc0 sc0 ls1 ws0">problem with this approach is that, particularly for</div><div class="t m0 xa h6 y3e ff3 fs3 fc0 sc0 lsa ws0">amplitude and frequenc y modulated signals, the</div><div class="t m0 xa h6 y3f ff3 fs3 fc0 sc0 ls1 ws0">resulting instantaneous frequency can be wildly erratic,</div><div class="t m0 xa h6 y40 ff3 fs3 fc0 sc0 ls1 ws0">even negative at times (Mandel 1974; Melville 1983;</div><div class="t m0 xa h6 y41 ff3 fs3 fc1 sc0 ls1 ws0">Boashash 1992). Such an instantaneous frequency is</div><div class="t m0 xa h6 y42 ff3 fs3 fc0 sc0 lsb ws0">extremely hard to interpret, and is of debatable</div><div class="t m0 xa h6 y43 ff3 fs3 fc0 sc0 ls1 ws0">physical signiﬁcance. Consequently, the analytic signal</div><div class="t m0 xa h6 y44 ff3 fs3 fc0 sc0 ls1 ws0">instantaneous frequency has been of limited practical</div><div class="t m0 xa h6 y45 ff3 fs3 fc0 sc0 ls1 ws0">utility. By contrast, the new LMD scheme is a robust,</div><div class="t m0 xa h6 y46 ff3 fs3 fc0 sc0 ls1 ws0">and conceptually simple way of analysing complicated</div><div class="t m0 xa h6 y47 ff3 fs3 fc0 sc0 ls1 ws0">data in terms of time-varying frequency, phase and</div><div class="t m0 xa h6 y48 ff3 fs3 fc0 sc0 ls1 ws0">energy. Most importantly, the results of the LMD</div><div class="t m0 xa h6 y49 ff3 fs3 fc0 sc0 ls1 ws0">analysis are physically plausible; the instantaneous</div><div class="t m0 xa h6 y4a ff3 fs3 fc0 sc0 lsc ws0">frequency generally cor responds to the oscillatory</div><div class="t m0 xa h6 y4b ff3 fs3 fc0 sc0 ls1 ws0">frequency we can actually see in the signal.</div><div class="t m0 xb h6 y4c ff3 fs3 fc0 sc0 ls1 ws0">The LMD method can be used to analyse any data,</div><div class="t m0 xa h6 y4d ff3 fs3 fc0 sc0 ls1 ws0">but is of particular relevance with regard to the analysis</div><div class="t m0 xa h6 y4e ff3 fs3 fc0 sc0 lsd ws0">of amplitude and frequency modulated biological</div><div class="t m0 xa h6 y4f ff3 fs3 fc0 sc0 ls1 ws0">signals. It could be used, for example, to analyse</div><div class="t m0 xa h6 y50 ff3 fs3 fc0 sc0 ls1 ws0">changes in blood pressure, or to detect small ﬂuctu-</div><div class="t m0 xa h6 y51 ff3 fs3 fc0 sc0 lse ws0">ations in the frequency of the electrocardiogram</div><div class="t m0 xc h8 y52 ff2 fs4 fc0 sc0 ls1 ws0">J. R. Soc. Interface (2005) 2, 443–454</div><div class="t m0 xd h8 y53 ff3 fs4 fc0 sc0 ls1 ws0">doi:10.1098/rsif.2005.0058</div><div class="t m0 xe h8 y54 ff2 fs4 fc0 sc0 ls1 ws0">Published online 28 July 2005</div><div class="t m2 x8 h9 y55 ff3 fs5 fc0 sc0 ls1 ws0">†</div><div class="t m0 xf ha y56 ff3 fs6 fc0 sc0 ls1 ws0">jsrsmith@neurotechno.co.uk.</div><div class="t m0 x8 ha y57 ff2 fs6 fc0 sc0 ls1 ws0">Received 17 May 2005</div><div class="t m0 x8 ha y58 ff2 fs6 fc0 sc0 ls1 ws0">Accepted 23 June 2005</div><div class="t m0 x10 h6 y59 ff3 fs3 fc0 sc0 lsf ws0">443 q 2005 The Royal Society</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625aa3d7be9ad24cfaec4b49/bg2.jpg"><div class="t m0 x11 h6 y5a ff3 fs3 fc0 sc0 ls1 ws0">(ECG). In neuroscience, potential applications include</div><div class="t m0 x11 h6 y5b ff3 fs3 fc0 sc0 ls1 ws0">the analysis of datasets as diverse as action potential</div><div class="t m0 x11 h6 y5c ff3 fs3 fc0 sc0 ls1 ws0">spike-trains and functional magnetic resonance imaging</div><div class="t m0 x11 h6 y5d ff3 fs3 fc0 sc0 ls1 ws0">(fMRI) data. Perhaps one of the mos t challenging</div><div class="t m0 x11 h6 y5e ff3 fs3 fc0 sc0 ls1 ws0">signals to analyse in terms of frequency and energ y is</div><div class="t m0 x11 h6 y5f ff3 fs3 fc0 sc0 ls1 ws0">the electroencephalogram (EEG) signal. The complex-</div><div class="t m0 x11 h6 y60 ff3 fs3 fc0 sc0 ls1 ws0">ity of the EEG makes it highly suitable as a test signal</div><div class="t m0 x11 h6 y61 ff3 fs3 fc0 sc0 ls1 ws0">for analysis using LMD. Furthermore, the decompo-</div><div class="t m0 x11 h6 y62 ff3 fs3 fc0 sc0 ls1 ws0">sition of EEG data using LMD into a set of product</div><div class="t m0 x11 h6 y63 ff3 fs3 fc0 sc0 ls1 ws0">functions with associated time-varying frequencies is of</div><div class="t m0 x11 h6 y64 ff3 fs3 fc0 sc0 ls1 ws0">particular interest, because various frequency bands of</div><div class="t m0 x11 h6 y65 ff3 fs3 fc0 sc0 ls1 ws0">the EEG have been correlated with particular cognitive</div><div class="t m0 x11 h6 y66 ff3 fs3 fc0 sc0 ls1 ws0">states ( Basar et al. 1999; Makeig et al. 1999; Kahana</div><div class="t m0 x11 h6 y67 ff2 fs3 fc1 sc0 ls1 ws0">et al. 2001). For this paper, the LMD method was</div><div class="t m0 x11 h6 y68 ff3 fs3 fc0 sc0 ls10 ws0">applied to a set of EEG visual perception data.</div><div class="t m0 x11 h6 y69 ff3 fs3 fc0 sc0 ls1 ws0">A subject viewed a screen showing a running ﬁgure</div><div class="t m0 x11 h6 y6a ff3 fs3 fc0 sc0 ls11 ws0">made up of black dots on a white background.</div><div class="t m0 x11 h6 y6b ff3 fs3 fc0 sc0 ls1 ws0">Depending on the numb er of di stractor dots, the</div><div class="t m0 x11 h6 y6c ff3 fs3 fc0 sc0 ls1 ws0">subject eit her perceived, or did not perceive the running</div><div class="t m0 x11 h6 y6d ff3 fs3 fc0 sc0 ls1 ws0">ﬁgure. Several issues regarding the nature of perception</div><div class="t m0 x11 h6 y6e ff3 fs3 fc0 sc0 ls1 ws0">are addressed using the LMD results. Recent research</div><div class="t m0 x11 h6 y6f ff3 fs3 fc0 sc0 ls1 ws0">has focussed on the role of gamma oscillations in</div><div class="t m0 x11 h6 y70 ff3 fs3 fc0 sc0 ls1 ws0">perception (Desmedt & Tomberg 1994; Tallon-Baudry</div><div class="t m0 x11 h6 y71 ff2 fs3 fc0 sc0 ls1 ws0">et al. 1996, 1997; Rodriguez et al. 1999). In this paper</div><div class="t m0 x11 h6 y72 ff3 fs3 fc0 sc0 ls1 ws0">the general EEG frequency–energy structure of the</div><div class="t m0 x11 h6 y73 ff3 fs3 fc0 sc0 ls1 ws0">response to the experimental stimulus is examined. In</div><div class="t m0 x11 h6 y74 ff3 fs3 fc0 sc0 ls1 ws0">addition, the signiﬁcance of differences in the degree of</div><div class="t m0 x11 h6 y75 ff3 fs3 fc0 sc0 ls1 ws0">instantaneous phase concentration between perception</div><div class="t m0 x11 h6 y76 ff3 fs3 fc0 sc0 ls1 ws0">and no perception trials are assessed statistically using</div><div class="t m0 x11 h6 y77 ff3 fs3 fc0 sc0 ls1 ws0">the ratio of the corresponding circular variances.</div><div class="t m0 x11 h7 y78 ff1 fs3 fc0 sc0 ls1 ws0">2. THE LOCAL MEAN DECOMPOSITION</div><div class="t m0 x11 h6 y79 ff3 fs3 fc0 sc0 ls1 ws0">Essentially the LMD scheme involves progressively</div><div class="t m0 x11 h6 y7a ff3 fs3 fc0 sc0 ls12 ws0">separating a frequency modulated signal from an</div><div class="t m0 x11 h6 y7b ff3 fs3 fc0 sc0 ls1 ws0">amplitude modulated envelope signal. This separation</div><div class="t m0 x11 h6 y7c ff3 fs3 fc0 sc0 ls1 ws0">is achieved by smoothing the original signal, subtract-</div><div class="t m0 x11 h6 y7d ff3 fs3 fc0 sc0 ls1 ws0">ing the smoothed signal from the original signal, and</div><div class="t m0 x11 h6 y7e ff3 fs3 fc0 sc0 ls13 ws0">then amplitude demodulating the result using an</div><div class="t m0 x11 h6 y7f ff3 fs3 fc0 sc0 ls1 ws0">envelope estimate. The envelope estimate and the</div><div class="t m0 x11 h6 y80 ff3 fs3 fc0 sc0 ls14 ws0">smoothed version of the original signal are both</div><div class="t m0 x11 h6 y81 ff3 fs3 fc0 sc0 ls1 ws0">obtained by using moving averaging weighte d by the</div><div class="t m0 x11 h6 y82 ff3 fs3 fc0 sc0 ls15 ws0">time-lapse between the successive extrema of t he</div><div class="t m0 x11 h6 y83 ff3 fs3 fc0 sc0 ls1 ws0">original signal. If the resulting signal does not have a</div><div class="t m0 x11 h6 y84 ff3 fs3 fc0 sc0 ls16 ws0">ﬂat envelope, the procedure is repeated until a</div><div class="t m0 x11 h6 y85 ff3 fs3 fc0 sc0 ls1 ws0">frequency modulated signal with a ﬂat envelope is</div><div class="t m0 x11 h6 y86 ff3 fs3 fc0 sc0 ls1 ws0">obtained. The instantaneous frequency can then be</div><div class="t m0 x11 h6 y87 ff3 fs3 fc0 sc0 ls1 ws0">calculated from the frequency modulated signal. The</div><div class="t m0 x11 h6 y88 ff3 fs3 fc0 sc0 ls17 ws0">corresponding envelope estimates are multiplied</div><div class="t m0 x11 h6 y89 ff3 fs3 fc0 sc0 ls1 ws0">together to form a ﬁnal envelope. This envelope is</div><div class="t m0 x11 h6 y8a ff3 fs3 fc0 sc0 ls1 ws0">then multiplied by the frequency modu lated signal to</div><div class="t m0 x11 h6 y8b ff3 fs3 fc0 sc0 ls1 ws0">form a product function, which is subtracted from the</div><div class="t m0 x11 h6 y8c ff3 fs3 fc0 sc0 ls1 ws0">original sign al. The whole process is then repeated on</div><div class="t m0 x11 h6 y8d ff3 fs3 fc0 sc0 ls1 ws0">the resulting signal, to produce a second product</div><div class="t m0 x11 h6 y8e ff3 fs3 fc0 sc0 ls1 ws0">function, with an a ssociated envelope and instan-</div><div class="t m0 x11 h6 y8f ff3 fs3 fc0 sc0 ls18 ws0">taneousfrequency.Thisdecomposition continues</div><div class="t m0 x11 h6 y90 ff3 fs3 fc0 sc0 ls1 ws0">until the remaining signal contai ns no more oscillations .</div><div class="t m0 x11 h6 y91 ff3 fs3 fc0 sc0 ls1 ws0">The resulting instantaneous frequency and envelope</div><div class="t m0 x11 h6 y92 ff3 fs3 fc0 sc0 ls1a ws0">values can be plotted together in the form of a</div><div class="t m0 x11 h6 y93 ff3 fs3 fc0 sc0 ls1 ws0">demodulated signal tfr.</div><div class="t m0 x12 h6 y94 ff3 fs3 fc0 sc0 ls1 ws0">The LMD of a signal is accomplished by progress-</div><div class="t m0 x11 h6 y95 ff3 fs3 fc0 sc0 ls1 ws0">ively smoothing the signal using moving averaging.</div><div class="t m0 x11 h6 y96 ff3 fs3 fc0 sc0 ls1 ws0">This averaging is weighted by the distance between the</div><div class="t m0 x13 h6 y97 ff3 fs3 fc0 sc0 ls1 ws0">successive extrema of the signal in the following way.</div><div class="t m0 x13 h6 y98 ff3 fs3 fc0 sc0 ls1 ws0">Consider the sample portion of EEG data shown in</div><div class="t m0 x13 h6 y99 ff3 fs3 fc1 sc0 ls1 ws0">ﬁgure 1a. The ﬁrst step of the decomposition involves</div><div class="t m0 x13 h6 y9a ff3 fs3 fc0 sc0 ls1 ws0">calculating the mean of the maximum and minimum</div><div class="t m0 x13 h6 y9b ff3 fs3 fc0 sc0 ls1 ws0">points of each half-wave oscillation of the signal. So the</div><div class="t m0 x13 h6 y9c ff2 fs3 fc0 sc0 ls1 ws0">ith mean value m</div><div class="t m4 x14 hb y9d ff2 fs7 fc0 sc0 ls1 ws0">i</div><div class="t m0 x15 h6 y9e ff3 fs3 fc0 sc0 ls1 ws0">of each two successive extrema n</div><div class="t m4 x16 hb y9d ff2 fs7 fc0 sc0 ls1 ws0">i</div><div class="t m0 x17 h6 y9e ff3 fs3 fc0 sc0 ls1 ws0">and</div><div class="t m0 x13 h5 y9f ff2 fs3 fc0 sc0 ls1 ws0">n</div><div class="t m4 x18 hc ya0 ff2 fs7 fc0 sc0 ls1 ws0">iC1</div><div class="t m0 x19 h6 ya1 ff3 fs3 fc0 sc0 ls1 ws0">is given by</div><div class="t m0 x1a h5 ya2 ff2 fs3 fc0 sc0 ls1 ws0">m</div><div class="t m0 x1b hd ya3 ff2 fs8 fc0 sc0 ls1 ws0">i</div><div class="t m0 x1c he ya4 ff6 fs3 fc0 sc0 ls1 ws0">Z</div><div class="t m0 x1d h5 ya5 ff2 fs3 fc0 sc0 ls1 ws0">n</div><div class="t m0 x1e hd ya6 ff2 fs8 fc0 sc0 ls1 ws0">i</div><div class="t m0 x1f h5 ya5 ff5 fs3 fc0 sc0 ls1 ws0">C n</div><div class="t m0 xe hf ya6 ff2 fs8 fc0 sc0 ls1 ws0">iC1</div><div class="t m0 x20 h6 ya7 ff3 fs3 fc0 sc0 ls1 ws0">2</div><div class="t m0 x21 h6 ya8 ff7 fs3 fc0 sc0 ls1 ws0">: ð2:1Þ</div><div class="t m0 x13 h6 ya9 ff3 fs3 fc0 sc0 ls1 ws0">These local means can be plotted as straight lines</div><div class="t m0 x13 h6 yaa ff3 fs3 fc0 sc0 ls1 ws0">extending between successive extrema (ﬁgure 1a). The</div><div class="t m0 x13 h6 yab ff3 fs3 fc0 sc0 ls1 ws0">local means are then smoothed using moving averaging</div><div class="t m0 x13 h6 yac ff3 fs3 fc0 sc0 ls1 ws0">to form a smoothly varying continuous local mean</div><div class="t m0 x13 h6 yad ff3 fs3 fc0 sc0 ls1 ws0">function m( t) (shown as the pink line in ﬁgure 1a).</div><div class="t m0 x13 h6 yae ff3 fs3 fc0 sc0 ls1 ws0">Details of the smoothing process can be found in the</div><div class="t m0 x13 h6 yaf ff3 fs3 fc0 sc0 ls1 ws0">electronic supplementary material. A corresponding</div><div class="t m0 x13 h6 yb0 ff3 fs3 fc0 sc0 ls1b ws0">envelope estimate can also be derived. The local</div><div class="t m0 x13 h6 yb1 ff3 fs3 fc0 sc0 ls1 ws0">magnitude of each half-wave oscillation is g iven by</div><div class="t m0 x22 h5 yb2 ff2 fs3 fc0 sc0 ls1 ws0">a</div><div class="t m0 x23 hd yb3 ff2 fs8 fc0 sc0 ls1 ws0">i</div><div class="t m0 x24 he yb4 ff6 fs3 fc0 sc0 ls1 ws0">Z</div><div class="t m0 x25 h6 yb5 ff8 fs3 fc0 sc0 ls1 ws0">jn</div><div class="t m0 x26 hd yb6 ff2 fs8 fc0 sc0 ls1 ws0">i</div><div class="t m0 x27 h5 yb5 ff5 fs3 fc0 sc0 ls1 ws0">K n</div><div class="t m0 x28 hf yb6 ff2 fs8 fc0 sc0 ls1 ws0">iC1</div><div class="t m0 x29 h6 yb5 ff8 fs3 fc0 sc0 ls1 ws0">j</div><div class="t m0 x2a h6 yb7 ff3 fs3 fc0 sc0 ls1 ws0">2</div><div class="t m0 x2b h6 yb8 ff7 fs3 fc0 sc0 ls1 ws0">: ð2:2Þ</div><div class="t m0 x13 h6 yb9 ff3 fs3 fc0 sc0 ls1 ws0">The local magnitudes are smoothed in the same way</div><div class="t m0 x13 h6 yba ff3 fs3 fc0 sc0 ls1 ws0">and to the same degree as the local means to form a</div><div class="t m0 x13 h6 ybb ff3 fs3 fc0 sc0 ls1 ws0">smoothly varying continuous envelope function a(t)</div><div class="t m0 x13 h6 ybc ff3 fs3 fc0 sc0 ls1 ws0">(shown in ﬁgure 1b). The LMD of most natural data is</div><div class="t m0 x13 h6 ybd ff3 fs3 fc0 sc0 ls1c ws0">an iterative process. As described above, having</div><div class="t m0 x13 h6 ybe ff3 fs3 fc0 sc0 ls1 ws0">obtained an envelope estimate and a corresponding</div><div class="t m0 x13 h6 ybf ff3 fs3 fc0 sc0 ls1 ws0">mean, the mean is subtracted from the original data,</div><div class="t m0 x13 h6 yc0 ff3 fs3 fc0 sc0 ls1 ws0">and the resulting signal, denoted by h(t), is then div ided</div><div class="t m0 x13 h6 yc1 ff3 fs3 fc0 sc0 ls1d ws0">bytheenvelopeestimate.Theobjectiveofthis</div><div class="t m0 x13 h6 yc2 ff3 fs3 fc0 sc0 ls1 ws0">procedure is to produce a purely frequency modulated</div><div class="t m0 x13 h6 yc3 ff3 fs3 fc0 sc0 ls1 ws0">signal (i.e. a signal with a ﬂat envelope) from which a</div><div class="t m0 x13 h6 yc4 ff3 fs3 fc0 sc0 ls1 ws0">positive instantaneous frequency can be derived. If the</div><div class="t m0 x13 h6 yc5 ff3 fs3 fc0 sc0 ls1 ws0">signal which is produced does not have a ﬂat envelope,</div><div class="t m0 x13 h6 yc6 ff3 fs3 fc0 sc0 ls1 ws0">then the procedure needs to be repeated. The number of</div><div class="t m0 x13 h6 yc7 ff3 fs3 fc0 sc0 ls1 ws0">times the procedure has been repeated (i.e. the number</div><div class="t m0 x13 h6 yc8 ff3 fs3 fc0 sc0 ls1 ws0">of iterations) will be denoted by the subscript q.In</div><div class="t m0 x13 h6 yc9 ff3 fs3 fc0 sc0 ls1 ws0">addition to deriving a frequency modulated signal, a</div><div class="t m0 x13 h6 yca ff3 fs3 fc0 sc0 ls1 ws0">corresponding envelope signal is derived by multiplying</div><div class="t m0 x13 h6 ycb ff3 fs3 fc0 sc0 ls1 ws0">together the successive envelope estimates obtained</div><div class="t m0 x13 h6 ycc ff3 fs3 fc0 sc0 ls1 ws0">during the derivation of the frequency modulated</div><div class="t m0 x13 h6 ycd ff3 fs3 fc0 sc0 ls1 ws0">signal. This ﬁnal envelope signal is then multiplied by</div><div class="t m0 x13 h6 yce ff3 fs3 fc0 sc0 ls1f ws0">the frequency modulated signal to form the ﬁrst</div><div class="t m0 x13 h6 ycf ff3 fs3 fc0 sc0 ls1 ws0">product function. The product function is then sub-</div><div class="t m0 x13 h6 yd0 ff3 fs3 fc0 sc0 ls1 ws0">tracted from the original signal, and the whole of the</div><div class="t m0 x13 h6 yd1 ff3 fs3 fc0 sc0 ls1 ws0">above procedure can then be repea ted on the resulting</div><div class="t m0 x13 h6 yd2 ff3 fs3 fc0 sc0 ls1 ws0">signal to form a second product function, and so on.</div><div class="t m0 x13 h6 yd3 ff3 fs3 fc0 sc0 ls1 ws0">The number of the product function will be denoted by</div><div class="t m0 x13 h6 yd4 ff3 fs3 fc0 sc0 ls1 ws0">the subscript p. The successive local mean functions</div><div class="t m0 x13 h5 yd5 ff2 fs3 fc0 sc0 ls1 ws0">m</div><div class="t m4 x2c hb yd6 ff2 fs7 fc0 sc0 ls1 ws0">pq</div><div class="t m0 x2d h6 yd7 ff3 fs3 fc0 sc0 ls1 ws0">(t), the successive envelope estimates a</div><div class="t m4 x2e hb yd6 ff2 fs7 fc0 sc0 ls1 ws0">pq</div><div class="t m0 x2f h6 yd7 ff3 fs3 fc0 sc0 ls1 ws0">(t), and</div><div class="t m0 x13 h6 yd8 ff3 fs3 fc0 sc0 ls1 ws0">the successive frequency modulated signal estimates</div><div class="t m0 x13 h5 yd9 ff2 fs3 fc0 sc0 ls1 ws0">s</div><div class="t m4 x30 hb yda ff2 fs7 fc0 sc0 ls1 ws0">pq</div><div class="t m0 x31 h6 ydb ff3 fs3 fc0 sc0 ls1 ws0">(t), are each labelled with two subscripts. The ﬁnal</div><div class="t m0 x13 h6 ydc ff3 fs3 fc0 sc0 ls1 ws0">envelope a</div><div class="t m4 x32 hb ydd ff2 fs7 fc0 sc0 ls1 ws0">p</div><div class="t m0 x33 h6 yde ff3 fs3 fc0 sc0 ls1 ws0">(t), the instantaneous phase 4</div><div class="t m4 x34 hb ydd ff2 fs7 fc0 sc0 ls1 ws0">p</div><div class="t m0 x35 h6 yde ff3 fs3 fc0 sc0 ls1 ws0">(t) and the</div><div class="t m0 x13 h6 ydf ff3 fs3 fc0 sc0 ls1 ws0">instantaneous frequency u</div><div class="t m4 x26 hb ye0 ff2 fs7 fc0 sc0 ls1 ws0">p</div><div class="t m0 x27 h6 ye1 ff3 fs3 fc0 sc0 ls1 ws0">(t) will all be labelled with a</div><div class="t m0 x13 h6 ye2 ff3 fs3 fc0 sc0 ls1 ws0">single subscript indicating the product function to</div><div class="t m0 x13 h6 ye3 ff3 fs3 fc0 sc0 ls1 ws0">which they correspond. The product functions, which</div><div class="t m0 x13 h6 ye4 ff3 fs3 fc0 sc0 ls1 ws0">are denoted by PF</div><div class="t m4 x36 hb ye5 ff2 fs7 fc0 sc0 ls1 ws0">p</div><div class="t m0 x37 h6 ye6 ff3 fs3 fc0 sc0 ls1 ws0">(t), also have a sing le subscript.</div><div class="t m0 x38 h6 ye7 ff3 fs3 fc0 sc0 ls1 ws0">So the initial envelope estimate shown in ﬁgure 1b is</div><div class="t m0 x13 h6 ye8 ff3 fs3 fc0 sc0 ls1 ws0">denoted by a</div><div class="t m4 x39 hc ye9 ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 x3a h6 yea ff3 fs3 fc0 sc0 ls1 ws0">(t), and the initial mean shown in</div><div class="t m0 x13 h6 yeb ff3 fs3 fc1 sc0 ls1 ws0">ﬁgure 1a is denoted by m</div><div class="t m4 x3b hc yec ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 x26 h6 yed ff3 fs3 fc0 sc0 ls1 ws0">(t). m</div><div class="t m4 x3c hc yec ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 x3d h6 yed ff3 fs3 fc0 sc0 ls1 ws0">(t) is subtr acted from</div><div class="t m0 x13 h6 yee ff3 fs3 fc0 sc0 ls1 ws0">the ori ginal data x (t),</div><div class="t m0 x3e h5 yef ff2 fs3 fc0 sc0 ls1 ws0">h</div><div class="t m0 x3f hf yf0 ff3 fs8 fc0 sc0 ls1 ws0">11</div><div class="t m0 x22 h6 yf1 ff8 fs3 fc0 sc0 ls1 ws0">ðtÞ Z xðtÞ K m</div><div class="t m0 x3d hf yf0 ff3 fs8 fc0 sc0 ls20 ws0">11</div><div class="t m0 x40 h6 yf1 ff8 fs3 fc0 sc0 ls1 ws0">ðtÞ: ð2:3Þ</div><div class="t m0 x11 h6 yf2 ff3 fs3 fc0 sc0 ls1 ws0">444 The local mean decomposition J. 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<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625aa3d7be9ad24cfaec4b49/bg3.jpg"><div class="t m0 x8 h5 yf3 ff2 fs3 fc0 sc0 ls1 ws0">h</div><div class="t m4 x41 hc yf4 ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 x42 h6 yf3 ff3 fs3 fc0 sc0 ls1 ws0">(t) is shown in ﬁgure 2b. h</div><div class="t m4 x43 hc yf4 ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 x44 h6 yf3 ff3 fs3 fc0 sc0 ls1 ws0">(t) is then amplitude</div><div class="t m0 x8 h6 yf5 ff3 fs3 fc0 sc0 ls1 ws0">demodulated by dividing it by a</div><div class="t m4 x45 hc yf6 ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 x46 h6 yf7 ff3 fs3 fc0 sc0 ls1 ws0">(t),</div><div class="t m0 x47 h5 yf8 ff2 fs3 fc0 sc0 ls1 ws0">s</div><div class="t m0 x48 hf yf9 ff3 fs8 fc0 sc0 ls20 ws0">11</div><div class="t m0 x49 h6 yfa ff8 fs3 fc0 sc0 ls1 ws0">ðtÞ Z</div><div class="t m0 x4a h5 yfb ff2 fs3 fc0 sc0 ls1 ws0">h</div><div class="t m0 x4b hf yfc ff3 fs8 fc0 sc0 ls1 ws0">11</div><div class="t m0 x43 h6 yfb ff8 fs3 fc0 sc0 ls1 ws0">ðtÞ</div><div class="t m0 x4a h5 yfd ff2 fs3 fc0 sc0 ls1 ws0">a</div><div class="t m0 x4b hf yf0 ff3 fs8 fc0 sc0 ls1 ws0">11</div><div class="t m0 x43 h6 yf1 ff8 fs3 fc0 sc0 ls1 ws0">ðtÞ</div><div class="t m0 x4c h6 yfe ff7 fs3 fc0 sc0 ls1 ws0">: ð2:4Þ</div><div class="t m0 xa h5 yff ff2 fs3 fc0 sc0 ls1 ws0">s</div><div class="t m4 x4d hc yf4 ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 xb h6 yf3 ff3 fs3 fc0 sc0 ls1 ws0">(t) is shown in ﬁgure 2a. The envelope a</div><div class="t m4 x4e hc yf4 ff3 fs7 fc0 sc0 ls1 ws0">12</div><div class="t m0 x4f h6 yf3 ff3 fs3 fc0 sc0 ls1 ws0">(t)of</div><div class="t m0 xa h5 y100 ff2 fs3 fc0 sc0 ls1 ws0">s</div><div class="t m4 x4d hc y101 ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 xb h6 y102 ff3 fs3 fc0 sc0 ls1 ws0">(t) can then be calculated. If a</div><div class="t m4 x40 hc y101 ff3 fs7 fc0 sc0 ls1 ws0">12</div><div class="t m0 x50 h6 y102 ff3 fs3 fc0 sc0 ls1 ws0">(t)s1, the procedure</div><div class="t m0 xa h6 y103 ff3 fs3 fc0 sc0 ls1 ws0">needs to be repeated for s</div><div class="t m4 x51 hc y104 ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 xd h6 y105 ff3 fs3 fc0 sc0 ls1 ws0">(t). A smoothed local</div><div class="t m0 xa h6 y106 ff3 fs3 fc0 sc0 ls1 ws0">mean m</div><div class="t m4 x33 hc y107 ff3 fs7 fc0 sc0 ls1 ws0">12</div><div class="t m0 x52 h6 y108 ff3 fs3 fc0 sc0 ls1 ws0">(t) is calculated for s</div><div class="t m4 x53 hc y107 ff3 fs7 fc0 sc0 ls1 ws0">11</div><div class="t m0 x54 h6 y108 ff3 fs3 fc0 sc0 ls1 ws0">(t), subtracted from</div><div class="t m0 xa h5 y109 ff2 fs3 fc0 sc0 ls1 ws0">s</div><div class="t m4 x4d hc y10a ff3 fs7 fc0 sc0 ls22 ws0">11</div><div class="t m0 xb h6 y10b ff3 fs3 fc0 sc0 ls1 ws0">(t), and the resulting function amplitude</div><div class="t m0 x8 h8 y10c ff3 fs4 fc0 sc0 ls1 ws0">Figure 2. The initial frequency modulated signal, the initial product function and the initial envelope estimate, the ﬁnal frequency</div><div class="t m0 x8 h8 y10d ff3 fs4 fc0 sc0 ls1 ws0">modulated signal and the ﬁnal product function and the associated envelope. (a) Initial estimate of the ﬁrst frequency modulated</div><div class="t m0 x8 h8 y10e ff3 fs4 fc0 sc0 ls1 ws0">signal. (b) Initial product function estimate obtained by subtracting the smoothed local mean shown in ﬁgure 1a from the EEG data.</div><div class="t m0 x8 h8 y10f ff3 fs4 fc0 sc0 ls1 ws0">The corresponding envelope estimate (from ﬁgure 1b) is shown as the green line. ( c) Final version of the ﬁrst frequency modulated</div><div class="t m0 x8 h8 y110 ff3 fs4 fc0 sc0 ls1 ws0">signal. (d ) Final version of the ﬁrst product function PF1, and its associated envelope.</div><div class="t m0 x8 h8 y111 ff3 fs4 fc0 sc0 ls1 ws0">Figure 1. Sample EEG data, local means, smoothed mean, local magnitudes and smoothed magnitude. (a) Sample portion of EEG</div><div class="t m0 x8 h8 y112 ff3 fs4 fc0 sc0 ls1 ws0">data as the thin solid line. The local means are shown as straight lines extending between successive extrema. The smoothed local</div><div class="t m0 x8 h8 y113 ff3 fs4 fc0 sc0 ls1 ws0">mean is shown in pink. The duration of the visual stimulus is shown by the black line extending from 0 to 0.5 s. (b) Local magnitudes</div><div class="t m0 x8 h8 y114 ff3 fs4 fc0 sc0 ls1 ws0">calculated from the same portion of EEG data. The local magnitudes are smoothed in the same way and to the same degree as the</div><div class="t m0 x8 h8 y115 ff3 fs4 fc0 sc0 ls1 ws0">local means to form a smoothly varying local magnitude function (the initial envelope estimate) shown as the green line.</div><div class="t m0 x55 h6 y116 ff2 fs3 fc0 sc0 ls1 ws0">The local mean decomposition J. S. Smith 445</div><div class="t m0 x8 ha y59 ff2 fs6 fc0 sc0 ls1 ws0">J. R. Soc. Interface (2005)</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>