NA-MEMD Is Actually What You Need for Computational Pulse Analysis

Time-frequency analysis is of necessity for wrist pulse signal due to its complexity, among which, empirical mode decomposition (EMD) algorithm and its improved noise-assisted versions (such as ensemble EMD, noise-assisted multivariate EMD (NA-MEMD) and very recently median EMD) are deemed to be the most representative ones. In this study, we provide an in-depth evaluation of these well-established noise-assisted EMD algorithms in computational pulse analysis for the first time. In particular, we compare the performance of the different algorithms systematically and quantitatively based on objective quantitative criteria: number and central frequency of intrinsic mode function (IMF) components, total orthogonality index and mode mixing. Rather than using synthetic signals with visual inspection in most existing literature, the wrist pulse signals used in the evaluation are real recorded samples acquired from both healthy and patient subjects. Through extensive experiments, we found that: 1) Advanced EMD algorithm that has the best performance in other areas may not be the most suitable method for pulse signal analysis, which indicates its high dependence on the type of analyzed signal; 2) Adding noise can improve algorithm performance significantly, but tends to produce physiologically irrelevant components, which however are usually neglected throughout the intelligent pulse diagnosis literature. Therefore, excluding redundant components before extracting features is expected to improve performance further. Together, currently NA-MEMD achieves a better performance consistently, potential to become a powerful tool for computational pulse analysis, but itself have not been applied in wrist pulse analysis before. We believe our works can bring up a new perspective to application of EMD-like algorithms in computational pulse analysis/diagnosis with effective information and guidance. Additionally, considering the similarity between physiological signals, especially such as photoplethysmogram/electrocardiogram, our research can be extended to wearable health monitoring technologies, including smart watches and fitness trackers, and their potential future applications, such as in heart rate estimation and evaluate various cardiovascular-related diseases. The present study underscored the necessity of evaluation noise-assisted EMDs or other adaptive decomposition algorithms based on real recorded signals with more objective measures. Especially, caution the possible redundant components that are introduced.

INDEX TERMS Computational pulse analysis, biomedical signal processing, noise-assisted EMD, objective evaluation.
The associate editor coordinating the review of this manuscript and approving it for publication was Wenbing Zhao . 28 For several thousands of years, pulse diagnosis has been one 29 of the most popular diagnostic methods in traditional chinese 30 medicine (TCM) community because of non-invasiveness 31 [10], [11] and cepstral analysis technique [12]), maybe ineffi-48 cient, necessitating a more advanced time-frequency analysis. 49 For decades, many efforts have been devoted to develop 50 time-frequency analysis methodologies (mainly short-time 51 fourier transform [13], [14], [15] and wavelet (packet) trans-52 form [16], [17], [18], etc.) to standardize and quantify wrist 53 pulse analysis. Despite their success, each of these-mentioned 54 methods still suffers from some inherent deficiencies. For 55 example, wavelet (packet) transform is signal non-adaptive 56 that usually require well-chosen prior kernels or basis func-57 tions [19], whereas all these methods cannot achieve arbitrary principle [20]. 61 Concurrently, on the other hand, an adaptive decomposi-62 tion technique called empirical mode decomposition (EMD) 63 algorithms [21], which overcome the limitations of the 64 traditional time-frequency methods, has attracted consid-65 erable attention in the past decade. Since its inception, 66 EMD has demonstrated its capabilities in many application 67 areas, including biomedical fields related to our research, 68 such as electroencephalogram (EEG) [22], electrocardiogram 69 (ECG) [23]. However, unlike ECG, application of EMD algo-70 rithms to wrist pulse signals are relatively lagged behind. 71 To our best knowledge, the earliest research can be traced 72 back to 2006, when Sun et al. [24] made the first attempt to 73 use the vanilla EMD algorithm to analyze the marginal spec-74 trum of pulse signals of normal people and patients with coro-75 nary heart disease, demonstating potential broad prospects 76 in pulse signal processing. Subsequently, studies have been 77 further advanced to employ the vanilla EMD for feature 78 extraction to better distinguish healthy subjects from patients 79 with certain diseases, such as hypertension [25], nephritis and 80 cholecystitis [17] and coronary heart disease [26], [27], [28]. 81 However, the vanilla EMD method still have to face some 82 problems, such as interpolation choice and noise sensitivity, 83 especially prone to mode mixing, which isn't uncommon in 84 practical recorded signals. To address these issues, a fam- 85 ily of noise-assisted EMD methods including the ensemble 86 EMD (EEMD) [29], the noise-assisted multivariate EMD 87 (NA-MEMD) [30], [31] and the very recently median EMD 88 (MEMD) [32], have shown appealing results in various fields. 89 For computational pulse analysis/diagnosis, one natural ques-90 tion arise: which of these well-established noise-assisted 91 EMD algorithms is the most suitable, especially considering 92 that aforementioned noise-assisted methods have not been 93 fully explored in the TCM community.

94
Based on this motivation, a comprehensive and systematic 95 understanding of vanilla EMD and three improved versions 96 named EEMD, MEMD and NA-MEMD, for wrist pulse 97 signal analysis, is presented in this paper. To be specific, 98 rather than using artificial signals with visual inspection in 99 most existing literature, we evaluate the performance of EMD 100 algorithms on large numbers of real pulse signals, which 101 are acquired from patient and health individuals. Moreover, 102 multiple quantitative measures such as number and central 103 frequency of intrinsic mode function (IMF) components, 104 total orthogonality index and mode mixing are employed to 105 obtain some reliable findings, guiding objectively to select 106 the appropriate EMD algorithm for computational pulse anal-107 ysis/diagnosis in TCM community.

108
The main contributions of this paper are summarized as 109 follows: 110 1) The performance of widely-used EMD algorithms are 111 compared systematically and quantitatively with various 112 measures on real pulse signals. For algorithm selection 113 in practical pulse analysis, some valuable conclusions 114 drawn from the results are available.

115
2) Currently, NA-MEMD shows a remarkable performance 116 improvement and is preferred over the others, which are 117 suitable otherwise but not for pulse signals. However, 118 there is still a gap with the ideal condition expected by 119 IMF, indicating these deficiencys of vanilla EMD algo-120 rithm can be reduced but not be avoided, at least for now. 121 3) Redundant components introduced in noise-assisted 122 EMD algorithms to improve performance should be 123 devoted more attention, rather than being frequently 124 unnoticed. This issue will likely result in serious per-125 formance degradation in classification diagnosis when 126 some IMF features involving no real signal information 127 but noise are used as features.

128
The remainder of this paper is structured as follows. Sig-129 nal acquisition experiment as well as signal pre-processing 130 are presented In Section II. Section III briefly review math-131 ematical theory of evaluated EMD algorithms, followed 132 by introducing four quantitative performance indicators in 133 Section IV. Section V evaluates the performance of pulse 134 decomposition with EMD algorithms and in terms of indi-135 cators. Results are discussed in Section VI and conclusions 136 are drawn in Section VII.  As a weak physiological signal, wrist pulse can be easily con-174 taminated by various kinds of interference, including power 175 frequency disturbances, amplitude oscillation caused breath-176 ing, muscle contraction and limb vibration [6], [33]. Previous 177 studies have shown that the wrist pulse signal is usually 178 located at low frequencies. 99% of the spectrum energy of 179 the normal signal is concentrated in 0-20 Hz whereas the 180 frequency range of pulse signal under abnormal condition is 181 higher but still not over 40 Hz [7]. Moreover, information 182 below 1 Hz can also be discarded, due to some uncontrol-183 lable movements of the subject's arm. For the preprocessing, 184 we follow the denoising and baseline drift correction methods 185 in [33]. Specifically, we first filter out the baseline drift and 186 the 50 Hz-frequency interference through a zero-phase shift 187 bandpass filter with a bandwidth of 1 Hz to 40 Hz, and then 188 remove the baseline wander by wavelet-based cascaded adap-189 tive filter [34]. After pre-processing, some pulse signals could 190 be further excluded by visual inspection due to technical 191 artifact.

192
Two raw samples and their preprocessed waveforms of 193 a typical healthy and patient subject are shown in Fig (2). 194 As illustrated, the wrist pulse signals in healthy conditions are 195 observed to have regular and smooth morphologies, whereas 196 in abnormal health conditions the pulse signals become irreg-197 ular, especially in the falling segment of the pulse. In practice, 198 the quantification of these irregularities can be helpful in 199 correlating with abnormal health conditions.

201
In this part, we will briefly review the EMD algorithm as 202 well as the improved noise-assisted versions of EMD used in 203 the study: EEMD, NA-MEMD and MEMD. We recommend 204 readers to see corresponding literatures and references for the 205 details.

206
Algorithm 1 Algorithm of EMD 1: Indentify all the locations of local extrema (both maxima and minima) of the input signal x(t). 2: Interpolate between all the minima (cf. maxima) to construct the lower (cf. upper) envelope e min (t) (cf. e max (t)).
3: Compute the local mean of the envelopes c(t) = (e min (t) + e max (t))/2 and subtract it from the signal to get the ''modulated oscillation'' and go to Step 1. 5: Subtract the derived IMF from the variable x(t) so that x(t) := x(t) − IMF m and repeat the above described process. 6: Stop the sifting process when the residual are monotonic-the trend r(t) and no longer IMF can be extracted.

207
A core innovative in vanilla EMD algorithm is introducing the 208 concept of so-called IMF function, which lend themselves to 209 conveying physically meaningful information with classical 210 VOLUME 10, 2022 FIGURE 2. Pulse waveform and its filtered results after pre-processing from the left wrist of two typical healthy (a) and patient (b) acquired through the pressure sensor. The top of each subgraph shows the original model and the bottom shows the processed waveform. Note that the wrist pulse signals in healthy conditions are observed to be more regular than that of abnormal health conditions.
Hilbert transform. Specifically, the procedure used to extract 211 an IMF from a signal can be described in Algorithms (1).

213
EEMD is the first noise-assisted method to enhance sift-214 ing [29], which is based on investigations of the statistical The key idea of NA-MEMD is to create an ''composite'' 228 space instead of directly adding noise to n-channel multivari-229 ate data. This space is a (n+l) -dimensional and consist of two 230 parts, one is n-dimensional signal subspace, the other is an 231 adjacent subspace of l-independent WGN realizations. With 232 the advantage of the filterbank property of multivariate EMD 233 for WGN, the subsequent decomposition produces (n + l) -234 variate coherent IMFs By discarding the l channels pertaining 235 to the noise subspace, the n-variate IMFs can be extracted 236 from the (n + l)-variate IMFs, which are corresponding to 237 the original signal. 1 Because of disjoint nature of the signal 238 and noise subspaces, residual noise and mode mixing can also 239 be reduced in the NA-MEMD. The corresponding specific 240 process of NA-MEMD is as follows in Algorithms (3). Algorithm 3 Algorithm of NA-MEMD 1: Creates l(≥ 1)-channel white Gaussian noise time series with the same length as that of n-channel input x(t) and add the generated noise to produce a new p (= l + n)dimensional signal y(t). 2: Create a suitable set of direction vector {X θ k } K k=1 on the (p−1) sphere with the aid of a sampling scheme based on the Hammersley sequence and Calculate the projections

IV. EVALUATION METRICS
the stop criteria, apply the above procedure to y(t) − c(t), else repeat for d(t). 6: Choose only the extracted IMFs corresponding to the input signal x(t) from the resulting (n + l)-variate IMFs, and discard the IMFs associated with the noise channels.
where f c2i and f c8i are central frequencies where 20% and 80% 288 of the energy of ith IMF (IMF i (t)) are reached, respectively. 289

290
In this section, the performance of EMD algorithms men-291 tioned in Section III are evaluated by using real pulse signals 292 and metrics described in Section IV. Followed by parameter 293 settings, the performance of the vanilla EMD is first evaluated 294 as a baseline for contrast purpose. Thereafter, for each noise-295 assisted EMD, five runs with pre-defined noise characteris-296 tics are performed to get statistically reliable performance 297 results. Since the degree of non-linearity is reported to be dif-298 ferent among heart disease group and healthy group in [9], for 299 better comparison, the results of healthy and patient groups 300 are shown individually.

302
Although there is no general principles for selecting EMD 303 parameters, some common instructions and rule of thumb can 304 be followed as described in [29], [37]. All possible combina-305 tions of the following parameters are listed as follows. shows that for all groups, MM is almost more 345 than 30% with a very large variance. It is observed that MM of 346 individuals is unpredictable and can be 0 or quite large. This 347 phenomenon demonstrates the instability of vanilla EMD.

348
From evaluation results in Figure 3, the issue of vanilla 349 EMD can be found, that is it can not guarante orthogonality, 350 which leads to a large mode confusion and some redundancy. 351 This is exactly the deficiency that following noise-assist ed 352 EMDs are intended to improve. Number of IMFs decomposed with noise-assisted EMDs 356 is summarized in Figure 4. It is clear that regardless of 357 noise-assisted EMDs used, more IMF componens are decom-358 posed than that using vanilla EMD.

359
More specifically, EEMD produces the most IMFs 360 (9 ∼ 10) while NA-MEMD provides the fewest IMFs 361 (7 ∼ 8), and MEMD is in the middle (8 ∼ 9). Although 362 number of IMFs difference among the three EMDs (about 363 2 ∼ 3) is not significant, NA-MEMD outperforms the 364 93986 VOLUME 10, 2022  For clarity, only the first 6 central frequency of IMF compo-375 nents is depicted in Figure 5. It is surprised that for EEMD 376 and MEMD, f c of the first IMF component is quite large and 377 beyond the effective frequency range. However, it can be seen 378 that the variance is also relatively large. It implies that the 379 first component is not always redundant, which should be 380 carefully considered in practice. Therefore, the effective pulse 381 signal is mainly concentrated in IMFs 2 ∼ 5. Considering NI 382   It appeares that f c of the first 5 IMFs using NA-MEMD 394 are all within the valid frequency range (see Figure 5(c) 395 and Figure 5(f)). Although f c of IMF 1 is much lower than 396 that of EEMD and MEMD, it is actually considered to be  condition for complete orthogonality, there still has much 450 room to improve the performance further. On the other hand, 451 the performance of different proposed algorithms is usually 452 only demonstrated by synthesis signals or with visual inspec-453 tion for signals in other fields. Our experimental results on 454 wrist pulse signals suggests that it will be more convincing for 455 other physiological signal analysis by using more real-world 456 signals and objective indicators in practice.

457
Secondly, introducing noise into EMD can indeed improve 458 performance on many metrics, but one of serious side effects 459 is introducing additional redundant components. To the best 460 of our knowledge, in TCM community, a lot of effort have 461 been made to the improve the performance by assisted 462 noise, but seldom mentioned the redundancy caused by those 463 assisted noise. Physiologically irrelevant redundancy mainly 464 comes from intrinsic activities such as residual respiration 465 or vasomotion components after removing interference for 466 vanilla EMD, and from residual noise for noise-assisted 467 EMDs. In terms of redundancy, vanilla EMD produces at 468 most 2 redundancies but it is up to 6 for noise-assisted EMDs. 469 In this aspect, the closest to vanilla EMD is NA-MEMD 470 with a redundancy of 3, because the noise and the signal 471 are combined in a channel way rather than being directly 472 aggregated. Actually proponents of NA-MEMD has pointed 473 out the fact that adding noise directly to the data could cause 474 residual noise to remain in IMFs [31]. Nowadays, EMD 475 algorithms are mostly used as signal decomposition methods 476 for feature extraction, and thus careful exclusion of redundant 477 components before extraction is expected to further improve 478 classification accuracy.

479
Finally, although this present study has elaborately pre-480 sented the comparative study of typical EMDs for pulse 481 signals, there are still some issues to make clear or worthy 482 of further in-depth research. 483 • In addition to the parameters specified in Section V-A, 484 several factors of the EMD algorithm itself, such as 485 interpolation and end effects, also contribute to the final 486 signal decomposition. Specialized fine-tuning of associ-487 ated parameters could be very useful.