Quickly Convert Photoplethysmography to Electrocardiogram Signals by a Banded Kernel Ensemble Learning Method for Heart Diseases Detection

Electrocardiography (ECG) is generally deemed the golden standard for diagnosing cardiovascular diseases and photoplethysmography (PPG) is unobtrusive, low-cost, and convenient for continuous monitoring. However, PPG contains insufficient information to diagnose diseases. In this study, we propose a novel method to accurately convert PPG to ECG. The banded kernel ensemble method converts a low-quality source (PPG) to a high-quality destination (ECG). Unlike neural network solutions, our algorithm requires no computation burden in the conversion task after a trained model is obtained. The proposed algorithm is then tested on a publicly available MIMIC III database. Our prediction shows excellent accuracy in the validation dataset. It offers the testing performance of under 0.314 and above 0.55 in rrmse (relative root mean squared error) and KGE (Kling–Gupta efficiency), respectively, under the scenarios of three prevalent heart diseases. The reconstructed ECG can be further used to perform heart disease detection and we obtained an average correctness rate of 81%. Our method can help a large population of high-risk, believed-healthy persons to walk into doctors’ offices before the situation becomes irreversible.


I. INTRODUCTION
EElectrocardiography (ECG) is the top preferred option for diagnosing cardiovascular diseases. This knowledge is also an essential part of the training of every medical student. However, detecting minute electrical currents on skin The associate editor coordinating the review of this manuscript and approving it for publication was Gyorgy Eigner . surfaces requires relatively expensive equipment operated by a professional. Therefore, monitoring cardiac activities for a healthy person by maintaining the wiring on for an extended period in a daily living space is difficult. Alternatively, noncontact photoplethysmography (PPG) is considered unobtrusive, low-cost, and convenient for continuous monitoring. PPGs are usually deemed low quality with random drifts. Consequently, the majority of PPG applications can only be VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ applied to simple tasks, such as heart rate counting, thereby producing low values in clinical applications. In this study, pervasive PPG signals, such as the ones from wristwatches or PPG imaging, are utilized to synthesize corresponding ECG signals (Fig. 1). Differing from those wiring devices, such as AppleWatch R [1] or AliveCor R [2], our solution does not need to prepare a clean contact or using both hands to acquire weak electric signals. Therefore, suspicious cardiology symptoms can be detected unobtrusively and ubiquitous, and the consequent early hospitalization can save many lives. The framework addresses the ill-posed inverse problem by applying a hybrid learning technique over a multi-band time-frequency domain with successive ridge domination (SRD) and generative pulse locking (GPL) algorithms.
We concentrate on PPGs because the device is low-cost and ubiquitous. PPG has a wide range of applications, from human physiology to psychology, because of its low-cost, ubiquitous, and unobtrusive nature. For example, PPG has been applied to vital sign monitoring [3], [4]. In professional sports training, simple PPG devices are apt to record history and optimize performance in the field [3]. In psychology, PPG can help humans outside laboratories understand stress responses and emotional communication [3], [5]. Despite the quality of the devices, recent advancement demonstrates a promising result in the accurate measurement of specific physiological parameters, e.g., estimating blood pressure from PPG based on specific haemodynamics [6].
Among PPG applications, medical diagnosis is scant, whereas the commonly used device is ECG [7], [8]. However, ECG devices require a good touch of electrodes on human skin, causing skin irritation and discomfort during long-term usage. As such, most consumer-grade products only perform occasional measures instead of continuous monitoring. For example, the wrist-worn Apple Watch can only collect signals when fingers touch the wristwatch [1]. Another consumer product, KardiaMobile, completely overlaps the market positioning with Apple Watch [2]. Other laboratory-grade products, such as a single-use patch with a 7-day lifespan [9] and a textile-based printed circuit in cloth [10], probably still need to overcome certain usability problems at the current stage.
This study offers a unique contribution that fills the current research gaps described in the existing literature. First, our hybrid band method can recover ECGs reliably at low bands without requiring a high-quality PPG input. This feature is essential because signals collected in a living space cannot guarantee good acquisition quality. Additionally, ECG information can be reserved at high-bands without affecting the stability of the recovery. The learning ensemble can adapt to different situations while maintaining the fundamental characteristic of ECGs because various heart activities reflect diversified PPG and ECG patterns. This feature fills a gap of not being capable of diagnosing PPG inputs. Therefore, doctors can automatically assess the heart condition of ordinary people through long-term surveillance in daily life.

II. RELATED WORKS
Research gaps exist in the existing literature. Although illposed problems have been studied for a long time, a large amount of active research keeps shedding new light in the field. Recently, approaches of optimization on l 1 have proven effective. Two strategies are usually applied to the inverse mapping problem [15]. Certain implicit constraints, such as smoothness or sparseness, can be imposed. Dimension reduction, such as the kernel-based method, can be applied. If the back-projection problem satisfies specific sparsity properties, the optimization, therefore, receives additional constraints [23]. Sparse domain techniques have been widely used in heart rate activity monitoring [32]. The space that possesses good sparsity is the one with timefrequency representation [33], [34]. Optimization performed on such multi-resolution space is considered effective [16].
The conversion from PPGs to ECGs has a solid theoretic ground in modern physiology. Studies have supported the relationship between PPG and ECG in electro-mechanics and blood fluid dynamics in heart activities. The transmembrane potential distribution can be constrained by a diffusionreaction model from cellular activation dynamics [11]. The electric current constraints from the ionic behavior of the individual cell, such as in [12], impose additional computational challenges due to millions of myocytes. Additionally, additional constraints also introduce more unknown parameters and the degree of freedom remains still high. However, a numerical problem exists because ECG signals contain more details than PPG signals do. To deal with the problem that physiological models and patient-specific body surface electrical data are always corrupted by noise and organ geometries may be constructed incorrectly. The inverse mapping problem reversely describes ECG, haemodynamics, and electro-fluid-mechanics in differential equations [11], [13], [14]. Therefore, reconstructing ECGs without involving differential equations is challenging.
Solving the ill-posed inverse problem that maps the surface ECG to the transmembrane potentials on the surface of the heart, the key adequately constrains the solution space to obtain a unique solution. Unfortunately, most regulation methods in the ill-posed situation can only condition the numerical difficulty from a mathematical point of view. However, the problem of multiple solutions must be addressed by restoring possible missing constraints. Continuous basis pursuit has been reported effective for sparse spike reconstruction in the sense of super-resolution [24]. In addition to sparsity constraints, other constraints, such as total variation with positivity constraint [25], smoothness on low-dimensional manifolds [26], constraints on the union of all subspaces [27], and block-sparsity constraints [28], have also proven effective.
The first attempt at the PPG-related problems starts from an easy one that reduces ECGs to PPGs, which only involves mapping from high to low dimensions [35]. Recently, many studies have reported success for the challenged ill-posed  inverse problem. For the back projection from PPGs to ECGs, an ordinary regulation on l 2 can only solve a part of numerical difficulty [17], [18]. Many studies resort to neural networks to bypass mathematical details [19]- [22]. The neural network approaches utilize a latent space in the auto-encoder or generative adversarial network framework, which is akin to the concept of partial least square. The key to these approaches is a proper design of the loss function, but they do not take advantage of the sparsity of ECG signals in l 1 space. Our study further improves the numerical ill-posedness and the wide tolerance of the quality of source PPGs.
A previous study takes a statistical approach but they need to know the subject's personal information in the regression and the input and output should be properly aligned; therefore pulse timing information is lost [17]. In another study, a compressive sensing technique is employed to improve the signal reconstruction [18]. However, the study has yet to clarify how discrete cosine transform and dictionary decompositionbased methods can resist the non-coherency distortion on PPGs. The recent neural network studies have achieved excellent progress, but they have yet to discover how the black box algorithm can handle low-quality PPGs and signals with disease patterns [19]- [22]. The massive computation of deep computation is also difficult to implement on consumer devices, such as smartphones [36].
Long-memory and fractal properties, such as coherency, usually exist between two time-series, particularly in the heart signals. That is, the coherency reflects the similarity between the semi-periodic heart activities. However, the coherency properties may change even within the proximal scale. Therefore, we should develop an elaborate method to estimate the model and ensure that the realizations of a stochastic process can correctly express the model. Therefore, a functional time series is a mathematical structure where the entire function becomes a value in Hilbert space [37]. We vectorize the signals into a functional time series such that the reconstructed signals depend not only on the immediate successors but also on the repetitive patterns of the previously reconstructed heartbeats. A functional time series possesses the advantage of modeling the long-range correlation in a semi-periodic wave. Therefore, the evolution of the continuous-time stochastic process of heart signals can be easily parametrized in a representative model. Functional time series have long been investigated for the coherency property. A subgroup covariance structure can be determined in functional auto-regression [29]. The coherency structure is proven effective in improving the performance of prediction [30]. The functional correspondence can also be effectively obtained through a matrix completion [31]. Therefore, our method also takes advantage of the coherency structure for the optimization of functional time series and swiftly reconstructs a high fidelity ECG. By upgrading the PPG to ECG, the automatic diagnosis of heart diseases can be performed in the consequence.

III. BANDED KERNEL ENSEMBLE METHOD
PPG signal s(t) and ECG signal z(t) originate from the same driving source but transmit through different media. Therefore, they should be similar in certain subspaces but dissimilar in other subspaces. The signal z(t) contains rich information, including smooth and non-smooth components. The versatile similarities and ill-posedness in each subspace VOLUME 10, 2022 should be treated differently. In certain smooth subspaces, an ordinary pseudo-inverse in l 2 is sufficient to reconstruct the signal. However, some components in ECG signals are difficult to handle and some extremely non-smooth ones must take special treatment in the destination manifolds.
A typical technique to solve an ill-posed problem is adding certain constraints with prior knowledge of the sparsity or smoothness in the original signals. However, no single set of constraints can perfectly solve the ill-posed problem at once. Appropriate methods should be applied to each subspace to pursue the most stable transformation results with different constraint sets. We need to decompose the signals to individual components in a particular subspace that inhere the prior knowledge. Multiple methods in the ensemble method are then applied to maximize and validate the accuracy of prediction. We develop an ensemble of methods in a series of operations ( Fig. 2). Our algorithm will find the best learner for each subspace in the inverse mapping.
In Fig. 2, with the original signal pair (z(t), s(t)) for ECG and PPG, we first transform the waves to banded functional t] for B bands, K kernels, M functional chunks with bandwidth n, respectively for using the kernel function k . The subscription b in z b (t) represents that the original signal z(t) has been separated into b = 1, . . . , B bands. The kernel operator for predictor functionals ·, · is defined in the Hilbert space such that the informative projection is prominent. We choose the kernel function k = k (t, t ) = k −1 e −γ ||(t−t )/k|| 2 for t ∈ [t,t] with a parameter γ such that the operator is given as We apply the functional kernel estimator (1) in accordance with previously described method to determine the functional bandwidth n [38].
t ] for B bands, K kernels, M functional chunks with bandwidth n. All the basic process unit X b,i ,k will send to each learner in the ensemble and produce Y l i for learner l = 1, . . . , L.
Let the process X b,i,k (t j ) satisfies a Hilbert-valued autoregressive process The sampling (x d ) d=1,...,N d is the realization sequence of a discrete-time stochastic process X b,i,k (t j ) j=1,...,N t . The estimationm n (x d , t j ) in Eq. (1) can be deemed a functional generalization of the Nadaraya-Watson regression estimator at any point x d and time t j for d = 1, . . . , N d and j = 1, . . . , N t with sampling sizes N d and N t , which gives the preceding data points more weight. (1) To determine the bandwidth n, we perform a grid search such that the square error At the first stage prediction, the back projection X → Y problem reduces to a standard form (2) that the parametric f (·) and the covariance matrix are estimated simultaneously. where the functional pair (Y , X ) represent the collection of Y b,i (t) and X b,i,k (t) for the b-th band and k-th kernel within the point i of frame t. The predicted functionals from each band at the second stage will be aggregated together into a single one through an adaptive band-boosting step.

A. DYNAMIC BAND SEPARATION AT THE FIRST PREDICTION STAGE
In the detailed operation at the first stage, as shown in Fig. 3, functionals have been separated into bands for a suitable prediction algorithm. The basic process unit X b,i,k has been sent to the learners in the stage 1 prediction process. The proposed successive ridge domination (SRD) splits signals into multiple bands dynamically according to their dominated wavelet ridges in the time-frequency domain. In Fig. 4, taking a sample of PPG and ECG signals, we depict their continuous wavelet for an any function w(τ ) such that the wavelet transform is defined as W w (a, t) = 1 √ a ξ τ −t a w(τ )dτ , with mother wavelet ξ performing dilation in scale a and shift in scale t.
The time-frequency representation easily reflects the local properties of the signal in the coefficients. For example, the continuous wavelet transformations (CWT) of the ECG in Fig. 4 indicated that the base frequency of heartbeats is around the period of 1 second (or 60 beats per minute). The vertical axis in the graph represents the frequency in a log scale from the top 0.5 Hz to the bottom 30Hz. To compute ridge points over the time-frequency representation, we need to define ridge curves. An amplitude ridge point and phase ridge point is the time/scale pair (a, t) with the strictly local maximum of {ln |W w (a, t)|} and ∂ ∂t {ln |W w (a, t)|} along the a and t axes, respectively, where the operators and retrieve the real and image part of a complex number, respectively, and W w (a, t) is a wavelet transform of an any function w(τ ). In other words, amplitude ridge points are defined as arg max (a,t) {ln |W w (a, t)|}. The sets of ridge points collectively become wavelet ridge curves. As depicted in Fig. 5, we split the bands based on the ridge curves. The ridge amplitudes are prominent in components around the vicinity of heart rates because of the semi-periodic nature of cardiac activities.
Therefore, we successively cut the components in the low band and evaluate the next dominated ridge over the residual coefficients.
The first subgraph in Fig. 5 is the band with the most dominated ridges, and the second subgraph becomes the successive dominated ridges. The time-frequency representation (a, t) forms some Riemannian submanifold M of R m×n . A manifold is a space having no global coordinate systems. However, locally at a point X ∈ M, the manifold is homeomorphic to a Euclidean space.
Therefore, we utilize the manifold to extend the residual subspace for accommodating possible hidden details. The components in the final band belong to non-smooth manifolds S x and S y for PPG and ECG. For the simplicity and consistency of notations, we still use the same X and Y to denote the non-differentiable functional in the non-smooth manifold.
The non-smooth manifolds S x and S y for signals X and Y are defined on a lattice of Laplacian , representing an n × n matrix. Let a functional correspondence T ∈ R n×n maps functions from Lebesgue space L 2 (S x ) to L 2 (S y ). The functional correspondence T can be approximated by the first k Laplacian eigenvectors as T ≈ x W w (a, t) T y in a sense subgradient, where W w (a, t) is a k × k matrix translating Fourier coefficients from the basis x to the basis y .
To avoid ''contaminating'' the good properties of components in the smooth manifolds, we keep extracting all lowfrequency smooth components according to the dominated wavelet ridges until reaching the final components, which are highly irregular and have sparse spikes. The final components cannot be discarded as noises because the mapping between the components determines the quality details of the destination signals. The estimation of the mapping function in the manifold is challenging because of the no-where differentiable property. However, a good estimation in the nonsmooth manifold is key to the inverse project in our target problem.
The filter bank obtained by our SRD algorithm guarantees a well-separation of heart activities, and the reconstructed waveform is loss-less (Fig. 6). We tried to demonstrate the difference between 3-band and 4-band decomposition. Fig. 6 graph shows the result of 4-band, which has no significant difference from the previous 3-band decomposition. We, therefore, choose to use 3-band decomposition hereinafter. The back-projection learning task in the first stage is an optimization process that evaluated the solutions to the illposed problem in each band b.
Considering that the application scenarios of wearable devices need to be accommodated to ordinary people's daily activities, the processing algorithm should overcome signals with a large portion of noise and artifacts. Wavelet methods are well-known for their excellent capability in removing non-coherent noises, artifacts, and drifting. Therefore, our SRD band splitting algorithm can obtain separated manifolds and rectify the input signals to an acceptable quality.

B. SPARSE KERNEL TRANSFORMATION FOR THE SOURCE PPG SIGNALS
Individual learner l = 1, . . . , L for the ensemble with L learners should adapt to the characteristics of each band. An effective way to highlight such characteristics is kernel transformation, k [39]. The representer theorem in the reproducing kernel Hilbert space (RKHS) guarantees that, over the training set, the minimizer of the regularized minimization problem will find a functional g for a target functional s such that min g ||s − g|| 2 + µ||g|| 2 H , µ ≥ 0, admits the representationĝ(X ) = n k=1 ν k X b,i,k (t)) over a Hilbert space H , where ν k ∈ R n denotes the unknown coefficients and µ is the Lagrange multiplier of the constraint.
The challenge of individual learners encountered in the illposed problem is to inverse a near-singular outer product, and it is often resolved through regularization methods with a positive perturbation. Therefore, the solution to the multivariate optimization problem becomes To avoid over-fitting, we further generalize the problem (3) as a generic elastic net (4) in the Least Absolute Shrinkage and Selection Operator (Lasso) term [40] with Lasso penalty α with l 1 and l 2 norms, || · || 1 , || · || 2 . max µ min g ||s − g|| 2 + µ 1−α 2 ||g|| 2 2 + α||g|| 1 .
Optimization (4) pushes the coefficients to zero if the covariates are insignificant due to the l 1 properties. The reconstruction efficiency increases when transformation manifests the sparsity properties [41]. The optimization (4) is regulated by the Lasso penalty (α = 1) or the ridge penalty (α = 0), and it takes advantage of the sparse l 1 norm in evaluating solutions for the ill-posed problem [42]. If α = 0, the optimization in (4) reduces to an ordinary generalized matrix inverse, which serves as a comparison basis. The leastsquare estimation in the objective creates a large variance when covariates exhibit multicollinearity. Ridge regression performs optimization to compensate for the problem of multicollinearity by finding a balance between variance and bias [43]. The ridge penalty effectively reduces the variance of the identified coefficients [44], [45].

C. GENERATIVE PULSE LOCKING FOR QRS COMPLEX
The mapping from PPG to ECG in high-frequency bands can be nearly singular because the PPG signals have insufficient high-frequency information. Instead of performing direct mapping, a generative method is exploited. Mirroring the concept of the phase-locked loop in radio communication, we estimate the transformation parameters based on a series of actively synthesized pulses from the low-bands of source PPGs (Fig. 7). The GPL pulses denoted as X B+1,i , are synchronized with the reference band in phases and amplitudes by maximizing cross-correlation and minimizing the covariance error. The way we maximize and minimize the two objectives is consistent with the principle of generative methods in the recent renowned research [46]. The generative method can effectively overcome the numerical difficulty in the ill-posed inverse mapping.

D. ADAPTIVE BAND BOOSTING AT THE SECOND AGGREGATION STAGE
At the second stage, as shown in Fig. 8, the multivariate regression (5) aggregate the predicted functionals Y l i , for which the superscript l of the Y (ECG) represents the output of the l-th learners in terms of the vectorized β l , the combiner function h(Y l ), and the independent residue η. The combined Y with combiner h(Y l ) becomes A numerical difficulty exists in (5) because multiple learners in the first stage increase the number of variables in multiple folds. While the un-trained records remain the same, the number of predictors is more than the number of observations in the regression. Therefore, ordinary regression fails in this case. Conventional solutions usually involve specific transformations to reduce the dimension of predictors. For example, principal component analysis and partial least square (PLS) regression extract components with a significant variation in the covariance matrix of input and input|output, respectively. We exploit the sparse partial least square (SPLS) regression to further limit the number of predictors [47], [48]. In comparison with PLS, the SPLS effectively keeps response and predictors uncorrelated in the regression with a large number of predictors.
In a conventional PLS, Y and X are transformed by a matrix V = XU ∈ R n×L , where L is the reduction dimension and U is the coefficient matrix. Many algorithms apply successive approach to solve V through finding each column vector in U . For example, simple partial least square (SIMPLS) [49] and multi-response SIMPLS [50] evaluate U = arg max U {U T XY T XY U }, s.t. U T U = I and u T j XX u j = 0 with covariances XX and XY for U = [u 1 , . . . , u L ], j = 1, . . . , L − 1, and identity matrix I. VOLUME 10, 2022 Assuming the latent space is far smaller than one of the predictors, we can then partition the space generated by X into two disjoint subspaces such that X = (X 1 , X 2 ) spanned by relevant (X 1 ) and irrelevant variables (X 2 ). The component w = [w 1 , . . . , w L ] can be obtained by adding an l 1 constraint in the optimization of SIMPLS such that where M = X T YY T X and λ is a parameter limiting the extent of sparsity. Similar to other sparsity algorithms, such as Lasso or e-net, a multi-fold cross-validation step is needed in the computation. We also follow the standard procedure to find related sparsity control parameters.
The two-stage computation steps and object construction are finally summarized in Fig. 9. The implementation is designed in a style of object-objected programming. Before being sent to the learning ensemble, a dataStore object is constructed to represent the data unit X b,i,k in sparse kernel transformation after extracting and matching records from MIMIC III patient database. The object dataLearner and dataCombiner are designed to accomplish stage-1 and -2 processes, respectively. The ensemble algorithm will choose the best learner and a regression model among the bag of candidates.

IV. SIGNAL RECONSTRUCTION AND ANALYSIS
The proposed algorithms have been tested on a publicly available MIMIC III database [51] of PhysioNet.com [52]. This multi-parametric dataset contains 25,328 admissions to intensive care units. A large number of testing waveforms were used to validate the algorithm. A large quantity of PPG/ECG waveform pairs was extracted for the experiment. There is no limit on the number of records we could use, but screening the waveforms with the selected diseases in the complex hospital information system and ICD-9 (international disease classification) code system was not easy. Of these pairs, 70% and 30% of data were used for training and testing, respectively.
From the MIMIC III database, we extracted 287 records without being diagnosed having heart disease. We also matched 60 acute myocardial infarction (AMI), 30 atrial fibrillation (AF), and 170 heart failure (HF) records in the dataset. According to the convention of the hospital registration system, each record represented a particular recording during the stay from one of the multiple admissions of a patient. Therefore, we followed the same managerial logic that ignored the patient number and concentrated only on each admission because different admissions might receive different ICD-9 codes. The recording lengths varied from a few seconds to several hours. We will not overuse a single patient and only kept a part of such a long record for keeping balanced sampling. The data structure in the experiment was divided into record, chunk, and segment. A segment is a basic unit with the bandwidth of the functional time series. A chunk contains 10∼50 segments, and a record consists of 5∼20 chunks. The part of recording outside the target range of chunks was discarded.
A semi-automatic pre-processing step was performed along with visual inspection to join multiple columns among patient subjectID, ICD9 code ranges for a group of diseases, admissionID, recordingID, and signal waveforms. Before the waveforms were entered for training, the records with all-zero elements or without complete 'PLETH'/'lead II' pair were excluded. Although our algorithm could accept a wide range of acquisition quality, flat zero signals still are unacceptable because they would mislead the algorithm. The high noise or irregular wave signals were preserved for testing our algorithm's robustness.

A. PREDICTION RESULTS IN EACH SRD BANDS
We iteratively employed a combination of learners in the first stage ensemble for each band in the experiments. The learners in the ensemble for this experiment were defined as [1, 2, 3, 4, 5, 6] = [t-l2, t-l1, w-l2, w-l1, dcnn, warp], for which represent [an ordinary generalized matrix inverse in l 2 , an ordinary generalized matrix inverse in l 1 , Lasso in l 2 , Lasso in l 1 , a 40-layer VGG19 model [53], and a waveform warping method], respectively. Many learners were excluded from the experiments because they did not perform well, either in the training or testing steps.
In Fig. 10 (a), the SRD filtered signals of low-band were smooth and congruent to each other. Therefore, the mapping in this band was easy and would guarantee the heart rate of the final transformation will be accurate.
In Fig. 10 (b), the mid-band of the SRD carried prominent signal information, and this band's accuracy dictated the quality of the final reconstructions. In Fig. 10 (c), in the highband of the SRD, most of the coefficients should be zero. The generative pulse locking for the QRS complex in this band also helped stabilize the quality of the mapping.

B. PREDICTION ACCURACY
Several metrics were adopted for evaluating performance, including standard ones, the relative root mean squared error (rrmse), and Pearson correlation coefficient (ρ). We also applied Kling-Gupta efficiency (KGE) values [54] to assess the goodness of fit for the similarity between two functional time series. Given the functionals of the simulated signal sim and observed signal obs, the statistics of sim and obs are FIGURE 11. The ECG signals were recovered from the source record set 4 and testing segment 17. The top two subgraphs showed the original PPGs and ECGs, and the third and fourth subgraphs showed the best ECG reconstruction using the appropriate learners (The learners are numbered as t-l2=1, t-l1=2, w-l2=3, w-l1=4, dcnn=5, warp=6.) prepared for the KGE measure. µ sim and µ obs are the means of sim. σ sim σ obs are the variances of obs.
where r is the Pearson correlation between the two functionals sim and obs. The KGE value implies that the larger the KGE is, the better the fit will be. Besides similarity, the dissimilarity between the simulated and all other non-target waveforms should be measured. Even though the simulation may not perfectly match the target, the simulation should not accidentally match any other targets. Therefore, we define a Kling-Gupta efficiency difference (KGED) by taking a uniform expectation E overall records in the dataset [30].
Positive values of KGED imply high goodness of fit for the specific targets.
The training result for the proposed ensemble approach was illustrated in Fig. 11. Based on the goodness-of-fit articulated in Table 2, the results of the prediction were demonstrated in the functional form. Even in the case of extremely poor acquisition quality of PPG, the predicted reconstruction still maintains robustness without losing statistical details.
Stable results are obtained through our algorithm. Based on the prediction accuracy in Table 2, we find that the reconstructed ECG can perform disease diagnosis with great accuracy. For patients with ICD-9 codes relating to AMI, AF, and HF, we obtained correctness rates of 85%, 80%, and 78%, respectively.
For comparison, in the individual learner, we also deployed a deep convolutional neural network (DCNN) in a wellknown 40 layers VGG19 model [53]. As shown in Fig. 13, the results did not perform as well as our method, either in the training or testing data. From the subgraph, we can observe that the performance of VGG19 significantly deviated from other methods. The original PPG and ECG are in the first two FIGURE 13. Comparison to other methods. A popular DCNN was used in the algorithm but the reconstructed signals did not perform as well as our method, compared to the results in Fig. 11. The first and the second subgraphs are the original PPG and ECG, respectively. The third (mid-band=w-l1,high-band=dcnn) and the fourth (mid-band=dcnn,high-band=dcnn) subgraphs are the DCNN prediction outputs.
subgraphs and the next two subgraphs show two samples of reconstructions by combinations of learners, which demonstrate an average reconstruction result.
In the subgraphs of Fig. 13, the third (mid-band=w-l1, high-band=dcnn) and the fourth (mid-band=dcnn,high-band=dcnn) were the outputs with DCNN learner. The reconstructions did not surpass the quality of other methods. A DCNN may not be a perfect choice in this application with time series contents.
The maximal similarity between the reconstructed and the original ECG occurred by using the methods SRD-GPL in l 2 and l 1 , whose performance is superior in both training and testing data, compared to other methods, such as DCNN. The proposed learning model improves prediction accuracy (KGE) and specificity (KGED).
Our algorithm shows excellent results in avoiding overfitting, and the prediction is made specifically for the target ECG, according to the indication of KGED. Our goal of prediction satisfies two criteria: statistical error should be small for the unseen ECG, and should not be similar to any other ECG (the error difference between target and non-target waves should be significant).

C. DISCUSSIONS
Our results show the feasibility of mapping from PPG to ECG signals based on the theoretical basis of electro-fluidmechanics and hemodynamics. The regularization in l 1 is rooted in the cellular activation dynamics.
The mapping results reveal that the low-band is relatively stable, the mid-band is informative for the waveform details, and the high-band is distinguishable for the similarity. We can spend less computation time in the low-band and still obtain a relatively stable prediction for the final mapping. We should spend most computation resources in the mid-band because the waveform details manifest in this band. Considering that  the QRS complex is the most distinguishable part of ECG, a conditioning step with generative pulses and a regularized l 1 penalization are effective in the computation of high-band. Using the trained model, we can quickly generate a highfidelity ECG solely based on easy-to-measure PPGs.
We demonstrate our conversion result by acquiring the PPG from an AMI patient. The reconstruction result from our ensemble was shown in Fig. 14. Even taking from a lowquality PPG signal, we still can synthesize an almost identical ECG signal based on such limited conditions. For the VOLUME 10, 2022 validation of our reconstruction of the AMI ECG, the original ECG signals are similar to the predicted ECG signal ( Fig. 12 (a)). Given an unseen PPG and ECG at the top 2 subgraphs of (a) (orig PPG=original PPG, orig ECG=original ECG), we obtain the predicted ECG (pred ECG=predicted ECG) at the bottom subgraph. We zoom in on the details in the subgraph (b) (from point 1 to point 250, corresponding to 2 seconds in the sampling rate of 125 Hz) of the predicted ECG (upper) and the original ECG (lower). The reconstructed prediction preserves the features of the original ECG even in the AMI case.

D. LIMITATIONS AND FUTURE WORKS
This study tried to extract matched records from the MIMIC III database. Because the waveforms and disease codes were stored in separated data sources, the matching process was not easy. The record relations between two parts of sources were so complex, and therefore writing an automatic process is not reliable. Currently, the community still relies on visual inspection in getting disease codes for a waveform.
Future development can focus on improving prior distribution specifications and initial conditions. We will additionally construct empirical distribution for all error terms to be a basis of the prior distribution. The algorithm will be continuously improved and parallelized for better computational efficiency.

V. CONCLUSION
Low-quality PPG signals were mainly used to count heartbeats and measure SpO 2 and were difficult to be convinced for disease diagnoses. This study contributes to cardiology applications by upgrading such low-cost, ubiquitous devices to an operational level for regular medical practice. Based on electro-fluid-mechanics and haemodynamics in cardiology, our algorithm achieves a substantial accuracy in the reconstruction performance, and the calculated ECGs sufficiently replicate the waveform details of the original ECG signals.
We found that the inverse mappings are effective with the time-frequency band decomposition in the kernel space with l 1 norm. To yield the maximal similarity between the reconstructed and the original ECG, we apply the most suitable mapping method for each band, which also adapts dynamically to the cardiac activities. To address the challenge of ill-posed inverse mapping, we accommodate a generative pulse train synchronizing with the base frequency. The lowband in the mapping results is always stable, the mid-band is informative, and the high band is distinguishable. Here, we should focus on the inverse mapping of the mid-bands for superior waveform details.
Our method can alert suspicious cardiovascular symptoms at home and help a large population of high-risk, believedhealthy persons walks in doctors' offices before their cardiovascular health becomes irreversible. Our study demonstrates a stable and accurate mapping from PPG to ECG signals. Once ECG waveforms are derived from PPGs, many diseases, such as acute atrial infarction or arrhythmia, can be prescreened in an ordinary surveillance camera. YENMING J. CHEN received the Ph.D. degree in systems science and mathematics from Washington University, St. Louis, USA, in 1998. He is currently a Distinguished Professor with the Information Management Department, National Kaohsiung University of Science and Technology, Taiwan. He is the author of several books, patents, and more than 75 SCI/SSCI articles. His research interests include the IoT/AI, sound/spectrum technologies, fault detection, precision medicines, and medical computations. YANYUN TAO received the Ph.D. degree from the East China University of Science and Technology. He was a Postdoctoral Researcher with Shanghai Jiaotong University. He is an Associate Professor with Soochow University. He has published 17 SCIE articles in machine learning for medical application. He was supported by the ''Talent of Colleges and Universities in Suzhou.'' He was also supported by the National Natural Science Foundation of China; the China Postdoctoral Science Foundation; and the State Key Laboratory of New Software Technology, Nanjing University. His research interests include computational intelligence, machine learning, and biomedical engineering. He is a member of the Young Scientist Association of the China Artificial Intelligence Society. He received the Second Prize of Scientific and Technological Progress from the Suzhou Computer Society.