In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if  Suppose that In particular, if d et al. , Howie , the collection of all groupoids defined on a non-empty set X. Given arbitrary groupoids X acts as an identity in L(R) denote the collection of all groupoids linear groupoid. They showed that Kim and Neggers  introducQ-algebra, and showed that every quadratic Q-algebra X is a field with Some researchers studied on linear groupoids and quadratic groupoids in several algebras. Neggers et al.  introducm and So  investigleft-twisted semigroup with respect to the map associator function of the groupoid Let dual left-twisted semigroup with respect to the map We may think of a dual equation of  as folloright-twisted semigroup with respect to the map Suppose we replace the Eq.  by3\docudual right-twisted semigroup with respect to the map We may think of a dual equation of  as follomiddle-twisted semigroup with respect to the map If we replace the Eq.  by5\docudual middle-twisted semigroup with respect to the map We may think of a dual equation of  as folloIf mentclass2pt{minimIfis a semigroup, then it is a  left-twisted semigroup.Consider Consider BCK-algebra which is also a right-twisted semigroup.Note that Example In Example Let There is no map such thatis a left-twisted semigroup.If we assume that there is a map Note that Proposition entclass1pt{minimaIf a left-twisted semigroup has a right identity element, then it is a semigroup.Let mentclass2pt{minimDually, if a right-twisted semigroup has a left identity element, then it is a semigroup.Notice that mentclass2pt{minimThe groupoidis a middle-twisted semigroup with respect toif and only iffor anyGiven If we let X as follows:linear groupoid over a field X. We define its associator function Let Let Letbe a linear groupoid over a fieldXand its associator function defined byIfis a dual left-twisted semigroup with respect tothen it has the formorSince In Proposition Let Note that Examples Letbe a linear groupoid over a fieldi.e.,for allwhereare not all zero inX. Ifis a right-twisted semigroup with respect tothen it has one of the forms: (i) andandX, there exist Since Letbe a linear groupoid over a field X. If we define a mapthenis a dual right-twisted semigroup with respect towhenStraightforward. Letbe a linear groupoid over a fieldX, i.e.,for alland letIfis both a left-twisted semigroup with respect to a mapand a right-twisted semigroup with respect to a mapthenis an additive group of the fieldX.X. Since it is a left-twisted semigroup, given mentclasspt{minimaLetbe a linear groupoid over a fieldX, i.e.,for allwhereare not all zero. If we define a mapwherethenis a middle-twisted semigroup with respect toStraightforward. Let Letbe a group and(fixed). Define a binary operation “G by If we define a mapwhereeis the identity ofG, thenis a dual left-twisted semigroup with respect toGiven n be a natural number. A group power property if there are x in G. For example, consider the dihedral group Let Letbe a group having thepower property. Define a binary operation “onGbyThenis not a left-twisted semigroup with respect to any mappingG such that Assume that Letbe a group having thepower property. Define a binary operation “onGbyThencan not be a right-twisted semigroup with respect to any mappingG. Since G such that Assume that Letbe a left-twisted semigroup with respect to a mapIfis anepimorphism of groupoids, i.e.,then there exists a mapsuch thatis a left-twisted semigroup with respect to the mapGiven f is an epimorphism, we haveGiven left-twisted-injective if X such that Let Letbe a left-twisted semigroup with respect to a mapIfis a left-twisted-injective epimorphism, then there exists a mapsuch thatis a left-twisted semigroup with respect to the mapf is onto, there exist f is left-twisted-injective, we haveFor any In Theorem f byUsing the notion of this concept, we introduce an equivalence relation on the Cartesian product d-algebra BCK-algebra Consider a If we define a map f is an epimorphism, we have Thus, let Letbe an epimorphism of groupoids, whereandare left-twisted semigroups with respect toandrespectively. IfthenGiven an epimorphism mentclasspt{minimaFor the other types of twisted semigroups we have similar situations. In the case of a dual left-twisted semigroup, we obtain the following results.Theorem 3Letbe a dual left-twisted semigroup with respect to a mapIfis anepimorphism of groupoids, i.e.,then there exists a mapsuch thatis a dual left-twisted semigroup with respect to the mapdual left-twisted-injective if X such that Let Proposition 11Letbe a dual left-twisted semigroup with respect toIfis a dual left-twisted-injective epimorphism, then there exists a mapsuch thatis a dual left-twisted semigroup with respect to the mapf byThe corresponding relationship on (1)right-twisted semigroups: (2)dual right-twisted semigroups: (3)middle-twisted semigroups: (4)dual middle-twisted semigroups: The other four types go in entirely the same way. We list the relevant information. X be equipped with the product binary operation given by the formula Given one of the six types of twisted semigroups, let a subgroupoidof a dual left-twisted semigroupneed not be a dual left-twisted semigroup.In order to obtain varieties we must be able to claim that if Z be the collection of all integers. Then v is an integer. Then k. This means that k. Hence there is no mapping In Example In this paper, we introduced the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Besides a number of examples and a discussion of homomorphisms, a class of groupoids of interest was the class of groupoids defined over a field In the investigation of “residual associativity” in groupoids one encounters a number of levels. The strongest version of such residual associativity is “associativity” itself. Making a study of twisted semigroups, besides being of interest in itself, is also relevant in several other ways. One may wish to determine as precisely as possible how different twisted semigroups may differ from semigroups. Accordingly we note that the study of twisted semigroups, commenced in this paper, will prove to be a rich area for research both in itself and as embedded in the area of the study of “general theory of groupoids”.


We start with the second-order Lane-Emden equation Also, in  several et al. [et al. employed a monotonicity formula-based approach and gave a complete classification of stable and finite Morse index (positive or sign-changing) solutions to (For the fourth-order Lane-Emden problem ons (see ). The suequality , this win>0; see . These rmentclasspt{minima [et al.  and furt [et al. . This apd Grunau . It shoup>1; see . Finallyp>1; see , Dávila Thanks to the Liouville-type theorem with finite Morse index in , the autished in , they reIn this work, we are concerned with Liouville-type theorems for the nonlinear elliptic equation  for \docThe paper is organized as follows. In Section f:In order to state our results, we present first some assumptions on the nonlinearity (1) We easily verify that the following functions satisfy f can have an exponential growth at infinity. Therefore, clearly an adequate behavior of f at zero is needed to obtain the Liouville theorem. The unique and important nonexistence result for stable solutions of the non-homogeneous second-order equation  and (2) show that equation  has beenequation  does nottions of  are trivxponents  and --3.30\docequation  by \docuu be a stable solution to (The proof of Theorem me as in , 6, 8. Nution to .Subcritical case:. By Proposition Critical case:. By Proposition u is not identically zero, then We now collect  and 3.2. By assuThe proof of Theorem equation  by \docuu gives The function x. Using  in the lng again , we get ution of , in the θ and combining it with (ϵ sufficiently small, we deduce Proof of 1. If u that the right hand side of (k large enough such that We proceed as in the proof of Proposition Proof of 2. If hen from  and by YWe can proceed as in the proof of Proposition


In the original article, one of the co-author’s names has been published incorrectly. The correct name should be K. Prabha.


Second-order sideband generation in an optomechanical system with the coupling between a mechanical resonator and a two-level system is discussed beyond the conventional linearized description of optomechanical interactions. The features of two-color second-order sideband generation are demonstrated in this hybrid system. We discovery that the switch between one- and two-color second-order sideband generation is easily realized by shifting the detuning between the control field and the cavity field or the transition frequency of the two-level system. The efficiency of two-color second-order sideband generation increases monotonously with the control field strength. The coupling strength between the mechanical resonator and the two-level system plays a decisive role in the appearance of the two-color second-order sidebands. The two-color second-order sideband generation may provide measurement with higher precision in new degrees of freedom. The subject of rapid developments brings about significant value in both fundamental physics and practical applications, such as achieving high precision measurement7, slowing and storage of light pulses9, information processing11 and optomechanical induced transparency(OMIT)13. Micro-mechanical resonators14 are in combination with optical cavities, which presents some important  phenomena, such as phonon blockade16, phonon-induced transparency17, and high-harmonic generation18.Cavity optomechanics22 emerge, such as high-order sidebands24 and chaos26 which are achieved in the bichromatic driving regime by making use of numerical calculation method. In the perturbative regime that the probe field is weaker than the control field, an analytical method has also been proposed to describe the nonlinear interactions in an optomechanical system29. Here we use the perturbation method to study the second-order sideband generation in a hybrid optomechanical system with a two-level system coupled to the mechanical resonator.Recently, due to the nonlinear optomechanical interactions between the optical cavities and the mechanical resonators, a lot of interesting phenomena30, superconducting qubit circuits31, two-level defects32 and so on. There are three situations in a hybrid optomechanical system for the coupling among an optical cavity, a mechanical resonator and another object: a two-level system is coupled to both the cavity field and the mechanical mode33; a two-level atom is only coupled to the cavity field30 and a two-level system is only coupled to the mechanical resonator34. For the third situation, the hybrid optomechanical system has been utilized to complete both the classical and quantal researches, such as the appearance of the two transparent windows34 and tunable photon blockade35. For other similar systems, the discovery of some new applications in the double-electromagnetic-induced transparency (EIT)37 and double-OMIT39 has been presented subsequently. All of these results inspire us to explore second-order upper sideband of double frequencies in output spectrum, which is called two-color second-order sideband generation here.The hybrid optomechanical system may consist of an optical cavity, a mechanical resonator and other objects including two-level atoms40 and provides measurement with higher precision in new degrees of freedom, such as charge measurement4 and force sensors41.In the present work, we show the two-color second-order sideband generation in a hybrid optomechanical system where a two-level system is only coupled to the mechanical resonator. The interpretation is given for the formation of two-color second-order sidebands and the features of two-color second-order generation are exhibited. The switch between the one- and two-color second-order sideband generation can be actualized by transforming the transition frequency of the two-level system compared to the mechanical-mode frequency, or the detuning between the cavity field and the control field, whose strength sets a direct influence on the efficiency of the second-order sideband generation. In addition, the coupling strength between the mechanical resonator and the two-level system determines whether two-color second-order sidebands can appear. The two-color second-order sideband generation may find potential applications in optical frequency combωa, ωb, respectively. The Pauli operator ωq is the transition frequency of the two-level system. The coupling parameter between the cavity field and the mechanical resonator, which is coupled to the two-level system with coupling strength g, is χ. Here we treat the interaction between the single-mode optical field and the mechanical resonator as radiation-pressure Hamiltonian. For simplicity, we assume that the mechanical resonator interacted with the two-level system is regarded as Jaynes-Cummings Hamiltonian, and we have no consideration on the direct interaction between the cavity field and the two-level system. The last term, where The hybrid optomechanical system consists of an optical cavity and a mechanical resonator coupled to a two-level system, which might be an intrinsic defect inside the mechanical resonator or a superconducting artificial atom, as shown in Fig. U = expAlthough the Hamiltonian mentclass2pt{minimωc. When the strong control field and a weak probe field with frequency ωp are applied to the optomechanical system, the whole Hamiltonian of the hybrid system can be written asε is the Rabi frequency of the probe field, which describes the coupling strength between the probe field and the cavity field. In a frame rotating at ωc, the Hamiltonian of Eq. , respectively.Considering the damping and environmental noise terms, the Heisenberg-Langevin equations of motion are read as follows:M0 which correspond to the values of the a, b, σ−, and σz, respectively.In this work, we consider the mean response of the hybrid system. By making use of the lowest-order correlation-truncation approximation, the operators in Eqs.  to 7) a a7) are To describe the process of the second-order sideband generation, we definefrom Eq.  by analy42 is not applied to expound the appearance of two transparent windows in the hybrid system where the mechanical resonator is coupled to a two-level system. For the sake of comprehending the physical processes of two transparent windows and two-color second-order sidebands, for simplicity, the single-photon and single-phonon excitations are taken advantage of exemplifying the transmission of photons in Fig. 34. A typical Λ-type three-level systems is composed of three states |0a, 0b〉, |0a, 1b〉, and |1a, 0b〉, where the subscripts a and b represent the photon and phonon states, respectively. However, the state |0a, 1b〉 is split into two dressed states43 |0a, 1b+〉 and |0a, 1b−〉 when the mechanical resonator is resonantly coupled to a two-level system, where the effective resonant frequency of the mechanical resonator becomes ωb − g and ωb + g from ωb which leads to two transparent windows due to the resonant matching between optical beat frequency Δ and the effective mechanical frequency. Here we consider the higher order upconverted processes42 of the control field that the control field absorbs two phonon or the anti-Stokes field absorbs a phonon to generate a second-order sideband which also exhibits two interference windows.The physical interpretation of the OMIT effecttp|2 and η vary with Δ by solving Eq.  resonant with the frequency ωq of the mechanical resonator, viz. Firstly, when we observe the case that the detuning between the transition frequency η of the second-order sideband generation is very low with maximum 3%. If we consider the Stark field from the down converted process of the control field exciting the mechanical resonator, the Stark sideband further absorbs a phonon to generate a second-order sideband, whose efficiency is higher. In general, the down converted process of the frequencies is easier than the up converted process of the frequencies. As shown in Fig. η of second-order sideband at Δ = 0.9ωb is higher than the efficiency η of second-order sideband at Δ = 0.9ωb. Regardless of the control field and the first-order sideband going through the up or down converted process to generate second-order sidebands, the condition that the two-level system is resonantly coupled to the mechanical resonator must be satisfied for the appearance of two-color second-order sidebands. Therefore, there is another variable the transition frequency ωq of the two-level system utilized to switch the one- and two-color second-order sideband generation.In Fig. η of the second-order sideband generation for different values of the coupling strength g. The blue curve shows the value of the η reaches its local minimum within a frequency window corresponding to about the line width of the cavity at Δ = ωb, which can also take place when the mechanical resonator is not coupled to a two-level system in an optomechanical system42. In fact, if the strength of the control field reduces appropriately, and the strength of the control field does not reach the condition of the appearance of OMIT, only a second-order sideband generates at Δ = ωb.In order to further verify the role of the coupling strength g between the two-level system and the mechanical resonator in the appearance of the two-color second-order sideband generation, Fig. g that results from the Jaynes-Cummings coupling between the mechanical mode and the two-level system. That is to say, by transforming the coupling strength g between the mechanical mode and the two-level system, we can get two-color second-order sidebands at different frequencies of the output spectra. In addition, the different values of the coupling strength g also have an effect on the efficiency η of two-color second-order sideband generation. For example, the efficiency η of two-color second-order sideband generation at the coupling strength g/2π = 10 MHz is the highest, which illustrates the coupling strength should be moderate.For other curves in Fig. ωq of the two-level system and the detuning Δa between the cavity field and the control field, whose strength also affects the efficiency η of two-color second-order sideband generation. The two-color second-order sidebands at different frequencies of the output spectrum can be observed from the second-order upper sidebands by controlling the coupling parameter g between the mechanical resonator and the the two-level system. Our finding of two-color second-order sideband generation maybe be an enlightenment of multi-color second-order sideband generation when the mechanical resonator is coupled to multi-level systems.In summary, we have expounded the second-order sideband generation in a hybrid system, consisting of an optical cavity and a mechanical resonator coupled to a two-level system via Jaynes-Cummings interaction. We give the explanation to two-color second-order sideband generation distinguished from one-color second-order sideband generation in the physical mechanism. It can be seen how to achieve the switch between one- and two-color second-order sideband generation by shifting the transition frequency ωc ± nΔ where n is positive integer. Instead of consideration on all frequencies of output fields, we are interested in the situation of the second-order sideband generation with n = 2. Here we use the perturbation method to deal with the Eqs. (It is very difficult for us to get an exactly analytical solution on nonlinear Eqs. –11. In tthe Eqs. –11, wherting Eq. –11, we cσz is a Hermitian operator, the relation In particular, it is pointed out that because d to Eq.  to simpl44After some derivation, we can obtain the values of ωc ± 2Δ is part of the output field with frequencies ωc ± nΔ and it has been studied in different optomechanical systems42. The term of The output field with frequencies


The approach relies on a pre-specified interval null hypothesis that represents the collection of effect sizes that are scientifically uninteresting or are practically null. The second-generation p-value is the proportion of data-supported hypotheses that are also null hypotheses. As such, second-generation p-values indicate when the data are compatible with null hypotheses (pδ = 1), or with alternative hypotheses (pδ = 0), or when the data are inconclusive (0 < pδ < 1). Moreover, second-generation p-values provide a proper scientific adjustment for multiple comparisons and reduce false discovery rates. This is an advance for environments rich in data, where traditional p-value adjustments are needlessly punitive. Second-generation p-values promote transparency, rigor and reproducibility of scientific results by a priori specifying which candidate hypotheses are practically meaningful and by providing a more reliable statistical summary of when the data are compatible with alternative or null hypotheses.Verifying that a statistically significant result is scientifically meaningful is not only good scientific practice, it is a natural way to control the Type I error rate. Here we introduce a novel extension of the  P-values abound in the scientific literature. They have become the researcher’s essential tool for summarizing when the data are incompatible with the null hypothesis. Although p-values are widely recognized as imperfect tools for this task, the impact of their flaws on scientific inference remains hotly debated  where a < b are real numbers, then its length is |I| = b − a. The second-generation p-value, denoted by pδ, is defined asI ∩ H0 is the intersection, or overlap, between intervals I and H0. The subscript δ signals the reliance of the second-generation p-value on the interval null. Numerically, δ represents the half-width of the interval null hypothesis. The value of δ is driven by scientific context and should be specified prior to conducting the experiment. The first term in p-value to indicate inconclusiveness when the observed precision is insufficient to permit valid scientific inferences about the null hypotheses.Let pδ is the length of the intersection between the two intervals, divided by the length of the interval estimate, multiplied by the correction factor. When the interval estimate is sufficiently precise, defined here as when |I| < 2|H0|, the second-generation p-value is just the overlap fraction, |I ∩ H0|/|I|. When the interval estimate is very wide, |I| > 2|H0|, the second-generation p-value reduces to 0.5 × |I ∩ H0|/|H0|, which is bounded by ½.As described here, I| would represent an area or volume. Neither interval is required to be symmetric or of finite length. Although the vast majority of intervals in the literature are ‘two-sided’ with finite length, pathologies are possible when neither interval has finite length. If intervals I and H0 overlap but neither has finite length, e.g., overlapping one-sided intervals, the second-generation p-value will be zero or one, depending respectively on whether |I ∩ H0| is finite or infinite (Remark 1 in Definition When measuring systolic blood pressure (SBP), the currently accepted recommendation is to report SBP to the nearest 2 mmHg when using analog or mercury devices ,18. Bloop-value based on δ = 2 mmHg, the maximum p-value over all possible null hypotheses in the range H0: 144 ≤ μ ≤ 148, and the traditional p-value for testing H0: μ = 146.μ0 ± δ = , has length |H0| = 2δ = 4 mmHg. So pδ isFor example, in Study 3, the upper confidence bound is 142.55 and the lower confidence bound is 147.45. The interval null, pδ of 0.7 means the data are inconclusive with a slight or weak favoring of null hypotheses. Intuitively, it means that 70% of the data-supported hypotheses are null hypotheses. Study 4 is an example where the correction factor comes into play. The length of the confidence interval, 8.82, is more than twice the length of the null interval, 4. Therefore, pδ is:A pδ of 0.5 means the data are strictly inconclusive. The role of the correction factor is discussed in Section 2.5.A p-values best describe pδ, and the maximum p-value seem to account for this paradox. However, pδ has more desirable attributes in other circumstances, such as in Studies 3 to 6. Study 4 has a traditional p-value and maximum p-value of 1, but the data are clearly inconclusive. The second-generation p-value reflects this, with a value of 0.5, allowing for a more nuanced interpretation of ‘inconclusive’. In studies where the point estimate falls in the null range, but the confidence interval extends beyond  the maximum p-value does not properly convey what the data are saying, and the traditional p-value does not account for the range of null values, or the precision of the measurement tool. Studies 7 and 8 illustrate the case when the confidence interval is fully beyond the null space.The second-generation H0: μa0 ≤ μ ≤ μb0. The second-generation p-value, pδ, has the following properties:Consider a general interval null hypothesis, say pδ is a proportion; a number between zero and one, inclusive.pδ is the fraction of data-supported hypotheses that are null hypotheses and therefore compatible with the null premise.pδ = 0, the data only support hypotheses that are scientifically or clinically meaningful, i.e., those that are meaningful alternative hypotheses.When pδ = 1, the data only support null hypotheses, i.e., those that are not scientifically or clinically meaningful.When pδ = ≈ 1/2, the data are strictly inconclusive. The degree of inconclusiveness is represented by pδ itself. For example, pδ = 1/8 and pδ = 7/8 both represent the same degree of inconclusiveness, but the balance between alternative and null hypotheses is reversed.When pδ has improved error rate control when the interval estimate I is a 100(1 − α) % CI. Under any null hypothesis within the interval null, the probability of observing pδ = 0 is less than or equal to α. This probability converges to zero as the sample size grows. Under any hypotheses beyond the interval null, the probability of observing pδ = 0 converges to one as the sample size grows.pδ may appear similar to that of a posterior probability of the null hypothesis. However, pδ is strictly not a posterior probability and it is not an estimate of the probability of the null hypothesis. Rather, pδ is simply the observed fraction of data-supported hypotheses that are null hypotheses; it is a descriptive statistic—a simple proportion. Essential elements of a proper posterior computation, namely knowledge that some hypotheses are better supported than others and the degree to which some hypotheses inside the interval null should be favored over others, are not needed. Specification of the latter is controversial because it does not depend on the data at hand. Second-generation p-values are descriptive; they indicate when the study has generated data that rule out null or alternative hypotheses.The interpretation of p-values; this is why statistical significance is not scientific significance. The fourth property is strictly false for classical p-values; a p-value larger than the Type I Error probability α is considered inconclusive. Large p-values never “support” the null hypothesis; they just indicate the lack of strong evidence against it. This is the absence of evidence is not evidence of absence conundrum. The fifth property is also not strictly true for traditional p-values; when non-significant, the p-value is to be interpreted as inconclusive despite the temptation to do otherwise. A welcome feature of second-generation p-values is that they distinguish between data that are inconclusive (0 < pδ < 1) and data that are compatible with null hypotheses (pδ = 1). This ability is sorely needed in practice.The third property does not hold for traditional p-value converges to zero or one as the sample size grows, which means the procedure is inferentially consistent in the limit . One way to do this is to use the delta-gap, which is the distance between the intervals in δ units. Recall that δ is the half-width of the interval null hypothesis. If the CI was shifted to the right of the null interval, the delta-gap would be δ makes it unit free. The delta-gap ranking favors extremes in effect size, which is a logical complement to the second-generation p-value. Remember that second-generation p-values provide a quick and easy marker of when a study reaches a natural endpoint, i.e., when the data are compatible with only null hypotheses (pδ = 1) or only alternative hypotheses (pδ = 0). Two studies with equal second-generation p-values do not necessarily represent equal amounts of statistical evidence. For example, their likelihood functions may not be proportional. The same is true of classical p-values, of course −1 and P(H1│Failed to Reject H0) = [1 + (1 − α)/βr]−1. The FDR and FCR for second-generation p-values are generally smaller than their hypothesis testing counterparts. Remark 20 in p-values decrease the FDR and FCR from classical p-values.In principle, the reliability of any inferential summary ought to be reported along with the summary itself. Neither is truly sufficient on its own. However, reporting the reliability can be difficult to do in practice for the reasons noted above. And while standards for reporting the FDR and FCR deserve further investigation and discussion, the key message is that second-generation We note that while estimates further from the point null are more reliable in the sense that their false discovery rates are lower, this does not mean that they are more ‘important’, more ‘relevant’ or more ‘meaningful’ in a scientific sense. A low false discovery rate means only that the results are more likely to replicate under identical conditions. The cause for this replication could be a true scientific finding or an unknown experimental flaw. Only careful investigation into the validity of the scientific experiment can determine which it is.p-value usage, despite its flaws, will persist. Science desires an easily digestible and reliable summary of whether the data are compatible with null or alternative hypotheses. Second-generation p-values are tailor made for this role. They are easy to apply and interpret; they can be used in conjunction with frequentist, Bayesian or Likelihood methods; and they exhibit excellent frequency properties. Moreover, they eliminate the haggling over ad-hoc adjustments to p-values that have become a real challenge in high-dimensional and data-rich settings.More than a century of experience tells us that p-values is that they can be small and yet also be associated with confidence intervals that include hypotheses or parameter values that are essentially null. In genomic studies with many thousands of variants, this has led to the detection of hundreds of variants with spurious significance. By introducing an interval null hypothesis of scientifically equivalent null hypotheses, second-generation p-values avoid the vast majority of the spuriously significant findings. By better reflecting the true nature of the null hypothesis in mathematical terms, statistical inference works better for science. Also, the width of the interval null need not be large, as the benefits of second-generation p-values will eventually be realized as the sample size grows.A well-known problem with classical p-values tend to have excellent behavior overall. Nonetheless, this should be explored in detail (See our “Frequently Asked Questions” in δ) will require re-calculation of the second-generation p-value. To facilitate this, we encourage the reporting and discussion of the interval estimates upon which the second-generation p-value is based. We remind the reader that the second-generation p-value is not intended to be the final product, but rather a quickly digested summary measure. Understanding the force and implication of statistical results will always require attention to the details of the statistical analysis.Some challenges remain. The statistical properties of this new tool need to be explored and detailed in a variety of settings. Our preliminary findings detailed here, and in the web-supplement , indicatexactly correct . In this paper, we assumed that the broader statistical model holds because this is a routine assumption for p-value based inference. However, the robustness of the second-generation p-value to model misspecification is an important topic that deserves attention.Regardless of the analysis, caution should always be used when interpreting statistical findings. Just because a result is statistically less likely to be a ‘false discovery’ does not mean that the result is more ‘important’, ‘relevant’ or ‘meaningful’ in a scientific sense. It means simply that the results are more reliable and more likely to replicate under identical circumstances, even if those circumstances are in some way flawed. A statistical hypothesis is a specific, precise mathematical statement about an unknown parameter. And statistical hypotheses assume a probability model to perform computations. If that model fails, then the associated statistical hypotheses often fail to be valid translations of the scientific hypotheses. This is, in fact, one of the most common criticisms of point null hypothesis testing; the point null is almost never p-values will be encouraging researchers to define a scientifically relevant finding prior to examining the data. While this is already routine in some areas of biomedicine , it is not even on the radar in others. Nevertheless, second-generation p-values are a clear improvement over classical p-values. They are between 0 and 1. They have a straightforward interpretation. A second-generation p-value of 0 is properly interpreted as favoring alternative hypotheses. A second-generation p-value of 1 is properly interpreted as favoring null hypotheses. A second-generation p-value between 0 and 1 favors both types of hypotheses and is properly interpreted as inconclusive. The error rates for second-generation p-values are bounded by α and converge to 0. Adjustments for multiple comparisons are obviated, and lower false discovery rates accompany observed results. In short, the second-generation p-value achieves the inferential properties that many scientists hope, or believe, are attributes of the classic p-value. Using second-generation p-values can only improve rigor, reproducibility and transparency across science.We anticipate that the major challenge in implementing second-generation S1 File(PDF)Click here for additional data file.S2 File(PDF)Click here for additional data file.


European J Pediatr Surg Repvol 5, issue 1. DOI of the original article is10.1055/s-0037-1599796. The names should be listed with "Sabine” as first name and “Vasseur Maurer" as family name.It has been brought to the Publisher's attention that the author's name "Sabine Vasseur Maurer" was not tagged correctly in the above-mentioned article, published in


Scientific Reports 10.1038/s41598-017-18413-8, published online 08 January 2018Correction to: lucius” was incorrectly given as “1ucius”. This has now been corrected in the PDF and HTML versions of the Article.The original version of this Article contained an error in the title of the paper, where the word “


In this paper, the idea of lacunary  The concept of statistical convergence  which isI-convergence is also an extended notion of statistical convergence ([A) is an ideal if I is additive, i.e.i.e.A is any non-empty set.Similarly, ergence  of realA lacunary sequence is an increasing integer sequence refer to –19 etc.θ will be determined by the interval Note: Throughout this paper, M if, for arbitrary i.e., A sequence I-convergent to A sequence et al. [For any lacunary sequence n et al. ) \documeThe concept of a Musielak-Orlicz function is defined as t}M (see )  .M in We define a sequence We write it as I be a non-trivial admissible ideal such that x as follows: Example: Let Then we have, for each Letbe a PNS. Ifis lacunarystatistical convergent, then it has a unique limit.X which has two limits, Suppose For Take the following sets: Since I, from which it is clear that I [Now, let If Since For arbitrary Letbe a PNS. Ifxis lacunarystatistical convergent, then it is lacunarystatistical convergent ifFor given Therefore, Since λ-statistically convergent to M with respect to ν as We define Letbe a PNS. Ifxis lacunaryλ-statistically convergent toM, then it is also lacunarystatistically convergent toM.IfthenIfthenLetbe a PNS. Ifxis lacunaryλ-statistical convergent toM, thenλ-statistically convergent to M, then, for every Let I is admissible, we have Since Letbe a PNS. Ifxis lacunaryλ-statistical convergent, then it has a unique limit.Letbe a PNS. Ifxis lacunaryλ-statistically convergent, then there exists a subsequenceofxsuch that it is also lacunaryλ-statistically convergent toM.


Protein crystal structures are potentially over-interpreted since they are routinely refined without any restraint on the upper limit of atomic B-factors. Consequently, some of their atoms, undetected in the electron density maps, are allowed to reach extremely large B-factors, even above 100 square Angstroms, and their final positions are purely speculative and not based on any experimental evidence.A strategy to define B-factors upper limits is described here, based on the analysis of protein crystal structures deposited in the Protein Data Bank prior 2008, when the tendency to allow B-factor to arbitrary inflate was limited. This B-factor upper limit (B_max) is determined by extrapolating the relationship between crystal structure average B-factor and percentage of crystal volume occupied by solvent (pcVol) to pcVol =100%, when, ab absurdo, the crystal contains only liquid solvent, the structure of which is, by definition, undetectable in electron density maps.It is thus possible to highlight structures with average B-factors larger than B_max, which should be considered with caution by the users of the information deposited in the Protein Data Bank, in order to avoid scientifically deleterious over-interpretations.The online version of this article (10.1186/s12859-018-2083-8) contains supplementary material, which is available to authorized users. In crystal structures, oscillation amplitudes of the atoms around their equilibrium positions are monitored by B-factors, which are related to the root-mean-square amplitude of oscillation  [Fenwick and co-workers designed a B-factor based order parameter, which approaches its maximal value equal to one for well ordered atoms, and approaches zero for highly disordered atoms. This order parameter, for single-conformer structural models, becomes unphysical (negative) when the sum of the B-factors of two atoms approaches 8π≈ 79 Å2) .Unfortunately, it became common in recent years to deposit in the Protein Data Bank , 23, B-fThis might have severe consequences on the bioinformatics use of the B-factors. For example the quantitative relationships between local stereochemistry and B-factor (and flexibility) would be misinterpreted by considering protein moieties with unreliable structures.Traditionally, when crystallographers are unable to position atoms or residues based on electron density maps, they insert a series of warnings in PDB files. The “REMARK 465”, “REMARK 470”, “REMARK 475”, “REMARK 480” lines were used to list (i) the residues that are present in the SEQRES records but are completely absent from the coordinate section (REMARK 465); (ii) non-hydrogen atoms of standard residues which are missing from the coordinates (REMARK 470); (iii) residues modeled with zero occupancy (REMARK 475); and (iv) non-hydrogen atoms of residues modeled with zero occupancy (REMARK 480).Obviously, this is the result of a rather subjective decision, based on the visual analysis of the electron density maps. Some crystallographers prefer to include these “invisible” protein moieties in their refinements, with the consequent inflation of the B-factors. An atom that cannot be precisely positioned tends to occupy a large portion of space to spread its electrons as much as possible.2 are often deposited in the Protein Data Bank and this has three major drawbacks. First, the occupancy and the B-factor fields merge in the PDB-formatted files, with consequent problem of readability: the string “1.00 99.99” becomes “1.00100.00” when the B-factor increases from 99.99 to 100.00 Å2. Second, at (sinθ/λ) = 2, the scattering power of at atom with B = 5, 20 or 80 Å2 is only 0.54, 0.08 or 0.0001 [2 is absolutely negligible. Ab absurdo, one might introduce several atoms of Fermium wherever possible in the asymmetric unit with B-factors extremely large without consequences on the R-factor, the free-R-factor and the electron density. Third, and this is a major concern, PDB file end users, which often are not crystallographers, might be induced to use structural data that are absolutely unreliable.Recently, B-factors larger than 100 År 0.0001  and, as It is therefore necessary to have an estimation, at least semi-quantitative, of the highest B-factor values that can be considered to be acceptable and over which atoms can be considered to be non-localizable. Apparently, only a careful analysis of the electron density map allow one to solve this problem. This is however impossible in most bioinformatics applications due to the large amount of data that must be examined.ab absurdo, the B-factor of a “liquid” atom in a protein crystal. In practice, in this article, I analyze the relationship between the average B-factors of protein crystal structures and the percentages of crystal volume occupied by solvent, which can be estimated through the Matthews volume [B_max). Based on the analysis of a large and well selected set of protein crystal structures, it can be predicted that at very high resolution (better than 1.5 Å), B_max is close to 25 Å2, which means that the average B-factor value should not be larger than 25 Å2 at that resolution, while larger values are observed at lower resolution. At very low resolution (worse than 3.3 Å), B_max grows up to 80 Å2, which means, again, that the average B-factor value should not be larger than 80 Å2 at that resolution.It is impossible to define a physically meaningful threshold value, over which B-factors can be safely be used to decide that an atom must be excluded in the refined model: it would be a mere arbitrary threshold, based on some subjective assumption, for example “less than 0.05%” of the structure factor. On the contrary, it is possible to try to identify, s volume . In thiswww.rcsb.org). It is however important to observe that other criteria, slightly more or less stringent, would produce similar results , given the extreme-value distribution of pairwise protein sequence similarity scores [Protein crystal structures were downloaded from the Protein Data Bank , 23 accoy scores . Eight dProtein atoms were divided into 24 classes according to Li and Nussinov : atom typcVol) were computed with the rwcontents program of the CCP4 software suite [pcVol is computed as3.Percentages of crystal volume occupied by solvent  in a protein crystal structure and the percentage of volume occupied by solvent in the crystal (pcVol). By plotting B versus pcVol, it is then possible to extrapolate, through linear regression, the value of B (B_max) at pcVol = 100%, when all the atoms contained in the crystal are, ab absurdo, in the liquid state.It is possible to compute the average B-factor  available for 2 and on the presence of missing atoms or residues . Therefore, it is possible to find four types of files: (i) those with large B-factors and with missing residues; (ii) those without large B-factors and without missing residues; (iii) those with large B-factors and without missing residues; and (iv) those without large B-factors and with missing residues.It is mandatory to use a set of protein crystal structures with a minimal fraction of atoms, the B-factors of which have been allowed to inflate – the common practice nowadays. For this reason, the entire PDB was analyzed and files were classified, based on the presence of atoms with B-factors larger than 100 ÅAs it can be seen in Fig. pcVol, assuming that these data are little affected by the modern tendency to allow B-factor inflation in the absence of interpretable electron density maps.It clearly appears that the tendency to allow B-factors to reach extremely high values became common only during the last decade. For this reason, only protein crystal structures deposited in the Protein Data Bank prior 2008 were used to analyze the relationships between B-factors and pcVol is shown in Fig. The relationship between B-factors and pcVol increases  depends on the B-factor according toI observe here that the dependence of the B-factor on resolution is absolutely obvious. In fact the atomic form factor , they are nearly always smaller than 0.5 for the 24 atom types listed in the Methods. This depends on the considerable scatter of the average B-factors within each resolution range, which is not reported in Fig. BN), defined asBav is the average B-factor of a protein crystal structure and Bs is its variance. However, this is not possible here since the average BN is equal to zero by definition and, as a consequence, it is invariant and cannot depend on any other variable. When one analyzes a specific type of protein atoms, for example the backbone carbonylic carbon atoms, its average BN value can be different from zero. However, individual average BN values are nearly independent on resolution and pcVol, since the standardization is performed within a single protein crystal structure – this is actually the reason why this standardization is performed.The correlation coefficients between B-factors and pcVol and resolution  are shown in Table B and pcVol are relatively small – they range between 0.23 and 0.51 – given the intrinsic variability of the B-factors; however, the slopes are statistically larger than zero for all atom types). B_max tend to increase if resolution decreases. In the highest resolution range (0.0–1.5 Å) and when all the atom types are considered together, the average B-factor at pcVol = 100% is only 25 Å2, while it is 80 Å2, in the lowest resolution range (3.3–4.0 Å).The B-factors expected when CI) of the B_max values are also reported in Table n is the number of points, pcVolave is the average pcVol value andBobs is the observed value of the B-factor, Bcalc is the computed value of the B-factors  CIs are definitely vary large (nearly 20 Å2).B_max values are smaller for main-chain atoms than for side-chain atoms. For example, they range from 23 to 24 Å2 in the 0.0–1.5 Å resolution range, with an average value of 23 Å2, while the analogous values range from 20 to 40 Å2 for side-chain atoms, with an average value of 28 Å2. In general, main-chain B_max values are about 5 Å2 smaller that side-chain B_max values.B_max values tend to increase in going from sulfur, to carbon, to nitrogen and to oxygen atoms. For example, in the 0.0–1.5 Å resolution range, they are equal, on average, to 24 Å2 for sulfur atoms, to 26 Å2 for carbon atoms, and to 29 Å2 for nitrogen and oxygen atoms. It must however be observed that these differences are often statistically insignificant, given the quite wide variety of atom types per each chemical element, with the partial exclusion of sulfur.B_max values are observed for the side-chain sulfur atom of cysteine (atom type 13) or the Nε1 nitrogen atom of tryptophan (atom type 16). Interestingly, the side-chain sulfur atoms of methionine (atom type 14), which is more distant from the backbone than the side-chain sulfur atom of cysteine, is also associated with larger B_max values. For example, in the 0.0–1.5 Å resolution range, it is equal to 27 Å2, while that of cysteine is equal to 22 Å2.Minimal B_max values are observed for the side-chain nitrogen atom of lysine (atom type 20) and for the side-chain carboxylic oxygen atoms of aspartate and glutamate (atom type 23), which are at the end of long side-chains and are also, in general, largely exposed to the solvent. Interestingly, the side-chain nitrogen atoms of arginine have, in general, B_max values lower than the side-chain nitrogen atom of lysine and this might depend on the fact that the guanidinium moiety of arginine is larger and may for more hydrogen-bonds than the ammonium moiety of lysine.Maximal B_max values for a charged residue (lysine), a polar residue (serine), and an apolar residue (methionine). It clearly appears that B_max values tend to increases if the distance from the backbone increases. Lysine, which has the longest side-chain that ends with a cationic group, has, as expected, the largest B_max values at its extremity.Figure B_max values were estimated from the analysis of protein crystal structures deposited until 2008 in the Protein Data Bank, when the tendency to allow B-factors to inflate arbitrarily was minimal. The analysis of a set of 2783 protein crystal structures deposited after 2014, assembled with the same criteria described in Methods, is summarized in Table B_max, at low resolution a large fraction of the structures have an average B-factor larger than B_max. This depends on the fact that at high resolution only few atoms are invisible in the electron density maps and, as a consequence, few atoms have unreasonably large B-factors. On the contrary, at low resolution, several atoms are invisible in the electron density maps and have, as a consequence, extremely large B-factors. From the perspective of structural bioinformatics and of the users of the Protein Data Bank information, this means that nearly one half of the crystal structures refined at low resolution should be considered with extreme caution, since many moieties of them are not experimentally determined, but just modeled computationally. It is also interesting to observe that even at very high resolution about 15% of the protein crystal structures are likely to be incomplete, given their large average B-factor.B_max).Based on well controlled data sets, it is possible to estimate the maximal B-factor average values that are compatible with the crystalline solid state


Fetal electrocardiogram (FECG) extraction is very important procedure for fetal health assessment. In this article, we propose a fast one-unit independent component analysis with reference (ICA-R) that is suitable to extract the FECG. Most previous ICA-R algorithms only focused on how to optimize the cost function of the ICA-R and payed little attention to the improvement of cost function. They did not fully take advantage of the prior information about the desired signal to improve the ICA-R. In this paper, we first use the kurtosis information of the desired FECG signal to simplify the non-Gaussian measurement function and then construct a new cost function by directly using a nonquadratic function of the extracted signal to measure its non-Gaussianity. The new cost function does not involve the computation of the difference between the function of the Gaussian random vector and that of the extracted signal, which is time consuming. Centering and whitening are also used to preprocess the observed signal to further reduce the computation complexity. While the proposed method has the same error performance as other improved one-unit ICA-R methods, it actually has lower computation complexity than those other methods. Simulations are performed separately on artificial and real-world electrocardiogram signals. The fetal electrocardiogram (FECG) contains much important information about the health and possible diseases of the fetus, which reflects the complete view of the heart activities. Additionally, it is more sensitive than color Doppler ultrasound in the case of fetal acidosis and anoxia. With the analysis of the FECG, the doctor can discover fetal hypoxemia, umbilical abnormality, and other fetal abnormality situations timely and take early effective actions to ensure the health of the fetus. However, the FECG signal is considerably weaker than the maternal electrocardiogram (MECG) and is often embedded in the noise, MECG, baseline wandering, power line interference, and so forth. Accordingly, it is quite difficult to extract the FECG signal and further reduce the diagnostic accuracy.A number of methods have been reported for extracting the FECG signal, like filtering method , singulaTo overcome this problem, the independent component analysis with reference (ICA-R) is applied to extract the FECG –14, whicThe one-unit ICA-R is a special case of the multiunit ICA-R. It only separates one desired source signal every time. The only difference between the one-unit ICA-R and the multiunit ICA-R is that the former only extracts one desired signal, and the latter is based on the former and uses the orthogonalization method to extract multiple desired signals. If we use the deflationary orthogonalization method or symmetric orthogonalization method, which are widely used in ICA  to extraAlthough the one-unit ICA-R and multiunit ICA-R have a little difference, they share the same cost function. Additionally, they both use the same non-Gaussian measurement function that is used in the classical ICA algorithm to measure the non-Gaussian component of the extracted signal. The negentropy approximation is needed to compute the difference between the function of the extracted signal and the function of a Gaussian vector that has the same mean and variance as the extracted signal. This will consume plenty of time. The classical ICA algorithm has no information about the source signals, except that the non-Gaussian measurements and statistics are independent. However, the ICA-R has prior information about the desired signal, but the previous methods did not use such information to reduce the computational complexity of the negentropy approximation. The aim of this paper is to reduce the computational complexity for the one-unit ICA-R by taking advantage of the prior information of the desired ECG to simplify the negentropy approximation.This paper is organized as follows. x(t) = T is the observed signal, A is the mixing matrix with the size (n × m), and s(t) = T is the source signal. The aim of the ICA is to estimate the source signal s(t) from the observed signal x(t) by computing the separating matrix W = T with size (m × n). The estimated source signal can be expressed as follows:y(t) = T is the estimated source signal, which is the same as s(t) under ideal condition. However, under real condition, the shape of the waveforms of y(t) is close to s(t), except for the amplitude and sequence of the waveforms. The source signals s(t) are considered as non-Gaussian and mutually statistically independent. Accordingly, it has strong non-Gaussian properties. The mixing signals x(t) have strong Gaussian characteristics according to the central limit theorem. When the estimated source signal is close to the source signal s(t), it will have strong non-Gaussian properties. Thus, the ICA uses a non-Gaussian measurement function of the estimated source signals as a cost function  is avoided in J(y) to eliminate E{F(v)} to avoid further computing [E{F(y)} − E{F(v)}]. J(y) is a part of the cost function of the one-unit ICA-R. As a result, this will reduce computation complexity of the one-unit ICA-R.As the previous one-unit ICA-R methods involve the computation of the F used in one-unit ICA-R is a positive even function, the terms E{F(v)} and E{F(y)} are both positive in } has a maximum value, this equals [E{F(y)} − E{F(v)}], which has a maximum value. On the contrary, when E{F(y)} has a minimum value, this equals −[E{F(y)} − E{F(v)}], which has maximum value.Since the nonlinear function itive in , and v hE{f′(si)} − E{sif(si)} < 0, E{F(y)} has maximum value. Therefore, when E{f′(si)} − E{sif(si)} < 0, we can determine that [E{F(y)} − E{F(v)}] has a maximum value based on the analysis described in the first paragraph of this subsection. Based on the above analysis, when E{f′(si)} − E{sif(si)} < 0, then } − E{sif(si)} > 0, the non-Gaussian measurement function can be simplified asc = E{sif(si)} − E{f′(si)}.Based on the analysis in  0, then  can be s0, then (Jy=ρEFy−Eunction (Jy=ρEFy.Olified asJy=−ρEFy.F(y) was proposed for extracting the super-Gaussian and sub-Gaussian signal, respectively.a, b ∈ R+. In [a = 1 in (y) and 1 − tanh2(y), respectively.In , the fol R+. In , Mi has ectively.Fsupy=1alR+. In [Fsupy=1alectively.Fsupy=1alR+. In [Fsupy=1alc in (si), which is described assi is a super-Gaussian signal with unit variance, E{si4} > 3. Then c < 0 in (If we use  as a nonc in  can be ec in (cEsifsi−Ec in (cEsifsi−Eterm in (cEsi2−si4c < 0 in  as(33)Jyy = wTx is the extracted signal, x is the centered and whitened signal, and w is the separating vector. ρ in  for the problem in , Newton's method is used to optimize L. The learning rule of the separating vector w is given by the following equation:η is the learning rate:The one-unit ICA-R is used to extract one desired signal. The prior information that the desired signal is a super-Gaussian or a sub-Gaussian signal is easily obtained. The FECG and MECG are both super-Gaussian signals in general. Therefore, in the extraction of FECG, we can directly use  instead min −Jy≈n (ρ^ in . ρ is jumin −Jy≈ρ in (ρ(E{F(y)} − E{F(v)}). In the original algorithm, we have to additionally compute (E{F(y)} − E{F(v)}) in every iteration, which considerably consumes much more time compared with the improved algorithm.The main difference between the improved algorithm in  and the ρ in  is just n Center the observed signals.(2)Whiten the centered signal.(3)γ, convergence threshold ς, parameter ρ, and the leaning rate η; generally let ρ = 1, η = 1.Choose a proper scalar penalty parameter (4)w0.Use  to initi(5)w0 ← w0/‖w0‖.Normalize (6)μ.Choose an initial value for (7)μ by (Update the Lagrange multiplier μ by .(8)w by (Update the separating vector w by .(9)w by Normalize the separating vector (10)wk+1 − wk‖, ‖wk+1 + wk‖) ≤ ς, otherwise go back to step (7).Until min , A is mixing matrix, B is the whiten matrix, w is the separating vector of mixed signals after whitening, wTB is the global separating vector, M is the number of mixing signals, and maxj | pj| is to find the maximum valued among the elements in the vector p = . The extracted signal by proposed method can be described asp = , pk ≠ 0, we will have the extracted signal y = pksk, which is just rescaled signals of sk. It is obvious from  = log cosh (y) as the nonquadratic function and ε = E{(y − r)2} as the closeness measure function. Additionally, g(y) = ε − ξ. We also use the corrected first derivative of g(y) as its first derivative. The corrected first derivative is given in [ρ, η, and γ all equal to one and perform 1000 trials for each data set.In the experiments, we select given in  as(42)∇wThe artificial signals include the power line interference with 50 Hz, Gaussian noise, FECG, and MECG. The FECG and MECG are generated by using the ECG toolbox of Sameni . The ref4A=0.89250To quantitatively compare the performance of the proposed method and that of Mi's method , the IPIThe real-world ECG data was contributed by De Moor  and has In the Ch1 recording, the strong and slow heart belongs to the mother, while the weak and fast belongs to the fetus. Therefore, we can use the information about fetus in the Ch1 recording to construct reference signal for extracting the FECG. We use impulse series whose occurrence time is the same as that of the subpeaks in the first channel signal Ch1 as reference for extracting the FECG. The impulse series whose occurrence time is the same as the peaks of any channel signal were used as reference for extracting the MECG.The reference signal and the FECG signal extracted by both methods are shown in In this paper a fast one-unit ICA-R algorithm for extracting ECG is proposed by simplifying the non-Gaussian measurement method of the extracted signal. In the proposed algorithm, the nonlinear function of the extracted signal is directly used as non-Gaussian measurement function. This avoids having to compute the difference between the function of the extracted signal and the function of the Gaussian random vector with the same mean and variance as the extracted signal. Centering and whitening are also used to preprocess the observed data to avoid computing the observed signal covariance matrix. As a result, the computation complexity for the one-unit ICA-R algorithm is greatly reduced.The validity of the proposed algorithm is tested and compared to Mi's algorithm using artificial ECG data and real-world ECG data. Both experimental results demonstrate that the convergence rate of the proposed algorithm is two times faster than Mi's algorithm, while both methods have the same error.


With a higher applied voltage in the c-LED, decreasing response time and rising time improve device performance, but a longer recombination time degrades luminescence efficiency. By using an m-LED with a stronger carrier localization and a low polarization effect, shorter response, rising, and recombination times provide more efficient carrier injection, transport, relaxation, and recombination. These advantages can be realized for high-power and high-speed flash LEDs. In addition, with a weaker carrier localization and a polarization effect in the c-LED, the slower radiative and faster nonradiative decay rates at a larger applied voltage result in the slower total decay rate and the lower luminescence efficiency. For the m-LED at a higher applied voltage, a slow decreasing nonradiative decay rate is beneficial to device performance, while the more slowly decreasing and overall faster radiative decay rate of the m-LED than that of the c-LED demonstrates that a stronger carrier localization and a reduced polarization effect are efficient for carrier recombination. The resulting recombination dynamics are correlated with the device characteristics and performance of the c- and m-LEDs.Based on time-resolved electroluminescence (TREL) measurement, more efficient carrier injection, transport, relaxation, and recombination associated with a stronger carrier localization and a low polarization effect in a nonpolar  The  current –13. Pola current , electro current , 10, poo current –13, and  current , 15 and  current  have alsd a-axes , 17, 18,d a-axes , and -c-n-polar)  can overcy droop . Withoutcy droop , 17, 18.nitrogen-polar InGaN/GaN LEDs with the opposite polarity were shown to provide the advantages of more efficient carrier relaxation and carrier recombination [m-LED are not well explored.Time-resolved electroluminescence (TREL) can explore the dynamic EL behaviors of carrier injection, carrier transport, carrier relaxation into active region, and carrier recombination at the excited states in LEDs . By TRELbination . Howeverm-LED in comparison with those of the c-LED by using currentz-voltage (I-V), electroluminescence (EL), external quantum efficiency (EQE), and TREL measurements. By using an m-LED to overcome the polarization effect, shorter response, rising, and recombination times demonstrate that a stronger carrier localization and a reduced polarization effect are efficient for carrier injection, transport, relaxation, and recombination. In addition, due to a weaker carrier localization and a polarization effect in the c-LED, the slower radiative and faster nonradiative decay rates at a larger applied voltage result in the slower total decay rate and the lower recombination efficiency. At a larger applied voltage, the more slowly decreasing and overall faster radiative decay rate of the m-LED than that of the c-LED demonstrates that a stronger carrier localization and a reduced polarization effect are efficient for carrier recombination.This study reports carrier transport and recombination dynamics associated with a stronger carrier localization and a low-polarization effect of the This paper is organized as follows. In section 2, methods are described. In section 3, results and discussions are reported. Finally, conclusions will be drawn in section 4.c- and m-LEDs were simultaneously grown on the c- and m-plane sapphire substrates, respectively, by a metal organic chemical vapor deposition (MOCVD) reactor. The LED structure comprises a 25-nm low-temperature (LT) GaN buffer layer, a 3.00-μm undoped GaN layer, a 1.50-μm n-type Si-doped GaN contact layer, 10-pair InGaN/GaN multiple quantum wells (MQWs) , an 15 nm AlGaN electron blocking layer, and a 0.25-μm Mg-doped p-GaN contact layer. The 25-nm LT-GaN buffers were grown on nitridized sapphires at 600 °C in H2. Except for the substrates, the structures of the two LEDs are the same.The I-V, EL spectrum, and EQE were measured with a source meter (Keithley 2614B), spectrometer , calibrated integrating sphere, and power meter. Without an ohmic contact, an indium dot melting on the sample surface was used as a contact for the probe station. For TREL measurement, a pulse generator (Tektronix AFG3152C) was used to generate 2.0–5.0 V, 0.5 μs pulse width, and 1 kHz repetition rate voltage pulses to the LEDs. The light output was focused and detected by a photosensor module containing a metal package PMT and a high-voltage power supply circuit (Hamamatsu H10721-210) operating directly on the surface of each LED. The same PMT voltages of the TREL measurement were applied for the two LEDs. The parasitic capacitance in the test circuit would introduce the same RC delay for the two LEDs. The transit EL signals were recorded by a digital oscilloscope (Agilent DSO 6052A) with a 500 MHz bandwidth. The overall resolution of TREL system is less than 2 ns. The detailed measurement was described in our previous study [us study .c- and m-LEDs, respectively. EL emission wavelength for the m-LED is much shorter than that for the c-LED. EL spectra of the c-LED just coincide with the lower energy shoulder of that for the m-LED. Both carrier localization effect [c-LED contribute to luminescence. The large difference of EL peak positions for the m-LED from that for the c-LED can’t be discussed by only one effect. It can be due to the weaker QCSE or lower indium contents in the MQWs for the m-LED. The larger bandwidth of EL and the lower energy shoulder in the m-LED represent a stronger indium composition fluctuation and a stronger carrier localization. The broader bandwidth of EL in the m-LED could be due to the luminescence contributions from different indium compositions and indium clusters.Figure n effect –23 and Qn effect  in the cm-LED could be due to the band-filling effect of higher energy states and a reduced polarization effect upon a forward bias.In addition, Fig. I-V characteristics and normalized EQE, respectively, of the two LEDs. The turn-on voltages of c- and m-LEDs are 2.5 and 3.5 V, respectively. Although a slightly larger threshold, smaller slope of current under forward bias, and lower EQE of the m-LED suggest poorer device quality, a stronger carrier localization and a reduced polarization effect help the m-LED to exhibit better carrier transport and recombination dynamics, as shown later.Figure c- and m-LEDs, respectively. After the voltage pulses are applied, different rising transit profiles for the two LEDs imply different behaviors of carrier injection and carrier transport. The rising part of the transit EL intensity of the c-LED shows a slower response and rise. Similar temporal behavior was observed in transition from free carrier to deep acceptor levels of a GaN/AlGaN LED [c-LED, the rising part of transit EL intensity for the m-LED shows a shorter response, steeply rising, and then decay within the first 100 ns. Similar temporal behavior, attributed to band-to-band emission, was observed in EL peak at 285 nm of a GaN/AlGaN LED [m-LED, carriers may transport from high energy states to lower energy states. It is noted that PMT has a better sensitivity and can amplify low signals. The intensity ratios of TREL transit profiles under the same bias for the two LEDs are different from those of EL spectra.Figure lGaN LED  and in tlGaN LED . BecauselGaN LED . Withoutτresponse) can be determined by the time delay between addressing the device with a short voltage pulse and the appearance of EL. The rising time (τrise) is defined by the intercept of the tangents. Delay time (τdelay) is the time delay in reaching both the maximum intensity of transit EL and the subsequent decay, when the voltage pulse is switched off . The recombination time (τrecombination) can be determined by fitting the decay profile with a single exponential.As shown in Fig. τresponse can be described as [As shown in Fig. ribed as :Fig. 5Reτinjection, τtransport, τrelaxation, and τ ' recombinaiton are the carrier injection time to the ohmic contact, carrier transport time from the ohmic contact to the active region, carrier relaxation time from the active region to the MQWs, and carrier recombination time in energy states, respectively. With the same contact material for the two samples, τinjection can be assumed to be the same. Also, due to the faster mobility of nonpolar m-plane GaN [τtransport of the m-LED should be shorter. Without polarization charges, a lower potential barrier makes the carrier relaxation efficiency of the m-LED better than that of the c-LED [τrelaxation of the m-LED should be shorter. The weaker polarization effects in the m-LED increases the overlap of electron and hole wavefunctions and the τ ' recombinaiton is expected to be shorter, as shown later. This suggests that a weaker polarization effect in the m-LED helps carrier relaxation into MQWs and a stronger carrier localization helps carrier recombination in higher energy states. Hence, the shorter τresponse of the m-LED should be due to shorter τtransport, τrelaxation, and τ ' recombinaiton.where lane GaN , τtranshe c-LED , so τrem-LED, shorter τresponse, τrise, τdelay, and τrecombination of the m-LED than those of the c-LED demonstrate that a stronger carrier localization and a reduced polarization effect of the m-LED are more efficient for carrier injection, transport, relaxation, and recombination. The advantages can be realized for applications of high-power and high-speed flash LEDs. Although a longer τrecombination at a higher applied voltage shows a slower carrier recombination rate, it is shorter than τresponse and τrise.At the same time, at a higher applied voltage in the Ga- and N-polar LEDs [c-LED and the reduced polarization effect in the m-LED could be compared. The polarization-associated recombination dynamics in the c- and m-plane LEDs were also revealed by using bias-dependent EL, EQE, and TREL measurements.In a previous study, bias-dependent EL, EQE, and TREL measurements were conducted to compare the effects of carrier localization and QCSE on carrier transport and recombination dynamics of lar LEDs . It is nc-LED at 3.0V and the m-LED at 6.0V, shorter τresponse, τrise, τdelay, and τrecombination of the m-LED than those of the c-LED can be confirmed to demonstrate that a stronger carrier localization and a reduced polarization effect are efficient for carrier injection, transport, relaxation, and recombination.In addition, the injected current density may affect the carrier transport and recombination dynamics in LEDs. Because the carrier recombination is mainly from the ground states in the quantum wells, the injected current density may not affect it much. Under the same current density of the τ) is the reciprocal of the total decay rate (κ = 1/τ), which is the sum of the radiative and nonradiative decay rates [The decay time  and (3).κ, κr, and κnr associated with recombination dynamics are shown in Fig. c-LED at a lower applied voltage, a smaller deviation between κ and κr shows that the recombination dynamics are dominated by the κr. With a higher applied voltage, the slower κr and faster κnr result in the slower κ.m-LED at a higher applied voltage, a steeply decreasing κnr is beneficial to device performance. In spite of the lower η of the m-LED, the slower decreasing κr and overall faster κr of the m-LED than that of the c-LED demonstrates that a stronger carrier localization and a reduced polarization effect are efficient for carrier recombination. Because the c- and m-LEDs were simultaneously grown on the c- and m-plane sapphire substrates, respectively, the un-optimized growth condition for the m-LED leads to a faster κnr than κr. The κr should be enhanced by optimizing growth conditions for the m-LED separately. Also, an m-LED can be fabricated on an m-plane GaN substrate [c-LED at 3.0V and the m-LED at 6.0V, the faster κr of the m-LED than that of the c-LED can be confirmed to demonstrate that a stronger carrier localization and a reduced polarization effect are efficient for carrier recombination, while the faster κnr of the m-LED can be due to an un-optimized growth condition and a higher defect density. The resulting recombination dynamics are correlated with the device characteristics and performance of the c- and m-LEDs.For the ubstrate . In addim-LED with a stronger carrier localization and a weaker polarization effect, shorter response, rising, and recombination times provide more efficient carrier injection, transport, relaxation, and recombination. By optimizing growth conditions to enhance the radiative recombination, the advantages in the m-LED can be realized for the applications of high-power and high-speed flash LEDs. In addition, due to a weaker carrier localization and a larger polarization effect in the c-LED, the slower radiative and faster nonradiative decay rates at a higher applied voltage together result in the slower total decay rate and the lower recombination efficiency. With a higher applied voltage in the m-LED, a slow nonradiative decay rate is beneficial to device performance, while the more slowly decreasing and overall faster radiative decay rate of the m-LED compared to that of the c-LED demonstrates that a stronger carrier localization and a reduced polarization effect are efficient for carrier recombination. The resulting recombination dynamics are correlated with the device characteristics and performance of the c- and m-LEDs.In summary, this study reveals that by using an


This paper presents a novel physics-driven spatiotemporal regularization (STRE) method for high-dimensional predictive modeling in complex healthcare systems. This model not only captures the physics-based interrelationship between time-varying explanatory and response variables that are distributed in the space, but also addresses the spatial and temporal regularizations to improve the prediction performance. The STRE model is implemented to predict the time-varying distribution of electric potentials on the heart surface based on the electrocardiogram (ECG) data from the distributed sensor network placed on the body surface. The model performance is evaluated and validated in both a simulated two-sphere geometry and a realistic torso-heart geometry. Experimental results show that the STRE model significantly outperforms other regularization models that are widely used in current practice such as Tikhonov zero-order, Tikhonov first-order and L1 first-order regularization methods. Linear regression has widespread applications in various fields such as engineering, healthcare, economics and social science, for predictive modeling, experimental design, or system optimization. Regression parameters are often estimated based on the static data set of explanatory and response variables. However, rapid advancement of distributed sensing and imaging technology brings the proliferation of high-dimensional spatiotemporal data, i.e., y = y and x = x in healthcare systems. Traditional regression is not generally applicable for predictive modeling in these complex structured systems.Linear regression is a widely used approach for modeling the relationship between explanatory variables y acquired by the ECG sensor network placed on the body surface, also named body surface potential mapping (BSPM)1x on the heart surface from BSPM y so as to investigate cardiac pathological activities 345For example, y = Rx + ε) poses several challenges includingIn general, high-dimensional predictive modeling  such as BSPMs. It is often difficult to directly measure heart-surface potential mappings x. As such, inferring x needs a better knowledge of parameter matrix R. Fortunately, physical laws define the mechanisms of electrical propagation from the heart to the body surface. This, in turn, enables the derivation of parameter matrix R using physics-based principles .(1) Ill-conditioned system: Linear systems involving high-dimensional data y and x are commonly ill-conditioned. This is partly caused by unobserved x, and partly due to the fact that parameter matrix R is rank deficient . The condition number of R  is also shown to be large in high-dimensional predictive modeling  Spatiotemporal regularization: Ill-conditioned systems make the prediction more sensitive to noise factors  and approximation errors in parameter matrix R. For example, measurement noises can potentially cause a small change Δy in the observed data y. Considering the estimation of x changes to x + Δx, we will have the changes in the solution expressed as Rcond, the pseudo-inverse solution of x and y are spatially distributed and dynamically evolving over time, they have spatial and temporal correlations. Very little has been done to develop new spatial regularization methods that handle approximation errors through spatial correlations of dynamic profiles on the complex geometry , as well as new temporal regularization methods to increase model robustness to measurement noises and other uncertainty factors.(3) This paper presents a new spatiotemporal regularization model to tackle these research challenges and address ill-condtioned problems in high-dimensional predictive modeling. Our contributions in the present investigation are as follows:R. We developed realistic models of torso-heart geometries, numerically discretized them with the boundary element method, and then utilized physical laws  to derive the parameter matrix.(1) High-dimensional systems involve complex geometries, which challenge the derivation of parameter matrix R with a spatiotemporal regularization (STRE) method to build the high-dimensional prediction model. This approach leverages data from actual experiments to improve spatial and temporal regularity of the solutions, thereby making the final prediction closer to reality.(2) As physics-based models are deterministic and do not account for real-world uncertainties, we developed a physical-statistical approach that integrates physics-derived parameter matrix x. As such, they are not generally applicable because the electric field involves both positive and negative potentials. We developed a new method of dipole multiplicative update, which is inspired by the dipole assumption in electrodynamic physics. This new idea overcomes the drawbacks of existing multiplicative update methods, and provides a generalized approach to solve spatiotemporal regularization problems.(3) The proposed STRE model involves quadratic programming and high-dimensional data, which cannot be solved analytically. Iterative algorithms are commonly used such as the multiplicative update method which, however, requires the nonnegative constraint of (4) Few, if any, previous works focused on both spatial and temporal regularizations in inverse and forward ECG problems. We evaluated and validated the proposed STRE model in simulation as well as a real-world case study to map electric potentials from the body to the heart surface. Experimental results show that our method not only effectively tackles the ill-conditioned problems in high-dimensional predictive modeling, but also outperforms those regularization models widely used in current practice . This research work provides a new and effective approach to investigate disease-altered electric potentials from the body to the heart surface.The remainder of this paper is organized as follows: Section II introduces the research background. Section III presents our research methodology. Section IV describes the experimental design. Experimental results are shown in section V. Section VI concludes this paper.y = Rx + ε, where x and y are spatiotemporal data, is generally ill-conditioned. For example, the inverse ECG problem in healthcare 7R  is a measure of relative sensitivity of the solution x to the observed data y . The pseudo-inverse solution of The high-dimensional predictive model, R, increase the model reliability and improve the prediction accuracy.Statistical regularization models such as Tikhonov and L1 regularization methods789The objective function of Tikhonov regularization is formulated aswhile the L1 regularization is formulated as2 and ||·||1 denote the L2- and L1-norm, respectively, λ is the regularization parameter, and Γ represents the mathematical operator constraining x. Note that Γ is the identity matrix in zero-order Tikhonov and L1 regularization methods . Note that Γ is a discretized gradient operator in the first-order regularization methods. One of the most commonly used gradient operators is a bidiagonal matrix9Zero-order regularization is effective to shrink unreliable components of the estimator and achieve sparse solutions for high-dimensional predictive modeling. However, they are limited in the ability to handle measurement noises or approximation errors in ill-conditioned systems. Therefore, first-order regularization methods were proposed to address such limitations by constraining the gradient of the solution x in one column as {x(s1|t), x(s2|t), …, x(sN|t)}T, and then applied the bidiagonal gradient matrix. Note that the alignment of spatiotemporal data in one column is not an effective way (maybe even incorrect) to compute the spatial gradients. As such, regularization results are not as satisfactory as expected.which is a central-difference approximation for the first-order derivative. However, this approximation does not account for the complex geometries of space-time dynamic systems, and is only effective for one-dimensional data. Most of previous works aligned x = ∂x/∂n, where x denotes the dynamic potential distribution on the heart surface and n denotes the surface-normal vector7x, but ignores the gradient component on the the heart surface /∂τ, where τ denotes the surface-tangent vector) and does not take into account the spatial correlations between adjacent regions. It is worth mentioning that spatiotemporal data from distributed sensing and imaging are generally spatially distributed and have spatial correlations13In the inverse ECG problem, another commonly used gradient operator is the normal derivative-operator of the potential distribution on the heart surface, Γet al.In addition, space-time systems are dynamically varying over time and have temporal correlations. For example, the human heart is a typical spatiotemporal system with cardiac electrical activities dynamically varying in both space and time15R by integrating the boundary element method with divergence theorem and Green’s theorem. Second, we investigate the spatial regularization that handles approximation errors through spatial correlations of dynamic profiles on the complex geometry , as well as the temporal regularization to increase model robustness to measurement noises. Finally, we develop a new generalized method of dipole multiplicative update to solve the objective function of the proposed STRE model.As shown in y are generally obtained from the surface of a complex structured system such as BSPMs. Inferring the internal dynamic variable x  of these systems depends on the high-dimensional predictive modelingThe observed data R is the parameter matrix characterizing the interrelationship between x and y.where SB and heart surface SH18x on the heart surface and y on the body surface are related by the Laplace’s equations derived from physics-based principles . Solving for the parameter matrix R involves tackling this Laplace’s equation and calculating complex surface integrations, which are difficult to solve analytically in realistic torso-heart geometry. Thus, boundary element method (BEM)20SB and SH into triangle meshes, and divide the surface integrals into a series of numerical integrations over the triangle elements. Thus, the parameter matrix R is expressed as1819In the human body system, the heart represents the bioelectric source, and the torso is modeled as a homogeneous and isotropic volume conductor whose boundary consists of body surface A's and M's depend entirely on the torso-heart geometry. The rows of ABB, ABH and MBH correspond to the locations of different nodes on the body triangle-mesh SB. Similarly, the rows of AHH, AHB and MHH represent the locations of different nodes on the heart triangle-mesh SH. The different columns of all the matrices correspond to locations of triangle elements on the surface of integration.where the coefficient matrices, x in complex structured systems is an ill-conditioned problem, because the parameter matrix R is often with a large condition-number7R. For examples, the human body is modeled as a homogeneous volume conductor, and geometrical variations over time are assumed to be negligible. These assumptions may not hold true in real-world situations and will introduce uncertainties when predicting x22However, inferring s to overcome the drawbacks in existing methods.The spatiotemporal data acquired by distributed sensing and imaging systems are generally distributed in the space and have spatial correlations. In existing regularization methods, the constraint operator Γ or the penalty term does not account for the spatial correlations or the geometries of complex systems, but rather align the mesh nodes in one column or take the normal derivative operator. As such, they are limited in the ability to improve the spatial regularity. In this investigation, we propose to define the constraint operator Γ to be a spatial Laplacian operator Δs is computed by determining the Laplacian at each mesh node. In a two-dimensional square lattice with a lattice constant d as shown in xi denotes the value of dynamic variable x at node pi = , where  are location coordinates. According to Taylor’s theory, xi is approximated as the sum of x0 and its derivatives at node p0 = :The matrix ΔAdding the above four equations yieldsx0 at node p0 is expressed asThus, the Laplacian of where xt(i) denotes the value of dynamic variable x at node pi at time t, and dij is the distance between pi and pj. Using linear interpolation, the value pi and pj, and pi, as shown in However, real-world geometries are complex and are generally discretized into irregularly triangulated meshes using the boundary element method20dij’s over the neighbor nodes pj’s of pi, and these neighbors pj’s are the vertices of the triangles that include pi as one of the vertices. Thus, the Laplacian of xt(i) at pi in a 3D triangle mesh is defined aswhere ni is the number of neighbor nodes pj’s of pi; pj’s. According to s aswhere pi is defined asTherefore, the spatial regularity of three-dimensional triangle mesh at node N is the total number of mesh nodes.where x and y are dynamically evolving over time and have temporal correlations. However, few, if any, previous works have effectively dealt with the temporal regularization for high-dimensional predictive modeling in space-time systems  = Rx + ε, the two-body dynamic prediction problem). Therefore, we propose to define the temporal regularity asIn addition, spatiotemporal data T denotes the length of the overall time span of the spatiotemporal data, and w is a time window. Temporal correlation is stronger when two time points are close to teach other, and electric potentials at two time points that are far away from each other tend to have bigger differences. Therefore, the time window w is often chosen to be a small number. Adding the temporal constraints in where Combining the parameter matrix, spatial and temporal regularization as described in previous subsections, we formulate our STRE model by the following objective functionλs and λt are the spatial and temporal regularization parameters, which can be chosen by the L-curve methodR, but also increase the model robustness to measurement noises in the time domain. Therefore, it is expected that the proposed STRE method will greatly improve the performance of high-dimensional predictive modeling in space-time systems.where x25xt is split into its positive part xt can be denoted as yt and xt to denote y and x here and later on. Then the term that only depends on vectxt in the objective function becomesThis objective function involves both spatial and temporal correlations, and is difficult to be solved analytically. Iterative algorithms are commonly used such as the multiplicative update method which, however, requires nonnegative constraint of I is an identity matrix whose dimension is the same as the Laplacian matrix Δs. We substitute where A+ and A− are the positive and negative parts of matrix A, whose definition is similar to that of where matrix In the present investigation, the proposed STRE model is implemented to predict the time-varying distribution of electric potentials on the heart surface from real-world sensor data of electric potentials on the body surface. The model performance is evaluated and validated in both a simulated two-sphere geometry and a realistic torso-heart geometry.R. A time-varying three-dimensional current dipole p(t) = (px(t), py(t), pz(t)) is placed at the center of the two-sphere geometry, which is defined ast ranges from 0 ms to 300 ms. Thus, the dynamic distributions of electric potentials on the inner surface x and outer surface y are calculated analytically by the equationswhere time σ = 1 is the electric conductivity inside the outer sphere, rH(s) and rB(s) denote the location vectors from the center to the inner and outer spheres, respectively, and rH = 1.0 and rB = 1.5 are the radii of the two spheres.where y on the outer sphere calculated by λs = 0.015 and λt = 0.5 are chosen by the L-curve methodw is specified to be 2. In our simulation studies, Gaussian noises with mean zero and variance y. Five different noise levels  are added at each time, which correspond to noises with standard deviations σε = 0.1, 0.2, 0.3, 0.4, 0.5, respectively. At each noise level, the predicted potentials on the inner sphere will be compared with the true data  calculated by The proposed STRE model is implemented to predict the electric potentials t ranges from 0ms to 1000 ms) on the heart and body surfaces, and the torso-heart geometry are obtained from the Center for Integrative Biomedical Computing (CIBC) at the University of Utahy are acquired from 367 sensors, which are located at 367 nodes on the body surface. Thus, a 367 × 257 parameter matrix R is generated. The STRE model is implemented to predict the potential distribution y. Regularization parameters λs = 2.0 and λt = 0.005 are chosen by the L-curve methodw is specified to be 2.Furthermore, we conduct experiments in the realistic torso-heart geometry, as shown in y to simulate the real-world uncertainties in this torso-heart geometry. The five noise levels are with standard deviations σε = 0.005, 0.01, 0.05, 0.1, 0.2, respectively. The estimated electric potentials on the heart surface from high-dimensional predictive modeling will be benchmarked with real-world sensor data of reference potentials.Similarly, five different noise levels  are added to the electric potentials on the body surface The performance metric, relative error (RE), is used to evaluate the model performance, i.e.,x denote the estimator and reference results, respectively. The performance of our STRE model is benchmarked with Tikhonov zero-order (Tikh_0th), Tikhonov first-order (Tikh_1st) and L1 first-order (L1_1st) regularization methods. In these first-order regularization methods, the matrix Γ is defined as the normal derivative operator of the electric potentials on the inner surface7where y of the outer sphere. Note that the proposed STRE model yields the RE of 0.006, which is significantly smaller than that obtained from Tikh_0th, Tikh_1st and L1_1st, which are 0.1475, 0.1026, and 0.1025, respectively.y of the outer sphere. In the present investigation, we replicated the experiment 20 times for each noise level, and thus the resulted RE is shown with a corresponding error bar . When the noise level increases from σε = 0.1 to σε = 0.5, the RE monotonically increases for all the methods. Specifically, the RE increases from (0.0670 ± 0.00057) to (0.0769 ± 0.0034) for the proposed STRE model, from (0.1557 ± 0.00058) to (0.2080 ± 0.005) for Tikh_0th, from (0.1037 ± 0.00031) to (0.1538 ± 0.0031) for Tikh_1st, and from (0.1046 ± 0.0004) to (0.1569 ± 0.0041) for L1_1st. Notably, the STRE model yields the smallest RE for all noise levels, and achieves the slowest increase of RE with respect to the noise level among various regularization methods.mV to 2.5 mV. Note that the potential distribution on the inner sphere is dynamically varying over time, and t = 150 ms. y of the outer sphere. Note that the predicted potential mapping by the STRE yields a smaller RE of 0.006 compared to that of Tikh_0th , Tikh_1st  and L1_1st , which achieves the best performance to predict the reference potential mapping shown in σε = 0.5 in y of the outer sphere. Notably, the predicted potential mappings by Tikh_0th, Tikh_1st and L1_1st under this noise level show different color patterns from the results under the condition of no noise, and their RE’s are 0.208, 0.1528 and 0.1569, respectively. However, the predicted mapping by the proposed STRE model closely preserves the color patterns of the results with no noise, and yields the smallest RE of 0.0769.Furthermore, As shown in y of the body surface. In the present investigation, our STRE model yields a much smaller RE of 0.0997 compared to that of Tikh_0th , Tikh_1st  and L1_1st . Note that the RE’s of all the methods in this realistic torso-heart geometry are relatively bigger compared to the results in the simulated two-sphere geometry when no extra noise is added to y. This is mainly due to the fact that y are real-world BSPM data with measurement noises and other uncertainty factors in the inverse ECG problem, while that in the simulated two-sphere geometry are clean data calculated analytically by y on the body surface, we added different levels of noises to increase the real-world uncertainties on y. In the present investigation, we also replicated the experiment 20 times for each noise level, and thus each resulted RE is shown with a corresponding error bar . When the noise level increases from σε = 0.005 to σε = 0.2, the RE monotonically increases for all the methods. Specifically, the RE increases from (0.2386 ± 0.0105) to (0.4933 ± 0.0175) for the proposed STRE model, from (0.5570 ± 0.0025) to (0.8521 ± 0.0086) for Tikh_0th, from (0.9720 ± 0.0115) to (2.8261 ± 0.1835) for Tikh_1st, and from (1.2481 ± 0.0082) to (2.8994 ± 0.1849) for L1_1st, respectively. It is worth mentioning that the RE’s increase dramatically when adding noises to y on the body surface, compared to the results in the simulated two-sphere geometry. This is mainly due to the fact that the realistic torso-heart geometry is much more complex and irregular. As such, the resulted high-dimensional prediction model tends to be more sensitive to noises. Nevertheless, our STRE model yields the smallest RE for all noise levels, and achieves the slowest increase of RE with respect to the noise level among various regularization methods in this realistic torso-heart geometry.mV to 15 mV. Note that the potential distribution on the heart surface is dynamically varying over time, and t = 50 ms. y of the body surface. Note that the proposed STRE yields the RE of 0.997, which is significantly smaller than that of Tikh_0th , Tikh_1st  and L1_1st , and yields the best performance to predict the reference potential mapping shown in σε = 0.005 in y on the body surface. It is worth mentioning that the predicted potential mappings by Tikh_0th, Tikh_1st and L1_1st under this noise level show significantly different color patterns from Furthermore, As shown in x and y. This poses significant challenges for high-dimensional predictive modeling  = Rx + ε) in complex systems . First, inferring x needs a better knowledge of parameter matrix R that characterizes the physics-based interrelationship between x and y. Second, ill-conditioned systems make the predictions more sensitive to measurement noises and approximation errors in R. Third, very little has been done to develop new spatial regularization methods that handle approximation errors, as well as new temporal regularization methods to increase model robustness to measurement noises. Thus, there is an urgent need to tackle these research challenges and address ill-conditioned problems in high-dimensional predictive modeling.Advanced sensing and imaging technology lead to the proliferation of spatiotemporal data R. Second, we developed a physical-statistical approach that integrates physics-derived parameter matrix R with a spatiotemporal regularization method to build the high-dimensional predictive model. Third, we designed a new method of dipole multiplicative update, inspired by the dipole assumption in electrodynamic physics, to solve the generalized spatiotemporal regularization problems.In this paper, we developed a physics-driven spatiotemporal regularization (STRE) model for predicting dynamic behaviors in space-time systems. First, we developed realistic models of torso-heart geometry, and utilized the boundary element method and physics-based principles  to derive the parameter matrix The proposed STRE model is implemented to predict potential distribution on the heart surface using BSPM data. The model performance is evaluated and validated in both a simulated two-sphere geometry and a realistic torso-heart geometry. Experimental results show that our method not only effectively tackles the ill-conditioned problems in high-dimensional predictive modeling, but also outperforms those regularization models widely used in current practice . The present research work provides a new and effective approach to investigate disease-altered electric potentials on the heart surface in healthcare systems.How to cite this article: Yao, B. and Yang, H. Physics-driven Spatiotemporal Regularization for High-dimensional Predictive Modeling: A Novel Approach to Solve the Inverse ECG Problem. Sci. Rep.6, 39012; doi: 10.1038/srep39012 (2016).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.