Unfortunately this is computationally impractical, particularly if data reconstruction is iterative or the acquisition protocol is dynamic. We develop a flexible statistical linear model analysis to be used with multi-frame PET image data to create valid bootstrap samples. The technique is illustrated using data from dynamic PET studies with fluoro-deoxyglucose (FDG) and fluoro-thymidine (FLT) in brain and breast cancer patients. As is often the case with dynamic PET studies, data have been archived without raw list-mode information. Using the bootstrapping technique maps of kinetic parameters and associated uncertainties are obtained. The quantitative performance of the approach is assessed by simulation. The proposed image-domain bootstrap is found to substantially match the projection-domain alternative. Analysis of results points to a close relation between relative uncertainty in voxel-level kinetic parameters and local reconstruction error. This is consistent with statistical theory.PET imaging is an important diagnostic tool for management of patients with cancer and other diseases. Medical decisions based on quantitative PET information could potentially benefit from the availability of tools for evaluation of associated uncertainties. Raw PET data can be viewed as a sample from an inhomogeneous Poisson process so there is the possibility to directly apply bootstrapping to raw projection-domain  The work here is stimulated by (Positron emission tomography (PET) is a well-established radiotracer imaging technique, extensively relied on in both secondary and tertiary clinical care settings, as well as in medical research. As the role of quantitative PET in clinical decision making evolves, it is likely that there will be an increasing interest in the availability of practical methods for evaluating uncertainty associated with the results reported for an individual patient. There is already a significant literature on variance assessment for PET. Much of this work has concentrated on the development of analytic approaches based on the linear approximations to the reconstruction process – see, for example, . The potlated by , who uselated by , with thlated by  proposede.g. dynamic PET studies. In this setting suitable modeling of the dynamic data enables us to identify an associated set of residuals that can be manipulated to construct a viable model-based image-domain bootstrapping procedure.The work here develops a more flexible procedure for image-domain bootstrapping. This new approach is applicable in situations where there may be a complex temporal structure in the measured PET data – a near-constant temporal structure is intrinsic to the method used in . In addiThe technical framework for the methodology is set out in 2.N × T array, {ijz; i = 1, ., N, j = 1, …, T} . Here N represents the number of voxels in the field of view and T is the number of time-frames in the PET acquisition. ijz is the reconstructed PET-measured tracer concentration value at the 3-D voxel co-ordinate, ix, at a time jt corresponding to the mid-point of the j’th time-frame of scanning. We begin by providing a formal mathematical description of the model, highlighting the various unknowns that must be estimated before it can be used for bootstrapping.The input data for the approach is a 4-D dynamic PET dataset represented by an 2.1.ijz be denoted ijμ and X = {kμ, k = 1, 2, …, K}, that have the ability to approximate local mean values, ijμ. The method used to identify X is described in ijz is expressed as a sum of systematic and random terms. The basic version is given byLet the true mean and variance of the PET measurement jx is the j’th row of X and iα is a vector of unknown coefficients. Note that the systematic part of ijz, here ijμ, is approximated by a linear form. The model is referred to as a generalized linear model because errors are allowed to be non-Gaussian and the scale factors, ijσ, are not assumed constant  and temporal (jϕ) factors. The constraint on ϕ is required for identifiability. A key part of ijϵ is related to a sub-ordinate Gaussian variable, ijη, by a (unknown) Q-transform. In statistical parlance, the inverse of Q defines the normal-quantile plot for the collection of measurement errors  = 1. Monotonicity of Q is a requirement for the distribution of the measurement errors to be well-defined; it does not restrict the flexibility of the model to adapt to arbitrary distributional forms for the error. Strict monotonicity implies that the quantile mapping is invertible. If the model errors have a continuous distribution - a very plausible approximation for PET and many other types of medical imaging data - then Q must be strictly monotone. The sub-ordinate Gaussian η-process, η = {ijη, i = 1, 2, …, N, j = 1, 2, …, T}, is assumed to have independent temporal components, matching the formal Poisson structure of PET ,t errors . We assue of PET , and a ce of PET . Note thblished.  providesijκ as a surrogate variable for representing deviations from the product form of the variance and the assumed distributional structure. This leads to a more elaborate generalized linear modelQ(·∣κ) is strictly monotone for each fixed κ-value and is assumed to be slowly varying as a function of κ. For identifiability, a scaling constraint involving Q is also required. For this with h2(κ) = var(Q(η∣κ)), we require Q does not vary with κ. If Q(·∣κ) corresponded to a scaled Gamma distribution then, based on 2 be proportional to ∣κ∣, iσ proportional to While the model in based on , κ wouldorted by . Thus thorted by . Regardlη can be written in the form of a Kronecker tensor product, ΣN ⊗ TI, where ΣN is a N × N matrix representing the spatial correlation pattern and TI is an T-dimensional identity matrix. Spatial stationarity implies that the covariance can be diagonalized using the Fourier transform  of detected events. If events are binned into count arrays, this is equivalent to drawing a random sample from a multinomial distribution with eN trials and probabilities proportional to the observed counts in binned array. Each simulated count array is processed to produce a bootstrap reconstruction, z*. Repeating the process BN times leads to a projection-domain bootstrap sample, list-mode re-sampling, we do not use it here. Note that as the count-rate increases, there will be little percentage difference between counts produced by either sampling method.This approach involves random sampling (with replacement) from the number  is readily expressed in terms of the associated model coefficient - i.e. iθ = f(i·μ) = f(ix′α) ≡ g(iα). Recall that a basic property of maximum likelihood is that if iα then i, c.f.θ  = ΣK = [X′WX]−1. iD is a spatially invariant transformation of the error process and also of the underlying sub-ordinate Gaussian process η. Thus for N × K array, D, with rows iD’s, is strictly stationary in a spatial sense. Now if the error in D has the Kronecker product form ΣN ⊗ ΣK, where ΣN is the spatial covariance of η. Thus simulated realizations of D can be produced by creating a N × K array whose rows are independent samples from a K-dimensional Gaussian with zero mean vector and covariance ΣK and then transforming the columns of the array by multiplication by the matrix N × K dimensional array using i’th row. An obvious modification of this would be to use the full set of We propose a novel 2.3.X, is specified and after that consider the other elements.As indicated earlier, the model in The goal in basis selection is to choose a configuration that is physiologically interpretable and has the ability to approximate the measured time-course data at all voxels in the volume. A clustering scheme is used to identify clusters of time-courses that have self-similar shapes. If a basis can be found to represent the average time-course in such clusters, then it can be expected that a simple scaling of the representation for the average will fit individual time-course data in the cluster. In light of this, the basis selection is optimized so that the set of cluster means are well represented. But because the number of clusters is quite small  relative to the number of voxels, evaluation of the objective function for assessment of any candidate basis set based on the reduced cluster-mean data is computationally efficient. A generalized cross-validation criterion is used as an objective function for basis set assessment. Backwards elimination, a well-established type of greedy algorithm in statistical model selection , is usedlC, l = 1, 2, … L}, with the property that the data in each cluster has a similar shape pattern: i.e. if i ∈ lC then for a suitable constant ia, ijz ≈ iμjla where j = 1, 2, …, T is the mean time-course for the l’th cluster. Step 2 takes the collection of mean time-course patterns associated with each of these clusters as an initial set of basis elements Detailed implementation of the above basis selection scheme substantially follows . The proX fixed, an unconstrained weighted least squares procedure, with a simple fixed temporal weighting scheme, is used for estimation of iα’s in ϕ-values areWith j’th time-frame and m is taken to be a fraction (0.1) of the maximum of these jdt and jζ−te, with ζ the decay constant for the radio-tracer isotope, is the standard tracer decay factor. While the estimates of iα based on these simple weights may not be optimal, under general conditions weighted least squares estimates are unbiased and also consistent, as the scale of the error diminishes  = −∞ and κl) seems to be quite adequate. As Q is piecewise constant, its variance is also piecewise constant - ijh = lh for lh given σ and ϕ isσ, ϕ and h. This defines K is the number of columns of X. This is to ensure there is adjustment for the bias arising from fitting the α-coefficients at each voxel. Such adjustments are standard in most parametric model fitting settings - see ; the second comes from a more recent breast cancer imaging trial with 18F-labeled Fluorothymidine (FLT) (We present two dynamic PET imaging studies of cancer patients. One is from a series reported in  and invone (FLT) . The studraws on .3.1.C, can be described by a convolution between the local arterial blood concentration supplied by the output of the left-ventricle (LV) of the heart, pC, and the corresponding impulse-response, or so-called tissue residue function R. ThusThe basic principle of most tracer imaging studies with PET is that the tracer’s interaction with the local tissue is linear and time-invariant. While there are situations where this assumption is not valid, it is very reasonable for FDG and FLT. The arterial supply is the primary system for transport of tracer to tissue, so linearity and time-invariance implies that the true tissue concentration, pC = pC (s − Δx) where pC(t) is the LV blood concentration and Δx is a suitable delay. From basic principles of blood-tissue exchange . This is a version of the form used in α-coefficients are constrained to be non-negative. In the NPRM setting, the backwards elimination scheme in LX by elements based on non-parametric residue modeling of the cluster mean data – i.e.There is a substantial literature on modeling techniques for PET time-course data. Most of this focuses on the analysis of region of interest data, see, for example, . But thecer flux .kR, k = 1, …, K}, i.e.As the basis elements are represented in terms of residue functions, Here a very short duration (less than 5 seconds) spiked residue is used to represent the AIF, a constant residue is used for the Patlak term.ET is the study duration and let BT for 0 < BT < ET represent a realistic upper-bound for large-vessel travel-time – for human PET studies with FLT and FDG, a value BT of around 5 to 10 seconds is reasonable physiologically. The residue over the observed time-frame of the study, , is decomposed in terms of vascular (BR), in-distribution (DR) and extraction (ER) components asIn the NPRM approach, decomposition of the voxel-level residue is used to create summaries variables for mapping tracer kinetics. Suppose iK(x). This is a measure of flux (see further BV(x), as well as the in-distribution flow, DK(x), and volume, DV(x), are recovered from BR and DR asThe apparent rate of extraction of tracer atoms by the tissue is measured by DV(x)/DK(x). However this does not take account of variation in the time of arrival of the tracer to the local tissue. To address this we use a flow-weighted average of mean transit-times associated individual sub-TACs – i.e. MTT(ix) = ΣkikwMTTk where MTTk = Δk + DkV/DkK with DkK and DkV obtained from k’th component residue (kR). Here the weights, which are normalized to sum to unity, are proportional to the flow contributions from the different component tissues represented at the i’th voxel, iki.e w ∝ ikKDkα for k = 1, 2, … K. Thus MTT is written asBy the central volume theorem , the meaw(ix) = ΣkikwΔk. In the case that the individual delays are all the same, Δw(ix) will constant across voxels.Δk1, k2, k3, k4). The impulse response function (a.k.a tissue residue) for the model is a mixture of exponentialsbf), this gives rise to a model representation of the tissue time-course (TC) asWhile a non-parametric approach is used for specification of sub-TAC residues, it is useful to record what the above summary parameters correspond to in the 2-compartmental model of Huang and Sokoloff, see . This mooR defined over the interval  bySuppose we introduce a simple linear residue BT → 0, oR becomes very spiked at 0. For small BT and the convolution of oR with the AIF is approximately bCpf(t). As a result, for sufficiently small BT, the 2-compartmental model can be given a residue representationCR(t) = oR(t) + (1 − bf)CI(t). Decomposing CR as described in K1 = (1 − bf)k1 and O(x) indicates that corresponding terms vanish as x → 0. If k4 = 0 then λ2 = 0. Here as BT → 0 with ET large (so eλ1ET− is negligible) the residue summaries for the 2-C model becomeNote k3 and k4 are zero, the 2-compartment model reduces to the 1-compartment Kety-Schmidt model (iK = 0 (no retention), BV = bf, DK = K1 and 15O-labeled water.When both dt model . Here thz). Individual bootstrap realizations e.g. such as those reviewed in  and T = 31 time-frames extending over a 90 minute period. The time-frame sequence is: 1(1 min) preinjection, 4(20 sec), 4(40 sec), 4(1 min), 4(3 min) and 8(5 min). The bootstrapping technique is used to evaluate sampling variation in computed metabolic images.PET studies with FDG play a major role in the diagnosis and management of many cancers . Spence reportediK) and extraction (iK/K1), in the tumor region. Standard errors demonstrate that variability is very much related to the scale of the metabolic variable. This pattern is likely a consequence of the overall pseudo-Poisson characteristic of PET data so areas with high metabolic values  also have greater absolute variance. We examine this more formally in iK/K1) values. As described in . The data considered here are from a patient studied at baseline (before chemotherapy). The study was conducted on a 74-plane scanner using a 3-D acquisition process and ML reconstruction. The 4-D PET data set consists of an imaging volume with N = 168 × 168 × 74 voxels (2.97 × 2.97 × 2.01mm3) and T = 45 time-frames of acquisition over one hour. The time-frame binning sequence was: 16(5 sec), 7(10 sec), 5(30 sec), 5(1 min), 5(3 min) and 7(5 min). Note that more than half of the temporal sampling is focused on the first 2.5 minutes of the 1-hour acquisition. This is in part because the kinetics of FLT  are faster than those of FDG. The left-ventricle of the heart was used to directly recover a blood time-course, which after approximate adjustment for metabolites provided an arterial input function, pC, used for kinetic analysis.Cellular DNA is replicated during cell division so that its concentration in rapidly proliferating tumor tissues can be expected to be higher than in normal tissue. PET FLT imaging has the potential to provide an approximate measure of DNA concentration and for this reason FLT imaging may be able helpful for diagnosis and treatment planning with certain cancers. Our data is from a multi-center American College of Radiology Imaging Network clinical imaging trial (ACRIN 6688) which conducted dynamic PET-FLT imaging of breast tumor before and during neoadjuvant chemotherapy. The goal was to see if PET-FLT imaging could give an early indication of the tumor response – trial results are reported in . The ACRBN = 500 image-domain bootstrap replicates were used for evaluation of sampling variation in computed metabolic images. Metabolic images and associated bootstrap estimates of their voxel-level standard errors are in dV), flow(Kd) and flux (iK) show significant enhancement in the tumor region. In a compartmental modeling framework, all three parameters have been suggested as appropriate ways to quantify FLT time-course data (iK/K1) appear quite stable as is the associated uncertainty measure. VOIs for tumor and contra-lateral normal breast were also accessed. The tumor region had 1280 voxels extending over 16 slices ; the normal VOI is not quite as large - 1054 voxels - but with a very similar shape. The bootstrap estimated histograms of the sampling distribution of the 95th percentile of the metabolic parameters in the VOIs are shown in bV). These differences are highly significant when formally assessed via the bootstrap information.Data were processed using the same methods as used for the brain study. rse data . Standarκ-bins shows that as κ-increases, there is increasing conformity to the Gaussian. As discussed in κ is proportional to the shape parameter and as that parameter increases the Gamma distribution formally converges to a Gaussian. Thus the data are in line with  are needed to evaluate the true target variance with reasonable accuracy. Bootstrapping techniques involve simulation and the number of such simulations (BN) also needs to be considered. Apart from the computation, in practice the storage requirements associated with retention of bootstrap samples may also be a factor. We will report studies in which both modest and large numbers of bootstraps are examined.The purpose here is to evaluate the performance of the novel image-domain bootstrapping technique for PET and make comparisons with the more computationally intensive, but fully non-parametric, projection-domain approach. Assessment of variance estimators is somewhat complicated because we do not have an analytic formula for the true target variance. Hence a number of replicate simulations  reconstruction only, but iterative ML and analytic reconstruction are considered in the 1-D case. Dynamic aspects of both 1-D and 2-D studies are based on results obtained for the FDG brain and FLT breast cancer data presented in 4.1.π], discretized to a 183 × 181 sinogram array of distances and angles. Note the π-periodicity of the parallel beam Radon transform restricts the angular extent of the projection domain. As shown in Rτ. This dose is specified so that the voxel-level noise in the reconstructed data matches the apparent voxel-level noise level of the real data. Independent Poisson count simulation from each element of the scaled sinogram array yields the synthetic projection-domain data (y). Each frame of the sinogram data is reconstructed analytically using a standard filtered backpojection (FBP) procedure with the raw ramp-filter result smoothed by convolution with a Gaussian resolution filter. The resolution filter bandwidth is required to be common across all time frames and by grid-search its value is selected in order to minimize the average squared error difference between the estimated and true activity summed over frames. This choice of bandwidth is to ensure that the uptake image is objectively adapted to the study dose , the image-domain and the approximate image-domain bootstraps are acquired. i = 1, 2, …, N. Here ipσ is the true standard deviation of the p’th metabolic parameter estimate for the i’th voxel; R2 exceeds 0.9 for all metabolic parameters and for both FDG and FLT simulations – values for flux (iK) are shown in Quantitative comparisons between the estimated and true standard deviations were assessed on a voxel-by-voxel basis using a simple linear regression analysis and also in terms of an overall root mean square error (RMSE) measure. Regression analysis considered the relation between the true standard deviation, estimated by replication, with the corresponding values evaluated by the bootstrap methods. These regression analysis models are expressed asa, b, c), in p’th metabolic parameter estimate at voxel i for the s’th replicate data for the bootstrap estimate, the RMSE is given byThe regression parameters,  but not for other parameters . RMSE values for the approximate image-domain bootstrap are consistently the largest. However it needs to be appreciated that, in light of the typical standard error of the true standard deviation, the significance of any of these differences is small. Standard errors are reported as The RMSE values for projection-domain and image-domain methods are denoted iK) is also shown in i.e. the voxel indicator (i) is replaced by an indicator of the ROI. Similar to the voxel case, bootstrap estimates perform very well. There is no evidence that these results are substantially impacted by the size-distribution or positioning of ROIs. The non-parametric and image-domain method (with BN = 25 replicates) have very similar RMSE reliabilities both for FDG and FLT. Both bootstraps tend to under-estimate the true ROI mean standard deviation. However, the amount of bias is small - on the order of 10–14% across the different kinetic parameters. Recycling is found to lead to more unreliable values - largely due to a greater systematic over-estimation of the true ROI standard deviation - on the order of 20% for most parameters. But the overall indication from the 2-D experiments is that image-domain bootstrapping scheme is very well aligned with the non-parametric projection domain approach and provides a viable mechanism for assessments of uncertainties at both the voxel and ROI level.For this analysis a nested sequence of grids were used to construct a range of ROIs with different sizes and tissue heterogeneity characteristics. 4.2.x = 1, 2, ., N (even) and T = 1, 2, …, T. Here 0 < xa ≤ 1 is a known attenuation factor and the matrix βK : NR → NR and βK has the form of a discrete convolution. Letting x ∈ NR, ν = ±1, ±2, …, N/2. With β > 0, the action of βK is to smooth the vector x. If y is a realization of a Poisson with mean τγ, then an unbiased estimate of the underlying source distribution λ is obtained by adjusting for the y-data by the attenuation factor and applying a least squares (LS) inversion procedure. This result is then smoothed to achieve consistent mean square error performance. As FBP is essentially equivalent to LS in the 2-D setting xt = xty/xa.Letting hS has a discrete Fourier representation, zfbp)( and z are efficiently computed using the 1-D FFT. Adapting (zml) estimation characteristic as a function of dose for the 1-D Poisson deconvolution model, and the corresponding MSE characteristic of 2-D PET reconstruction.If Adapting , the EM Toeplitz . In d-diorted in  show thaN = 128. The temporal patterns are matched to those arising in the 2-D simulations. Spatial patterns are indicated in βαjK’s, and the attenuation pattern. Reference dose-values Rτ were again chosen so that the qualitative variability of simulated 1-D data matched that seen at the voxel-level in the real FDG and FLT data. Five dose levels, τ = Rτ/5, Rτ/2.5, Rτ, 2.5Rτ, 5Rτ were examined with FDG and FLT. In the 1-D setting a more extensive bootstrapping process was used . Simulated data were reconstructed using FBP and iterative ML techniques. Raw frame-by-frame reconstructions were smoothed by convolution with a Gaussian kernel. Similar to the 2-D case, bandwidth was common across all frames, and its value was optimized according to the mean square deviation of the estimated total uptake from the true known source. Due to the implicit regularization associated with raw ML reconstruction , whose standard deviation by the image-domain methods are consistently lower than reported by the projection-domain bootstrap, there is remarkably close alignment between the methods.The estimates of the α-coefficients should be approximately W is the diagonal matrix with elements j = 1, 2 …, T. The NPRM kinetic analysis procedure involves fitting the model α-coefficients are non-negative. By θ = , are simple functions of the constrained α-coefficients. Assuming the unconstrained α-coefficients are sufficient for the constrained values, θ can also be regarded as a function of the unconstrained coefficients. Hence we can consider the estimated kinetic parameters as functions of the unconstrained α-coefficients g : KR → PR (P = 6). By application of the delta method, e.g.  that have a linear dependence on the constrained α-coefficients. Hence αℓ may depend on the local α-value. In spite of this, the analysis suggests a relation between the voxel-level parameter standard deviation and the standard deviation of the measurement. In the simulation setting, the square-root of the weighted mean square reconstruction error can be used as an assessment of measurement error: 1γp1 as well as the quality of fit of the model measured by the R2 statistic. The model fits are remarkably good (most in excess of 80%) particularly for . With MTT the model pattern is continues to be remarkably accurate for FLT; but not for FDG. On the other hand the model variance pattern for extraction (iK/K1) is still very reasonable for FDG but not for FLT. While these analyses give an understanding of the behaviour of voxel-level kinetic parameter variability, they also help to provide some underpinning for the basic theoretical heuristic for the approximation used to recycle the image-domain bootstrapping. Further studies were conducted in 1-D in order to evaluate the accuracy of the bootstrapping techniques as a function of the size of the imaging domain (i.e.K/T). But in PET, dose constraints mean that reconstruction error will increase with temporal sampling. So the ability to manipulate relative RMSE performance merely by increased temporal sampling would be unrealistic.The relation between voxel-level parameter standard deviation and reconstruction error was also examined. Our analysis is motivated by the approximation used in constructing the recycling process for the simplified image-domain bootstrap in od, e.g. , the covg domain . Remarkag domain , the pri5.The work has presented a novel image-domain approach to bootstrapping PET data having a dynamic component. The method is based on a novel general linear model approximation of the dynamic source distribution in which the error is described in terms of a sub-ordinate Gaussian process that is assumed independent across time-frames and stationary in the imaging space. In addition the distribution of error is allowed to adapt to the local skewness of the data. Thus there is no requirement for the method to be modified depending on whether an analytic or iterative data reconstruction process is used. This removes a number of potentially limiting assumptions used in . The booOur methodology is illustrated by application to real examples involving the use of dynamic PET imaging for the purpose of mapping metabolic parameters of tissues in the field of view. The general linear modelling analysis technique enables us to create bootstrap replicate data sets for analysis. Application of the NPRM kinetic mapping technique to the bootstrap data provides voxel-level estimates of metabolic parameters and their associated uncertainties (SEs). Numerical studies motivated by these examples compare the image-domain bootstrapping approach to the more computationally demanding but fully non-parametric projection-domain approach . Image-dhttps://cran.r-project.org) in the near future. In comparison to projection-domain bootstrapping, in which each bootstrap replicate requires reconstruction of simulated list-mode data; computation of image-domain bootstrap replicates are negligible. But of course significant computation is required to setup the image-domain bootstrapping model. For the 3-D data sets analyzed in The current analysis is implemented in  - an opee.g. National Cancer Institute (https://www.cancerimagingarchive.net), do not to our knowledge include any list-mode information. Thus the analysis of image uncertainty using data from such archives can only be based on a suitable image-domain bootstrapping approach, as we have described here.Of course as computing capabilities grow and it becomes standard to retain list-mode data for dynamic PET studies, non-parametric projection-domain bootstrapping may become practical as well. Indeed this would be an ideal circumstance because the theory underpinning the non-parametric approach is very well developed . But curFor situations where retention of extensive bootstrap samples is prohibitive, we have proposed a novel recycling process as an approximate image-domain bootstrapping approach. While the RMSE performance of this approximation is not as good as a full bootstrap, the results are still quite reasonable. Importantly our studies indicate that the performance of the image-domain simulation techniques are not impacted by whether or not the reconstruction methodology is analytic (FBP) or iterative (ML).It would be interesting to use the methods here to examine multiple studies with a similar anatomy on the same scanner, in order to develop a practical scanner-specific understanding of the study-to-study stability of the image-domain model estimates. If list-mode data were available, it would also be possible to use the projection-domain bootstrap to gain some insight into this. Specifically such a bootstrap could be used to create replicate projection-domain scanning data that could be used to estimate image-domain model parameters. The collection of model parameters obtained across bootstrap replicates would then provide a direct assessment of their sampling characteristics.There are a number of other medical imaging modalities where subject-specific assessment of uncertainty in quantitated imaging measurements are not currently available and the techniques here might be useful. For example, dynamic imaging with MR  or CT are routine in the clinical management of cancer and stroke and the structure of datasets is substantially similar to dynamic PET. This will be a focus for future work.


Nature Communications 10.1038/s41467-020-15812-w, published online 20 April 2020.Correction to: The original version of this Article contained errors in Figure 2b, where the y-axis was labelled “cm” when the unit should have been “mm”, and the y-axis error bars should have been omitted. These have been corrected in both the PDF and HTML versions of the Article.


Communications Biology 10.1038/s42003-021-02017-0, published online 27 April 2021.Correction to: In the original version of the Article, reference 61 was incorrectly given as: “Nikolai. Trait-based range expansion aided in the global radiation of Crocodylidae (2019).”Glob. Ecol. Biogeogr. 28, 1244–1258 (2019).”This has been corrected in the HTML and PDF versions of the Article to: “Nicolaï, M. P. J. and Matzke, N. J. Trait-based range expansion aided in the global radiation of Crocodylidae. In addition, the reference to “Nikolai and Matzke” in the third paragraph of the Discussion has been corrected to “Nicolaï and Matzke” in the text in the HTML and PDF versions of the Article.


The deep Q-network (DQN) is one of the most successful reinforcement learning algorithms, but it has some drawbacks such as slow convergence and instability. In contrast, the traditional reinforcement learning algorithms with linear function approximation usually have faster convergence and better stability, although they easily suffer from the curse of dimensionality. In recent years, many improvements to DQN have been made, but they seldom make use of the advantage of traditional algorithms to improve DQN. In this paper, we propose a novel Q-learning algorithm with linear function approximation, called the minibatch recursive least squares Q-learning (MRLS-Q). Different from the traditional Q-learning algorithm with linear function approximation, the learning mechanism and model structure of MRLS-Q are more similar to those of DQNs with only one input layer and one linear output layer. It uses the experience replay and the minibatch training mode and uses the agent's states rather than the agent's state-action pairs as the inputs. As a result, it can be used alone for low-dimensional problems and can be seamlessly integrated into DQN as the last layer for high-dimensional problems as well. In addition, MRLS-Q uses our proposed average RLS optimization technique, so that it can achieve better convergence performance whether it is used alone or integrated with DQN. At the end of this paper, we demonstrate the effectiveness of MRLS-Q on the CartPole problem and four Atari games and investigate the influences of its hyperparameters experimentally. Reinforcement learning (RL) is an important machine learning methodology for solving sequential decision-making problems. In theory, by interacting with an initially unknown environment, the RL agent can learn the optimal action policies at different states to maximize the cumulative expected return . UnfortuCurrently, there are three main categories of research work on improving DQN. The first category mainly focuses on how to estimate action values accurately. For example, Hasselt et al.  proposedIn traditional RL, Q-learning algorithms often use linear functions to approximate action values, which have better stability and fewer hyperparameters to be trained than DQNs . In partIn our previous work , we propThe remainder of this paper is organized as follows. In this section, we briefly review the related theories and algorithms of our MRLS-Q, including the Markov decision process (MDP), DQN, and LS-DQN. In addition, we also describe some notations that will be used throughout this paper.𝒮, 𝒜, P, r, γ〉, where 𝒮 is the state space, 𝒜 is the action space, P ∈  and r ∈ ℛ are the state-transition probability and the immediate reward from the state st to the next state st′ by taking the action at, and γ ∈ , i.e.,J(π) is usually defined in the form of discount return [s0 is the initial state, whereas J(π) is hardly calculated by using the above equation directly, since P is unknown in RL, and st′ and rt can only be obtained by the agent's interaction with the environment.For a given MDP, the goal of RL is to learn the optimal policy π), i.e.,π∗=argmaxQπ to measure the performance of π when the initial state and action are st and at. In this paper, we assume that 𝒮 is continuous and 𝒜 is discrete. For this kind of MDP problems, to overcome the curse of dimensionality, Qπ is often approximated by linear function approximators or deep neural networks.To tackle this problem, RL usually resorts to estimating the action value DQN is probably the most important algorithm in deep RL. It combines the Q-Learning algorithm with deep neural networks and uses the experience relay for breaking the correlation among samples and training network parameters.t, the agent firstly uses the ϵ-greedy policy to select the action at asϵ is the exploration factor, Θt−1 is the policy network parameter, and Q is approximated by this network. Then, after taking at, the agent moves to the next state st′, obtains the reward rt, and stores  into the experience replay buffer 𝒟, where dt ∈ {0,1} denotes that st′ is the terminal state or not. Next, by using the minibatch 𝒟 randomly, the algorithm calculates the loss of the policy network ast−1 to Θt. For example, by using the SGD method [t−1 is updated asα is the learning rate and ∇t−1Θ denotes ∂L(Θt−1)/∂Θt−1.The DQN algorithm can be summarized as follows. At the current step D method , Θt−1 isLS-DQN is a hybrid approach, which combines the traditional LSPI or FQI algorithm with the DQN algorithm. By enjoying the stability and efficiency of LSPI or FQI, it can obtain better performance than DQN.tL in the policy network astL is the ith column vector of ΘtL. Besides, A and b are defined as follows:tL by using FQI, since it is a batch shallow RL algorithm that computes iterative approximations of the Q-function using regression. For brevity, we will not discuss the FQI algorithm in this paper.The LS-DQN algorithm can be briefly summarized as follows. Whenever the agent runs DQN some steps, it uses LSPI or FQI to retrain the last layer of the policy network once. The retraining consists of the following three substeps. Firstly, by recalculating all samples in the experience replay buffer with the current network parameters, the policy network generates a new dataset eir work , the algIn this section, we will introduce the detailed derivation and the practical implementation of our proposed algorithm, respectively. Our algorithm, the MRLS-Q algorithm, can be used not only alone but also as the last layer of DQN.MRLS-Q is a new Q-learning algorithm with linear function approximation, but it is more similar to the DQN algorithm rather than the traditional Q-learning algorithm. It uses the experience replay and the minibatch training mode, separates the linear function approximator into a policy approximator and a target approximator, and uses the state features rather than the state-action features. Besides, it uses an average RLS method for updating parameters.t, the agent also uses the ϵ-greedy policy to select the action at as equation  is approximated by the policy approximator asϕ(st) ∈ ℛN is the feature vector of st, Θt−1 ∈ ℛN × |𝒜| is the policy approximator parameter, in(a) denotes the index of a in 𝒜, and Θt−1) is the in(a)th column vector of Θt−1. Then, the agent takes at, moves to st′, obtains rt, and stores  into the experience replay buffer 𝒟.First, we introduce the agent's interaction with the environment. At the current step equation . Note he𝒟 at the nth step, and let λ ∈ 22) as foPt=At−1. By using the Sherman–Morrison formula for 27) into t−1Θ.As reviewed in If the loss function of MRLS-Q is defined by equation , by usinRecall the fact that we once used quations  and 24)kϕ¯tϕt¯T quations  and 20)kϕ¯tϕt¯T In addition, from equation , we can Using equation  yields11), the Using equations  and 40)40), we cTherefore, we can rewrite equation  as(42)Θtt in MRLS-Q.It shows that L1 regularization term into the above equation, i.e.,η is the regularization factor and sgn(·) is the sign function.However, although RLS has a fast convergence rate, it often suffers from overfitting. In recent years, there has been extensive research on this problem. Based on Ekşioğlu's work , we add t−1Θ can be calculated by the automatic differentiation package of PyTorch or TensorFlow directly. Besides being used alone, MRLS-Q can also be used as the last layer of DQN, since it uses the same loss function and experience replay as DQN. However, there is still an obstacle to the combination of MRLS-Q and DQN. As the training goes on, the parameters of the DQN network are continuously changing, and the outputs of the same inputs are changing as well. Thus, we cannot use the inputs of the DQN's last layer as the features of MRLS-Q directly. In order to alleviate this kind of change and integrate MRLS-Q into DQN, we present a new method to define the feature function of MRLS-Q asXtL ∈ ℛM×NL−1 is the output matrix of the DQN's penultimate layer and ν is a small hyperparameter to prevent the denominator becoming zero.Based on the above derivation, the pseudocode of MRLS-Q is summarized in In this section, we use two sets of experiments to demonstrate the effectiveness of MRLS-Q. Our experiments are divided into two sections. In In this set of experiments, we firstly verify the performance of MRLS-Q on the CartPole-v0 problem, which is from the OpenAI Gym. For comparison purposes, we build a new algorithm called Adam-Q, by replacing equation  with the4 random samples in the CartPole's state space, by using eight scikit-learn RBFSamplers [ϵ is initialized to 0.95 and is gradually decreased to 0.01 over 1000 steps. (3) The discount factor γ is 0.99. (4) The capacity of the experience replay buffer 𝒟 is 104, and the minibatch size is 32. The learning starts when the number in 𝒟 reaches the minibatch size. (5) The policy approximator parameter Θ0 and the target approximator parameter t each step. Note that the performances of both algorithms will get worse if we increase the update steps, since the CartPole problem is very simple and both algorithms converge fast. (7) The max norm of ∇t−1Θ is clipped to 1 by the L2 norm. (8) The two algorithms run five times and 100 episodes for each time. In each episode, each algorithm runs 200 steps at most. (9) Besides, in Adam-Q, the learning rate, β1, and β2 of Adam are 0.001, 0.9, and 0.999; in MRLS-Q, the initialization P0, the forgetting factor λ, the regularization factor η, and the approximation factor k are 0.5I, 1, 10−5, and 1/32, respectively. The average result of this experiment is shown in To compare the performance between MRLS-Q and Adam-Q, the experimental settings are summarized as follows. (1) Both algorithms use 400 radial basis functions (RBFs) for action-value approximation. These RBFs are generated from 10Samplers  with ker𝒟| ∈ {500,1000,5000,10000}, P0 ∈ {0.1I, 0.2I, 0.5I, I}, and k ∈ {(1/2), (1/8), (1/32), (1/128)}, respectively. The other settings of these experiments are the same as what we did for MRLS-Q in the previous experiment. The average results of these experiments are presented in Figures 𝒟 has a significant influence on the performance of MRLS-Q. The larger capacity will result in the better performance, since big 𝒟 is helpful to remove the correlation between the observed transitions. From P0, whereas too big P0 will make MRLS-Q become unstable and too small P0 will make MRLS-Q converge slowly. From k also has a significant influence on MRLS-Q. From equation (k will make Pt update with higher strength. If state feature values change greatly, k should be set to a big value.To investigate hyperparameter influences on MRLS-Q, we test |equation , bigger β1, and β2 of Adam are 0.0000625, 0.9, and 0.999; in the last layer of Hybrid-DQN, the initialization P0, the forgetting factor λ, the regularization factor η, the approximation factor k, and ν are 0.1I, 1, 10−8, 1/2, and 10−12, respectively. Note that here we use a big k to update Pt for adapting to the feature change.In this set of experiments, we verify MRLS-Q as the last layer of DQN on four Atari games: Pong-v0, Breakout-v0, SpaceInvaders-v0, and RiverRaid-v0, which are from the OpenAI Gym. Here we choose the traditional DQN algorithm with the Adam optimizer for comparison. For Adam-DQN and in the second to fifth layers of Hybrid-DQN, the learning rate, The average evaluation results are presented in L1 regularization technique to prevent overfitting, and propose a new method to define the feature function for alleviating the feature change of the same state and integrating MRLS-Q into DQN. Experiment results on the CartPole problem demonstrate that MRLS-Q has better convergence than Adam-Q and reveal the hyperparameter influences on MRLS-Q. In addition, experiment results on four Atari games demonstrate that DQN can improve convergence and stability by integrating with MRLS-Q.How to improve convergence and stability of the DQN algorithm is one of the key issues in deep RL. In this paper, we propose MRLS-Q, a linear RLS function approximation algorithm with the similar learning mechanism to DQN. MRLS-Q can be used not only alone but also as the last layer of DQN. Similar to LS-DQN, the Hybrid-DQN with MRLS-Q can enjoy rich representations from deep RL networks as well as stability and data efficiency of the RLS method, but it can seamlessly integrate MRLS-Q and thus is easier to use. In MRLS-Q, we use the experience replay to break the correlation between training samples, present an average RLS optimization method to improve the convergence performance and reduce the computational complexity, employ an


Data analysis in real life often relies mainly on statistical probability distributions. However, data arising from different fields such as environmental, financial, biomedical sciences and other areas may not fit the classical distributions. Therefore, the need arises for developing new distributions that would capture high degree of skewness and kurtosis and enhance the goodness-of-fit in empirical distribution. In this paper, we introduce a novel family of distributions which can extend some popular classes of distributions to include different new versions of the baseline distributions. The proposed family of distributions is referred as the Marshall-Olkin Weibull generated family. The proposed family of distributions is a combination of Marshall-Olkin transformation and the Weibull generated family. Two special members of the proposed family are investigated. A variety of shapes for the densities and hazard rate are presented of the considered sub-models. Some of the main mathematical properties of this family are derived. The estimation for the parameters is obtained via the maximum likelihood method. Moreover, the performance of the estimators for the considered members is examined through simulation studies in terms of bias and root mean square error. Besides, based on the new generated family, the log Marshall-Olkin Weibull-Weibull regression model for censored data is proposed. Finally, COVID-19 data and three lifetime data sets are used to demonstrate the importance of the newly proposed family. Through such an applications, it is shown that this family of distributions provides a better fit when compared with other competitive distributions. Selecting a suitable statistical distribution for modeling and analyzing data is of great importance in order to achieve more accurate decisions. Over the years, many statistical distributions have been proposed to fit different shapes of the data. Applying classical distributions to fit these data sets may lead to inaccurate results. Hence, the need for modifying the standard distributions is clearly evident.Recently, many methods have been proposed to obtain more flexible distributions that can reflect the data behavior in many situations more accurately. Adding a new parameter to existing distributions is one popular method that has attracted more attention in literature.  proposedX from a baseline distribution and such variable has a cdf G. Then, for the MO family, its cdf and its probability density function (pdf) are, respectively, given byα > 0 is the shape (tilt) parameter, τ is a vector of parameters for the baseline model. The MO transformation has some attractable characteristics which can add more flexibility to any baseline distribution. This encouraged many researchers to apply the MO transformation to common classical distributions such as the Pareto distribution by −1 and G))c(i+1)−1, the pdf for MOW-G family with α > 1 is obtained byNow by using Pt,m is defined by , with a parameter c can be respectively defined for an arbitrary G(x) asAt this point, the cdf and pdf for the exponentiated-G (Exp-G) distribution, denoted by Then, a linear combination of the Exp-G density functions serves as a way of expressing the pdf of the MOW-G family. Hence, starting from the properties of the Exp-G class of distributions, some features of the MOW-G family of distributions can be obtained. Such features have been investigated in several studies such as, –36, amonYj is a random variable from Exp-G distribution with parameter (c(i + 1) + m + r), then the rth moment of the MOW-G family is given byωj is defined in Let Yj is a random variable from Exp-G distribution with parameter (c(i + 1) + m + r) and ωj is defined in The moment generating function of the MOW-G family can be derived in term of the Exp-G distribution as followssth incomplete moment for MOW-G family is obtained as followssth moment for the Exp-G distribution with parameter (c(i + 1) + m + r) and ωj is defined in The Q(p), 0 < p < 1, of the MOW-G family is as followsThe quantile function, Therefore, the median is given byThe Rényi entropy for the MOW-G family is given byα < 1, (fMOWG(x))R can be obtained using For 0 < Therefore,Now, using G)R− and ))R(c−1)+ci, the Rényi entropy for the MOW-G family for α ∈ , is expressed asPt,m is defined in Applying Eqs  and 12)12), respα > 1 and using Now for A is defined in Applying α > 1, the Rényi entropy for the MOW-G family is written asPt,m is defined in Thus, for x1, x2, …, xn from MOW-G distribution with order statistics xn1: < xn2: < … < xn:n. Then the distribution of the qth order statistics of MOW-G distribution is given byConsider the random sample α ∈  and after some algebra, the distribution of the qth order statistics of the MOW-G is expressed as followsThen, for for α > 1, we can obtainz is defined in Similarly, for c(i + 1) + m + r). Thus, many proprieties of these order statistics can be achieved from the proprieties of the Exp-G.From Eqs  and 25)25), we cx1, x2, …, xn follows a distribution associated to the MOW-G class, with a parameters vector φ = , where τ represents the vector of parameters of the baseline distribution G, then the log-likelihood function (ℓ) is defined byTo obtain the estimators for the parameters of any member from the MOW-G family, the maximum likelihood method can be applied. That is, if we have The complexity of the above log-likelihood makes it difficult to solve analytically. Therefore, we take the derivative of φ can be acquired by obtaining the solutions in iterative way of the above nonlinear equations using the well-known Newton-Raphson method or any other numerical method. Instead, employing any standard non-linear optimization technique, the log-likelihood in Thus, the MLEs of the parameters vector γ and scale parameter δ. For each distribution, the simulation is performed over the number of iterations, nsimu = 1000. Four distinct sample sizes n = 25, 50, 100, 200, 500 are considered with the following cases for the true parameters, θtru.For MOW-E:For MOW-W:For evaluating the performance of the MLEs, some simulation studies is conducted for the two particular members of the MOW-G family namely; MOW-E distribution with rate parameter λ and MOW-W distribution with shape parameter We compute the mean square errors (MSEs) and biases of the MLEs of the parameters based on 1000 iterations, whereThe R programming language  has beenn increases, which indicates that the estimates are consistent. This provides evidence that the maximum likelihood method has a good performance when estimating the parameters of the proposed family.The “optim” function in R software is used to achieve all estimation results. The results of the simulation study are reported in Tables Four real-world data sets with the exponential and Weibull distributions as the G, are considered to investigate the flexibility of the proposed MOW-G family. The first data set is fitted to MOW-E distribution and compared with Marshall Olkin-G defined by , 38 and ℓ), Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), Cramér-von Mises (W*), Anderson-Darling (A*), Kolmogorov-Smirnov (KS) and P-value statistics. These statistics or criteria are widely used to assess the performance of a distribution in modeling a data set. Smaller values of these statistics, indicate a better fit.The proposed model is compared with other models according to some criteria, include the value of the log likelihood function  will have the LMOW-W distribution with the following pdf and survival function respectivelyα, c, β are defined in σ > 0 is the scale parameter and The regression model can be obtained by considering the log MOW-W (LMOW-W) distribution in which the location parameter depends on some explanatory variables For the standardized random variable Xi of the ith individual in the sample, for i = 1, …, n, and considering a set of p covariates such that vi = T, where the 1 is for the intercept term, the location-scale regression model relating to the response variable yi = log(xi) with the explanatory variable vector v can be mathematically described from the LMOW-W as followszi has density function in σ, α, c, β > 0 and τ = T are the unknown regression coefficients of the p explanatory variables. To illustrate, yi has the location parameter μ = T of the location parameter is described by the linear model μ = τV, such that V = T is a known model matrix.Assuming the survival time n observations (y1v1), …, , assume that F and C be the sets of individuals for which the response variable yi is the log-lifetime and log-censoring, respectively. Considering the non-informative censoring such that the observed lifetimes and censoring times are independent, the random response is determined by yi = {log(xi), log(ci)}. Then, the log-likelihood function for the vector of parameters ζ = T from model (f(yi) is the density function in S(yi) is the survival function in Yi. Thus, the log-likelihood function for ζ defined asr is the number of uncensored observations (failures) and For an independent sample of om model  can be oζ. For this case, numerical non-linear optimization procedures are commonly applied. The optim function in R can be used in order to find the MLEs.The log-likelihood function in v1 which represents the two-arms . Therefore, the LMOW-W regression model is considered asz1, …, z96 are independent random variables with pdf in The LMOW-W regression model is applied to fit data from an earlier study by , which cThe results from LMOW-W model are compared with some alternative regression models, namely; the log-Weibull (LW) in , the logResults in Weibull-G family introduced in  has advaS1 Appendix(PDF)Click here for additional data file.S1 Text(TXT)Click here for additional data file.


To the Editor:I enjoyed very much reading the article by Revilla-Martí et al. , describ


Introduction: The omnipolar electrogram method was recently proposed to try to generate orientation-independent electrograms. It estimates the electric field from the bipolar electrograms of a clique, under the assumption of locally plane and homogeneous propagation. The local electric field evolution over time describes a loop trajectory from which omnipolar signals in the propagation direction, substrate and propagation features, are derived. In this work, we propose substrate and conduction velocity mapping modalities based on a modified version of the omnipolar electrogram method, which aims to reduce orientation-dependent residual components in the standard approach.Methods: A simulated electrical propagation in 2D, with a tissue including a circular patch of diffuse fibrosis, was used for validation. Unipolar electrograms were calculated in a multi-electrode array, also deriving bipolar electrograms along the two main directions of the grid. Simulated bipolar electrograms were also contaminated with real noise, to assess the robustness of the mapping strategies against noise. The performance of the maps in identifying fibrosis and in reproducing unipolar reference voltage maps was evaluated. Bipolar voltage maps were also considered for performance comparison.Results: Results show that the modified omnipolar mapping strategies are more accurate and robust against noise than bipolar and standard omnipolar maps in fibrosis detection . They present better correlation with unipolar reference voltage maps than bipolar and original omnipolar maps .Conclusion: The modified omnipolar method improves fibrosis detection, characterization of substrate and propagation, also reducing the residual sensitivity to directionality over the standard approach and improving robustness against noise. Nevertheless, studies with real electrograms will elucidate its impact in catheter ablation interventions. Diagnosis and treatment of a wide range of atrial arrhythmias, including atrial fibrillation, require extraction of substrate and propagation features from intracardiac electrograms (EGMs) . This strategy identifies as fibrosis  method tries to cope with the problems affecting unipolar and bipolar measurements  by using two clique configurations: square and triangular cliques. These modifications are hypothesized to improve the accuracy and robustness of the voltage and CV estimates and to reduce the error induced by the b-EGMs dependence on catheter orientation, which is not fully compensated by the OP-EGM approach. Specifically, the aims of this study are: (1) to assess the estimates of CV and propagation direction within each clique with the proposed MOP-EGM method as compare to the ones obtained by the original OP-EGM; (2) to propose mapping strategies based on omnipolar EGMs estimations to characterize the atrial substrate; and (3) to assess the ability of the proposed maps in detecting a fibrotic patch and in reproducing unipolar voltage maps.IKr) was selected 1.6-fold greater than the original right atrial model , the L-type calcium current (ICaL), the inward rectifier potassium current (IK1), the ultrarapid outward potassium current (IKur) and the slow delayed rectifier potassium current (IKs) (see APD90) (248 vs. 111 ms), in concordance with experimental data . Simulations were run in ELVIRA software  conditions was simulated by dividing it into adjacent square elements whose centers were separated 0.1 mm. Within the tissue slice, a circular patch with a diameter of 2 cm was defined, whose center was at the center of the 2D tissue. Inside this circular patch, a diffuse fibrosis pattern was randomly defined following a uniform distribution. All the non-fibrotic nodes were assigned the Courtemanche cellular model stabilized during 1 min at basic cycle length (BCL) 500 ms  see , as in pIKs) see . Twenty IKs) see . The intIKs) see . AlthougIKs) see , therefoN × N high-density multi-electrode array (MEA) (N = 15), where the inter-electrode distance was d = 2 mm. The simulated tissue geometry, including the MEA location and the fibrotic patch, is shown in i, j) within the MEA, is referred to as electrode 1. The rest electrodes of the clique are numbered from left to right and bottom to top. Therefore, electrodes 2, 3, and 4 correspond to locations , , and , respectively.Recording electrodes were distributed over the simulated tissue, mimicking a The simulated electrode grid is located so that its central electrode corresponds to the center of both the tissue slice and the fibrotic patch, and the MEA is rotated by an angle Ψ with respect to the tissue fiber orientation. Three different MEA orientations with respect to the tissue preferential direction were considered: Ψ = 0°, Ψ = 30°, and Ψ = 45°, which are illustrated in z = 1 mm distance from the tissue surface. This represents a plausible compromise between recording realistic amplitude EGMs and the real clinical setting situation where there is no guarantee of maintaining a perfect electrode-tissue contact during the mapping.The MEA grid was placed at ui, j(t) were simulated as generated by the propagation wave when passing by electrodes located at sites ,  of the grid. They were computed in a volumetric tissue-blood model with a temporal resolution of 1 ms, as in Martinez-Mateu et al. . Bipolar EGMs Bipolar signals, as defined in Equation (1), are used to compute bipolar voltage maps for performance comparison, as described in subsections 2.12, 2.13.i, j) in its lower left corner, i, j) electrode position, see black arrows direction variations in The omnipolar methodological developments assume a locally plane wave propagation within a clique, with a velocity vector Examples of unipolar and bipolar EGMs used in this study, corresponding to the case when the global propagation direction of the wave and MEA direction form an angle Ψ = 0°, are shown in tc be the reference time at which the center c of the selected square is activated. Assuming constant velocity within the arrangement, the activation times at the four electrodes are:Let us consider any four-electrode clique within the MEA, like the one in t) be the unipolar voltage waveform generated by a plane wave when passing by a given position at time t = 0. When one or more electrodes within the square of t) passage, the following u-EGMs and b-EGMs are modeled as:Let ϕ(x direction (b12(t) and b34(t)), y direction (b13(t) and b24(t)), as well as along the directions of its diagonals (b14(t) and b23(t)). Depending on the waveform arrival time to each electrode tm, the activation time of each b-EGM bmn(t) corresponds to the passing of the wave by the middle point between electrodes m and n. Therefore, relative delays  occur between the different bmn(t).It should be noted that bipolar signals in Equation (4) are derived within each clique along um(t) = ϕ(t − tm) around tm = tc, we can approximate:By using the Taylor's series expansion of tmn = tn−tm are the differences between activation times at electrodes m and n, m, n ∈ {1, 2, 3, 4}, and Δtm = tm−tc are the differences between activation time at electrode m and the reference activation time of the center of the square. By replacing Equation (2) in Equation (6) and operating, the following b-EGMs expressions are derived:where Δbmn(t), showing the previously mentioned delay between the different bipolar signals. If we relate b12(t) and b13(t), measured in different x and y directions, we obtain:From these expressions we can relate the second order approximations of the different pairs of b-EGMs y-direction (1–3) is a scaled version of the b-EGM along x-direction (1–2) by a factor x-direction), b13(t) = 0, while for θ = 45°, the delay is τ = 0, meaning that both bipolar EGMs in the x- and y- directions are activated at the same time. Similarly, if we relate b12(t) with b34(t), both along the x-direction, we obtain:Equation (13) shows that the b-EGM along d/v, when the propagation is orthogonal to the direction of the electrode pairs (θ = 0°).Equation (14) shows that b-EGMs along parallel directions have the same amplitudes (as expected), and the delay t), and the electric field (E-field) at the extracellular-myocardial interface. Let us consider a locally plane myocardial surface, defined by its normal unit vector The purpose of the OP-EGM method is to obtain EGM amplitude and propagation features which are invariant to the relative angle between the propagation direction and the catheter  is a vector containing the available b-EGMs, D is a matrix of interelectrode distances, and ϱ ∈ {□,△} denotes the clique type, square (□) or triangular (△), respectively. In case of square clique, as the one illustrated in bϱ(t) ≡ b□(t) contains six b-EGMs, Dϱ ≡ D□ is a 2 × 6 matrix, reported below. Similarly, in case of triangular configurations, as the clique 1 in bϱ(t) ≡ b△, 1(t) contains three b-EGMs, Dϱ ≡ D△, 1 is a 2 × 3 matrix, also shown below.where D△, 1 were obtained for the other three-electrode configurations of the clique shown in Expressions similar to From Equation (15) we see that the projection of the E-field along the inter-electrode distance results in the corresponding b-EGM. Since measured b-EGMs will be affected by noise and model mismatch errors, we can use the least-squares criterion to estimate the electric field c of each ϱ-type clique, describes a loop trajectory in the 2D plane. Under the assumption of locally plane and homogeneous wave, E⊥(t) ≈ 0 within the clique, so that The evolution over time of the N × N is the size of the MEA, there will be (N − 1) × (N − 1) = N2 − 2N + 1 estimates of x(t) and Êy(t) of the electric field can be expressed as linear combinations of the two b-EGMs along x- and y- directions, respectively:If where the approximated terms make use of Equations (7–10).v. Such delay represents the difference in the wavefront arrival time to the middle point of each bipole.Equations (18–22) show that the two electric field components within each clique have a different amplitude, depending on the propagation angle θ. In addition, for triangular cliques, there is also a delay between both components, which is also a function of the propagation angle θ and the conduction velocity magnitude In the formulation of Equation (15), no delay is considered among the b-EGMs. However, when estimating the electric fields making use of measured b-EGMs, the different bipolar signals are activated at their corresponding wavefront arrival times, thus having a delay among them, as expressed in Equations (7–12).x(t) and Êy(t), for plane wave propagation and to that level of approximation. Consequently, the x(t) and Êy(t) in Equation (18) is the summation of two terms which have a delay between them: Equations (7–10), respectively. It is well known that the summation of delayed components result in smoothed estimates,  we found that there are no delays between Êx(t) and Êy(t) components. This misalignment will be reflected at the x(t) and Êy(t), in this study we propose a modified version of the least squares estimator of Eϱ(t), where the b-EMGs in the clique are previously time-aligned. For that purpose, both in square and in triangular configurations, the time delay τmn between each independent bipolar signal bmn(t), mn ∈ {12, 34, 13, 24} and the b-EGM with the highest amplitude within the clique bmax(t) was estimated by maximizing their cross-correlation:For the triangular cliques, Equations (19–22) show that, in general, there is a delay between Êbmn(t) was then advanced by τmn so that it is aligned with bmax(t). Figure 5 shows examples of E-field estimated with and without previous alignment.Each b-EGM c, and the projection of the electric field in that direction Êp(t). The unipolar voltage signal c: E(t) is expressed as E(t) along the propagation direction is given by:OP-EGM approach estimates propagation direction by finding the angle E(t) with its estimate according to Equations (18–22) and solving:The estimated conduction angle c maximizing the previous cross-correlation, the propagation angle estimate is:If For square cliques, considering the estimates of the E-field components for plane wave propagation in Equation (18), the expressions in Equation (3) and the fact that Using a similar procedure for triangular cliques, as clique 1 in x(t) and Êy(t) are estimated from properly aligned bipolar signals, then we obtain This result points out that the estimate of θ is biased in general for triangular cliques. Nevertheless, if the electric field components Êuc(t) = u1(t). For triangular cliques, the signal from the electrode corresponding to the vertex of the right angle is chosen, so that uc(t) = uk(t), k = {1, 2, 3, 4} for each of the four triangles indicated with the same k-th index, respectively, in In the reference omnipolar method introduced by Deno et al. , it is nAlternatively, we propose in this work a more robust approach, where the estimation of umax(t), by computing the maximum cross-correlation between them. In this way, the four aligned u-EGMs uk(t − τk), k ∈ {1, 2, 3, 4}, are obtained;each of the four u-EGMs is aligned with respect to the one with the highest amplitude, denoted as t) is calculated by averaging the four aligned u-EGMs uk(t − τk);the average signal ū(t) is computed and used as estimate of the time derivative of ū, as they are expected to be proportional under the assumption of locally plane wave: v can be estimated by computing the ratio of the peak-to-peak (pp) amplitudes of p(t), as proposed by Deno et al.  of p(t):In order to avoid the large sensitivity to noise that be expected from an estimation based on peak-to-peak amplitudes, we propose here a more robust estimate of For square cliques, by using Equation (18) in Equation (24) and if the propagation direction is well estimated (x(t) and Êy(t), and assuming that the propagation direction is well estimated , using the approximations with delays in Equation (19) for ÊThis equation shows that the misalignment of i, j, (q)(t), where q ∈ {1, 2, 3, 4} refers to each of the four possible triangles when using triangular clique configuration, were defined by projecting t) within the corresponding clique onto the following directions:In this work, OP-EGM signals φme) direction was derived as: Emax of the electric field, proposed by Deno et al.  of the loop described by t) within the depolarization interval. The resulting OP-EGM signal The principal direction Let us consider y(t) and Êx(t) components with the delay approximation in Equation (19) and time aligning them, we obtain the aligned version It should be noted that there are no second-order terms in Equation (32), unlike the expressions of b-EGMs (see Equations 7–12). Again, alignment prevents attenuation at the Figure 6, for both clique configurations and the three MEA orientations, with and without the previous alignment of b-EGMs.whose amplitude is properly estimated. Examples of the omnipolar EGM signals here introduced are observed in s ∈ {me, pca, pca⊥}, we computed the peak-to-peak amplitude pca approach, we also created maps based on the root sum square of the voltage in the first and second principal components, For each OP-EGM signal introduced in this work r) of both values, as well as their maximum (m) were also considered so as to have a performance reference based on bipolar signals:We also built bipolar maps based on the peak-to-peak amplitudes All the voltage maps were computed for each orientation of the MEA, and by using both clique configurations analyzed in this study. Omnipolar voltage maps were also performed from aligned b-EGMs. We obtained color-coded maps where a pixel corresponds to a clique .v-values and arrows representing angular direction θ of the estimated velocity vectors v in each clique. Plotted v-values were estimated by using both OP-EGM approach of the Equation (28) proposed by Deno et al.  measurement configurations  within the MEA, maps performed with this configuration present higher spatial resolution.It should be noted that maps performed with square cliques present the same resolution as bipolar maps, both providing ® catheter  in 27 different mapping points at intervals with no recorded EGMs. All noise segments were normalized in order to guarantee the same power level, coinciding with the observed average power. One hundred different realizations of this recorded noise were randomly added to each of the simulated b-EGMs x and y directions of the MEA. As a result, we generated one hundred noisy bipolar signals along each of the two directions, n, the different modalities of voltage and conduction velocity maps were computed, obtaining 100 different maps from noisy b-EGMs. These mapping strategies were tested regarding fibrosis detection, as well as voltage map reproducibility. The previous procedure was repeated for six different noise levels, scaling the noise realizations so that they had standard deviations σn ∈ {3, 6, 14, 28, 42, 55} μV. The standard deviation of the recorded noise ranged from 1.2 μV to 11 μV, with an average value of 2.4 μV, lower than those reported in Unger et al. , receiver operating characteristic (ROC) curves were used. Two ground-truth masks of fibrosis were created for bipolar and omnipolar maps, having the resolution of square and triangular cliques and indicating whether a clique lay within the fibrotic or the normal tissue. A 14 × 14 ground-truth mask was used to evaluate bipolar and omnipolar maps with square cliques. A value of 1 was assigned if the four electrodes defining the square clique lay in the fibrotic area, and a value of 0 was assigned if all of them lay in the non-fibrotic area. Cliques with some electrodes inside and some outside the fibrotic patch were not considered in the evaluation. Similarly, a 28 × 28 ground-truth mask was used to assess omnipolar maps performed with triangular cliques, considering if the three electrodes defining the triangular clique lay in fibrotic or non-fibrotic tissue.true positive denotes the number of cliques correctly identified as fibrotic, false negative represents the number of missed cliques in the fibrotic area, true negative stands for the number of cliques correctly identified as non-fibrotic and false positive is the number of cliques incorrectly detected as fibrotic. The accuracy (ACC) was defined as the number of cliques correctly identified as fibrotic or non-fibrotic divided by the total number of cliques. The maximum ACC was used as a measure of the overall fibrosis detection ability of a given map. In addition, the sensitivity and specificity with the threshold achieving the maximum ACC were also computed.Both ground-truth masks used in this study are illustrated in p and Spearman's rank ρs correlation coefficients. A p ≤ 0.05 was required to have statistical significance. Reference voltage maps were obtained by following two sequential steps:The ability of the voltage mapping strategies in reproducing the spatial distribution of reference maps was also evaluated. We created peak-to-peak amplitude maps of u-EGMs, and resampled them to match the spatial resolutions of bipolar and omnipolar pixel maps (with both square and triangular cliques). To quantify the agreement of each voltage map with the substrate characterization given by the reference map, we used both Pearson's ρN × N peak-to-peak amplitude map of u-EGMs ui, j(t),  ∈ {1, …, N}, was performed placing three interpolated values between any two adjacent electrodes, along both MEA directions, thus increasing the total number of pixels to (N + 3 · (N − 1)) × (N + 3 · (N − 1)). The original and interpolated maps for N = 15 are shown in A bicubic interpolation of the N − 1) × (N − 1) and 2·(N − 1) × 2·(N − 1) reference maps, matching the dimensions of bipolar and omnipolar maps with square cliques, and omnipolar maps with triangular cliques, respectively. Resulting resampled pixel maps from N = 15, for the case when Ψ = 0°, are illustrated in The interpolated pixel map was later resampled so as to obtain  between each noisy map and the respective noise-free map. In case of voltage maps, we calculated:s ∈ {b-m; b-r; o-me; o-me, a; o-pca-r; o-pca-r, a}. Analogously, for velocity maps, we computed:where s ∈ {o; o-m; o-m, a}.with o and the modified θo-m omnipolar method. Angle differences, ϵθ, between propagation direction maps obtained from noisy realizations and those obtained from noise-free signals (considered as the reference) were computed for each omnipolar approach and noise level. The mean and standard deviation of the angle differences in each map were obtained and averaged for all noise realizations.We also studied the effect of noise on the estimation of the propagation direction with both the original θAll performance measurements were reported jointly for the three MEA orientations considered in this study, thus assuming that the relative angle of the propagation direction with respect to the catheter is not known a priori.i, j) electrode positions. All the E-field loops were estimated with and without previous time alignment of b-EGMs, as proposed in subsection 2.6 (denoted as Ea(t) and E(t), respectively). In the top three rows, cliques were chosen in a non-fibrotic area with plane wavefront. Rows A–C correspond to the clique  =  for Ψ = 0°, Ψ = 30° and Ψ = 45°, respectively. The estimated local propagation angles with the proposed MOP-EGM method were i, j) =  with Ψ = 0° in i, j) =  with Ψ = 30° in i, j) =  with Ψ = 45° in (B), (C), (D) and (E)].The effect of the proposed alignment is evident in both upper and lower panels. In square cliques, the E-field peak value for the aligned versions is clearly higher than for the non-aligned ones. On the other hand, in triangular cliques the alignment of b-EGMs proves to improve the estimation of the propagation direction. These is clearly seen especially in those areas where the local propagation direction differs from the dominant directions of the MEA [see panels i, j) as in s ∈ {me, pca, pca⊥}, in square and triangular cliques, with and without alignment of b-EGMs. These results show that the OP-EGM signals produced by the two modalities are quite similar in most of the cases. Square cliques reveal that n = 28μV (right columns). In the latter case, only one of the 100 noisy realizations was shown. The omnipolar maps are presented without and with the alignment of b-EGMs. These results reveal that combined bipolar maps obtained as the root sum square or the maximum voltage of both MEA directions, Vb-r and Vb-m, shown at row A, present better fibrosis detection performance than omnipolar maps without alignment, illustrated at upper and lower rows in me and pca, respectively. However, omnipolar maps with previous time alignment of b-EGMs (upper and lower rows in The different bipolar and omnipolar voltage mapping strategies are illustrated in A) and with the proposed approach (without (row B) and with (row C) alignment of b-EGMs) are presented in CVo-m and CVo-m, a) than the standard approach (CVo). Moreover, the standard approach tends to overestimate conduction velocity values, when compared with the MOP-EGM method.Velocity maps obtained with the reference OP-EGM method  achieve values ρp = 0.87 and ρs = 0.87 when square cliques are used and b-EGMs alignment is performed. As a reference, Vb-r and Vb-m present correlations of ρp = 0.86 and ρp = 0.83, respectively. Regarding CV maps, the MOP-EGM version proposed in this work  presents indeed comparable performance in fibrosis detection to combined bipolar voltage maps , and better performance than both omnipolar voltage maps and than the original OP-EGM approach to estimate conduction velocity (CVo).AUC, ACC, Pearson ρp, A, omnipolar voltage and CV maps are less affected by noise than bipolar maps for noise levels greater than or equal to σn = 28μV. Note that the proposed time alignment of b-EGMs improves fibrosis detection performance and robustness to noise of the MOP-EGM maps, when comparing them to their unaligned versions. The highest ACC was obtained by CV maps computed with the MOP-EGM method in square cliques, including prior alignment of b-EGMs (CVo) proved to be unstable with noise, both in square and triangular clique configurations. As shown in panel B, voltage maps performed with the MOP-EGM method, including b-EGMs alignment, and with square cliques, Similar results are observed in The RMSE between maps computed from noisy b-EGMS and those computed with noise-free b-EGMs are depicted in o and θo-m for square and triangular cliques, considering jointly all three MEA orientations. These results show that propagation direction maps performed with square cliques are less affected by noise than those performed with triangular cliques, regardless of the omnipolar approach. However, for noise levels lower than σn = 28μV, direction maps with the MOP-EGM approach θo-m exhibit a smaller mean bias than those using the original method θo.Vo-me, Vo-pca-r, Vo-me, a, Vo-pca-r, a, CVo-m and CVo-m, a) performed with square cliques. The accuracy in fibrotic patch detection reached the values ACC = 91, 99, and 100% using Vo-me; ACC = 91, 100, and 100% using Vo-pca-r; and ACC = 100, 91, and 91% using CVo-m, when Ψ = 0°, Ψ = 30° and Ψ = 45°, respectively. When previous time alignment of b-EGMs is performed, accuracy achieves values ACC = 100, 91, and 92% for both Vo-me, a and Vo-pca-r, a, while ACC = 93, 100, and 100% when using CVo-m, a, for the same catheter orientations.Finally, In this work, we investigated the performance of different substrate and velocity mapping modalities based on the OP-EGM method, and on the here proposed MOP-EGM approach, in characterizing and detecting fibrotic areas. For that purpose, we used a 2D multi-electrode array over an atrial tissue simulated with the Courtemanche cellular model.ex vivo  always improves the fibrosis detection ability of the omnipolar voltage and conduction velocity maps. The latter provide a characterization of the atrial substrate comparable to bipolar voltage maps combining b-EGMs along Most studies to detect fibrosis rely on bipolar voltage mapping , we observe a reduction of the omnipolar voltage values. In addition, in tissue areas where the wavefront propagation is curved, the electric field is overestimated when the local propagation direction becomes parallel to one of the two main directions of the catheter (as when Ψ = 45°). As this happens in an area of healthy tissue, it causes a higher overall accuracy in fibrotic patch detection. The reverse occurs when these curved areas are oblique to the MEA (as in Ψ = 0°), providing underestimated voltages and decreased fibrotic detection accuracy.v, in Equation (29), has an inverse relationship with the magnitude of the estimated E-field, while in voltage estimation (equations in subsection 2.9), the dependence is direct. In addition, in those areas where the wave propagation is curved, velocity is overestimated when the MEA is in the same direction as the tissue (Ψ = 0°). Therefore, the fibrotic patch is globally better detected for this configuration. On the other hand, if catheter becomes oblique to the tissue (Ψ ≠ 0°), areas of curved propagation appear underestimated, thus reducing the overall fibrosis detection performance for those configurations. When b-EGMs are time-aligned, both MOP-EGM based voltage and velocity maps continue to be sensitive to the orientation between MEA and tissue, but to a lesser extent if compared to their unaligned versions. As a result, improved fibrosis detection, better voltage estimations and less degradation by noise are observed in average when using the MOP-EGM approach. The remaining dependence with Ψ is not predicted by the mathematical model based on the plane-wave assumption, but the sampling frequency of the b-EGMs (1 kHz) could be behind this observation.As for the orientation dependence of velocity maps, the opposite behavior is observed, with velocity estimation increasing for Ψ ≠ 0°. This is explained by the fact that The basic assumption of plane and homogeneous propagation within each clique represents an intrinsic limitation of the omnipolar methodology, requiring the use of high spatial resolution multi-electrode catheters (like the one considered in this study). Moreover, the hypothesis of a plane myocardial surface in the area defined by the clique is questionable in non-compacted myocardial regions or in the atrial appendage. Nevertheless, those areas will be identified with 3D electroanatomic mapping systems . Moreover, the proposed MOP-EGM maps need to be evaluated over real data, where the effect of electrode size, inter-electrode distance and variable tissue-electrode contact will be present. The validation of these methods In this work we showed that voltage mapping strategies based on the MOP-EGM method are able to discriminate fibrotic from healthy tissue. For low noise levels, they attain comparable performance to maps obtained by combining the b-EGMs amplitudes along the two directions of the catheter, but clearly outperform bipolar maps when noise is added. On the other hand, omnipolar CV maps reveal higher fibrosis detection accuracy than voltage maps, specially when the modifications proposed in this work were applied. The fibrosis detection performance of voltage and velocity maps benefit from prior intra-clique time alignment of b-EGMs as proposed in the MOP-EGM approach. In addition, the use of square cliques outperforms the triangular ones.me) and the one based on principal component analysis (pca) yield similar results, in terms of detection accuracy and correlation with unipolar voltage maps. Similar conclusions are drawn when b-EGMs are affected by noise extracted from real signals.As for the different approaches to estimate the omnipolar voltage from the E-fiel loop, both the one based on maximal excursion  Ethics Committee. The patients/participants provided their written informed consent to participate in the study.JR, AA, PL, and JM designed the study and analyzed the results. JR, PL, and JM wrote most of the manuscript. SR, LM-M, and JS performed the electrophysiological simulations. SL provided the clinical data. PL and JM supervised and formalized the project. JR developed the software and performed the required computations. All authors critically revised the manuscript and approved the submitted version.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state S-probability. If a set P of S-probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic P is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of S-probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of S-probabilities are characterized and related to each other and descriptions based on sets of states are derived.With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let  In axiomatic quantum mechanics, orthomodular partially ordered sets and generalizations of them are considered as “quantum logics” that determine the behaviour of a physical system. In particular, if the quantum logic is a Boolean algebra, then one will have reason to assume that one deals with a classical physical system. The elements of a quantum logic can also be interpreted as events, and a Boolean algebra then as the equivalent of a classical field of events as known from probability theory.numerical event .S be a set of states of a physical system and p(s) the probability of the occurrence of an event, when the system is in state p from S to  is called a numerical event, or alternatively more precisely an S-probability. If ordered by the order S-probabilities becomes a partially ordered set (poset) that can be conceived as a quantum logic. In this paper, we study different kinds of such quantum logics, especially those that are not far away from being Boolean algebras. For this end, we provide the following notions.Let P be a set of S-probabilities including the constant functions 0 and 1, partially ordered by the order of functions. We will call disjoint, in symbols p and q, respectively, considered as real functions.Let P of S-probabilities is called specific if if if A set R of subsets of some set R, one has R and R. If A and B are disjoint, R having characteristic 2, i.e. Condition (2) seems natural in respect to dealing with probabilities, and as for (3), this condition is motivated by classical fields of events (yet for the time being limited to considering the sum of disjoint events). Conceiving such a field of events as a Boolean ring S-probabilities p and q are called orthogonal, in symbols S-probabilities in general are not), this property is known to be Boolean . We extP with an antitone involution is called Boolean, if A poset S-probabilities are Boolean posets.According to this definition, specific sets of p, q of a set P of S-probabilities and denoting their infimum by Writing P be a set of S-probabilities satisfying (1) and (2) and let P in the following sense: If P then P and P then P and Let P. Then, P and Let Finally, we recall the definitions of two structures of numerical events which we will later relate to specific sets of numerical events.P of S-probabilities is called a generalized field of events (in short GFE) (cf. Dorninger ((4)if If a GFE satisfies (5)if then it is called an algebra ofS-probabilities (cf. Beltrametti and Ma̧czyński (A set orninger ), if it ̧czyński  and Belţczyński ).r to be 0.Condition (4) is a special case of condition (5)—just assume S-probabilities, investigate their interrelations and closeness to Boolean algebras, and indicate when they will actually be Boolean algebras. Moreover, we will consider the question whether  sets of S-probabilities will belong to a Boolean subalgebra of a specific set of S-probabilities and we will characterize specific sets of S-probabilities by states.The goal of this paper is to characterize various classes of specific sets of S-probability p is called varying, if p is neither An S-probabilities are varying , the elS-probability is varying often comes up independently or can be achieved by adding further experimental data directed to this purpose.As for data won by experiments: That an S-probabilities that are varying. An S-probability is called complementary if P of S-probabilities with 0 and 1 will be called complemented if all of its elements are complementary. Further, we recall that a poset P with complementation orthoposet.Now, we will turn our attention to specific sets of P of varying S-probabilities has the following properties: (i)P is complemented and hence an orthoposet,(ii)if (iii)P is a GFE.A specific set (i)r being a varying S-probability. Therefore, If (ii)P being complemented, If (iii)If Let P be a set of S-probabilities satisfying (1) and (2). Then, all elements of P are varying if and only if P is complemented.Let P are varying, then P is complemented according to the proof of Proposition P to be complemented. Let P is varying.If all elements of if if is called a concrete logic .S-probabilities are exactly the complemented Boolean GFEs. They all are concrete logics.The specific sets of varying P be a set of S-probabilities and P to be a specific set of varying S-probabilities. By Proposition P is a complemented GFE. Conversely, assume P to be a complemented Boolean GFE. Then, as mentioned in Remark P are varying. Further, if S-probabilities are orthoposets and hence, Boolean GFEs are Boolean orthoposets. As mentioned in Tkadlec .S-probabilities is a concrete logic, its elements can be represented by functions which have only the values 0 or 1. So, these S-probabilities must be varying from the outset. If S is finite, Theorem S-probabilities are Boolean algebras, since finite Boolean orthoposets are Boolean algebras. So to distinguish between a classical and a quantum mechanical behaviour by measurements in the form of numerical events, one would need data from S-probabilities for a continuous set S of states.Since any specific set of varying S-probabilities and algebras of S-probabilities.Next, we turn our attention towards the connection of specific sets of varying S-probabilities that are Boolean.The complemented Boolean GFEs are exactly the algebras of S-probabilities was already shown in Dorninger .S-probabilities are orthomodular posets with a full set of states, and vice versa . Furthe Traczyk ). To be S-probabilities are exactly the infimum faithful algebras of S-probabilities.The specific sets of varying S-probabilities by Boolean rings, in line with Theorems Returning to the motivation of the definition of specific sets of S-probabilities which is a Boolean algebra can be conceived as a Boolean ring if one extends S-probabilities p and q by assuming within the pointwise addition of the functions p and q that An infimum faithful algebra of P be a set of S-probabilities. We consider the following conditions: (6)If (7)if (8)if Condition (6) can be motivated by regarding a classical field of events as a Boolean ring R for which it is the case that S-probabilities that satisfy condition (6) as specific (join-specific) sets of S-probabilities. If (1), (2) and (7) hold, P is called a structured set ofS-probabilities (cf. Dorninger and Länger (P is known as a weakly structured set ofS-probabilities (cf. Dorninger and Länger (Let d Länger ), and ifd Länger ).S-probabilities:S-probabilities,S-probabilities (satisfying (6)),S-probabilities (for which (7) is distinctive),S-probabilities (characterized by (8)).Now, we define the following classes of sets of We have P be a set of S-probabilities and P is a specific set of S-probabilities. If Let We have P be a specific set of S-probabilities which is also a weakly structured set of S-probabilities and assume Let S-probabilities is a proper subclass of the class of S-probabilities which on its part is a proper subclass of the class of weakly structured sets of S-probabilities unless one assumes that only specific sets of S-probabilities are taken into account.The class of structured sets of As for the inclusions to be proper, in agreement with Lemma S-probabilities are exactly the elements of First, assume S-probabilities are away from being Boolean algebras. A first reference to this can be a subclass of Next we will discuss the question how far S-probabilities are exactly the infimum faithful algebras of S-probabilities.(cf. Dorninger and Länger ) The comS-probabilities P is finite, it is a Boolean algebra, because, as already mentioned, finite Boolean orthoposets are Boolean algebras. Further, P is a Boolean algebra if it is orthocomplete (cf. Tkadlec (P has to belong to P.) Moreover, if P is a lattice . (To be Tkadlec ). That Pd Länger , a strucS-probabilities are characterized to be Boolean algebras by specifying some structural properties—for an overview of these papers see Dorninger  classes of algebras of orninger —and therlukowski , Navara lukowski , Pták .S-probabilities P is a Boolean algebra but if a  subset of P belongs to a Boolean subalgebra of P. If this were the case, this would indicate that one locally deals with a classical physical system. To answer this question, the existence of some further S-probabilities in P will have to be asked for, but the knowledge of P in detail will not be important.Sometimes it is not of interest if a whole algebra of S-probabilities P is given. If P wherein S-probabilities is orthomodular. So, let us suppose that P.So let us assume that a subset r et al. ), and asr et al. , every aP can only assume the values 0 and 1 .P with an antitone involution.We begin by extending the commonly known notion of a state to the class of bounded posets specific state on a bounded poset s from P to  satisfying the following conditions for all if if If for P, then a specific state on P satisfying (S5)if is called a pseudostate on P (cf. Dorninger and Länger (A d Länger ).setTof specific states on P is called full if for setTof specific states on P is called uniform if for disjoint r.  with Assume From this, we can conclude that f is an isomorphism from P onto f(P). Hence, P is isomorphic to a member of Conversely, let S-probabilities are exactly the bounded posets with an antitone involution in which the join of two disjoint elements exists and which have a full set of pseudostates, which in the light of Lemma As shown in Dorninger and Länger  up to isS-probabilities are exactly the bounded posets with an antitone involution in which the sum of two disjoint elements equals their join and which have a full set of pseudostates.Up to isomorphism, the S-probabilities are exactly the orthomodular posets having a full set of states.Theorems To find out whether quantum mechanical effects can occur in a physical or technical system, various classes of sets of numerical events were studied, i.e. data-sets obtained by surveying the system which then form or contribute to the underlying quantum logic. In particular, sets of numerical events, called specific sets, were analyzed that are located at the threshold between classical and quantum behavior, a situation that can be characterized by the fact that in case of a classical physical system the underlying logic is a Boolean algebra. The interrelations of specific sets of numerical events and their closeness to Boolean algebras were ascertained and the question was answered when an unstructured set of numerical events can be a part of a Boolean algebra. Moreover, specific sets of numerical events were characterized by the states they admit.


The action of Steenrod algebra on the cohomology of Given an  Khovanov homology -module M we define the Borel equivariant cohomology of XX using the convention described above. In particular, we have In order to state and prove the next result, we need to set up some notation and recall some basic facts from homological algebra. If G-map reduced Borel cohomology of X, to beThere is a natural m-periodic link diagram of a link M, the equivariant Khovanov homology Let To begin with, observe thatM. We haveNext, let mentclasspt{minimamentclasspt{minimamentclasspt{minimantly, by \documentWe remark that the same argument as in the proof of Theorem m-periodic annular link and let m-periodic diagram of M, it holdsLet k is a family of natural transformations between functors Given two generalized cohomology theories The Steenrod algebra For a prime Coming back to Khovanov homology we make the following observation, see , 35.k over L and Let knotkit package ). package  implemen package . We remaThe next statement shows that Steenrod operations commute with group action.m-periodic link and Let m-periodic diagram of Cohomology operations are natural, so they commute with the group action on cohomology of p. The cohomology ring of Z. Write X, if Y, when Recall that X be a p a prime. There exists an isomorphism of graded , 40 Let As an immediate corollary of Theorems p-periodic link with L denoting the quotient link. For any Let p, the modpSteenrod algebra, i.e. the algebra of stable Let I is admissible if degree of I, excess of I, I we associate the cohomology operation I is admissible if degree of I, excess of I, I we can associate a cohomology operation For unstable elements, For a graded Equivariant annular Khovanov homology p be a prime. p-periodic link and L is the quotient link, then for any If p-periodic link L denotes the quotient link.For a Let This is an immediate corollary of [Smith inequalities given in Theorems We have the following chain of inequalities