We develop an auto-reservoir computing framework, Auto-Reservoir Neural Network (ARNN), to efficiently and accurately make multi-step-ahead predictions based on a short-term high-dimensional time series. Different from traditional reservoir computing whose reservoir is an external dynamical system irrelevant to the target system, ARNN directly transforms the observed high-dimensional dynamics as its reservoir, which maps the high-dimensional/spatial data to the future temporal values of a target variable based on our spatiotemporal information (STI) transformation. Thus, the multi-step prediction of the target variable is achieved in an accurate and computationally efficient manner. ARNN is successfully applied to both representative models and real-world datasets, all of which show satisfactory performance in the multi-step-ahead prediction, even when the data are perturbed by noise and when the system is time-varying. Actually, such ARNN transformation equivalently expands the sample size and thus has great potential in practical applications in artificial intelligence and machine learning. Predicting future values of a short time series remains a challenge. Here, the authors propose an auto-reservoir computing framework, which achieved accurate and robust multistep ahead prediction by transforming high-dimensional data into temporal dynamics. On the other hand, neural networks including the RNN and LSTM can theoretically learn the nonlinear dynamics from the observed data18. However, these methods typically suffer from the overfitting problem due to the lack of sufficiently many samples when only a single short-term time series is available to train the networks. In addition, training a neural network sometimes costs considerable time and computing resources20, which also preclude the traditional neural networks from being applied to many real-world systems.It is a challenging task to make multistep-ahead predictions of a nonlinear dynamical system based only on a short-term time series due to its complicated nonlinearity and insufficient information. Although many methods including statistical regression , and an output or readout layer22. RC has drawn much attention because of its dynamical property and easy scalability since the recurrent connections in the network are  fixed in advance rather than trained. In other words, training is performed only at the readout stage due to the fixed/random reservoir, thus significantly reducing the training parameters and the computational cost. RC has been applied to a series of real-world studies, such as Great Lakes water level prediction23, handwritten digit image recognition24, NARMA time-series prediction25, limit cycle generation26, and temporal parity tasks27. However, the current RC framework requires knowledge of the external/additional dynamics that generates the reservoir, which is unrelated to the observed/target system.Reservoir computing (RC) is an extension of neural networks recently developed following RNN frameworks that is suitable for temporal/sequential information processing30. Nevertheless, the most recent short-term time series usually contains more information on its immediate future evolution than the remote-past time series owing to the time-varying non-stationary nature of many real-world dynamical systems34. Thus, even if long-term data are measured, prediction effectiveness depends mainly on recent short-term data. On the other hand, notably short-term but high-dimensional data possess rich information due to the high-dimensional variables, whose dynamics is intertwined and thus can be exploited for the prediction. Therefore, it is natural and important to predict future states on the basis of a short-term high-dimensional series, which is also widely available in real-world cases. Actually, by assuming that the steady state is contained in a low-dimensional manifold even for a high-dimensional system that is generally satisfied for dissipative real-world systems, the spatiotemporal information (STI) transformation has theoretically been derived from the delay-embedding theory37. This method can transform the spatial information of high-dimensional data to the temporal dynamics of any target variable. Based on the STI transformation, the randomly distributed embedding (RDE) framework has been developed for one-step-ahead prediction from the short-term time series by separately constructing multiple maps using the original primary STI equations or the linearized STI equations36. Furthermore, the multistep-ahead prediction was also performed by adopting a multilayer neural network as the STI transformation37. However, there are two issues that remain unsolved for the prediction: one is the computation cost, and the other is the robustness.Compared with many existing studies on predictions based on long-term time-series data, there have been only a few studies on predictions from short-term but high-dimensional dataF as a reservoir structure based on both the primary and conjugate forms of the STI equations, thus constructing the ARNN-based equations. Based on such equations, ARNN encodes F(X)t to Yt and decodes Yt to F(Xt), where Yt is the temporal  dynamics across multiple time points and Xt is the spatial  information at one time point. Hence, ARNN transforms the known spatial information of high-dimensional data to the unknown temporal/dynamical information or prediction of any target variable. On the other hand, the reservoir of ARNN is a multilayer neural network F in which the weights among neurons are randomly given and fixed in advance, while the weights of the output are determined directly from solving the ARNN-based STI equations with the dropout scheme38. In other words, there is no traditional training process for the neural network in the ARNN algorithm. Unlike traditional RC using external/additional dynamics (irrelevant to the target system), ARNN transforms the dynamics of the observed high-dimensional data as the reservoir, therefore exploiting the intrinsic dynamics of the observed/target system. ARNN actually has a similar form to the autoencoder , as illustrated in “Methods”. Such a transformation makes ARNN robust.In this study, by taking advantage of both the RC structure and STI transformation, we propose a novel auto-reservoir computing approach, namely, the Auto-Reservoir Neural Network (ARNN), to achieve an accurate, robust and computationally efficient multistep-ahead prediction with short-term high-dimensional data. In particular, we take a nonlinear function To validate ARNN, it is applied to a representative model, i.e., a 90-dimensional coupled Lorenz system under different noise and parametric conditions. Furthermore, ARNN is applied to predict many real-world systems. The results show that ARNN achieves multistep-ahead prediction with only a short-term series, which is better than other existing methods in terms of accuracy, efficiency and robustness.D variables and t = 1, 2, …, m, we construct a corresponding delayed vector Yt = , there are actually L − 1 unknown future values of the target variable y, i.e., {Yt . For each observed high-dimensional state ion Fig. , where s, m Fig. . Thus, tion Fig. . In Eq. mentclass2pt{minim Yt Fig. . However Yt Fig.  at t = 1F as a reservoir structure based on both the primary and the conjugate forms of the STI equations , thus g(see Eq.  or 9) i iF as a . Note that we can also directly adopt (Yt−1) as the input of f instead of B.Note that n of Eq.  can be rn of Eq.  can be rA with F maps/encodes the spatial information to the temporal information in the primary STI equation, whereas the matrix B with F maps/decodes the encoded temporal information to the original spatial information in the conjugate STI equation . It shoion (Eq. . The det43G(·) is the nonlinear function set of the Lorenz system with P is a parameter vector. The exact Lorenz system and detailed description are provided in Supplementary Note To illustrate the mechanism and the basic idea of the ARNN framework, a 90-dimensional coupled Lorenz modely1, y2, and y3, each of which is randomly selected from {x1, x2, …, x90}. In one prediction, we use the 90-dimensional data from the initial 50 steps as known information/input, and ARNN outputs 18-step-ahead prediction for the target variables, i.e., D = 90, m = 50, and L − 1 = 18. For all the noise-free cases . The performance of ARNN and that of the traditional prediction methods were compared. Specifically, three cases are selected in Fig. σ = 1), ARNN remains robust and is capable of providing accurate prediction in both future states and dynamical tendency with RMSE = 0.333 in Fig. Second, we discuss the situation when there is additive noise. ARNN (Eq. ) and eigm = 50 steps as the input and a prediction L − 1 = 18 steps ahead as the output), ARNN was applied to a 90D time-varying/time-switching Lorenz system,G(·) is the nonlinear function of the Lorenz system with P(t) is the time-varying/time-switching parameter vector. In other words, the parameters of the Lorenz system change as time evolves; that is, when the time variable t moves forward every 10 units, the parameters P(t) change once. The exact expression of the time-varying Lorenz system and other detailed information are provided in Supplementary Note Third, with the same setting , its shσ = 1, and (iii) time varying and noise free. Under each condition, the performance of nine prediction methods is compared based on short-term series with parameter sets m = 50, L − 1 = 18 and m = 15, L − 1 = 6.In Supplementary Table m = 50 and the prediction length is L − 1 = 18, the average normalized RMSE of ARNN is 0.397, which is better than those of the other prediction methods with RMSE ∈ . When the known length becomes even shorter (m = 15), the performance of ARNN (RMSE = 0.168) is still better than that of the other methods with RMSE ∈ ; that is, ARNN achieves at least 42% more accuracy than the other methods and is at least 31% faster than the traditional neural network methods. In particular, when the known length is only 15, the LSTM shows a poorer result (RMSE = 0.538) but still incurs three times the running cost of ARNN  in Wakkanai, Japan, provided by the Japan Meteorological Agency44. The wind speed was recorded every 10 min during the total period of 138,600 min in 2011. We take the length of the known series as m = 110. Based on ARNN, the wind speed of one randomly selected site  and that of the real data (in cyan) are all above 0.90  dataset of wind speed generated from 155 sampling sites  with different prediction steps Fig. . When th 50 Fig. , the ove 30 Fig.  and 10 . The correlation that reflects the consistency between the real and predicted dynamical trends by ARNN reaches 0.961.The second real-world dataset contains solar irradiance data generated from min Fig. . Each prD = 72) recorded every 1 h, collected from 1998 to 2004 in the Houston, Galveston and Brazoria areas45. The sea-level pressure (SLP) and average temperature were predicted as shown in Fig. m = 60 steps, and the outputs were the 25-step-ahead values of a target index (L − 1 = 25).Next, ARNN was applied to the third dataset, a 72-dimensional ground meteorological dataset . The dataset is composed of a series of 241 cloud images from 15 March 2018 to 24 March 2018 with one image taken per hour. There are D = 2402 variables in each image. Thus, the 241 images can be regarded as a time series within a period of 241 h. For each prediction, the initial m = 50 images were regarded as known information, and ARNN was applied to forecast the central position of the tropical cyclone for the next L − 1 = 21 time points, i.e., 21-step-ahead prediction in one output. The predicted results are shown in Fig. The fifth dataset, satellite cloud image records of typhoon Marcus, comes from the National Institute of Informatics  from 1 May 2018 to 22 November 2018. Due to the linkage effect in the stock market, different sectors of the stock market interact internally and form a complex system. By applying ARNN with m = 50 (days) and L − 1 = 20 (days), we predicted the B-Share index of the Shanghai Stock Exchange  values of nge Fig. .47. According to the high correlation between the cardiovascular inpatients and air pollutants48, ARNN was applied to forecast the inpatient number based on a set of air pollutant indices. Considering the delay effect of every potential factor as well as a dummy vector of the weekday effect48, we obtained a 48-dimensional system (D = 48). Based on the daily concentrations of pollutants, ARNN predicted the short-term dynamical trend of the daily cardiovascular disease admissions  and L − 1 = 60 (days).The prediction accuracy of ARNN was also validated in a real-world dataset that contains several time series, including the index series of air pollutants and the number series of cardiovascular inpatients in major hospitals in Hong Kongons Fig. , leadingons Fig. , where mD = 207 loop detectors in Highway 134 of Los Angeles County49. Each detector was regarded as a variable. By applying ARNN, a multistep prediction (L − 1 = 30 time points ahead), was obtained from the high-dimensional data with m = 80 time points at four adjacent locations  based on a dataset collected from ons Fig. . SupplemClearly, these results show that ARNN accurately predicted the dynamical behaviors for non-periodic and highly-fluctuating cases based on only short-term data.Xt and temporal vector Yt) correspond to each other one-by-one via a smooth map36, and thus we obtain the primary and conjugate STI equations  to the temporal information of a target variable Yt, while the conjugate equation decodes/recovers the (encoded) temporal information Yt to the high-dimensional variables F(Xt), i.e.,  In this study, we propose the ARNN framework to make multistep-ahead predictions based on short-term high-dimensional data in an accurate, efficient, and robust manner. The delay-embedding theorem ensures that two vectors , i.e., ARNN is computationally efficient and accurate because it incorporates both the STI transformation and the RC structure. On the one hand, by the STI transformation, ARNN transforms the spatial information of high-dimensional data to the temporal information of any target variable, thus equivalently expanding the sample size and alleviating the small sample size problem. On the other hand, by the RC structure, ARNN requires fewer parameters to train, thereby avoiding the overfitting problem. Moreover, in contrast to the external dynamics used in traditional RC, ARNN takes the inherent dynamics of the high-dimensional data themselves as the reservoir.51 of all datasets used in this work in Supplementary Table Notably, most long-term data, such as expression data from biological systems and interest-rate swaps data from financial systems, may also be regarded as short-term data because those systems are generally not stationary but highly time-varying with many hidden variables. Therefore, to characterize or forecast their future states, it is more reliable to employ recent short-term data, than a long-term series of past data. Therefore, ARNN is a general method suitable for many real-world complex systems even when only recent short-term data are available. To check the assumption of STI equations, i.e., the low dimensionality of the underlying attractors, we estimated the box-counting dimensions52. On the other hand, the critical transitions resulting from bifurcation can be detected by dynamical network marker methods55. In addition, ARNN cannot make accurate predictions for strongly noisy data because its theoretical framework is mainly based on deterministic dynamics. Besides, given a low-dimensional sequence, the prediction of ARNN may be similar to the traditional approaches in terms of accuracy due to insufficient spatial information to be transformed.One limitation of ARNN is that it is unable to accurately forecast sudden changes or critical transitions in real-world systems. The critical transitions in complex systems are often led by changes in external factors, whose information is generally not included in the measured dataIn summary, compared with traditional prediction methods, ARNN possesses the following advantages. First, ARNN achieves multistep-ahead prediction even with only short-term data due to the transformation from high-dimensional spatial information into temporal information. In contrast to many deep learning methods that suffer from overfitting problems when a large number of parameters are to be trained but with only a single short-time series, ARNN has much fewer parameters due to its reservoir structure where most of the parameters are randomly given. Thus, in practical applications, ARNN requires less computing resources. In particular, ARNN takes the observed high-dimensional variables as the reservoir which represents the inherent dynamics of the target variable, rather than the external dynamics. Third, by simultaneously solving a conjugated pair of STI equations (similar to an autoencoder with both encoding and decoding), ARNN is highly robust and performs well in both noise-perturbed and time-varying-parameter cases, which widely exist in real-world systems. In addition, ARNN has a solid theoretical background based on the delay-embedding theorem. The results for the applications to a variety of real-world problems demonstrate the effectiveness and efficiency of our method. Therefore, ARNN paves a new way for short-term prediction in terms of computationally efficiency, accuracy, and robustness, which is of high potential in real-world applications.The descriptions on the parameters and variables in ARNN framework are summarized in Supplementary Table 57, derived from independently proposed RNN models, such as the echo state network (ESN)58 and the liquid state machine (LSM)59. Generally, ESN is the widely studied RC framework.Reservoir computing (RC) is a unified computational framework58. When feedback from the output to the reservoir is absent, the time evolution of the neuronal states in the reservoir is described as follows58:t denotes the discrete time, rt is the state vector of the reservoir units, Xt is the input vector, Win is the weight matrix for the input-reservoir connections, and W is the weight matrix for the recurrent connections in the reservoir. The function fk among f =  represents the kth elementwise activation function of the reservoir units, which is typically a sigmoid-type activation function. Equation .ESN uses an RNN-based reservoir consisting of discrete-time artificial neuronsn-dimensional state space t where symbol “′” is the transpose of a vector, and any time interval between two consecutive time points is equal. After a sufficiently long time, all of states are converged into a compact manifold 40.For a general discrete-time dissipative system, the dynamics can be defined asd, for a smooth diffeomorphism L > 2d, that is,X = Xt and h(Xt) = yt where Xt is n-dimensional variables here, but sometimes it is used as D-dimensional variables (D ≤ n) in this work.If mentclass2pt{minimmentclass2pt{minim37 has theoretically been derived from the delay-embedding theory40, which can transform the spatial information of high-dimensional data to the temporal information of any target variable. The related description is given in Supplementary Note L > 2d where d is the box-counting dimension of the attractor, the STI equations  or one-Actually, all of the above conditions are generally approximately satisfied for a real-world system. The conditions above are sufficient conditions to ensure a one-to-one mapping. In practice, even if the conditions are not fully satisfied, e.g., the system is not in a steady state but in a transient state, ARNN also gives an accurate prediction for many cases, since usually the transient dynamics of a dynamical system is also constrained to a lower dimensional space or manifold.AB = I, A and B are L × D and D × L matrices, respectively, and I represents an L × L identity matrix. Clearly, the first and second equations represent the linearized primary and conjugate forms, respectively. We can use the matrix form to represent these equations, i.e., AX = Y and X = BY where X and Y are defined asY represents the unknown/future information y by estimating the maps Φ and Ψ via A and B, respectively, but the accumulated error drastically increases with the prediction horizon. The prediction performance of the linear method based on the linearized STI equations was provided in Supplementary Figs. Generally, Φ and Ψ are nonlinear functions, which can be linearized as Eq.  Fig. 1b1b at \doXt as an input and Yt as an output, i.e.,AB = I, A is an B is a I represents an L × L identity matrix. Note that ARNN can be represented by the ARNN-based STI equations Eq.  Fig. 1c1c at \doentclass1pt{minimaAF(X) = Y and F(X) = BY where Xt or F(Xt) with t = 1, 2, …, m, we can obtain the unknown future values A and B.Equations above or Eq.  can alsoons (Eq. ) for givXt or F(Xt), we solve the ARNN-based STI equations Eq. -step-ahead prediction, rather than the one-step-ahead prediction. It should be noted that when we solve the ARNN equations, {y1, …, ym} are known but {ym+1, …, ym+L−1} are unknown future values of the target variable because of the delay embedding. Therefore, it is similar to a semi-supervised learning process. The detailed algorithm of ARNN is given in Supplementary Note It is clear that the future states including ARNN Eq.  with then-dimensional variables y1, y2, …, ym}, we select the most correlated variables  with the target variable y, i.e., variables with the high mutual information with y, and obtain the following input vectors and matrixD is the number of the selected variables with n ≥ D, and m is the length of the known time series.Although high-dimensional data have rich information, they may also have a noisy effect on the prediction if some of the high-dimensional variables contain little information on the target variable. Thus, choosing relevant variables or eliminating irrelevant variables to the target variable from the high-dimensional data may significantly enhance the performance of ARNN in practical applications. Given a time series of Compared with traditional neural network method, ARNN takes much less time and computing resources in decoding the intertwined information among massive variables of a complex system, for the future value prediction of the target variable. A comparison between the ARNN and traditional neural network method on the computational complexity is demonstrated in Supplementary Note Supplementary Information


More explicitly, we consider consequences of f being We are interested in how stronger convexity properties of of Sason , who demf-divergence inequality is Pinsker’s inequality, which bounds the square of the total variation above by a constant multiple of the relative entropy. That is for probability measures P and Q, Perhaps the most well known example of an les, see ,10,11). les, see  and Topsf is restricted to a sub-interval of the real line. This observation is especially relevant to the situation in which one wishes to study f-divergences realized as linear combinations of skewed divergences.Aside from the total variation, most divergences of interest have stronger than affine convexity, at least when vergence  based onvergence ,14, callvergence  and its vergence  based onvergence  and its vergence  give exaf. As a consequence, we obtain bounds between the mean square total variation distance of a set of distributions from its barycenter, and the average f-divergence from the set to the barycenter.Let us outline the paper. In f-divergences. In particular, we introduce the skew-symmetrization of an f-divergence, which recovers the Jensen–Shannon divergence and the Vincze–Le Cam divergences as special cases. We also show that a scaling of the Vincze–Le Cam divergence is minimal among skew-symmetrizations of In  Nielsen   We use this decomposition to sharpen, for f-divergences. We then demonstrate explicitly, using an argument of Topsøe, how our sharpening of Guntuboyina’s inequality gives a new sharpening of Pinsker’s inequality in terms of the convex decomposition induced by the Bayes estimator.In tuboyina  that genf denotes a convex function P and Q on a Polish space f-divergence from P to Q, via densities p for P and q for Q with respect to a common reference measure Throughout, We note that this representation is independent of f-divergenceX and a set A, we denote the probability that X takes a value in A by A, we write For Definition 1.A f is twice differentiable,  (see ); in ourThe next lemma is a restatement of Jensen’s inequality.Lemma 1.If f is κ-convex on the range of X,Proof. Apply Jensen’s inequality to f such that For a convex function Definition 3.For .,21.20,2f-divergences , that the inequality characterizes the “joint range”, that is, the range of the function We mention the work of Harremos and Vadja , in whicLemma 2.For a κ-convex f,Proof. f. Thus, we may assume without loss of generality that f is Define a f about 1, Nielsen and Nock ≈Theorem 2.For measures P and Q, and a κ convex divergence Proof. By Lemma 2 and then Theorem 1,f-divergence inequalities.The proof of Lemma 2 uses a pointwise inequality between convex functions to derive an inequality between their respective divergences. This simple technique was shown to have useful implications by Sason and Verdu in , where iTheorem 3.For divergences defined by g and f with Moreover, if (Sason–Verdu ).For diCorollary 1.For a smooth κ-convex divergence f, the inequalityis sharp multiplicatively in the sense thatif f-divergence is defined as an f-divergence satisfying the normalization f-divergence that is 1-convex. In particular, the lower bound in Lemma 2 complimenting the estimate =1 (see ). Thus, estimate  is shownProof. Without loss of generality, we assume that Thus,  follows.Proposition 2.Θ such that Θ, thenWhen In particular, when Proof. Let ty givesDf(P||Q)≤To obtain the total variation bound, one needs only to apply Jensen’s inequality,f-divergence from the set of distribution to their barycenter, by the mean square total variation of the set of distributions to the barycenter,Observe that, taking An alternative proof of this can be obtained by applying f strongly convex, Pinsker type inequalities can never be reversed,The next result shows that, for Proposition 3.Given f strongly convex and Proof. By f divergence, f-divergence can be bounded above by a constant multiple of a the total variation, if and only if In fact, building on the work of Basu-Shioya-Park  and Vadjf on f-divergences. The mapping If we denote f-divergences and leverage f-divergence inequalities in this section.Moreover, as mentioned in the Introduction, skewing imposes boundedness of the Radon–Nikodym derivative f-divergence preserves its status as such. This guarantees that the generalized skew divergences of this section are indeed f-divergences. A proof is given in the The following appears as Theorem III.1 in the preprint . It statTheorem 4.For is an f-divergence as well..For t,Definition 4.For an f-divergence, its skew symmetrization,f Topsøe  providesDefinition 5.When If one denotes the skew symmetrization of the ily from  that Δχ2Corollary 2.For an f-divergence such that f is a κ-convex on with equality when the Proof. Applying Proposition 2When nce .For P Proof of Theorem 4.If We recover the classical bound ,16 of thCorollary 3.For probability measure P and Q,Proof.  Since Proposition 4 gives a sharpening of Lemma 1 of Nielsen , who proDefinition 6.For p and q densities with respect to a reference measure μ, where (Nielsen ).For p Note that, when Theorem 5.For p and q densities with respect to a reference measure μ, where w average of the Note that, since e et al.  for the Theorem 6.For where  thatProposition 6.When Proof. Since Information theoretic bounds to control the Bayes and minimax risk have an extensive literature  that 2nObserve that the f, quantified by the parameter f-divergence inequalities, and as a consequence achieve a sharpening of Pinsker’s inequality.In this article, we begin a systematic study of strongly convex divergences, and how the strength of convexity of a divergence generator


Scientific Reportshttps://doi.org/10.1038/s41598-020-74770-x, published online 23 October 2020Correction to: The original version of this Article contained an error in the title of the paper, where the phrase “Disease - associated” was incorrectly given as “Disease: associated”. This has now been corrected in the PDF and HTML versions of the Article.


Correction to: Molecular Psychiatry10.1038/s41380-018-0112-7 published online 14 August 2018Following publication, the authors noticed that ‘Laura Cantwell’, ‘Otto Valladares’, and ‘Li-San Wang’ were inadvertently omitted from the author list. These authors have now been added to the author list in 21st, 77th, and 79th position, respectively. This has been corrected in both the PDF and HTML versions of the article.


Some studies have suggested that early-life exposure to these compounds could lead to increased body mass index (BMI) during childhood. Our aim was to assess whether children’s exposure during the first 2 years of life is associated with BMI z-score in Japanese children at 42 months of age.Children are exposed to n = 290) of the Tohoku Study of Child Development. p,p’-DDT and p,p’-DDE levels were measured in breast milk samples collected 1 month after birth, and levels in children were estimated using a toxicokinetic model for three exposure periods . Associations between exposure estimates and BMI z-score at 42 months of age were assessed using multivariate linear regression models.We used data from a birth cohort  for the 0–6 months exposure period, 0.26  for the 6–12 months exposure period, and 0.24  for the 12–24 months exposure period.We found no significant association between levels of p,p’-DDE levels were associated with increased BMI z-score at 42 months of age, mostly in girls. These results are in line with previous studies supporting that early-life exposure to p,p’-DDE may be associated with higher BMI during childhood.In this study of Japanese children, estimated postnatal  Although it was banned in most countries decades ago, it is still used in some countries for malaria vector control [p,p’-DDT and its metabolites, such as p,p’-dichlorodiphenyldichloroethylene , are widespread in the environment [ control . p,p’-DDironment . Both ofironment –6.p,p’-DDT and p,p’-DDE through placental transfer and breastfeeding [p,p’-DDT and p,p’-DDE accumulate in the mother’s adipose tissues [p,p’-DDT and p,p’-DDE mainly comes from diet [Children are exposed to tfeeding –9. Becau tissues . These c tissues . In mostrom diet . They arrom diet , the latp,p’-DDT and p,p’-DDE, there has been an increasing interest in their relationship to obesity. A systematic review and meta-analysis conducted by Cano-Sancho et al. [p,p’-DDT and p,p’-DDE and obesity. In vitro studies have shown that p,p’-DDT can impair adipose tissue homoeostasis and induce endocrine disruptions; results on p,p’-DDE were inconsistent. The in vivo studies found that perinatal exposure to p,p’-DDT can induce thermogenesis and lipid metabolism disruptions, especially in female offspring [p,p’-DDE exposure and higher BMI z-scores. Associations with other weight-related outcomes were less consistently reported, like associations between p,p’-DDT and BMI z-scores. The authors expressed the need for more epidemiologic and experimental studies, and classified these contaminants as “presumed” human obesogens.Among all the health impacts of o et al.  gatheredffspring , and traffspring . The metp,p’-DDT and p,p’-DDE with children’s weight focused on prenatal exposure, little is known about the potential impacts of exposure through breast milk. We only identified three studies on lactational exposure to p,p’-DDE and children’s weight [p,p’-DDE in seven European birth cohorts. They found an association between prenatal p,p’-DDE exposure and increased infant growth, but no association with postnatal exposure estimates. Given the limitations of previous studies and the paucity of data the association between lactational exposure to p,p’-DDT and p,p’-DDE and children’s weight, further investigation is warranted.Because most epidemiologic studies on the association of s weight , 16, 17.s weight  used bres weight  and Pan s weight –21 to esp,p’-DDT and p,p’-DDE, and BMI z-score in 42-month-olds participating in the Tohoku Study of Child Development. Because no postnatal exposure assessment was conducted in this study, we used a validated toxicokinetic model to estimate children’s levels throughout their first two years of life.The aim of our study was to evaluate the association between measured breast milk levels and estimated children’s postnatal levels of The Tohoku Study of Child Development is a prospective birth cohort study that comprises two birth cohorts from an urban area and a coastal area cohort of northeastern Japan. The protocol and research areas have been described previously , 23. ThePregnant women were recruited through their obstetrical units in two hospitals in the Sendai city . Women wn = 54), because they moved to another region (n = 18), or because they dropped out or weren’t reachable (n = 16) [p,p’-DDT and p,p’-DDE, had data on total breastfeeding duration , and had data on child weight and height at 42 months , leaving 290 mother-child pairs.Of the 1500 women who were met between January 2001 and September 2003, 687 gave their consent to participate in the study (participation rate: 46%). During their pregnancy, 88 women had to withdraw from the study, either because their baby did not meet the inclusion criteria ((n = 16) , 23. ThiAll the procedures of this study were approved by the Medical Ethics Committee of the Tohoku University Graduate School of Medicine and of the National Institute for Environmental Studies . In addiData on demographics, lifestyle, medical history, and fish consumption was obtained through questionnaires administered four days after delivery . BreastfMothers were asked to send a breast milk sample of more than 50 ml in a clean glass bottle one month after delivery. Samples were aliquoted in 10 ml centrifuge tubes and immediately frozen at − 80 °C until analyses.p,p’-DDT and p,p’-DDE were then measured using a gas chromatography/high-resolution mass spectrometry (HRGS/HRMS) system. The quality control was performed in accordance with the German external quality assessment scheme (G-EQUAS). The method limit of detection  was calculated using the Currie et al. method [All manipulations were made by IDEA Consultants, Inc. laboratory in Tokyo . A sampl. method . Becausep,p’-DDT and p,p’-DDE) developed by Verner et al. [p,p’-DDT: ≈ 5 years [p,p’-DDE: 13 years [p,p’-DDT and p,p’-DDE levels for each participating mother-child pair. For each pair, the model incorporated individual-specific data on mother’s age at delivery, the total breastfeeding duration, mother’s body weight before pregnancy and one month after delivery, weight gain during pregnancy, and child’s sex. Available information did not allow estimating the duration of exclusive breastfeeding precisely. For that reason, only the total duration of breastfeeding was used, and daily breast milk intake was based on equations derived from studies including exclusively and partially breastfed children. For each pair and each chemical, iterative model simulations were performed to calibrate maternal lifetime daily oral intake until the breast milk level estimated with the model matched the breast milk level measured in the study. This maternal daily dose was then used in the model to estimate children’s areas under the concentration vs. time curve (AUCs) for three periods: 0 to 6 months, 6 to 12 months, and 12 to 24 months of age. Average children’s levels during these periods were calculated by dividing AUCs by the integration period. We performed toxicokinetic model simulations using acslX .We used the toxicokinetic model of prenatal and lactational exposure to lipophilic persistent organic pollutants . Exposu13 years , and fro13 years  for the p-value < 0.05) but most of the chi-squared tests did not. Little’s MCAR test showed statistical significance. Thus, we concluded that the missing data was not missing completely at random (MCAR), but was missing at random (MAR) as there might be systematic differences between the missing and observed values, but these can be entirely explained by other observed variables [Data was missing for 11% of the weight gain during pregnancy and 26% of the maternal weights one month after delivery. Missing data was due to non-response to the questionnaires. We tested the Missing at random assumption with t-tests for numeric variables, chi-squared test for categorical variables and an overall test of randomness (Little’s MCAR test). Most results for t-tests showed statistical significance , an approach that reduces the degree of bias that can occur when measuring causal relationships between exposure and effect   to assess multicollinearity, and verified that the residual plots did not display unwanted patterns. Statistical analyses were performed in R, version 1.1.456 [We calculated sex and age-specific BMI z-scores using the World Health Organization’s growth charts, which is an international standard taking into account children from different ethnic and cultural backgrounds . Summary 1.1.456 .p,p’-DDT concentrations in breast milk in the excluded group made it impossible for us to assess the difference between included and excluded pairs for these variables.The median maternal age at delivery was 31 (range: 20–41) years, and the median total breastfeeding duration was 1.4 (range: 0.1–4.0) years  were estimated using the toxicokinetic model previously presented.Table p,p’-DDT and p,p’-DDE levels and BMI z-scores at 42 months are shown in Table p,p’-DDT levels were not significantly associated with higher BMI z-scores in either model. Only one significant association was found with p,p’-DDE when including boys and girls; BMI z-score was positively associated with estimated children’s levels during the 12–24 months period. Additional statistically significant associations were revealed with sex stratification, with p,p’-DDE being associated with greater BMI z-score in girls for the three exposure periods in both Model 1 and Model 2. Overall, when significant associations were found, they were slightly stronger in Model 2. Beta coefficients for different time periods were similar, which may be related to the fact that exposure estimates were strongly correlated across periods, with correlation coefficients ranging from 0.87 to 0.98 for p,p’-DDT and 0.90 to 0.99 for p,p’-DDE. In all regression models, VIF values were below 1.2. Residuals were normally distributed, and showed no pattern indicative of non-linearity.Associations between p,p’-DDT and p,p’-DDE is associated with BMI z-score in 42-month-old children participating in the Tohoku Study of Child Development in Japan. We found associations between children’s estimated p,p’-DDE levels and higher BMI z-scores (mostly in girls), but none with p,p’-DDT.In this study, we evaluated whether postnatal exposure to p,p’-DDE. This difference between chemicals could be due to a stronger obesogenic potential of p,p’-DDE than p,p’-DDT. However, this could also be due to the fact that measured p,p’-DDE levels of this contaminant were higher , and that toxicokinetic model predictions were more precise and accurate for p,p’-DDE than p,p’-DDT in this cohort [p,p’-DDT and p,p’-DDE on BMI increase remains uncertain. In vitro studies suggested that these chemicals may be endocrine disruptors [p,p’-DDT and p,p’-DDE on the endocrine system, as the former is an estrogen and androgen receptor antagonist, whereas the latter is an androgen and progesterone antagonist [p,p’-DDT and p,p’-DDE, the reasoning behind this phenomenon still remains uncertain.We selected BMI z-scores as our outcome because studies have shown that childhood obesity and/or overweight is associated with obesity, cardiometabolic morbidities and cancer in adulthood –42. Assos cohort . The mecsruptors , and thasruptors . In rat sruptors , and cousruptors . The diftagonist . Given tp,p’-DDT and p,p’-DDE levels during different time windows  and BMI z-score. Because the strength of the associations was similar from one time period to another, we weren’t able to identify a specific critical exposure window. This could be interpreted as the lack of a specific window of vulnerability to the obesogenic impacts of these chemicals during the first two years of life. Another explanation could be that estimated exposure metrics, being highly correlated across periods, did not allow distinguishing time-specific differences in strength of exposure-outcome associations.In this study, we examined associations between estimated p,p’-DDE. We found slightly stronger associations in Model 2 and with later exposure periods in general, although the differences were mostly negligible. In the meta-analysis, three studies stratified by sex to examine potential differences between boys and girls regarding associations with obesity indicators [p,p’-DDT and p,p’-DDE and higher BMI z-score at nine years of age in boys [p,p’-DDE and greater waist circumference and weight/height ratio in girls [Our effect estimates for postnatal exposure were slightly stronger than the meta-analytic effect estimate reported by Cano-Sancho et al. , but condicators –47. One dicators . Another in boys . The thip,p’-DDT and p,p’-DDE. Given the complex changes in physiology and exposure during infancy, spot measurements are unlikely to provide comprehensive information on children’s exposure profile. Using our toxicokinetic model, we were able to reconstruct children’s levels from birth to two years, and to estimate cumulative exposure over three different periods . This allowed us to assess exposure-outcome associations during different potential windows of vulnerability.Our study has some important strengths. First of all, the longitudinal study design is ideal when evaluating associations between risk factors and developmental outcomes. Anthropometric data were collected directly by trained physicians or research coordinators using a certified electronic scale, which reduces measurement error. Typically, epidemiologic studies rely on one or few blood or breast milk levels to assess children’s early-life exposure to persistent organic pollutants like p,p’-DDE and p,p’-DDT levels due to analytical measurement error and intra- and inter-feed variability may have impacted precision in exposure estimates. Calculating BMI z-scores based on WHO data may not be optimal for the Japanese population [Some limitations are also worth mentioning. First of all, when stratifying by sex, our sample size was cut by more than half, going from 290 to 134 mother-child pairs for analyses in girls. Also, estimations using the toxicokinetic model are not perfectly precise, and the error in level estimates likely impacted the precision in exposure-outcome models. Precise information on exclusive breastfeeding duration data wasn’t available, which could have potentially helped increase the precision of the estimates. Furthermore, uncertainty in measured breast milk pulation ; howeverp,p’-DDE levels and higher BMI z-scores in girls. These results suggest that prenatal exposure may not be the only sensitive time window when looking at chemical exposures and weight-related outcomes later in life. The strength of our associations concurred with that of Cano-Sancho et al.’s meta-analysis [Our study suggested an association between estimated children’s analysis . PostnatAdditional file 1: Fig. S1. Directed acyclic graph (DAG) for the association between p,p’-DDT/E exposure metrics and BMI z-score at 42 months of age. Table S1. Correlations between breastfeeding duration and ln-transformed exposure metrics for p,p’-DDT. Table S2. Correlations between breastfeeding duration and ln-transformed exposure metrics for p,p’-DDE.


Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. Due to the second characterization, we construct a function F that maps any n-bit partial Boolean function to some integer, and if an n-bit partial Boolean function f depends on k bits and can be computed exactly by a 1-query quantum algorithm, then n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n.We provide two sufficient and necessary conditions to characterize any  In classical computation, a simplest model of computation is the decision tree . Corresum model . Roughlyf, quantum advantage can be investigated by comparing the exact quantum query complexity In quantum computation, the hope is to find out many problems whose computational complexity in quantum computer is less than the computational complexity in classical computer, i.e., finding out many problems that have the quantum advantage. For a function ity D(f) , where Dry model ,18,19,20ry model  proved tAmbainis , and theAmbainis  presentef can be computed exactly by a 1-query quantum algorithm if and only if f can be computed by the Deutsch-Jozsa algorithm [t queries. However, the result in [t-query quantum algorithms in terms of the unit ball of a space of degree-For any partial Boolean function, the best separation is still achieved by Deutsch-Jozsa algorithm ,17. And,lgorithm . In the lgorithm  showed algorithm  investigesult in   Boolean functions. Naturally, what is the characterization of partial Boolean functions with exact quantum query complexity 1?The partial Boolean functions can be regarded as a generalization of the total Boolean functions. Actually, Deutsch’s algorithm  computesIn the field of quantum computation, it is a fundamental and interesting subject to evaluate the computational power of the 1-query quantum model, and is also critical for discovering quantum advantage. Specifically, the number of partial Boolean functions with exact quantum query complexity 1 shows the power and advantage of the exact 1-query quantum model. So, how many partial Boolean functions can be computed exactly by 1-query quantum algorithms?The rest of the paper is organized as follows. In In this section, we introduce some basic notations and recall some basic knowledge of partial Boolean functions and the exact quantum query model. For the details, we can refer to Refs. ,23,30,31D (or promised set) to denote a subset of x is denoted by S, the notation S and the notation S. For any finite set S, the notation S. For a complex matrix A, A, and A. Obviously, A. Furthermore, the notation a. In particular, As usual, notations f computed by Deutsch’s algorithm [n-bit partial Boolean function f computed by Deutsch-Jozsa algorithm [In this paper, we mainly concern partial functions lgorithm  can be g),f(01),f,f(11))=xi ,30. In pxi ,30. Ier basis ,30. If axi ,30. I all x∈D . Unlike j is the auxiliary space. As a result, a t-query quantum algorithm can be determined by an initial state In the quantum query model, for every input fined byOx|i,j〉=n-bit partial Boolean function f, if some equations n-bit partial Boolean function n-bit partial Boolean function g defined byIf the exact quantum query complexity of an This subsection considers a special class of all partial Boolean functions using Theorem 1.First, we introduce the following definition ,31 A ba.Definition 1.,31. An nn, the n-bit partial Boolean function computed by Deutsch-Jozsa algorithm [By Definition 1, the two-bit total Boolean function computed by Deutsch’s algorithm  depends lgorithm  depends For partial Boolean functions depending on all bits, the result is stated in the following.Theorem 2.For any n-bit partial Boolean function Proof. n-bit partial Boolean function f can be written ask-th bit being flipped. With an argument, if f depends on n bits , then there must exist at least n input pairs k, if f depends on n bits).By Theorem 1, there exists a non-negative vector f can be given by The case n-bit partial Boolean functions into Since there are many partial Boolean functions with exact quantum query complexity 1, it is necessary to divide them into some classes. In 2015, Montanaro et al. ,24 studin-bit partial Boolean functions depending on 1 and 2 bits, the problem is trivial. For n-bit partial Boolean functions depending on n bits, the result is put into Theorem 2. By a trivial (not direct) argument, the statement that a partial Boolean function f depends on all n bits is consistent with previous definition. This is a key hint in the proof. Intuitively, this implication seems obvious. However, since the implication does not hold for For all n-bit total Boolean function depending on kk-bit total Boolean function depending on k bits. Therefore, Theorem 2 generalizes the result on total Boolean functions [n-bit partial Boolean functions depending on n bits is quite big. This fact is implied by the following lemma.Theorem 2 clarifies all partial Boolean functions that depend on all bits and can be computed exactly by a 1-query quantum algorithm. Observing the unique multilinear polynomial of any total Boolean function, any unctions  to partiLemma 1.Let n-bit total Boolean functions  is 22n u et al. ,24) is 3n-bit partial Boolean functions depending on As a result, many This section introduces and proves the second characterization and two related results.f and Following the discussions of Lemma 7 and Theorem 17 in [First, the following lemma follows the discussions of Lemma 7 and Theorem 17 in .Lemma 2.. If therIn order to give the second characterization, we introduce the following definition. This definition is also used in . More reDefinition 2.,37,38,39hold, then the is called a degree-1 SOS complex representation matrix of f and Equation .S the column index. In a 1-query quantum algorithm, the number pair S traverses As we know, for the state By Definition 2 and the coefficient matrix, the second characterization is presented in the following.Theorem 3.Any n-bit non-constant partial Boolean function Proof. i-th block matrixf and According to Lemma 2 and Definition 2, the coefficient matrix Equation  holds.f and f can be determined by As a result, the three states f and In order to use Theorem 3, we need to find a pair of SOS real representations of unctions ,37,38,39t-query quantum algorithms in terms of the unit ball of a space of degree-To some extent, Theorem 3 provides a different style for the characterization of partial Boolean function with exact quantum query complexity 1. On one hand, similar to Ref.  =∑l=1tisfyingf(x)=∑l=1Equation  holds.n-bit partial Boolean function f with exact quantum query complexity 1, there exists an SOS complex representations matrix f and x such that On the other hand, by Theorem 3, for an Equation , we can Equation  for x suf given by the two-tuple The inverse direction of Theorem 4 is not always hold. For example, a three-bit partial Boolean function  of Ref. , QE(f)≥2n-bit partial Boolean function depends on k bits and can be computed exactly by a 1-query quantum algorithm. That is, the function Theorem 4 gives a necessary condition on the case that an f is bigger than 1 by Theorem 4, then the result is exact.Compared with Theorem 1, Theorem 4 provides an approximate method on deciding the exact quantum query complexity of a partial Boolean function. Most importantly, the method in Theorem 4 is more efficient than Theorem 1. In order to construct the linear system in Theorem 1, we should observe at most n-bit partial Boolean functions which depend on k bits and can be computed exactly by a 1-query quantum algorithm.In this subsection, let us evaluate the number As a preparation, the following lemma is necessary.Lemma 4.Let the vector function By Theorem 4, the fifth result is the following.Theorem 5.Let Proof. n-bit partial Boolean functions satisfying n-bit partial Boolean function can be determined using the following two steps.Recall that all columns in the matrix In the first step, we choose n-bit partial Boolean function f is determined as follows. x in the set x in the set In the second step, we add some other vectors Equation  and Lemmn-bit partial Boolean functions, as each n-bit partial Boolean function corresponds to a string n-bit partial Boolean function is in the set n-bit partial Boolean functions which depend on k bits and can be computed exactly by a j-query quantum algorithm. Thus, all n-bit partial Boolean functions with exact quantum query complexity 1 only make up a very tiny proportion of all n-bit partial Boolean functions.The most important contribution of Theorem 5 is to give an estimate on the number of partial Boolean functions with exact quantum 1-query complexity. This is the first non-trivial upper bound on this problem. In contrast, Furthermore, corresponding to Fact 1 in Ref. , the folf and g satisfying f can be computed exactly by a 1-query quantum algorithm, then g can also be computed exactly by this 1-query quantum algorithm.g defined byn, n-bit partial Boolean functions with exact quantum query complexity 1. In general, given an n-bit partial Boolean function with exact quantum query complexity 1, we can find out trivially at least As we know, the partial Boolean function computed by Deutsch-Jozsa algorithm can be written asFinally, the gap between the upper bound and the lower bound comes from the following two aspects. The first aspect is that the upper bound is obtained from a necessary condition  and an approximate counting argument. The second aspect is that there exist many unknown partial Boolean functions with exact quantum query complexity 1.n-bit partial Boolean functions with exact quantum query complexity 1, and one necessary condition for characterizing n-bit partial Boolean functions that depend on kn-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm  of the actual number of partial Boolean functions in this class is quite big. On the other hand, known non-trivial n-bit partial Boolean functions in this class are still fairly rare [Find all (or some) non-trivial rly rare ,22,23,24n-bit partial Boolean functions can be computed exactly (or with bounded-error) by k-query quantum algorithms for all k-query quantum model. In contrast, the result of Ambainis et al. [How many s et al.  is a qua


As of May 25, 2020, the novel coronavirus disease  spread to more than 185 countries/regions with more than 348,000 deaths and more than 5,550,000 confirmed cases. In the bioinformatics area, one of the crucial points is the analysis of the virus nucleotide sequences using approaches such as data stream techniques and algorithms. However, to make feasible this approach, it is necessary to transform the nucleotide sequences string to numerical stream representation. Thus, the dataset provides four kinds of data stream representation (DSR) of SARS-CoV-2 virus nucleotide sequences. The dataset provides the DSR of 1557 instances of SARS-CoV-2 virus, 11540 other instances of other viruses from the Virus-Host DB dataset, and three instances of Riboviria viruses from NCBI . With this, it is possible to use data stream algorithms.•All researchers in bioinformatics, computing science, and computing engineering disciplines can benefit from these data because by using this numeric representation, they can apply several stream algorithms and techniques such as TEDA , TEDA-Cloud, TEDA-Cluster and Teda-Class in genomic information.•Data experiments that use analytic stream techniques in SARS-CoV-2 virus genomic information can be used with this dataset.•These data represent an simple way to evaluate the SARS-CoV-2 virus genome with stream algorithms.•Differently of the conventional bioinformatics techniques in which are based on dynamic programming (such as BLAST and other), this approach allows the utilization of different techniques (techniques commons in other areas) to find similarities between genome sequences.1k-mers mapping (kMersM) and k-mers mapping with CGR (kMersM-CGR). k-mers is a frequency count metric used in Bioinformatics. Other k-mers datasets are presented in This work presents a dataset of data stream representation (DSR) of SARS-CoV-2 virus nucleotide sequences. The dataset contains two kinds of data, the raw data, and the processing data. The raw data is composed of the 1557 instances of the SARS-CoV-2 virus genome collected from the National Center for Biotechnology Information (NCBI) k-mers representation k-mers also can be used with a signature of the genome sequence. However, in this manuscript, the genome sequence is transformed into a linear stream data, and this type of transformation can be used with stream algorithms. Another important aspect of this dataset is associated with applied CGR not in all sequences but just in each k bases (with mers or not). This strategy maintains the statistical characteristics and reduces the size of the stream.In the Chaos Game Representation (CGR) •k.mat”;For DM, the DSR was generated for •k.mat”;For DM-CGR, the DSR was generated for •k.mat”;For kMersM, the DSR was generated for ••k.mat”;In the directories “Other viruses data” and “SARS-CoV-2 data”, the DSR was generated for •k.mat”;In the “Virus-Host DB data”, the DSR was generated for For kMersM-CGR:The data is organized into three main directories: “SARS-CoV-2 data”, “Virus-Host DB data” and “Other viruses data”. Each main directory contains three files called “RawDataTable.mat”, “RawData.mat” and “RawData.xlsx”, and four sub-directories named “DirectMapping”, “DirectMappingCGR”, “kmersMapping” and “kmersMappingCGR”. “RawDataTable.mat”, “RawData.mat” and “RawData.xlsx” files store the raw data information from viruses database; they have the same information, however in the “RawDataTable.mat” the attributes are stored in Matlab table format (after 2013b version), in the “RawData.mat” the attributes are stored in Matlab cell arrays format, and in the “RawData.xlsx” the attributes are stored in a Microsoft Excel file. In the sub-directories “DirectMapping”, “DirectMappingCGR”, “kmersMapping” and “kmersMappingCGR” are stored the DM, DM-CGR, kMersM and kMersM-CGR data stream representation, respectively. Inside each sub-directory the files are called:i-th file is called “PointsData_k_k=k.mat” for sub-directory “DirectMapping”, “PointsDataCGR_i_k=k.mat” for DM-CGR, “PointsDatakmers_i_k=k.mat” for kMersM and “PointsDatakmersCGR_i_k=k.mat” for kMersM-CGR.For the main directory “Virus-Host DB data”, the values are stored in 10 files where each 2s, expressed asN is the length of sequence and ns is the nth nucleotide of the sequence.The streams were based in nucleotide sequence, s, are grouped in sub-sequences of the k bases. The group of sub-sequences can be expressed asi-th vector bi is a i-th group of the k nucleotides, that isFor DM and DM-CGR, the nucleotide sequence, B, are transformed in a sequence of the integer values expressed asc is the DM stream stored in dataset. The DM stream, c, calculus can be expressed asfmap( · ) is the mapping function expressed byFor DM, the group of sup-sequences, stored in matrix a expressed asia is the i-th value of CGR. In CGR  is the mapping processing characterized by z expressed asiz is the i-th value of CGR. Each ith element iz is a bi-dimensional value expressed asH, that isFor kMersM and kMersM-CGR, the nucleotide sequence, x(·) see –(14)) anThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


We have read the article titled „Quality of life 10 years after cardiac surgery in adults: a long-term follow-up study “ by Perrotti A. et al. published in your distinguished journal with great interest. Unfortunately, we found few errors in this article. Dear Editor,I am contacting you in the name of the group of authors who published “Health-related quality of life five years after coronary artery bypass graft surgery” in the International Journal of Cardiology . We read


In the 1950s, Crick proposed the concept of so-called comma-free codes as an answer to the frame-shift problem that biologists have encountered when studying the process of translating a sequence of nucleotide bases into a protein. A little later it turned out that this proposal unfortunately does not correspond to biological reality. However, in the mid-90s, a weaker version of comma-free codes, so-called circular codes, was discovered in nature in J Theor Biol 182:45–58, 1996. Circular codes allow to retrieve the reading frame during the translational process in the ribosome and surprisingly the circular code discovered in nature is even circular in all three possible reading-frames ( Crick et al.  proposedX—had been found by extensive statistical investigations in large samples of genetic data of archaea, plasmids and viruses, in addition to bacteria and eukaryotes  or even 24  . As usual, v a prefix of w if v a suffix of w if i-th circular shift ofw for Let We recall a few classical definitions of codes as follows:l-letter code and X is k-circularl-letter code if for any l-tuple from X there is only one partition into l-tuple from X when read on a circle. In other words, for any a circular l-letter code if it is a k-circular l-letter code for all a strong comma-free code if no X. In other words, given any two non-necessarily distinct elements X, for every a comma-free code if for any two elements X, we have a maximalcircular l-letter code if it is not contained in a larger a maximumcircular l-letter code or, equivalently, code of maximal size if Y is a l-letter code over a Let l-letter code ((k-)circular, comma-free, strong comma-free), namely the maximal size of a 1-circular l-letter code over complete classes l. The number of such complete classes is given byl-letter codes over Obviously, any strong comma-free code is also comma-free and hence also circular. Moreover, a maximum code is certainly also maximal. There is a general upper bound for the size of a maximum l et al.  that alll-letter code. We say that X is Let An interesting class of codes is also given by the so-called X are also circular.A circular code A stands for adenine, C stands for cytosine, G stands for guanine and T stands for thymine  if fixed point of a bijection We will also consider the so-called If the cardinality of G and a subset G, the centralizerS in G is the set of all elements of G that commute with all elements of S, i.e. normalizer of S in G is the set of all elements g of G that commute with S as a set but not necessarily point wise, i.e. We will need some further notations from general group theory -comma freeness and also the Letbe a finite alphabet anda circular . This representation had many implications on the study of these codes since the codes from the same equivalence class have the same error-detecting properties th rows for j)th columns for n of them will have the shape Without loss of generality we may assume thatWe would like to remark that there are alternative descriptions of the group (outer) semidirect product with respect to wreath product. It is an interesting investigation in itself to get involved with the group. However, this is outside the scope of the article.The group n and c. It was pointed out in “c-self-complementarity. It is thus natural to define an equivalence relation on codes by settingX and X will be denoted by [X].  strong comma-free codes.Let us consider the class of maximal . Without loss of generality, assume that x starts with 0 and ends with 1. However, if X would also contain c(x) which starts with 1 and ends with 0—contradiction to strong comma-freeness. Let c-self-complementary strong comma-free diletter codes over an alphabet of even cardinality:Before we continue our considerations about equivalence classes, we have to close a gap and count the number of maximal Letbe a finite alphabet withfor someand letbe the centralizerof some involutioncwithout fixed points. Then, there aredifferent maximal (=maximum)c-self-complementary strong comma-free diletter codes overc-self-complementary code, we have to ensure that for every x and c(x) belong to different sets. There are n pairs of complementary letters, thus, we have In Fimmel et al. , the strIn general, it can be assumed that in the case of maximum strong comma-free codes, the equivalence classes can be smaller than Let l-letter  strong comma-free codes, some Letbe a finite alphabet withfor someand letbe the centralizer of some involutioncwithout fix points. Moreover, letThen, the action ofoninduces some equivalence classes of size strictly less thanwhereis one of the following classes of codes: The class of all maximal (=maximum) strong comma-freel-letter codes;The class of all maximal (=maximum)c-self-complementary strong comma-freel-letter codes forc as well as l be given as stated in the theorem. We start with the proof of (1), so let l-letter codes over For Let us now consider the case l et al. , it is sLet us consider now the case  Note that partition means in particular that the sets By . Assume now that c-self-complementary strong comma-free 1-letter code, namely For c-self-complementary strong comma-free diletter codes over Let us now consider the case Let us consider now the case Again we fix c and hence For instance, if r odd equality above would imply that r is even, then equality would imply that X is also c-self-complementary. Therefore, we choose Orb(x) under c. So also K into two disjoint parts r-letter words and end with a code We proceed as in the proof of (1) in order to construct a code Let n in case (1). It is not clear if not all of the codes are invariant under some permutation.We would like to remark that the above proof certainly also works for c-self-complementary strong comma-free triletter codes:For The following example shows how the construction from Theorem We give an example how the construction in Theorem Finally, one putsX is a strong comma-free code which is c-self-complementary and invariant under For However, the result of Theorem X is not invariant under any permutation from X generates an equivalence class of size equal to the size of For X is 18 and it is strong comma-free since all triletters in X start with either A, C or G, T or A appears exactly nine times in the first position, C exactly six times and X invariant must fix these three letters and a similar argument on the last letters shows that it must also fix G, T and We shortly give an argument why the code in the above Example is not invariant under any permutation from c-self-complementary) strong comma-free triletter codes, the negative result can also be obtained more easily. The numbers of maximum (c-self-complementary) strong comma-free triletter codes calculated in Fimmel et al.  For The code In the following, we will only need the sufficient condition of Theorem Letbe a finite alphabet witha triletter code overin that no initial diletter ever occurs as a final diletter. Then,Xis comma-free.X beginning with X is comma-free. Let Letbe the symmetric group acting on the elements of the alphabet withan involutory bijection without fix points. Then, there is a maximum triletter comma-free codewith1. Casemis even ormis odd andIn this case, we choose m and because not all three positions Without loss of generality, we may assume thatThen, 2. Casemis odd and2.1There areIn this case, we choose 2.2For allc(xyz) has the same shape again and no initial diletter ever occurs as a final diletter.In this case, we choose The so constructed code xy with xy cannot appear as final diletter in every tuple from the first case, because m. The constellation m. In summary, in the constructed code, an initial diletter can never appear as the final diletter, so the code is comma-free.Let’ us take a look at X is also maximum k is the smallest natural number s such that Let (ii)Since (iii)l-letter X or one of the X provided Any (iv)l-letter such that for some i we have x has only one pair of identical consecutive letters with the convention that x generates a complete equivalence class .If We now write i) to (iii) from above hold when considered on l-letter word from We try to prove the above theorem by showing that for any Case 1:k is even.In this case, let iv), it follows that x generates a complete equivalence class. Hence, we may assume without loss of generality that iii) , the word x generates a complete equivalence class , we may assume that x contradicting circularity of X.Let We now distinguish cases:Case 2:k is odd.l. l copies of x is a concatenation of k words of length l, say iii). Moreover, x gives X.Choose a) applies and yields a contradiction. Note that also in this case, the In this case, the same construction as in Case m copies of x generates a complete equivalence class and by (iii) we may assume that x and Let We need to distinguish cases again according to the size of A first corollary is immediate.Letbe a finite alphabet or even size 2nand letbe the class of allmaximumc-self-complementaryl-lettercodes for some natural numberlwhereis an involution without fix points. Then, the action ofinduces equally sized equivalence classes onFollows directly from Theorem We would like to remark that it is an open question if maximum and Letbe the genetic alphabet andas well asLetbe the class of all maximalc-self-complementary trilettercodes. Then, the action ofinduces equally sized equivalence classes . It was speculated that nature may use this mechanism in order to encode not only one set of information in DNA but 8 (the size of L) or even 24  genetic code, the case of the binary alphabet is especially important for applications in signal processing. It proves to be a special case for classes of maximal strong comma-free and maximum self-complementary comma-free trinucleotide codes. Namely, only in this case, the corresponding equivalence classes all have the maximum possible size.Our results clarify completely the situation for several classes of codes showing the (non-) existence of equivalence classes of size |Moreover, the results of the present investigation suggest that the code properties responsible for the maximal size of equivalence classes are that the codes are maximally large and retain their error-detecting properties in all frames (


To enhance the utility of acceleration time (AcT) in the diagnosis of internal carotid artery (ICA) stenosis, we assessed the value of AcT measurements with different waveform patterns.Ninety-three patients with acute atherothrombotic cerebral infarction were enrolled, and they underwent both carotid ultrasonography and digital subtraction angiography (DSA). AcT was determined by a conventional procedure (using the first peak point or the bending point) and the peak systolic velocity (PSV) procedure. The AcT ratio was calculated as (AcT of ICA)/. We evaluated the correlation of stenosis rate as assessed by the North American Symptomatic Carotid Endarterectomy Trial method using DSA (DSA-NASCET) with the AcT of ICA (ICA-AcT), the AcT ratio measured by the conventional procedure , and the AcT ratio measured by the PSV procedure (PSV AcT ratio). The area under receiver operating characteristic curves (AUC) for DSA-NASCET was calculated based on the ICA-AcT and AcT ratio.r = 0.723), conventional ICA-AcT (r = 0.638), and PSV AcT ratio (r = 0.245). The corresponding AUCs for ICA stenosis ≥ 50% were 0.971, 0.886, and 0.572, respectively.Forty-five vessels had 50% or greater ICA stenosis. DSA-NASCET was positively correlated with the conventional AcT ratio (We demonstrated the usefulness of the conventional procedure for diagnosing stenosis of ICA origin using AcT and showed that the AcT ratio was a more beneficial parameter than AcT. Carotid artery ultrasonography is capable of noninvasive evaluation of the internal carotid artery (ICA). The peak systolic velocity (PSV) , 2, as wIn addition, the acceleration time (AcT) of the ICA, which can be measured by pulsed-wave Doppler, has been reported useful in cases, where measurement of the PSV of stenosis is difficult –8. The AThe systolic pulsed Doppler waveform is different in each case. The commonly observed waveform has a monomodal or bimodal peak pattern, where the time from minimum flow velocity, namely initiation of upstroke, to the PSV (time-to-PSV) is consistent with the acceleration time, and this time-to-PSV has often been defined as AT . HoweverThere is a study comparing digital subtraction angiography (DSA) with the ICA-AcT, in which the time from initiation of the upstroke to the first peak of the bimodal peak pattern was considered to be the AcT, and the time from initiation of the upstroke to the bending point in the monomodal peak pattern was considered to be the ICA-AcT . In the This is why it was considered easier to define the time-to-PSV as the AcT. Furthermore, there have been few reports looking at the relationship between stenosis rate on DSA and AcT , 8.In the present study, we evaluated the usefulness of the time-to-PSV, being defined as AcT, in the diagnosis of stenosis using DSA, and we compared it to previous methods.Our subjects were 93 consecutive patients  who were hospitalized in the Dokkyo Medical University Hospital for acute atherothrombotic cerebral infarction and evaluated by both carotid artery ultrasonography and DSA between April 1, 2015 and March 31, 2018.In 68 vessels, pulsed Doppler waveform patterns of the CCA and ipsilateral ICA were identical, showing a bimodal peak pattern without a bending point in which the initial peak of the wave coincides with the PSV. Therefore, only time-to-PSV was calculated as AcT, and those vessels were excluded. In addition, 20 vessels were excluded, because the ICA was obstructed at its origin. Ultimately, 98 vessels were retrospectively analyzed.KI, HT, AS, YT, HI, MO, and AI performed carotid artery ultrasonography using the SSA-770A , TUS-AI800, and TUS-A500 .Carotid artery ultrasonography was performed, while the subject was in the supine position, with the head turned away from the scanned side and the neck slightly extended.The pulsed Doppler waveform of the CCA was measured approximately 2 cm from the carotid sinus using a linear-array probe . The pulsed Doppler waveform of the ICA was measured approximately 3.5 cm (3.56 ± 1.00 cm) from the origin of the ICA using a convex-array probe .The sample volume of the pulsed-wave Doppler was set at 1/2 or 2/3 of the vessel diameter, and the Doppler insonation angle against the direction of blood flow was 60° or smaller. Sweep speed was fixed at 43.8 mm/sec (SSA-770A and TUS-A500) and 60.0 mm/sec (TUS-AI800).The measurement was continued for five beats or more, until the same waveform was plotted by the pulsed-wave Doppler.The stenosis ratio of the ICA origin diagnosed via DSA was evaluated by DT, SK, and TN using the North American Symptomatic Carotid Endarterectomy Trial method (DSA-NASCET) .Type A) A bimodal peak pattern, where the initial peak of the wave coincides with the PSV;Type B) A waveform, where the second peak coincides with the PSV of the bimodal peak pattern;Type C) A waveform of a monomodal peak pattern and a clear bending point; andType D) A monomodal peak pattern without a clear bending point.Pulsed Doppler waveforms were classified into the following four types Fig. :Fig. 1MeWe measured “conventional AcT” of the CCA  and ICA  according to Takekawa’s method . BrieflyThe AcT ratio, which is the ratio of the ICA-AcT to the CCA-AcT, was calculated.In other words, the “conventional AcT ratio” was derived from the “conventional ICA-AcT”/“ipsilateral conventional CCA-AcT”. At the same time, the “PSV AcT ratio” was derived from the “PSV ICA-AcT”/ipsilateral “PSV CCA-AcT”.The AcT used an average of five heartbeats, and the AcT and AcT ratio were measured by KI, HT, and AI, without taking into account the DSA-NASCET results.We evaluated the correlation of the DSA-NASCET with the “conventional ICA-AcT”, “PSV ICA-AcT”, “conventional AcT ratio”, and “PSV AcT ratio”, using the Pearson correlation coefficient. Then, we evaluated whether the DSA-NASCET was predictable for each item, using single regression analysis. We also calculated the area under the curve (AUC) based on the “conventional ICA-AcT”, “PSV ICA-AcT”, “conventional AcT ratio”, and “PSV AcT ratio”, using a receiver operating characteristic (ROC) curve.We calculated the inter-rater reliability for the AcT and the DSA-NASCET of five vessels, using the intraclass correlation coefficients (ICC).p < 0.05 was considered statistically significant.IBM SPSS  was used for statistical processing and plotting, and HT, KS, and HK were involved in overseeing the entire study.The ICCs for the “conventional ICA-AcT”, “PSV ICA-AcT”, and DSA-NASCET were 0.714, 0.994, and 0.998, respectively.The CCA pulsed Doppler waveforms were Type A, 87 vessels; Type B, 8 vessels; Type C, 2 vessels; and Type D, one vessel. The ICA waveforms were Type B, 39 vessels; Type C, 41 vessels; and Type D, 18 vessels, and there was no Type A (Table The median DSA-NASCET was 38.6% (range 0–85.2); stenosis ≥ 50% was identified in 45 subjects (45.9%), and of these, 29 subjects had stenosis of ≥ 70%.p < 0.0001), “conventional AcT ratio” (p < 0.0001), and “PSV AcT ratio” (p = 0.0150), but no correlation was identified for “PSV ICA-AcT” (p = 0.629).DSA-NASCET showed significant correlation with the “conventional ICA-AcT” , and six vessels (54.5%) were ≥ 50%.p = 0.00344), there was no correlation with the “conventional ICA-AcT” (p = 0.0722), “PSV ICA-AcT” (p = 0.101), and “PSV AcT ratio” (p = 0.903), whereas the correlation coefficient with the “conventional AcT ratio” was 0.795 and the AUC was 1.00.In relation to DSA-NASCET, although a significant correlation was seen with the “conventional AcT ratio”  pre“PSV ICA-AcT” and “PSV AcT ratio” may have low utility as the pulsed Doppler waveform changes under the influence of various factors, such as age, heart disease, and arteriosclerosis. The results of our study point to the possibility that the “conventional AcT”, especially “conventional AcT ratio”, is a useful indicator for diagnosing stenosis of ICA origin in patients in whom measurements of the PSV of the maximum stenosis point and ICA PSV/CCA PSV are difficult.Our study had several limitations. First, we excluded 68 vessels in which pulsed Doppler waveforms showed Type A patterns in both the CCA and ipsilateral ICA. Therefore, the usefulness of AcT in vessels that had identical patterns of “conventional AcT” and “PSV ICA-AcT” could not be evaluated. In addition, because only 11 vessels showed patterns of Type B, C, or D in the CCA, further evaluations for those patterns other than Type A are needed. Finally, pulsed Doppler waveforms were measured at a constant sweep speed, but a change in the sweep speed could result in different patterns in pulsed Doppler waveforms. Determining the optimal sweep speed for AcT measurements requires further studies.We studied the value of AcT measurement in stenosis of ICA origin. Our study demonstrated that the proper measurement method for AcT was the “conventional AcT” method, measuring the time from initiation of upstroke to the initial peak or to the bending point. According to our inferences, the “conventional AcT ratio” has a still greater utility compared with the “conventional ICA-AcT”. There are few reports, where DSA-NASCET and AcT are compared. Multicenter studies should be conducted in the future, where “conventional AcT” or the “conventional AcT ratio” and DSA-NASCET are studied. Guidelines for the diagnosis of stenosis of ICA origin need to be created based on the knowledge of its relationship to AcT or the AcT ratio.


Labels are a way to add some information on a text, such as functional annotations such as genes on a DNA sequences. V(D)J recombinations are DNA recombinations involving two or three short genes in lymphocytes. Sequencing this short region (500 bp or less) produces labeled sequences and brings insight in the lymphocyte repertoire for onco-hematology or immunology studies.BW-index). Both indexes need a space related to the entropy of the labeled text.We present two indexes for a text with non-overlapping labels. They store the text in a Burrows–Wheeler transform (BWT) and a compressed label sequence in a Wavelet Tree. The label sequence is taken in the order of the text (TL-index) or in the order of the BWT (TLBW-index has an overhead on simple label queries but is very efficient on combined pattern–label queries. We implemented the indexes in C++ and compared them against a baseline solution on pseudo-random as well as on V(D)J labeled texts.These indexes allow efficient text–label queries to count and find labeled patterns. The TLNew indexes such as the ones we proposed improve the way we index and query labeled texts as, for instance, lymphocyte repertoire for hematological and immunological studies. Labels are a way to add some information on a text, as the semantics of words on an English sentence or functional annotations such as genes on a DNA sequences. 1 in the positions 12–41 and the label L2 in the positions 56–96?Can we build an index that saves a labeled text like ACGCC…TTGA (of size 96), which have the label L We consider here the case where a same label can be given to different (but similar) patterns and can occur several times in the text. We introduce two indexes which store a labeled text and answers to position–label association queries. Those indexes share some ideas with the RL-FMI (BW-indexes (text–label indexes) and their associated queries. The last section presents experimental results on simulated and real genomic data.e RL-FMI  which usT = t0t1 … tn − 1 be a text of length n over an alphabet of size σ. The text may be composed of several sequences, each sequence ending with the symbol $. Let L = {L0, L1 … Ll − 1} be a set of labels. A labeled text  is here a text with non-overlapping labels: a letter should have at most one label. Each position i of the text is labeled by exactly one label ia ∈ L ∪ {ε}, where the special ε label is put on every letter without label. A = a0a1 … an − 1 is the label string. Let bit vectorB = B[0]B[1] … B[n − 1], where each B[i] is either 0 or 1, we define as B the vector B[i]B[i + 1] … B[j]. Let us call rank the number of times the bit b ∈ {0, 1} appears in the prefix B and select  the position i of the jth appearance of the bit b in B. Such a bit vector B can be stored in nH0(B) + o(n) bits to support rank and select in O(1), where H0(B) is the zeroth order entropy of B (Given a opy of B .BWT (BWT(T), is the concatenation of the last letters of the lexicographically sorted rotations of the text. BWT(T) is easier to compress and can be stored using knH(T) + o(n) bits, where kH is the kth order empirical entropy of the text T. The FM-index  giving the number of occurrences of α in BWT(T). The FM-index allows to efficiently search a pattern using backward search.The BWT  is a revFM-index  uses theWT  + o(a log l) bits when bit vectors are zero-order compressed (A WT  is a bina⌉) time  and requmpressed .The usual full-text indexes such as the FM-index  or the Lnon-overlapping labels and look for ways to efficiently store such labels along with sequences as well as to query them. The basic idea is that consecutive labels in A are often the same: we will thus store compressed label sequences.We focus here on T, A), we define the TL-index as, using a FM-index built on a BWT U to index the text, a bit vector AB marking the positions in the text where the labels change, and a WT AW indexing a compressed label sequence , where Q is the set of nodes of AW and q0 ∈ Q is the root node. Each node q ∈ Q is q = , where q.label ∈ L ∪ {ε} is the label of q and q.parent is the parent node of q . Both q.left and q.right are child nodes in Q ∪ {⊥}. A leaf is a node q where q.right and q.left are ⊥. Each leaf q is linked to a label q.label ∈ L ∪ {ε}. The l + 1 leaves are exactly the labels of L ∪ {ε} and we define leaf (q.label) = q. On a leaf q, we have q.val = ⊥. Let q be a non-leaf node: we have q.label = q.val is the bit vector rooted at q in the WT. We explain in “Shaping the WT for a Label Hierarchy” how AW can be further shaped depending on a label hierarchy.AW is part of the index. This WT is used to answer efficiently bidimensional range queries where the labels and the label positions are the two dimensions  time. This is a classical query within a WT. Given a label xL ∈ L, the function selectW〈xL, i〉 gives the position of the ith xL label in A′ in O(log l) time. The accessor −1W〈xL〉 gives the list of positions in A′ where a′i = xL. It runs in O(log l × occ) time, with occ the number of occurrences of xL in A′.mensions . A balanU.The BWT tends to store text repetitions consecutively. As those repetitions may have the same labels, it would be interesting that the labels benefit from the BWT reordering. Hence, labels can also be stored in the order of T, A), the TLBW-index is defined as   . The BWTThe indexes allow the following classical queries.label(it) )—Which label is on the letter it? This query is done is O(log l) time in the TL-index, and in O(log1+εn + log l) time in the TLBW-index since it has to convert positions in U order (see rder see .findP(P) P—Which are the occurrences of a pattern P? It is solved with the FM-index alone (O(|P| + occ × log1+εn) time in both indexes, where occ is the number of occurrences of P in T.ex alone . This qufindL(xL)—Which text positions are labeled xL? The query runs in O(y × log l) time in the TL-index, and in O(y(log l + log1+εn)) time in the TLBW-index, where Y = −1W〈xL〉 and y = |Y|. See The three previous queries are well known in text indexes. The two next queries search for a pattern and a label at the same time.countPL—How many text positions are labeled xL and begin an occurrence of a pattern P? As in the findL(xL) query, the occurrences of P are found in U, in the positions from i to j.–occ occurrences to have the corresponding positions in the text. For each of them, we run the query label(it). The total time is O(|P| + occ (log n1+ε + log l)).TL-index: We translate all these –BW-index: See i and j correspond to the positions i′ = rank and j′ = rank in the root q0 of DW. We then use an accessor customized from the rangeLocate function of  time. This accessor first traverses the WT from the root to the leaf xL and then traverses it back to the root to find the positions in q0.val which match the label in the given range. For every position found, we find the corresponding positions in DB and expand them to the following 0-bits in DB. This query runs in O(|P| + |Z| × log l) time.TLction of , simulatfindPL—Which text positions are labeled xL and begin an occurrence of a pattern P?–countPL.TL-index: This query is exactly the same as –BW-index: We use the countPL query detailed in cnt with a list Y holding the matching positions in U. The positions are converted at the end in the text order. This query runs in O(|P| + |Z| × log l + y × log1+εn) time, with |Y| = y.TLZ| ≤ y ≤ occ, the countPL and the findPL queries may thus be faster on the TLBW-index, |Z| depending of the compression DB can do on Y.As |countPL query could be faster if the WT was directly built on the labels, without the intermediate bit vector DB (AB): the answer would be known while reaching the leaf of the WT in O(|P| + log l) time. We chose to favor the execution time of the findPL query, as well as the size of the structure (when A can be compressed).Note that the findPL can vary to find the patterns P which have the label Lx on the ith position of the pattern, or in any pattern’s position. Adapting this queries is easy in the TL-index as we find the positions of the pattern in the BWT, translate them in the text order and then find the label of the ith position following each of them. In the TLBW-index, we find the patterns’ positions in the BWT, access to the ith letter (we need to sample the BWT to read the patterns in the forward way), and find the label as usual. To have the label in any pattern’s position, in the TL-index we need to find a label xL between the first and last letter of the pattern (with only two access in the WT) but in the TLBW-index, we look for the label of all the pattern’s letters.The T, A) is of length n and is labeled with l unique labels. As defined above, the indexes store U in knH(T) + o(n) bits. The TL-index stores the bit vector with rank and select capabilities in nH0(AB) + o(n) bits. The size of AW depends on the compressed label sequence A′, of length a. AW takes a H0(A′) + o(a log l) bits. Similarly, the TLBW-index stores DB in nH0(DB) + o(n) bits and DW takes d H0(D′) + o(d log l) bits, where d is the length of D′. The BWT can be built in linear time while using little space (AB is built while reading A in O(n) time. To make DB, we need to read the labels in the order of the original data file in O(n) time. To make AW, we find the occurrence of each label, corresponding to a 1-bit in AB, in O(a) time. Then we form the shape of AW in O(l). The labels corresponding to a 1-bit are extracted to make the WT’s root q0. For each node containing at least two labels, we separate them by following the shape previously calculated, in DW the same way. The TL-index has thus a size of knH(T) + nH0(AB) + a H0(A′) + o(n log l) bits, assuming σ = O(l), and is built in BW-index has a size of knH(T) + d H0(D′) + o(n log l) bits and is built in We recall that the text  will be shaped according to the hierarchy, and internal nodes q of W may have non-empty q.label values belonging to F. For example, in q1.label = L1, where L1 is the label family gathering L1.1, L1.2 and L1.3.Labels may be further organized into a families . Both TLfindL and findPL queries naturally extend to label families. With the hierarchy depicted in findL(L1) has to find the positions that have a label L1.1, L1.2 or L1.3. Such a query does not need to be iterated on every label of the family L1, but rather directly starts at the corresponding internal node (q1 on The de q1 on .W for a label hierarchy may increase the height w of the WT to O(l) in the worst case. To have a lower space consumption and a faster average query time, one can give a Huffman shape to W (H0(A′) in the best case while in the worst case it could be O(l). If no label hierarchy is given, the average height w will be H0(A′) (Shaping ape to W . A leaf e H0(A′) .BW-indexes with a baseline solution called HT-index, indexing the text T with a BWT. The labels are stored in a map linking each label to the list of its  positions. We also store the labels in the text order with the compressed bit vector AB and, stored in plain form, A′. This enables the findL(xL) query in O(y), where Y′ is the list of pairs  which represent the occurrences and y = |Y′|. Note that Y (in the TL-index) and Y′ represent the same information as the labels are stored in the text order in both indexes. The label(it) query runs in O(1) time. This solution is not space-efficient with repeated labels: it needs knH(T) + nH0(AB) + a + l′ + o(n) bits, where a is the size of A′ and l′ the number of labeled factors. The query times are summarized in We compared the TL- and TLO(n log n) time on small DNA sequences, as it is very efficient for storing and appending sequences corresponding to our projected application (O(log1+εn) steps, we use sequences of length 50, which is similar to the sampling distance usually chosen in practice. The real files have longer sequences, thus longer sampling distances. The queries relying on the BWT will be slower and therefore cannot be compared between real and simulated files. We build the AB (or DB) bit vectors, compress them using the rrr_vector class of SDSL-Lite, and finally build AW (or DW) using a shape we added in SDSL-Lite, which depends of the label hierarchy.The three indexes were implemented in C++. We used the SDSL-Lite library  to buildlication . As RopefindP(P) behaving similarly in all indexes and countPL being very similar to findPL. The three indexes were tested on various datasets of labeled texts, each with 100 M characters .Simulated files, with random sequences but fixed labels. Here a given label is associated to the same pattern, possibly with some variation, and we alter the proportion of labeled letters (5–100%), the variation in the label’s pattern , the number of unique labels , the length of the labels (5–100 letters). The dataset has 546 files, two of those files are shown in Genomic sequences with immunologic labels. A person’s immune system can be described with a set of labeled DNA sequences with V(D)J recombinations. The dataset, detailed below, uses 838 K labels from 355 unique labels, with d ∼ 0.26n.lymphocytes, are under maturation, their DNA undergo a recombination which allows several billions possibilities from a register of a thousand genes J recombinations . These sThe tested DNA sequences come from patients 09, 12, 14 and 63 from a public dataset on a study on acute lymphoblastic leukemia . They haIndex sizes. The U, B (either AB or DB) and W (either AW or DW) grows linearly with the number of indexed elements (data not shown). The TL-index is the smallest index, and the TLBW-index is generally slightly larger. The compression is directly related to a and d. The file with random labels (d = 0.77n) is hard to compress, whereas the files with a low d/n ratio give a 2× to 7× compression. T or in U, thus more 1-bits in B. Note that when there are more labeled letters, the text is more similar (as labels are placed on similar substrings), hence a decrease in the BWT size . W increases while the number of unique labels increases (D), the height of W increasing logarithmically with the number of unique labels.Build time. Most of the build time of TLBW-index is devoted to build D′. For TL-index, the building of U takes most of the total time.Queries. The label query is faster on the HT-index. As expected, the TLBW-index needs more time for label and findL, as it needs to translate positions from the BWT back to the text. Note that locating the positions in the text takes about the same time as label(it) in the TL-index. However, for the complex findPL query, the TLBW-index is the fastest solution because the position translation is only done on the letters which have both the label and the pattern. For the TL-index and HT-index, the actual time of the findPL queries is more affected by the number of pattern occurrences than the number of final occurrences (between 0 and 100 K depending on the file).BW-index suffers here even more from the missing suffix array sampling of the implementation for queries label and findL. However, on the findPL query, the other indexes are penalized due to the sparsity of the sampling, bringing a more than 30× difference with TLBW-index.On the genomic dataset, the sequences are longer: The TLBW-index store a labeled text and allow efficient queries. They can be constructed from files of some MB in a few seconds. Experiments confirm that the indexes stay small when the text is redundant , when each label describes a pattern with few variations , and when few letters are labeled . However, the TL-index and TLBW-index are robust even to non-redundant text with almost random labels everywhere. The TLBW-index needs more memory space than the TL-index but is more efficient in combined label/pattern queries. Those structures might be used on any labeled data, such as DNA sequences, but also on natural language texts or on music sheets with some semantic annotations.The TL-index and TLsd_vector bit vector instead of the rrr_vector bit vector which should improve space consumption when the bit vectors are very sparse. However, this would only minimally improve the global space consumption of the index. We plan to use one of the indexes in a clone database for hematological malignancies: It will allow comparison of V(D)J recombinations between different samples of a patient or between several patients.Perspectives include improvement of the implementation, with label families queries or parameterizing the distance between samples in the FM-index to offer a space-time trade off. Within SDSL we could use the 10.7717/peerj-cs.148/supp-1Supplemental Information 1Click here for additional data file.10.7717/peerj-cs.148/supp-2Supplemental Information 2Click here for additional data file.


Scientific Reportshttps://doi.org/10.1038/s41598-020-64654-5, published online 15 May 2020Correction to: The original version of this Article contained multiple errors.The incorrect image was used for Figure 13.Furthermore, there was a typographical error in the legend of Figure 14.a) and in part (b–i) the 2D distribution of the corresponding elements is represented.”“Images show a radiation-induced dissolution of complex precipitation. The HAADF image is represented in part (now reads:a) and in part (b–c) the 2D distribution of the corresponding elements is represented.”“Images show a radiation-induced dissolution of complex precipitation. The HAADF image is represented in part (Finally, a Supplementary File containing Tables S1 and S2 was omitted from the original version of this Article.These errors have now been corrected in the HTML and PDF versions of the Article.