We established a universality of logarithmic loss over a finite alphabet as a distortion criterion in fixed-length lossy compression. For any fixed-length lossy-compression problem under an arbitrary distortion criterion, we show that there is an equivalent lossy-compression problem under logarithmic loss. The equivalence is in the strong sense that we show that finding good schemes in corresponding lossy compression under logarithmic loss is essentially equivalent to finding good schemes in the original problem. This equivalence relation also provides an algebraic structure in the reconstruction alphabet, which allows us to use known techniques in the clustering literature. Furthermore, our result naturally suggests a new clustering algorithm in the categorical data-clustering problem. Logarithmic loss is a unique distortion measure in the sense that it allows a “soft” estimation (or reconstruction) of the source. Although logarithmic loss plays a crucial role in learning theory, not much work has been published regarding lossy compression until recently. A few exceptions are a line of work on multiterminal source coding ,2,3, theoptimal schemes for the two problems are the same; anda good scheme for one problem is also a good scheme for the other.In this paper, we present a new universal property of logarithmic loss in fixed-length lossy-compression problems. Consider an arbitrary fixed-length lossy-compression problem, where source and reconstruction alphabets 𝒳 and We are more precise about the “optimal” and “goodness” of the scheme in later sections. This finding essentially implies that it is enough to consider the lossy-compression problem under logarithmic loss.The above correspondence provides new insights into the fixed-length lossy-compression problem. In general, the reconstruction alphabet in the lossy-compression problem does not have any well-defined operations. However, in the corresponding lossy compression under logarithmic loss, reconstruction symbols are probability distributions that have their own algebraic structure. Thus, under the corresponding setting, we can apply various techniques, such as the information geometric approach, clustering with Bregman divergence, and relaxation of the optimization problem. Furthermore, the equivalence relation suggests a new algorithm in the categorical data-clustering problem, where data are not in the continuous space.The remainder of the paper is organized as follows. In Notation: Uppercase X denotes a random variable, where 𝒳 denotes a set of alphabet. On the other hand, lowercase x denotes a specific possible realization of random variable X, i.e., n-dimensional random vector Suppose 𝒳 is a finite set of discrete symbols, and X with finite alphabet M messages. On the other side, a decoder In this section, we briefly introduce the basic settings of the fixed-length lossy-compression problem . In a fiFirst, we can define the code that the expected distortion is lower than a given distortion level.Definition 1.An (Average distortion criterion)The minimum number of codewords required to achieve average distortion not exceeding D is defined bySimilarly, we can define the minimum achievable average distortion given number of codewords M.One may consider a stronger criterion that restricts the probability of exceeding a given distortion level.Definition 2.An (Excess distortion criterion)The minimum number of codewords required to achieve excess distortion probability ϵ, and distortion D is defined bySimilarly, we can define the minimum achievable excess distortion probability given target distortion D and number of codewords M.D and Given target distortion There exists a unique rate-distortion function achieving conditional distribution We assume that x, then, there is no difference between If We make the following benign assumptions:Define the information density of joint distribution D-tilted information that plays a key role in fixed-length lossy compression.Then, we are ready to define Definition 3.The D-tilted information in where the expectation is with respect to the marginal distribution of ]≥H(X|X^⋆)X and In the above theorem, distortion Equation  is the “X and reconstruction M, the minimal achievable distortion is given by X and X and On the other hand, we viewed the “information” rate-distortion function differently. We considered the one-shot setting where source Remark 1.In the corresponding fixed-length lossy-compression problem under logarithmic loss, the minimal achievable average distortion given number of codewords M iswhere the conditional entropy is with respect to distribution Remark 2.From now on, we denote the original lossy-compression problem under given distortion measure n-dimensional binary vector where n is fixed, so the problem is in the one-shot setting. Distortion measure d is separable Hamming distortion, i.e.,M be the number of messages. Then, we are interested in optimal encoding and decoding schemes that achieve distortion In this section, we consider the memoryless Bernoulli source under Hamming distortion measure as an example of the above equivalence. Let In this scenario, the information rate-distortion function is not hard to compute :7)R(D)=R(D)=(7)RThen, the corresponding problem is the rate-distortion problem under logarithmic loss where the set of reconstruction symbols isRemark 3.We can rewrite Equation  in this The above equation explicitly shows the correspondence between logarithmic loss and the original distortion measure.Theorem 2 implies that, for any fixed-length lossy-compression problem, we can find an equivalent problem under logarithmic loss where optimal encoding schemes are the same. Thus, without loss of generality, we can restrict our attention to the problem under logarithmic loss with reconstruction alphabet f and g are a suboptimal encoder and decoder for the original fixed-length lossy-compression problem. Then, the theorem impliesSuppose The left-hand side of Equation (9) is the cost of suboptimality for the corresponding lossy-compression problem. On the other hand, the right-hand side is proportional to the cost of suboptimality for the original problem. In In general, reconstruction alphabet M. Note that this is a single-letter version of  We alssion of  coincides with Equation (10), we haveRemark 4.In extend the reconstruction set from 𝒴 to find the measure The above result (13) implies that In the previous section, we obtained the optimal reconstruction symbol from the extended reconstruction alphabet, and projected it to the feasible set. In this section, instead of direct projection to 𝒴, we propose another slight extension of 𝒴, namely, log-convex hull. As we show in the following sections, the log-convex hull has interesting properties.p and q be probability distributions in p and q is given byBefore defining the log-convex hull, we need to define the log-convex combination of probability distributions. Let r is a weight vector (i.e., It is clear to see that Instead of having projection of convex, (, [TheorerI-projection of Projection rI-projection satisfies the following inequality for all Csiszár and Matúš (, [TheoreOn the other hand, the log-convex combination of probability measures measures . The autThe above result holds for any Remark 5.The above result is similar to the projection to polytope in Euclidean space. Suppose vectors where As we saw in the previous section, we want to find Since the first term is not a function of Thus, minimizing Since the objective function is a convex function of In the corresponding lossy-compression problem under logarithmic loss, reconstruction symbols are probability measures that have a natural algebraic structure, as we discussed in k-means clustering to a lossy-compression problem [k-means clustering is only available when there exists a well-defined operation in k-means clustering requires computing the mean of data points, which is the center of each cluster. In general lossy-compression problems, reconstruction alphabet k-medoidlike clustering [k-medoidlike algorithm in the context of lossy compression is shown in Algorithm 1.Lossy compression is closely related to the clustering problem ,20,21. M problem ,23,24, w problem ,26. Howeustering , where tAlgorithm 1k-medoidlike clustering in lossy compression. repeat  Set   fordo     end for  forMdo     end for until converge Randomly initialize k-meanslike clustering algorithm, as shown in Algorithm 2.On the other hand, in the corresponding problem, the reconstruction alphabet is the set of probability distributions where operations such as log-convex combinations are well-defined. This allows us to propose a Algorithm 2k-meanslike clustering in lossy compression. repeat  Set   fordo      Set   end for  forMdo     end for until converge Randomly initialize k-means clustering [The main idea of the above algorithm is that log-convex combination ustering ,29. The mean are not well-defined in this case, it is hard to apply known data-clustering algorithms in continuous space. The key idea is that the equivalence relation with logarithmic loss allows the algebraic structure on any set. More precisely, we can transform any clustering problem to the clustering problem in continuous space and apply known techniques such as variations of k-means.The idea of the previous section can be applied to an actual clustering problem. We mainly focus on clustering categorical data where data points are not in continuous space ,32,33,34M clusters.A more rigorous definition of the problem is given below. Assume that we have a finite set of data points 𝒳, and each data point has its weight M. Let k-means to the corresponding problem. For example, Algorithm 2 can be applied to the corresponding problem.If we let Equation . Then, wRemark 6.Note that it is hard to have an exact analytic formula for k-meanslike clustering algorithm in categorical data-clustering problems.To conclude our discussion, we summarize our main contributions. We showed that for any fixed-length lossy-compression problem under an arbitrary distortion measure, there exists a corresponding lossy-compression problem under logarithmic loss where optimal schemes coincide. We also proved that a good scheme for one lossy-compression problem is also good for another problem. This equivalence provides an algebraic structure on any reconstruction alphabet that allows using various optimization techniques in lossy-compression problems, such as log-convex relaxation. Furthermore, our results naturally suggest a


With the great significance of biomolecular flexibility in biomolecular dynamics and functional analysis, various experimental and theoretical models are developed. Experimentally, Debye-Waller factor, also known as B-factor, measures atomic mean-square displacement and is usually considered as an important measurement for flexibility. Theoretically, elastic network models, Gaussian network model, flexibility-rigidity model, and other computational models have been proposed for flexibility analysis by shedding light on the biomolecular inner topological structures. Recently, a topology-based machine learning model has been proposed. By using the features from persistent homology, this model achieves a remarkable high Pearson correlation coefficient (PCC) in protein B-factor prediction. Motivated by its success, we propose weighted-persistent-homology (WPH)-based machine learning (WPHML) models for RNA flexibility analysis. Our WPH is a newly-proposed model, which incorporate physical, chemical and biological information into topological measurements using a weight function. In particular, we use local persistent homology (LPH) to focus on the topological information of local regions. Our WPHML model is validated on a well-established RNA dataset, and numerical experiments show that our model can achieve a PCC of up to 0.5822. The comparison with the previous sequence-information-based learning models shows that a consistent improvement in performance by at least 10% is achieved in our current model. Biomolecular functions usually can be analyzed by their structural properties through quantitative structure-property relationship (QSPR) models (or quantitative structure-activity relationship (QSAR) models). Among all the structural properties, biomolecular flexibility is of unique importance, as it can be directly or indirectly measured by experimental tools. Debye-Waller factor or B-factor, which is the atomic mean-square displacement, provides a quantitative characterization of the flexibility and rigidity of biomolecular structures. With the strong relationship between structure flexibility and functions, various theoretical and computational methods have been proposed to model the flexibility of a biomolecular. Such methods include molecular dynamics (MD) , normal Other than the above deterministic models, data-driven machine learning models are also considered in flexibility analysis –29, thanMore recently, a persistent-homology (PH)-based machine learning model is proposed . In thisk-simplexes with k starting from 0. In particular, by assigning a weight value of 0 or 1 to each point, we can naturally arrive at a local PH model and element-specific PH model  is divided into N bins of equal size f. The number of barcodes which are located on each bin are then counted and used as feature vector [i is defined as:bj and dj are referring to birth and death of bar j. B is referring to the collections of barcodes with α referring to the selection of atoms and D referring to the dimension of the Betti numbers. Essentially, for each C1 atom, we have a N * 1 topological vector for each element and dimension of the Betti numbers.In this paper, we only consider topological features constructed using a binning approach . More spe vector , 55. MorAfter the topological features are represented as a feature vector, it can serve as input to predict B-factor values with ML algorithms. We consider four main ML models, namely regularized linear regression, tree-based methods (including random forest and extreme gradient boosting), support vector regression, and artificial neural networks. All our ML algorithms are implemented in Python (packages mentioned below refer to the packages in Python).n data ith sample , ith sample, and p is the number of structured features. Conventionally, we denote by ŷ the predicted normalized B-factor value of a sample.In the following descriptions of the ML models, we assume that we train our models with x by ŷ [Linear regression is a straightforward yet efficient approach to model the relationship between a quantitative response and features. The incorporation of regularization can effectively address the high dimensionality setting where the number of features is larger than the sample size. The variable selection feature of the regularized linear regression makes it particularly suitable for our task as our feature vector is usually lengthy. The general formulation of regularized linear regression can be read as the following regularized minimization problem:acy of ŷ –60.α‖β‖1), where α is the tuning parameter that strikes the balance between efficiency and regularization. The regression problem with these two types of regularization are also known as Ridge regression [least absolute shrinkage and selection operator (LASSO) [scikit-learn” [In our study, we consider the two typical choices of gression  and leas (LASSO) , respectt-learn” .Random Forest (RF) [Extreme Gradient Boosting (XGBoost) [Classification And Regression Tree (CART)  or decisest (RF) , 64 and XGBoost) .RH. RH is an ensemble learning method that creates a variety of decision (regression) trees independently during training, where each decision tree is constructed using a random subset of the features as split candidates. During the training of each tree, the split at each node is determined by the least-square method. In other words, for each region of each tree, we predict the B-factor value with the average of the B-factor values of the samples fallen in the region. In a regression RF, the final prediction is the average of the predicted values of all individual trees. In the implementation of ensemble trees, the number of trees, minimum number of samples at each leaf node, and the number of split candidates in each splitting, i.e., parameter mtry, are all tuning parameters. In our application of RF, we choose scikit-learn” [ Breiman  and tunet-learn” .XGBoost. Has been one of the popular ML tools used by the winning teams of many ML challenges and competitions, such as the Netflix prize [xgboost” [ix prize  and varixgboost” .β0 + x⊤β that has at most ϵ deviation from the actual target values yi for all the training data while trying to be as flat as possible [β0, β) is determined by the following minimization problem:ξ and C determines the trade-off between the efficiency and the amount up to which deviation larger than ϵ is tolerable. Typically, we adopt kernel methods to transform the input features from a lower to a higher dimensional space, where a linear fit is feasible. Common choices of kernel include polynomial kernel, Gaussian kernel, and radial basis function (RBF) kernel. In our study, we have opted to use RBF kernel, i.e., scikit-learn” [SVR , as a vepossible . Sometimpossible . The SVRt-learn” .ANN has been proved to be capable of learning to recognize patterns or categorize input data after training on a set of sample data from the domain . The abikeras” [In our study, the number of hidden layers, number of nodes in each hidden layer, and the number of epochs are treated as hyperparameters. The hidden and output activation functions are set as sigmoid and leaky ReLU functions respectively. The dropout rate is set to 20% and the remaining hyperparameters are set to default values as defined by the package. ANN is implemented with the package “keras” .We consider the same RNA dataset and data preprocessing by Guruge et al. . The chaThe values of B-factors may differ significantly from chain to chain due to reasons such as a relatively small number of residues in a protein chain or differences in refinement methods used . Thus, tE, F/E ratio, and bin size f are the hyperparameters to be optimized. We chose the value of E to be in the range from 10 Å to 45 Å with a stepsize of 5 Å, i.e., E = {10 Å, 15 Å, 20 Å, 25 Å, 30 Å, 35 Å, 40 Å, 45 Å}. The filtration interval F is defined such that the ratio of F/E is between 0.5 to 1.0 with a stepsize of 0.1, i.e., F/E = {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}. Bin size f is chosen to be in the range from 0.15 Å to 1.50 Å, i.e., f = {0.15 Å, 0.50 Å, 1.00 Å, 1.50 Å}. A total of 32,823 PBs are generated based on the Vietoris-Rips complex for each combination of element type, E and F/E ratio. Both “GUDHI” [Dionysus” [In our dataset, cut-off distance  “GUDHI”  and “Dioionysus”  packagesTo determine the optimal hyperparameter values for each ML model, we conduct a five-fold cross validation (CV) using the training set. Specifically, the training set is randomly divided into five folds with a similar number of chains. In each fold, for each combination of the hyperparameters, we find the predicted B-factor values for the left-out training set with the ML model trained by the remaining training set. The optimal hyperparameter set maximizes the out-of-sample PCC between the predicted and actual values across all folds. The optimal hyperparameter values for each ML model can be found in i-th B-factor value, Once the hyperparameter values of the dataset and models have been optimized, the trained models are evaluated using a test set that was non-overlapping with the training set. The PCC between the predicted and actual normalized B-factor values in the test set is calculated for each modelIn this section, we demonstrate the performance of our WPHML model. For both single-element and four-element-combined models, it can be seen that WPHML models are able to consistently outperform the evolution-based method (PSSM) by at least approximately 10% with only the exception of linear regression models (Ridge and LASSO). Among all the models, RF achieves the best result with PCC = 0.5788 (15.1% improvement). Moreover, the performance of the RF model further improves to 0.5822 when the topological features for all four elements were used, which is about 15.8% improvement.The comparison between the results from single-element and four-element-combined models shows that generally there is no significant improvement. In fact, SVM improves only slightly (approximately 0.8%), while XGBoost and ANN models even show some small reduction of accuracy (1.8% and 2.4% respectively). The results seem to be different from previous studies that concluded that element-specific models always deliver better results , 36–38. Comparably speaking, RNA structures are more regular and relatively simple. Similar topological features may be embedded in different types of element models. In this way, the additional features do not incorporate new information, instead they will contribute more noises, which causes the drop in performances. Noted that the best test performance of all the models except linear regression using a single element are all based on element P.One of the reasons that larger cut-off distance delivers good results is that our predicted PCC values are predominantly determined by the several larger-sized RNAs. From In this paper, we propose the weighted-persistent-homology-based machine learning (WPHML) models and use them in the RNA B-factor prediction. We found that our WPHML models can consistently deliver a better performance than the evolution-based learning models. In particular, local persistent homology and element-specific persistent homology are considered for topological feature generation. These topological-feature-based random forest models can deliver a PCC up to 0.5822, which is 15.8% increase as compared to the performance of the previous model. Our WPHML models are suitable for any biomolecular-structure-based data analysis. Note that more sophisticated feature engineering of sequence-based information can further improve the accuracy to 0.61 . This agS1 Table(PDF)Click here for additional data file.S2 Table(PDF)Click here for additional data file.


In the original article, there was an error in the title. Instead of “Hypernetwork Construction and Feature Fusion Analysis Based on Sparse Group Lasso Method on Functional fMRI Dataset” it should be “Hypernetwork Construction and Feature Fusion Analysis Based on Sparse Group Lasso Method on fMRI Dataset”.Additionally, in Equations (4) and (7) the parameters mentioned and its explanations were wrong. “n” should be superscript and not subscript. Also in Equation (4), “x” should not be in italics and “α” should be in italics. A correction has been made to the following sections:The Materials and Methods section, subsection Construction of Hypernetwork, sub-subsection Sparse Linear Regression Model, paragraph 2:m-th ROI for n-th subject, m-th ROI for n-th subjects, with T being the number of time points in the time series; m-th ROI ; m-th ROI; and m-th ROI.”“The average time series of The Materials and Methods section, subsection Construction of Hypernetwork, Construction of Hypernetworks Based on the Sparse Group Lasso Method, paragraph 3:“Similar to the gLasso method, clustering was adopted before creating the hyperedge, and then the sgLasso method was used to construct the hyperedge by solving the sparse linear regression model. The method is represented by the optimization objective function:Gi is a node with tree structure. λ1 and λ2 are regression parameters, with λ1 being used to adjust the sparsity of intra-groups to control the number of non-zero coefficients in non-zero groups, and λ2 being used to adjust group-level sparsity  of all three models, when λ2was equal to 0.4. .”The authors apologize for these errors and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.


As of April 16, 2020, the novel coronavirus disease  spread to more than 185 countries/regions with more than 142,000 deaths and more than 2,000,000 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, digital signal processing, and machine learning techniques and algorithms. However, to make feasible this approach, it is necessary to transform the nucleotide sequences string to numerical values representation. Thus, the dataset provides a chaos game representation (CGR) of SARS-CoV-2 virus nucleotide sequences. The dataset provides the CGR of 100 instances of SARS-CoV-2 virus, 11540 instances of other viruses from the Virus-Host DB dataset, and three instances of Riboviria viruses from NCBI . With this form of the data, it is possible to use data stream, digital signal processing, and machine learning algorithms.•All researchers in bioinformatics, computing science, and computing engineering field can benefit from these data because by using this numeric representation they can apply several techniques such as machine learning and digital signal processing in genomic information.•Data experiments that use clustering and classification techniques in SARS-CoV-2 virus genomic information can be used with this dataset.•These data represent an easy way to evaluate the SARS-CoV-2 virus genome.1This work presents a new dataset of a chaos game representation (CGR) 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 100 instances of the SARS-CoV-2 virus genome collected from the National Center for Biotechnology Information (NCBI) The dataset provides two groups of formats files for all data. In the first group, all data are stored in Matlab file format (.mat), and in the second group, part of the data is stored in Microsoft Excel (.xlsx) and another part in the text file (.txt). The two groups have the same information. The data is organized into three main directories: “SARS-CoV-2 data”, “Virus-Host DB data” and “Other viruses data.” Each main directory is formed by two sub-directories: “Matlab” and “Excel and txt.”k-th file is called “RawData_k.mat.”Each sub-directory “Matlab” contains three files called “RawDataTable.mat”, “RawData.mat” and “CGRData.mat”. “RawDataTable.mat” and “RawData.mat” files store the raw data information from the viruses database; they have the same information, however in the “RawDataTable.mat” the attributes are stored in Matlab table format (after 2013b version) and in “RawData.mat” the attributes are stored in Matlab cell arrays format. Each “CGRData.mat” file stores the CGR values of all viruses presented in each “RawDataTable.mat” and “RawData.mat” file. For the main directory “Virus-Host DB data”, the CGR values are stored in 10 files where each COD.txt” where COD is the code (locus name) associated with the virus in Genbank Each sub-directory “Excel and txt” is composed of a file and another sub-directory called “RawData.xlsx” and “CGRData”, respectively. Each “RawData.xlsx” file has the raw data information from the viruses database, and each “CGRData” has the CGR of viruses presented in each “RawData.xlsx” file. The points of the CGR associated with each virus are stored in a text file called “LocusName_2The Chaos Game Representation (CGR), proposed by H. Joel Jeffrey in s, expressed asN is the length of sequence and ns is the n-th nucleotide of the sequence. Each n-th nucleotide, ns, is mapped to bi-dimensional symbol (xs(n), ys(n)) and it can be expressed asn-th symbol (xs(n), ys(n)) is transformed in CGR values by equations expressed asxp(n), yp(n)) from dataset presented in this work.The CGR has with input the nucleotide sequence, ressed ass=[s1,⋯,s


Recently X. Fang et al. using a method given in previous research, where several classes of new MDS self-dual codes were constructed through (extended) generalized Reed-Solomon codes, in this paper, based on the method given in we achieve several classes of MDS self-dual codes.Maximum distance separable (MDS) self-dual codes have useful properties due to their optimality with respect to the Singleton bound and its self-duality. MDS self-dual codes are completely determined by the length  A q-ary k-dimensional subspace of Let The study of MDS self-dual codes has attracted a great deal of attention in recent years due to its theoretical and practical importance. The center of the study of MDS codes includes the existence of MDS codes , classifn, the main interest here is to determine the existence and give the construction of q-ary MDS self-dual codes for various lengths. The problem is completely solved for the case where q is even [As the parameters of an MDS self-dual code are completely determined by the code’s length  is even . Many MD is even ,12,13,14In , Jin andIn this section we introduce some basic notations of generalized Reed-Solomon codes and extended generalized Reed-Solomon codes. For more details, the reader is referred to .q is a prime power, q elements and let n be a positive integer with Throughout this paper, n distinct elements of n nonzero elements k-dimensional generalized Reed-Solomon code (GRS for short) of length n associated with Let q-ary It is well known that the code .g., see ).q-ary Furthermore, the extended generalized Reed-Solomon code .g., see ).Put Lemma 1.The solution space of the equation system .The soWe defineThe conclusion of the following lemma is straightforward. For completeness, we provide its proof.Lemma 2.Let n be an even number, if there exists .Let n Proof. Let This implies that H. Yan  observedLemma 3.Let n be an even integer and .Let n Lemma 4.Let .Let m∣r is odd prime power, q elements. Suppose m-th root of unity and Let Theorem 1.Let Proof. m-th root of unity and Let t distinct elements, such that Let Then the entries of It is known that Let So g be a generator of Let q-ary This implies there exists a Example 1.Let  13,932.Theorem 2.Let Proof. t distinct even number Let v is a square element of q-ary Let Example 2.Let Theorem 3.Let Proof. t distinct even number Let For any t is odd, so q-ary Since Example 3.Let Theorem 4.Let Proof. Let For any Case 1: If m is even, t is odd.Case 2: If m and t are even, A is an even integer. It follows that Case 3: If m is odd, t is even.(1)If (2)If (1)If (2)If  □We can extend the Theorem 1 to a more general case.Theorem 5.Let Proof. m-th root of unity and Let t distinct elements, such that Let Similar with Theorem 1, we gets-th primitive root of unity. So Since So g be a generator of Let Case 1: If m odd and t even, we can take Case 2: If m even and q-ary MDS self-dual code with length So there exists a In this paper, based on the method from , we cons


R tensor admitting a Q-order Canonical Polyadic Decomposition (CPD) with large factors of size s-divergence, the Bhattacharyya Upper Bound (BUB) -based binary classification problems of interest. Under the alternative hypothesis, i.e., for a non-zero SNR, the observed signals are either a noisy rank- N-dimensional measurement vector N) relied on the exponential rate of s-divergence over parameter s. To circumvent this difficulty, a simplified case of s is often called s-divergence is the so-called Chernoff Evaluating the performance limit for the “Gaussian information plus noise” binary classification problem is a challenging research topic, see for instance ,4,5,6,7.ds ratio  since anractable ,8. To cin theory ,11,12,13n theory ,15 is asetection , sparse etection , energy etection , multi-ietection ,20, netwetection , angularetection , detectietection , just to problem  and oftevergence .P-order tensor is equal to the minimal positive integer, say R, of unit rank tensors that must be summed up for perfect recovery. A unit rank tensor is the outer product of P vectors. In addition, the CPD has remarkable uniqueness properties [R. Unfortunately, unlike the matrix case, the set of tensors with fixed  rank is not close [p-th mode. Following the Eckart-Young theorem at each mode level [The tensor decomposition theory is a timely and prominent research topic ,26. Confot close ,30. Thisot close . Note thot close . The TKDot close  and the ot close  are two de level , this code level  or adaptde level , it is sde level . More prThe classification performance of a multilinear tensor following the CPD and TKD can be derived and studied. It is interesting to note that the classification theory for tensors is very under studied. Based on our knowledge on the topic, only the publication  tackles R tensor admitting a Q-order CPD with large factors of size More precisely, we consider two cases where the observations are either (1) a noisy rank-s-divergence, the Bhattacharyya Upper Bound (BUB)—Chernoff Information calculated at Under the null hypothesis ibutions ,38,39 foWe note that Random Matrix Theory (RMT) has attracted both mathematicians and physicists since they were first introduced in mathematical statistics by Wishart in 1928 . When Wis-divergence, Chernoff Upper Bound, Fisher Information, etc.), but completes [Finally, let us underline that many arguments of this paper differ from the works presented in ,48. In , we tackThe organization of the paper is as follows: In the second section, we introduce some definitions, tensor models, and the Marchenko-Pastur distribution from random matrix theory. The third section is devoted to present Chernoff Information for binary hypothesis test. The fourth section gives the main results on Fisher Information and the Chernoff bound. The numerical simulation results are given in the fifth section. We conclude our work by giving some perspectives in the In this section, we introduce some useful definitions from tensor algebra and from the spectral theory of large random matrices.Definition 1.The Kronecker product of matrices We have Definition 2.The vectorization where Definition 3.The q-mode product denoted by where Definition 4.The q-mode unfolding matrix of size where R CPD of order Q is defined according toThe rank- product , ϕr(q)∈Rq-mode product defined in Definition 3 isq-th factor matrix of size An equivalent formulation using the q-mode unfolding matrix for tensor The  product .Q is defined according toThe Tucker tensor model of order q-mode product of q-mode unfolding matrix for tensor q-mode unfolding matrix of tensor The Following the definitions, we note that the CPD and TKD scenarios imply that vector M-dimensional vectors for which M remains fixed, matrix The Marchenko-Pastur distribution was introduced half a century ago  in 1967,Theorem 1 The empirical eigenvalue value distribution . The emN large enough, all the eigenvalues are located in a neighbourhood of interval We also observe that Theorem 1 remains valid if ee e.g., ). TheoreWe denote by We note that the estimated hypothesis ce (KLD) . The expce (KLD) . Howeversed-form ,8.The minimal Bayes’ error probability conditionally to vector s-divergence for X. The error exponent, denoted by According to , the relrding to −limN→∞lAs parameter Finally, using Equations (5) and (7), the Chernoff Upper Bound (CUB) is obtained. Instead of solving Equation (7), the Bhattacharyya Upper Bound (BUB) is calculated by Equation (5) and by fixing Lemma 1.The log-moment generating function given by Equation (6) for test of Equation (4) is given byProof. See From now on, to simplify the presentation and the numerical results later on, we denote byRemark 1.The functions s-divergence in the small In the small deviation regime, we assume that Lemma 2.The s-divergence in the small deviation regime can be approximated according towhere the Fisher information  is givenProof. See s-value at low s-value for larger According to Lemma 2, the optimal Lemma 3.In case of large where Proof. See Q-order tensor of size Q-order CPD with a canonical rank of M, we haveWhen tensor Q-order TKD of multilinear rank of When tensor The measurement tensor follows a noisy We recall that in the CPD case, matrix Result 1.In the asymptotic regime where with with Proof. See Remark 2.In , the CenIn this section, we assume that Result 2.In the small Proof. Using Lemma 2, we can notice thatinstance ). ◻Note that Result 3.In case of large Proof. It is straightforward to notice thatThe last equality can be obtained as in . Using LRemark 3.It is interesting to note that for as long as Proof. It is straightforward to note thatUsing Equation (14) and condition R is supposed to be small compared to N, it is realistic to assume In the case of low rank CPD where its rank Result 4.Under this regime, the error exponent can be approximated as follows:Proof. See It is easy to notice that the second-order derivative of At low Equation s⋆≈21+1SResult 1 and the above approximation allow us to get the best error exponent at low Contrarily, when The two following scenarios can be considered:In the TKD case, we recall that matrix Result 5.In the asymptotic regime where where Proof. See Remark 4.We can notice that for where ee e.g., ). As a cResult 6.In case of large Proof. We have thatUsing Lemma 3, we get immediately Equation (17). ◻Under this regime, we have the following resultsResult 7.For small Proof. Using Lemma 2, we can notice thatEach term in the product converges a.s towards the second moment of Marchenko-Pastur distributions Remark 5.Contrary to the Remark 3, it is interesting to note that for which does not depend on Q, andwhich depends on Q.In practice, when c is close to 1, we have to carefully check if Q is in the neighbourhood of In this section, we consider cubic tensors of order s-value, L = 10,000 the number of samples for Monte–Carlo process and where Firstly, for the CPD model, in s-divergences: s-divergence obtained by fixing In c goes to zero as predicted by our theoretical analysis.In For the TKD scenario, we follow the same methodology as above for CPD, s-value.For TKD scenario, the mean square relative error is in mean of order We can also notice that the convergence of In this work, we derived and studied the limit performance in terms of minimal Bayes’ error probability for the binary classification of high-dimensional random tensors using both the tools of Information Geometry (IG) and of Random Matrix Theory (RMT). The main results on Chernoff Bounds and Fisher Information are illustrated by Monte–Carlo simulations that corroborated our theoretical analysis.For future work, we would like to study the rate of convergence and the fluctuation of the statistics


Many-valued -algebras are algebraic systems that generalize Boolean algebras. The MV-algebraic probability theory involves the notions of the state and observable, which abstract the probability measure and the random variable, both considered in the Kolmogorov probability theory. Within the MV-algebraic probability theory, many important theorems  have been recently studied and proven. In particular, the counterpart of the Kolmogorov strong law of large numbers (SLLN) for sequences of independent observables has been considered. In this paper, we prove generalized MV-algebraic versions of the SLLN, i.e., counterparts of the Marcinkiewicz–Zygmund and Brunk–Prokhorov SLLN for independent observables, as well as the Korchevsky SLLN, where the independence of observables is not assumed. To this end, we apply the classical probability theory and some measure-theoretic methods. We also analyze examples of applications of the proven theorems. Our results open new directions of development of the MV-algebraic probability theory. They can also be applied to the problem of entropy estimation. MV-algebras, being generalizations of Boolean algebras, were introduced by Chang  and usedM is a normalized Carathéodory defined the basic notions of point-free probability, replacing Kolmogorovian probability measures on e, e.g., ). The MV Mundici  as a nor Mundici ) and RieM is weakly Main theorems of the MV-algebraic probability theory, including the basic version of the central limit theorem (CLT), laws of large numbers, and the individual ergodic theorem, can be found in ,8,9. In yniewicz . It is iThe MV-algebraic probability theory was also applied in the Atanassov intuitionistic fuzzy sets and interval-valued fuzzy sets settings , are deWe present three examples of the applications of the Marcinkiewicz–Zygmund, Brunk–Prokhorov, and Korchevsky SLLN for sequences of observables with convergent scaled sums. In particular, independent identically continuously distributed, as well as both independent and dependent discretely-distributed observables are considered.The problem of entropy estimation is important from both theoretical and practical points of view. Classical versions of the law of large numbers are used in this field. In particular, the authors of ,26,27 apThe paper is organized as follows. We present some notations that are used in the paper.We denote by Let X on a probability space X and by X with respect to P (if they exist).For each real-valued random variable The foundations of the theory of MV-algebras can be found in . In thisDefinition 1.An MV-algebra⊕ is associative and commutative with zero as the neutral element,is an algebra, where M is a non-empty set, the operation and for each:In an MV-algebra, a partial order is defined by the relation:The underlying lattice of M is the distributive latticewith the least element zero and the greatest element one, where the join and the meet are defined as follows:for each.Definition 2.An MV-algebra M is called σ-complete (complete) if every sequence  of elements of M has the supremum in M.X and an MV-algebra M, we introduce the notations:for for for For a non-empty set Definition 3.Given a σ-complete MV-algebra M, a functionis called a state on M if it satisfies the following conditions for eachand:(i) (ii) if, then(iii) if, thenA state m on M is called faithful iffor each non-zero element x of M.Definition 4.A probability MV-algebra is a pairconsisting of a σ-complete MV-algebra M and a faithful state m on M.M is complete (see Theorem 13.8 in [If  13.8 in ).Definition 5.Given a σ-complete MV-algebra M, a functionis called an n-dimensional observable in M if it satisfies the following conditions:(i) (ii) andfor eachsuch that(iii) for eachTheorem 1.Given M a σ-complete MV-algebra, an n-dimensional observable, and a state m on M, the functiondescribed by:is a probability measure on.The proof of the above theorem can be found in .Definition 6.Letbe a probability MV-algebra. We call an observableintegrable inif the expectationexists. Moreover, we writeforifwhereis the absolutemoment of x. If, then the variance of x is given by the formula.Definition 7.Observablesin a probability MV-algebraare said to be independent (with respect to m) if there exists an n-dimensional observable such that for arbitrary:Remark 1.Letbe independent observables in a probability MV-algebraandbe their joint observable. Then, for arbitrary Borel measurable function, the mapping given by:is an observable.Convergence almost everywhere of observables in a probability MV-algebra was defined by Riečan and Mundici .Definition 8.A sequenceof observables in a probability MV-algebrais said to converge to zero m-almost everywhere (m-a.e.), if:In the further part of this section, we will use the Kolmogorov probability space of observables considered by Riečan and Mundici .Let Kolmogorov probability space of the observablesWe call the triplet For each Proposition 1.Letbe a probability MV-algebra andbe a sequence of independent observables in, withbeing the joint observable of. Letbe the Kolmogorov probability space of the observablesin. For eachletbe an arbitrary Borel function. Let further the observablebe defined byand the random variablebe described by. Then:and the convergence ofto zero P-a.s.implies the convergence ofto zero m-a.e.In this section, we formulate and prove MV-algebraic counterparts of the Marcinkiewicz–Zygmund and Brunk–Prokhorov SLLN see .In the following part of the paper, in formulas containing integrals, we will assume that image measureLet 3.2 from ), which Lemma 1.Letbe a probability MV-algebra and. Then, for any-valued Borel functionand observablesin, the expected valueexists if and only if:Furthermore, if the above condition is satisfied, then:Proof. Let 16.12 in ):(5)∫XTn as follows.The MV-algebraic version of the Marcinkiewicz–Zygmund SLLN concerns the case of independent observables belonging to Theorem 2.Given a probability MV-algebra, letbe an independent sequence of observables inhaving the same distribution. Let,,forandfor. Then:converges to zero m-a.e.Proof. m-a.e. □Let The following MV-algebraic version of the Brunk–Prokhorov SLLN concerns sequences of observables that are not necessarily identically distributed.Theorem 3.Let. Given a probability MV-algebra, letbe an independent sequence of observables insuch that,for each. Let:Then:Proof. We denote by fore, by  and TheoIn this subsection, we consider the convergence of observables in a probability MV-algebra with product Remark 2.Letbe observables in a probability MV-algebra with product. By Proposition 2.4 from [such that for arbitrary:called the joint observable of. Moreover, for arbitrary Borel measurable function, the formula:defines a one-dimensional observable.Kolmogorov probability space of the observablesLet given by . We callThe following proposition is a consequence of Theorem 3.17 and Proposition 3.16 from .Proposition 2.Letbe a probability MV-algebra with product andbe a sequence of observables in, withbeing the joint observable of. Letbe the Kolmogorov probability space of the observablesin. For eachletbe an arbitrary Borel function. Let further the observablebe defined byand the random variablebe described by. Then, the convergence ofto zero P-a.s. implies the convergence ofto zero m-a.e.We formulate and prove the main theorem of this subsection. For its classical counterpart and some notations, we refer the reader to Theorem 4.Let observablesin a probability MV-algebra with productbe non-negative, i.e.,and their absolute moments of some orderbe finite. Letbe a non-decreasing unbounded sequence of positive numbers. If, where:and:where ψ is a function belonging to, then:Proof. Let We also present two special cases of the above theorem, corresponding to Theorem 2 and 3, formulated by Korchevsky  within tTheorem 5.Let observablesin a probability MV-algebra with productbe non-negative and have finite variances. Letbe a non-decreasing unbounded sequence of positive numbers. Ifand:for some function, then:Theorem 6.Let observablesin a probability MV-algebra with productbe non-negative and have finite absolute moments of some order. Letbe a sequence of positive numbers,,and:for some function. Then:Theorem 5 follows from Theorem 4 applied for We analyze the asymptotic behavior of scaled sums of three sequences of observables. They are independent identically continuously distributed in the first sequence and independent identically discretely distributed in the second one, whereas the third example concerns non-negative dependent observables. The MV-algebraic version of the Kolmogorov SLLN, presented in , cannot m of the form We consider observables taking values in the probability MV-algebra Let Let for each ented in , does noLet n, the joint observable We fix S-measurable functions full tribe, equipped with the operations c on We consider a probability MV-algebra with the product, presented in . Let Ω,Suct (see ).Let This paper is devoted to the development of MV-algebraic probability theory. We formulate and prove three generalized versions of the strong law of large numbers. The first two versions of the strong law, i.e., the MV-algebraic Marcinkiewicz–Zygmund SLLN and Brunk–Prokhorov SLLN, describe the asymptotic behavior of the sums of independent observables, whereas the third one, i.e., the Korchevsky SLLN, concerns the case of dependent observables. Their proofs require an application of the Kolmogorov theory of probability and some measure-theoretic techniques. To illustrate our theoretical results, we also present and analyze some examples of sequences of observables in a probability MV-algebra. We believe that our results open new possibilities for further development of the MV-algebraic probability theory in the non-Kolmogorovian setting. In particular, they can be used for the future development of the theory of fuzzy, intuitionistic fuzzy, and interval-valued fuzzy random events in complex spaces. We would like to apply the proven theorems to the estimation of logical entropy, as well as other types of entropy in the case of intuitionistic fuzzy random events. This requires, among other things, the definition of entropy for observables in MV-algebras.


Scientific Reports 10.1038/s41598-020-57777-2, published online 22 January 2020Correction to: The original version of this Article contained an error in the title of the paper, where the word “spectrophotometry” was incorrectly given as “|spectrophotometry”. This has now been corrected in the PDF and HTML versions of the Article.


Nature Communications 10.1038/s41467-020-16154-3, published online 8 May 2020.Correction to: Nucleic Acids Res.44, 6994–7005 (2016), when describing construction of the blue-light activation tool (BLAT). The sentence “The construction of BLAT and BLRT follows a previously reported strategy38” has been added to the beginning of the second paragraph of the “Results” subsection “Constructing the basic tools for powering cell division”. Reference 38 has also been added to the sentence following: “To construct a BLAT, two units were designed and assembled : a blue optogenetics unit (BOU) to express light-sensitive protein EL222, and an activation reporter unit to replace the LuxR-binding region with an EL222-binding region by overlapping the -35 region (E. coli consensus -35 and -10 regions) of the LuxI promoter38.”The original version of this Article omitted a reference to previous work in reference 38, Jayaraman et al. Blue light-mediated transcriptional activation and repression of gene expression in bacteria. 38” has been added to the legend of Fig. 4 to clarify this.Furthermore, Fig. 4, panels a and b, are adapted from reference 38 Figure 1, panels a and c, and this was not clear in the figure legend. The sentence “Panels a and b are adapted from Jayaraman et al.These have been corrected in both the PDF and HTML versions of the Article.


This approach increases exponentially, in the number of variables, the search space of the state-of-the-art tDBN algorithm. Concerning worst-case time complexity, given a Markov lag m, a set of n random variables ranging over r values, and a set of observations of N individuals over T time steps, the bcDBN algorithm is linear in N, T and m; polynomial in n and r; and exponential in p and k. We assess the bcDBN algorithm on simulated data against tDBN, revealing that it performs well throughout different experiments.Dynamic Bayesian networks (DBN) are powerful probabilistic representations that model stochastic processes. They consist of a prior network, representing the distribution over the initial variables, and a set of transition networks, representing the transition distribution between variables over time. It was shown that learning complex transition networks, considering both intra- and inter-slice connections, is NP-hard. Therefore, the community has searched for the largest subclass of DBNs for which there is an efficient learning algorithm. We introduce a new polynomial-time algorithm for learning optimal DBNs consistent with a breadth-first search (BFS) order, named bcDBN. The proposed algorithm considers the set of networks such that each transition network has a bounded in-degree, allowing for  Bayesian networks (BN) represent, in an efficient and accurate way, the joint probability of a set of random variables . Dynamicmth-order Markov DBNs such that each transition network has inter- and intra-slice connections. More recently, a polynomial-time algorithm was proposed that learns both the inter- and intra-slice connections in a transition network [The problem of learning a BN given data consists in finding the network that best fits the data. In a score-based approach, a scoring criterion is considered, which measured how well the network fits the data ,7,8,9,10 network . The seaBy looking into lower-bound complexity results for learning BNs, it is known that learning tree-like structures is polynomial . HoweverIn this paper, we propose a generalization of the tDBN algorithm, considering DBNs such that each transition network is consistent with the order induced by the BFS order of the optimal branching of the tDBN network, that we call bcDBN. Furthermore, we prove that the search space increases exponentially, in the number of attributes, comparing with the tDBN algorithm, while running in polynomial time.We start by reviewing the basic concepts of Bayesian networks, dynamic Bayesian networks and their learning algorithms. Then, we present the proposed algorithm and the experimental results. The paper concludes with a brief discussion and directions for future work.X denote a discrete random variable that takes values over a finite set n-dimensional random vector, where each n random variables Let , let X=X,…,Xn repDefinition 1 (Bayesian Network).A n-dimensional Bayesian Network (BN) is a triple where B induces a unique joint probability distribution over A BN D of size N, where variable D where the set of parents Let Observe that,Intuitively, the graph of a BN can be viewed as a network structure that provides the skeleton for representing the joint probability compactly in a factorized way, and making inferences in the probabilistic graphical model provides the mechanism for gluing all these components back together in a probabilistic coherent manner .M—Maintenance problems; S—Severe weather; F—Flight delay; O—Overnight accommodation; and C—Cash compensation. In this simple example, all variables are Bernoulli (ranging over T and F). Inside the callouts only the CPTs for variables taking the value T are given.An example of a BN is depicted in G is given, and our goal is to estimate the set of parameters of the network G, given only the training data. We assume data is complete, i.e., each instance is fully observed, there are no missing values nor hidden variables, and the training set N i.i.d. instances. Using general results of the maximum likelihood estimate in a multinomial distribution we get the following estimate for the parameters of a BN B:Learning a Bayesian network is two-fold: parameter learning and structure learning. When learning the parameters, we assume the underlying graph B fits the data D  and Minimum Description Length (MDL) . InformaThe LL is given by:This criterion favours complete network structures, and does not generalize well, leading to the overfitting of the model to the training data. The MDL criterion, proposed by Rissanen , imposesThe penalty introduced by MDL creates a trade off between fitness and model complexity, providing a model selection criterion robust to overfitting.n random variables by The structure learning reduces to an optimization problem: given a scoring function, a data set, a search space and a search procedure, find the network that maximizes this score. Denote the set of BNs with Definition 2 (Learning a Bayesian Network).Given a data n nodes has a superexponential number of structures, k-graphs (CkG) [k-graphs (BCkG) [kG and BCkG graphs are exponentially larger, in the number of variables, when compared with branchings [The space of all Bayesian networks with  problem ,21,22. Hructures ,23 or toructures , it is phs (CkG)  and breas (BCkG)  network anchings ,17.Definition 3 (k-graph).A k-graph is a graph where each node has in-degree at most k.Definition 4 (Consistent k-graph).Given a branching R over a set of nodes V, a graph Definition 5 (BFS-consistent k-graph).Given a branching R over a set of nodes V, a graph k-graph, there can only exist an edge from R. We assume that if R reaches Observe that the order induced by the optimal branching might be partial, while its BFS order is always total (and refines it). Given a BFS-consistent t. Furthermore, let T sequential instants of time. The set of observations is represented as n attributes, measured at time t and referring to individual h.Dynamic Bayesian networks (DBN) model the stochastic evolution of a set of random variables over time . Considet, In the setting of DBNs the goal is to define a probability joint distribution over all possible trajectories, i.e., possible values for each attribute t, the probability distribution is considered constant, i.e., Observations are viewed as i.i.d. samples of a sequence of probability distributions Definition 6 (mth-Order Markov assumption).A stochastic process over In this case m is called the Markov lag of the process.mth-order Markov process.If all conditional probabilities in Equation  are invaDefinition 7 (First-order Markov DBN).A non-stationary first-order Markov DBN consists of:A prior network A set of transition networks We denote by p parents from the previous time slices.Learning DBNs, considering no hidden variables or missing values, i.e., considering a fully observable process, reduces simply to applying the methods described for BNs for each transition of time . SeveralDefinition 8 (Tree-augmented DBN).A dynamic Bayesian network is called tree-augmented (tDBN) if for each transition network k and is consistent with the BFS order of the tDBN; the inter-slice network has in-degree of at most p. The main idea of this approach is to add dependencies that were lost due to the tree-augmented restriction of the tDBN and, furthermore, remove irrelevant ones that might be present because a connected graph was imposed. Moreover, we also consider the BFS order of the intra-slice network as an heuristic for a causality order between variables. We make this concept rigorous with the following definition.We introduce a polynomial-time algorithm for learning DBNs such that: the intra-slice network has in-degree at most Definition 9 (BFS-consistent k-graph DBN).A dynamic Bayesian network is called BFS-consistent k-graph (bcDBN) if for each intra-slice network Given an optimal branching Moreover, each node Before we present the learning algorithm, we need to introduce some notation, namely, the concept of ancestors of a node.Definition 10 (Ancestors of a node).The ancestors of a node mth-order bcDBN. Let p. For each node We will now describe briefly the proposed algorithm for learning a transition network of a p parents from past time slices, and the one in Equation (p parents from the past time slices when We note that the set quations  and 9)  Xps[t−m+quations  refers tk. Note that, if A complete directed graph is built such that each edge  network . In ordeD, a Markov lag m, a decomposable scoring function p and a maximum number of intra-slice parents k.The pseudo-code of the proposed algorithm is given in Algorithm 1. As parameters, the algorithm needs: a dataset Algorithm 1 Learning optimal mth-order Markov bcDBN1:for each transition do2: Build a complete directed graph in 3: Weight all edges Equation   in the intra-slice network the graph is consistent with a total order (so no cycle can occur); and (ii) in the inter-slice network there are only dependencies from previous time slices to the present one (and not on the other way). ☐Let Algorithm 2 Compute all the weights 1:for all nodes do2: Let 3:fordo 4:ifthen  5:   Let 6:end if  7:end for 8:for all nodes do 9:  Let 10:fordo  11:ifthen   12:    Let 13:end if   14:end for  15:end for 16: Let 17:end forAlgorithm 3 Compute the set of parents of 1:Let 2:fordo3:fordo 4:ifthen  5:   Let max = 6:   Let the parents of 7:end if  8:end for 9:end forTheorem 2.Algorithm 1 takes timegiven a decomposable scoring function ϕ, a Markov lag m, a set of n random variables, a bounded in-degree of each intra-slice transition network of k, a bounded in-degree of each inter-slice transition network of p and a set of observations of N individuals over T time steps.Proof. p elements is given by:For each time transition r, and that each variable has at most p from the past and 1 in p elements from the past and k elements from the present is upper bounded by Calculating the score of each parent set (Step 11 of Algorithm 2), considering that the maximum number of states a variable may take is Theorem 3.There are at least Proof. i-th node of V with at most k lower nodes. When V with at most k lower nodes exist. Therefore, there are at leastk-graphs.Consider without loss of generality the time transition k-graphs.Let There are https://margaridanarsousa.github.io/learn_cDBN/). The experiments were run on an Intel® Core™ i5-3320M CPU @ 2.60GHz×4 machine.We assess the merits of the proposed algorithm comparing it with one state-of-the-art DBN learning algorithm, tDBN . Our algN. The parameters p and k were taken to be the maximum in-degree of the inter and intra-slice network, respectively, of the transition network considered.We analyze the performance of the proposed algorithm for synthetic data generated from stationary first-order Markov bcDBNs. Five bcDBN structures were determined, parameters were generated arbitrarily, and observations were sampled from the networks, for a given number of observations one intra-slice complete bcDBN network with me slice a;one incomplete bcDBN network, such that each node in nd p+k≤4 b;two incomplete intra-slice bcDBN network , with higher precision scores and similar recall, comparing with tDBN+MDL.For both algorithms, in general, the LL gives raise to better results, when considering a complete network structure and a lower number of instances, whereas taking an incomplete network structure and a higher number of instances, the MDL outperforms LL. The complexity penalization term of MDL prevents the algorithms of choosing false positive edges and gives raise to higher precision scores. The LL selects more complex structures, such that each node has exactly N. In order to understand the number of instances N needed to fully recover the initial transition network, we designed two new experiments where five samples where generated from the first-order Markov transition networks depicted in We stress that in all settings considered both algorithms improve their performance when increasing the number of observations k, the number of necessary observations to totally recover the initial structure increases significantly.The number of observations needed for the bcDBN+MDL to recover the aforementioned networks are 0±478.18 a and 290±1134.77 b, with ak-structures, the bcDBN algorithm achieved consistently significantly higher When considering more complex BFS-consistent The bcDBN learning algorithm has polynomial-time complexity with respect to the number of attributes and can be applied to stationary and non-stationary Markov processes. The proposed algorithm increases the search space exponentially, in the number of attributes, comparing with the state-of-the-art tDBN algorithm. When considering more complex structures, the bcDBN is a good alternative to the tDBN. Although a higher number of observations are necessary to fully recover the transition network structure, bcDBN is able to recover a significantly larger number of dependencies and surpasses, in all experiments, the tDBN algorithm in terms of A possible line of future research is to consider hidden variables and incorporate a structural Expectation-Maximization procedure in order to generalize hidden Markov models. Another possible path to follow is to consider mixtures of bcDBNs, both for classification and clustering.


The frequency distribution of the h-index rank follows an exponential decay curve with the following formula y = 152.94e−0.0312x, where x is the rank of the entry and y is the h-index, and R2 = 0.9812 . The remaining 141 countries have an h-index > 2, 78 of those have an h-index > 11, just 53 of them have an h-index > 22 and 36 have an h-index > 44 , with the exception of Iran . We have also included three other entries in this list, namely Argentina , Estonia  and Hungary  on the basis of their h-index performance (h-index ranks of those can be examined in comparison to a group of 30 countries that produce >98% of the world’s highly cited (top 1%) papers  (h-index per million inhabitants) , climbing to 0.964 if Greece, Iran, Italy and Russia are excluded  is missing , raising questions about barriers to entry, and despite a wealth of opportunities for international collaboration, that will need to be addressed in the future.To examine whether the use of the formance . Interesformance . The h-ied here)  and two bitants) . These 4milarity . The ran missing . Dispari missing . It is wAs the field of bioinformatics has expanded across all of the life sciences , the prebtaa132_Supplementary_DataClick here for additional data file.


Moreover, we provide a polynomial-time algorithm to identify all u-fp orthology assignments in a BMG. Simulations show that at least not on any a priori knowledge about underlying gene or species trees.Genome-scale orthology assignments are usually based on reciprocal best matches. In the absence of horizontal gene transfer (HGT), every pair of orthologs forms a reciprocal best match. Incorrect orthology assignments therefore are always false positives in the reciprocal best match graph. We consider duplication/loss scenarios and characterize unambiguous false-positive ( Orthology is one of the key concepts in evolutionary biology: Two genes are orthologs if their last common ancestor was a speciation event Fitch . Distingevolutionary scenario comprising a gene tree T and a species tree S together with a reconciliation mapT to S. The map In this contribution, we consider exclusively duplication/loss scenarios, i.e., we explicitly exclude horizontal gene transfer. Characterizations of reconciliation maps are given e.g. in Górecki and Tiuryn inaccuracies in the assignment of best matches from sequence similarity data Stadler et al. , and(ii)limits in the reconstruction of the “true” orthology relation from best match graphs Geiß et al. .We consider best matches as an evolutionary concept: A gene y in species s is a best match of a gene x from species s contains no gene x. That is, best matches capture the idea of phylogenetically most closely related genes. Maybe surprisingly, the combinatorial structure of best matches has become a focus only very recently Geiß et al.  and  in the BMG. Here, we aim to identify false-positive edges from the structure of the BMG itself.Of course, the ß et al.  showed tß et al.  below. Oxy is a reciprocal best match. If there are no other descendants that harbor genes witnessing the duplication event, then the framework of best matches provides no information to recognize xy as a false-positive assignment.We first note that false positives cannot be avoided altogether, i.e., not all false positives can be identified from a BMG alone. The simplest example, Fig. On the other hand, RBMGs and thus BMGs contain at least some information on false positives. Since the orthology relation forms a cograph but RBMGs are not cographs in general Geiß et al. , incorrez et al.  noted thz et al. . The remz et al. . Here, wz et al.  to a comGood quartets cannot be defined on RBMGs because information on non-reciprocal best matches is also needed explicitly. This suggests to consider BMGs rather than RBMGs as the first step in graph-based orthology detection methods. In practice, best matches are approximated by sequence similarity and thus are subject to noise and biases Stadler et al. . The empunambiguous false-positive (u-fp) edges, i.e., reciprocal best matches that cannot be orthologs w.r.t. to any gene tree explaining the BMG. Sec. We provide a full characterization of unambiguous false-positive orthology assignments in BMGs.We provide a polynomial-time algorithm to determine all unambiguous false-positive orthology assignments in BMGs.In Sec. Sec. Since the material is extensive and very technical, we subdivide our presentation into a main narrative part Secs. –6 and a xy is an edge in undirected. A graph subgraph of symmetric part of a directed graph induced, denoted by vertex-colored graphs M of colors. A vertex-coloring is called proper if x, y) in We consider finite, directed graphs path (of length) in a directed graph G is a subgraph induced by a nonempty sequence of pairwise distinct vertices A treesundirected, planted and phylogenetic, that is, they satisfy (i) the root L(T) for the leaves  the subtree of T rooted in v. The set of clusters of a tree T is TSemple and Steel  best matches of the same color clusters of orthologous groups (COGs) concept  if there exists a leaf-colored tree explainsreciprocal best match graph (RBMG) if it is the symmetric part of a BMG An arbitrary vertex-colored graph x and y are reciprocal best matches in For the symmetric part of the BMG ß et al. .evolutionary scenario extends the map L(T) only contains extant genes. Inner vertices in the gene tree T that designate speciations have their correspondence in inner vertices of the species tree. In contrast, gene duplications occur independently of speciations and thus belong to edges of the species tree. The embedding of T into S is formalized byAn (Reconciliation Map) Let S is a map (R0)Root Constraint.(R1)Leaf Constraint. If (R2)Ancestor Preservation. If (R3)Speciation Constraints. Suppose (i)x in T.(ii)S for any two distinct children x in T.Several alternative definitions of reconciliation maps for duplication/loss scenarios have been proposed in the literature, many of which have been shown to be equivalent. This type of reconciliation map has been established in Geiß et al. . MoreoveS with leaf set  is the map Given a reconciliation map The following result is a simple but useful consequence of combining the axioms of the reconciliation map with the event labeling of Def. S and suppose that  endowed with a map Lemma ß et al. , Lemma 2extremal event labeling of T is the map Let T and a reconciliation of T with a species tree S, a duplication vertex of T is an AD vertex if its two subtrees have at least one color in common. In contrast, it is a non-apparent duplication (NAD) vertex if the color sets of its subtrees are disjoint. This notion is useful for a variety of parsimony problems that usually aim to avoid or minimize the number of NAD vertices Swenson et al.  has vertex set L(T) and edges The The orthology graph is often referred to as the orthology relation. Orthology graphs coincide with a well-known graph class:G is an orthology graph for some event-labeled tree , i.e. G is a cograph. for any given reconciliation map connecting a gene with a species tree.S. Then  Let xy in false-positive, or fp for short, if for every reconciliation map S we have xy is called fp whenever x and y cannot be orthologous w.r.t. any possible reconciliation fp s can be identified without considering reconciliation maps explicitly.In other words, xy be an edge in xy is fp.The edge There are two children For the extremal labeling Let fp can be verified in polynomial time for any given gene tree fp edges coincide with the edges xy in Lemma entclass1pt{minimaxz, yz are fp but not fp. Since we assume that no information on a priori, it is natural to consider the set of edges that are false positives for all trees explaining a given BMG.As shown in Fig.  Let xy in unambiguous false-positive (u-fp) if for all trees xy is fp.xy in u-fp, then it is in particular fp in the true history that explains u-fp edges are always correctly identified as false positives. Not all “correct” false-positive edges are u-fp, however. It is possible that, for an edge xy in xy is not fp for some gene tree xy is not fp and therefore not u-fp.Hence, if an edge entclass1pt{minimaentclass1pt{minimax and y for which T rooted in the children of u-fp edges in a BMG.In order to adapt the concept of AD vertices for our purposes, we introduce the color-intersection mentclass2pt{minimxy in a BMG fp for every tree u-fp.Every edge As we shall see below, the converse of Prop. xy in xy is fp.The edge xy is u-fp.The edge Let Prop. ab, bc, and cd.Since every orthology graph is a cograph Thm.  and thus Let L, and let Q, resp., the induced subgraph good quartet if (i) a bad quartet if (i) a ugly quartet if an The edge xy in a good quartet middle edge. The edge zx of an ugly quartet first edge. First edges in ugly quartets are uniquely determined due to the colors. In bad quartets, this is not the case and therefore, the edge r, s, and t must be pairwise distinct. Note that (R)BMGs may also contain induced The three different types of quartets are shown in Fig. mentclasspt{minimaz and u-fp edges. We collect here the main results of Sec. Good quartets are characteristic of a complementary gene loss yz is u-fp.If (ii)xy and its middle edge yz are u-fp.If (iii)xy and zw are u-fp.If Let Not surprisingly, quartets are intimately linked to color-intersections:xy. Then, xy is either the middle edge of some good quartet or the first edge of some ugly quartet, which in turn implies that xy is u-fp.Let u-fp edges xy with xy must be u-fp since u-fp. Moreover, xy, on the other hand, is the first edge of the ugly quartet All mentclass2pt{minimxy is the middle edge of a good quartet, then u-fp edges that are not identified with the help of the color-intersection u-fp edges. The edge xy in the BMG shown in Fig. u-fp, but it is not contained in a good, bad, or ugly quartet.Furthermore, if an edge u-fp edges that are not identified by quartets, we first introduce an additional motif that may occur in vertex-colored graphs.In order to characterize the (Hourglass) An hourglass in a proper vertex-colored graph xy and Note that Condition (i) rules out arcs between Every hourglass is a BMG since it can be explained by a tree as shown in Fig. Hourglasses are not necessarily part of an induced xy with Hourglasses xy is u-fp.If a BMG u-fp edges that are not contained in a quartet, see Fig. Prop. (Hourglass chain) An hourglass chainA vertex z is called a left  tail of the hourglass chain tailed if it has a left or right tail.u which is common to all hourglasses in the chain  Let augmented tree ofIn particular, the augmented tree preserves the best match relation:For every leaf-colored tree We now have everything in place to present the main results of this section.Let u-fp edges  On S, we use a variant of the well-known constant-rate birth-death process with a given age  A planted species tree nd Klemm , which mbona fide multifurcations of species Kliman et al. . Multifurcating gene trees were produced for p and T, the true BMG G are known, we can also determine the setu-fp edges as well as the subsets u-fp edges that are middle edges of a good or first edges of an ugly quartet, respectively. Note that in general we have u-fp edges in good and ugly quartets increases moderately for larger p.The simulated data set of evolutionary scenarios comprises species trees with 10 to 30 species (drawn uniformly). The time difference between the planted root and the leaves of u-fp edges in Fig. u-fp edges that appear only as first edges of bad quartets is even smaller; only 2-3% of the u-fp edges associated with hourglass chains, i.e., less than 0.15% of all u-fp edges are of this type. The overwhelming majority of u-fp edges associated with quartets thus appear  as middle edges of good quartets. This observation provides an explanation for the excellent performance of removing the u-fp edges, which are completely covered by First, we note that, consistent with Geiß et al. , Stadlerß et al. . In partß et al. , there iu-fp edges in terms of underlying subgraphs and refinements of trees. Since the best match graph contains only false positives, we have obtained a characterization of all unambiguously incorrect orthology assignments. Simulations showed that the majority of false positives comprises middle edges of good quartets, while u-fp edges that appear only as first edges of an ugly quartet are rare. Not surprisingly, the hourglass-related u-fp edges become important in gene trees with many multifurcations. They do not appear in scenarios derived from binary gene trees. For the theory developed here, it makes no difference whether polytomies in the gene tree appear as genuine features, or whether limited accuracy of the approximation from underlying sequence data produced the equivalent of a soft polytomy in the BMG.We have shown here how all unambiguously false-positive orthology assignments can be identified in polynomial time provided that all best matches are known. In particular, we have provided several characterizations for The augmented tree d et al. , i.e., dd et al. , Lafond d et al.  with besT and species tree S, we ask here about the information contained in the BMG implies a set S Hernandez-Rosales et al.  hits into evolutionary best matches in a systematic manner Stadler et al. , estimatclusters of orthologous groups (COGs) in an empirically estimated RBMG Tatusov et al. (u-fp edges and thus, it is the closest orthology graph. However, Orthology prediction tools intended for large data sets often do not attempt to infer the orthology graph, but instead are content with summarizing the information as v et al. , Roth etv et al. . Formall