While the h-index is determined by the maximum square that fits under the citation curve of an author when plotting the number of citations in decreasing order, the χ-index is determined by the maximum area rectangle that fits under the curve. The height of the maximum rectangle is the number of citations ck to the kth most-cited publication, where k is the width of the rectangle. The χ-index is then defined as h-index and other similar indices. We present a comprehensive empirical comparison between the χ-index and other bibliometric indices, focusing on a comparison with the h-index, by analysing two datasets—a large set of Google Scholar profiles and a small set of Nobel prize winners. Our results show that, although the χ and h indices are strongly correlated, they do exhibit significant differences. In particular, we show that, for these data sets, there are a substantial number of profiles for which χ is significantly larger than h. Furthermore, restricting these profiles to the cases when ck > k or ck < k corresponds to, respectively, classifying researchers as either tending to influential, i.e. having many more than h citations, or tending to prolific, i.e. having many more than h publications.We propose the  P) as a measure of quantity and the total number of citations to these publications (C) as a perceived measure of quality; several variants of these, such as the average number of citations per publication, the number of citations to the top or the 10th most cited publication, and the number of publications with at least 10 citations, have also been suggested [h-index [g-index [The debate in bibliometrics on quality versus quantity in evaluating academic research performance is still an ongoing concern . One peruggested . Althoug[h-index , the g-i[g-index  and gene[g-index , combineh-index and some of its variants was provided by Egghe in [h-index was given by Bornmann et al. in [h-index and its variants, and proposed a family of bibliometric indicators that do not suffer from these inconsistency problems. Of particular interest are extensions of the h-index, which take into account the full publication list of a researcher such as the taperedh-index [h-index continue to appear, for example [An extensive review of the Egghe in , and, a t al. in . In addit al. in  discusseh-index . Proposa example –13, as d example , 15.h-index and its variants do not normally take into account the full citation list of a researcher. This could be perceived as a drawback; however, the total citation count has the disadvantage of biasing the index in favour of researchers with very highly-cited top publications or very many publication with a relatively small number of citations. We now review the h-index and some of its variants, and then introduce the χ-index, a new index that addresses some of the drawbacks mentioned.Nevertheless, the h-index of a researcher is the maximum number h of the researcher’s publications such that each has at least h citations [citation vector, 〈c1, c2, …, cn〉 of a researcher, where the ci, the number of citations to publication i, are sorted in descending order, i.e. ci ≥ cj if i < j. Here we assume that for all i, ci > 0, and that h will be zero in the absence of any citations; this is consistent with defining the value of a bibliometric index of a researcher to be zero if none of the researcher’s publications have been cited [h-index is thus the largest rank h for which ch ≥ h. The h-index is completely insensitive to the fact that a researcher’s top few publications may be very highly cited, and conversely also to a researcher having a fair number of publications whose number of citations is less than but close to h [h-index, which gives extra weight to highly cited publications, is the g-index. The g-index of a researcher is the largest rank g for which g ≥ h. A problem with the g-index is that it may still be biased since, if a researcher has a few publications that are very highly cited and the rest have very few citations, the g-index will still be high. This is because the g-index is equal to the largest rank g such that the average number of citations up until that rank is at least g. Suppose the h-index of a researcher is h, then the h-core is the set of the h most highly-cited publications for this researcher. The A-index, which is the average number of citations to the publications in the h-core, i.e. h-index does not take into account the total number of citations to publications in the h-core [A-index suffers from the fact that taking an average will, all other things being equal, often favour authors with fewer publications when they are highly cited. To remedy this issue, the R-index has been proposed, where h ≤ R ≤ A. Nevertheless, the A and R indices, and to a lesser extent the g-index, ignore the effect of publications outside the h-core, which are also part of a researcher’s output. A recent proposal is the Euclidean-index [E-index), designed to take account of the full list of an author’s cited publications; it is defined as the Euclidean norm of the citation vector, i.e. The itations . Equivalen cited . The h-iose to h . A sugge=1gci≥g2 ; it is ee h-core . However=1hci=Ah . It is ean-index   1 publication with 100 citations, (ii) 10 publications with 10 citations each, and (iii) 100 publications with 1 citation each, are considered as equally desirable. So the χ-index is essentially the largest product ici where 1 ≤ i ≤ n; however, for comparison purposes with the h-index, we will actually define the χ-index to be the square root of this, i.e. χ-index of the researcher is 10; see k denote the value of i that maximises ici, we see that in all three cases, the researcher has exactly k publications with ck or more citations. It is clear that the h-index cannot be larger than the χ-index, since In order to motivate the χ-index with other indices, along the lines of the two-variable metrics examined in [A possible future line of research would be to investigate pairwise combinations of the mined in .χ-index is formally introduced in Section 2, generalising the h-index by allowing the interplay between k  and ck . We also list some properties of the χ-index, which could form the basis of its axiomatisation  when k is significantly greater than h and (iii) when k is significantly less than h. We further partition that data according to whether χ is approximately equal to h or significantly larger than h to get a sense of when these two indices differ. Membership of the classes is determined by a basic bootstrap percentile method [The ion cf. , 20), an, anχ-inde method , Sectioncitation curve is the curve resulting from plotting the number of citations against the ranking of the publications, as specified by the citation vector. The χ-index is the square root of the maximum area rectangle that can fit under the citation curve .The onotonic , 19, in χ-index is scale-invariant [h-index is not scale-invariant).The nvariant , in the χ-index is not independent [χ indices of both 〈2, 2〉 and 〈1, 1, 1, 1〉 are 2, however the χ-index of 〈2, 2, 1〉 is still 2 but the χ-index of 〈1, 1, 1, 1, 1〉 is h-index is also not independent).The ependent , since aχ-index, comparing it to the citation indices mentioned in the introduction, however, focusing our attention on the comparison of the χ-index and the h-index. We make use of a large data set compiled by Radicchi and Castellano from Google Scholar [In the following sections we carry out an empirical analysis of the  Scholar , and alsP, the number of cited publications. We carry out a more in-depth comparison of the χ and h indices in Section 4, by separating authors whose χ and h indices are approximately the same from those for which they are significantly different.Our initial comparison between the indices is carried out using the Spearman rank-correlation coefficient , which dbootstrap method [basic bootstrap percentile method [h-index for each resample, and then compute a 99% one-sided confidence interval for the h-index values, starting from the lowest one from the 1000 resamples. This allows us to determine for a given author whether k is approximately equal to h and, additionally, whether χ is approximately equal to h by checking whether k or χ are in the confidence interval or not.We make use of the p method , which ie method , Sectione method , which ak ≈ h, (ii) k > h, or (iii) k < h, where ≈ means approximately equals. The second and third classes capture a tendency of an author towards being prolific when k > h, or influential when k < h. . We further partition each class according to whether χ ≈ h or χ > h to see when the indices differ, and to get a sense of the proportion of researchers for which χ ≈ h. Finally, we also consider the subclasses of χ > h, depending on whether ck > k or ck < k.We thus first partition the authors into three classes, according to whether (i) We now introduce the two data sets, provide some basic statistics of these data sets, and compute the correlations between various indices for the researchers concerned. In Subsection 3.1 we consider the Google Scholar data set and in Subsection 3.2 we consider a data set of Nobel prize winners.For our main analysis, we made use of a large data set of Google Scholar profiles compiled and made available by Radicchi and Castellano . The fulh, g, A, R, χ, P, stand for the h-index, the g-index, the A-index, the R-index, the square root of the Euclidian-index, the χ-index, the square root of the total number of citations and the number of publications, respectively. . We obsh and χ indices, h being the most commonly employed index, and leave detailed comparison to other indices for future work.From now on, we will concentrate on comparing the h and χ indices both follow log-normal distributions [Jensen-Shannon divergence (JSD) [p = (pi) and q = (qi), where i = 1, 2, …, n.We start by showing, as was done in , that thibutions , 29. To ce (JSD) , which iJSD, which is a symmetric version of the Kullback-Leibler divergence and is based on Shannon’s entropy [pi = 0 or qi = 0, or both, 0 ln 0 and 0 ln (0/0) are both defined to be 0. . We observe that the JSD is equal to 0 when p = q.The formal definition of the  entropy , is giveμ, and standard deviation σ of the log-normal distributions fitted by the maximum likelihood method, and the JSD between the empirical distributions of the h and χ indices and the fitted log-normal distributions. The low JSD values indicate good fits for both indices. We also note that the means and standard deviations are quite close.In A is more of an outlier than P. On the other hand, the correlations are comparable to the Google Scholar data set, although, on average lower.For our second data set, we collected the citation vectors of 99 Nobel prize winners across a variety of disciplines from the Web of Science platform . We inclh and χ indices and the fitted log-normal distributions. As for the Google Scholar data set, the low JSD values indicate good fits for both indices. We again note that the means and standard deviations are quite close.In k ≈ h or not, or whether χ ≈ h or not. In Subsection 4.1 we analyse the Google Scholar data set, and in Subsection 4.2 we analyse the Nobel prize winners data set.We now analyse the data sets introduced in Section 3, with the aim of revealing how authors are separated into classes depending on whether k ≈ h, (ii) k > h, or (iii) k < h, exhibiting the geometry of the h and χ indices. When k > h there are many publications, each with fewer than h citations (tending towards prolific), and when k < h therefore fewer publications, each with more than h citations .In k < h is the largest class, the other two comprising just over 53.50% of the data set. It is also apparent that, within the class k < h, there are by some margin, more authors for which χ > h. What this means is that, when χ is significantly larger than h, we expect that k will be significantly smaller than h, i.e. we expect the author to have several publications with more than h citations, contributing to χ being larger h; this can be justified from the data in χ-index addresses a problem of the h-index that it does not sufficiently take into account highly cited publications. The statistics in χ-index, showing higher average values for the χ-index when k < h.In χ is significantly larger than h, clearly demonstrating the potential of the χ-index to separate authors that may have similar h indices. In addition, the statistics shown in χ > h. The breakdown of the χ > h class, when ck > k and ck < k, can be seen in ck > k, are much higher than those for the smaller subclass, ck < k.Moreover, it can be seen in h values and therefore also very high χ values. In χ-index for this data set. These examples can be contrasted to the ones shown in χ-index when k > h or k < h.The Nobel prize winners data set looks at the extreme case of researchers having, on average, very high k < h and, of those, for over 75% of the authors χ > h. As expected, this implies that, overall, Nobel prize winners are influential. Looking at the statistics in k < h, on average, the χ values of researchers are much larger than the h values. This is due to publications with a large number citations, significantly more than h. An interesting observation is that unlike χ-index are the highest when k < h, in χ is highest for the smaller class when k > h. This is most likely due to a long tail of highly cited publications for these few laureates.In χ > h for over 60% of laureates. However, as the statistics in h-index for those Nobel prize winners with χ ≈ h, is actually, on average, higher than both the h and χ indices of the laureates with χ > h. This may indicate that for very influential researchers, such as Nobel laureates, when χ > h the h-index undervalues their contribution. The breakdown of the χ > h class, when ck > k and ck < k, can be seen in ck > k are, in fact, much lower than those for the subclass ck < k. This latter class is quite small as there are only three such Nobel prize winners; see In contrast to χ-index, which addresses some shortcomings of the h-index in terms of the balance between number of citations and number of publications. The χ-index has a simple geometric characterisation in terms of the largest area rectangle that fits under the citation curve; this generalises the h-index for which the rectangle is constrained to be a square.We have presented a new citation index, the χ-index. First, as with many of the citation indices that combine number of citations  with number of publications (quantity), the χ-index correlates strongly with the square root of the total number of citations, yet it is selective in its choice of publications to include in the index. Second, as we have seen from our analysis, there are many researchers whose χ-index is significantly larger than their h-index due to their tendency to be influential, in the case k < h, or prolific in the case k > h. We believe that this property of the χ-index is beneficial and could lead to a more satisfactory ranking of researchers than that obtained using the h-index.We have analysed two data sets, a large one from Google Scholar and a small one of Nobel prize winners. Studying these data sets clearly shows the utility of the


Scientific Reports 10.1038/s41598-018-25048-w, published online 30 April 2018Correction to: The original version of this Article contained an error in the title of the paper, where the word “Reservoir” was incorrectly given as “Reservior”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supporting Information file.


Nature Communications; 10.1038/s41467-017-01713-y, published online 23 Nov 2017.Correction to: The original version of this Article contained an error in Fig. 1. In panel d, the model on the right of the panel was incorrectly labeled ‘+Heme’, and should have read ‘− Heme’. This has now been corrected in both the PDF and HTML versions of the Article.


Neuroscience of Consciousness, Volume 2019, Issue 1, 2019, https://doi.org/10.1093/nc/niz009In the original version of this article, the section ‘The reason for metacognitive inflation’ contained a small error. The corrected sentence reads as follows:Therefore, metacognitive noise likely contributed to the increase in metacognitive efficiency scores (meta-d′/d′ and meta-d′–d′) in the current analyses but not to the increase in metacognitive sensitivity scores  .The authors would like to apologize for any confusion this may have caused.


Automated machine-learning systems are able to de-identify electronic medical records, including free-text clinical notes. Use of such systems would greatly boost the amount of data available to researchers, yet their deployment has been limited due to uncertainty about their performance when applied to new datasets.We present practical options for clinical note de-identification, assessing performance of machine learning systems ranging from off-the-shelf to fully customized.We implement a state-of-the-art machine learning de-identification system, training and testing on pairs of datasets that match the deployment scenarios. We use clinical notes from two i2b2 competition corpora, the Physionet Gold Standard corpus, and parts of the MIMIC-III dataset.Fully customized systems remove 97–99% of personally identifying information. Performance of off-the-shelf systems varies by dataset, with performance mostly above 90%. Providing a small labeled dataset or large unlabeled dataset allows for fine-tuning that improves performance over off-the-shelf systems.Health organizations should be aware of the levels of customization available when selecting a de-identification deployment solution, in order to choose the one that best matches their resources and target performance level. Over the past several years, health service researchers have significantly expanded their use of free text in medical research . For exaDe-identified clinical datasets are created by labeling all words and phrases that could identify an individual, and replacing them with surrogate data or context-specific labels. For example, “John London complains of chest pain that started on January 1st 2012” becomes “[PersonNameTag] complains of chest pain that started on [DateTag]”. The de-identification process needs to have high recall (sensitivity) since publicly releasing text containing protected health information (PHI) represents a legal and ethical liability. On the other hand, it also needs to have reasonable precision, because unnecessarily removing non-identifying text limits the data’s usefulness to researchers . Notice Automatic de-identification systems have not been widely adopted on a commercial level, despite the fact that their performance already surpasses that of human annotators: fully customized de-identification systems achieve precision and recall of 97% or higher , 6, whilguarantee performance on all medical text it will ever encounter. Structured data such as forms are easy to de-identify, e.g., by removing “Name” and “Date” fields; however, free-text clinical notes vary widely with note purpose and institutional conventions, and include PHI in ways that are challenging to identify and redact.A blocking factor is that, like many other tools based on machine learning, a de-identification system cannot This challenge may necessitate customizing the de-identification system in order to teach it about the formatting and jargon used in a particular organization. Consider a note with the line “John referred to Alzheimer’s clinic ...”. An off-the-shelf system knows “Alzheimer’s” as a medical condition and de-identifies to “[PersonNameTag] referred to Alzheimer’s clinic...”; a system customized on a target organization’s labeled data where Dr. Alzheimer and his clinic appear frequently, would correctly give “[PersonNameTag] referred to [PersonNameTag]’s clinic...”.As reviews of de-identification strategies in healthcare concluded, “True anonymization is challenging, and further work is needed in the areas of de-identification of datasets”  and “de-Therefore, in such a privacy-sensitive field, healthcare organizations need to employ de-identification systems in a controlled manner, with performance assurances specific to each deployment. In order to help organizations make an informed decision, we use publicly available clinical note datasets to assess the performance of automated de-identification systems in several deployment scenarios.Our first scenario is a fully customized system: a healthcare organization employs human annotators to label a sufficiently large number of PHI examples for training a new machine learning model to perform automated de-identification. We assess performance in this scenario using 3 medical datasets, training the model on part of a dataset and evaluating on the remainder of the dataset.Our second scenario is off-the-shelf use: the organization provides no labeled data, instead using a pre-trained model as-is. We replicate this scenario by training custom models on 3 datasets and testing each model on all other datasets.some labeled data. Since labeled data is an expensive resource, requiring the work of human annotators, we study how many labeled examples are needed to improve performance over the off-the-shelf scenario, and how many are required to obtain results equivalent to the fully customized scenario.Our third scenario is partial system customization with labeled data: the organization has the resources to provide unlabeled data. With this data we create a custom token embedding (data representation) for the machine learning system.Our last scenario is partial system customization with unlabeled data: the organization can avoid the legal and privacy concerns involved with annotating data, and instead improve performance using a large set of its The first automated approaches for medical free-text de-identification were proposed in the late 1990s and were mainly rule-based , 11. SubOur partial customization scenario with labeled examples is an example of semi-supervised transfer learning/domain adaptation; we build on the work of Lee JY et al. in neural networks . Lee H-JThe US HIPAA de-identification standard specifies the use of either “Expert Determination” or the “Safe Harbor” method to de-identify data . In the From the i2b2 National Center for Biomedical Computing for the NLP Shared Tasks Challenges, we use the i2b2-2006  and i2b2We also use the PhysioNet gold standard corpus of de-identified medical text , containFinally, we use the Medical Information Mart for Intensive Care III (MIMIC-III) dataset . This daThe i2b2-2006, i2b2-2014, and Mimic-III de-identification guidelines vary regarding which entities are considered PHI. When dataset pairs in an experiment were annotated with the same guideline, we report results on all PHI types in the guideline; in cross analyses, we use the Name PHI, which is labeled consistently across all guidelines. The train/test splits were made by patient. The i2b2 datasets were released with a supplied partition, and the remaining datasets were split randomly. Descriptive statistics are given in Table Our machine learning model implements the state-of-the-art in de-identification of medical notes , 6 and nPatient, prescribed, 50, mg, ...).A Tokenizer, breaking down the input text into tokens, e.g. “Patient prescribed 50mg...” is split into , and the number of spaces and line breaks before the token.A Named Entity Sequence Tagger, responsible for converting a token sequence to a tag sequence while taking into account context information. For example, in the sentence “Mr. Jack London was examined”, “London” should be tagged as a person’s name. The Tagger consists of the following:A Token BiRNN, adding context information to the extracted token information.A Tag prediction layer, projecting the 200 dimensional BiRNN into a probability distribution over the PHI tags , including the “not PHI” tag.A Conditional Random Field, imposing an additional structured prediction layer to make sure that PHI labels make sense as a sequence.Figure Model training was carried out using cross entropy loss over the given set of labeled examples as the loss function, and applying Adagrad  as the sEvaluation results are given for recall , precision (percent of detected PHI which was indeed PHI), and F1 score . We can tune the balance between recall and precision by setting the activation bias for “not PHI” prior to the Conditional Random Field block. In an off-the-shelf system, this tuning would be accomplished using heuristics or manual monitoring.p = 85%), while not adjusting if precision is less than this cutoff. As shown in Table Because recall is of highest importance for patient privacy, we compare system recall using R@P as described in, e.g., Eban et al. , 35 The We perform four experiments using the text de-identification system described in the previous section.We follow the standard machine learning setup of training and testing on the same dataset “A”.We follow the standard machine learning setup of training on a dataset “A” and testing on a different dataset “B”.Train from scratch: A new model is trained using these “n” datapoints. We call this model “only B”.Further tune an existing model: A model pre-trained on “A” receives further training on “B”. We call this model “A then B”.Jointly train a new model: A model is trained from scratch using an even mixture of “A” and “B”. We call this model “A mix B”.We train the system on “A” augmented with “n” labeled PHI instances from “B”, and test on “B”. We consider three ways to make use of the additional “n” samples from “B”:We train the system on “A” only, using a custom token embedding that is generated using unlabeled dataset “B”. A token embedding maps a discrete word to a floating-point vector; vectors corresponding to similar words cluster together, thus providing information about the language to the machine learning system. Token embeddings are built using large unlabeled text corpora; in some settings, using domain-specific corpora improves system performance , 37. We We go through our four scenarios, presenting results and discussing their implications.We train custom models on the three datasets that contain sufficient PHI for a fully trained model. The fully customized results are given in Table To illustrate the challenges remaining in even the best-performing de-identification scenario, we consider the errors on the i2b2-2014 model. From the 15,201 PHI elements in the evaluation set, the model classified 15 K true positives, 116 false negatives, and ~ 2.5 K false positives.We focus on the errors in Name, as the most identifying field. Name had 14 false negatives, i.e. undiscovered PHI: 3 doctor initials, 1 patient single-letter suffix (“I” as in “John Smith I”), 1 dataset-mislabeled apostrophe-s, and 9 names. All 9 names were dictionary words , showing remaining challenges in automated de-identification.False positives remove information useful to researchers; they are worth reviewing both to see what is being unnecessarily lost, and to get an intuition for the workings of the algorithm. The Name false positives included medical terms similar to names , which could be corrected using heuristics based on medical terminology; errors due to overreliance on sentence structure, e.g. the second word after a title being labeled a name (“awoke” was labeled in “Ms Mitchel awoke feeling...”); and misspellings creating non-dictionary words (“Interesetd (sic) in quitting smoking”).Our next deployment scenario is an organization using an off-the-shelf system with no customization. We use our custom models from the previous section. We test each model on the datasets with compatible labeling schemes, reporting recall/precision/F1 for all PHI types combined. We then present a full cross-dataset analysis using Name only.The i2b2-2014 model tested on physionet yields 76.6/60.5/67.6. Error analysis shows that 272 of the 441 false negatives (i.e. missed PHI) are of type Location, and consist mainly of “MICU”, “PMICU”, “cath lab”, and similar. Investigation revealed that these initials appear only in physionet, not i2b2-2014, thus providing a good example of an off-the-shelf system missing local jargon. Dropping Location from the analysis yields an improved 89.1/59.8/71.6; this improvement shows that a real deployment could consider using an off-the-shelf model together with heuristics  gleaned from a manual error analysis.The mimic-discharge model tested on mimic-radiology yields 65.7/90.9/76.2. Error analysis shows that 595 of the 597 false negatives are of type Id; of these errors, 577 are the 7-digit record number at the top of every note. This error is again dataset-specific and easily fixed with a simple heuristic. Including the heuristic yields 99.4/95.2/97.3, on par with a custom model.The mimic-discharge model test on mimic-echo yields 99.7/98.7/99.2, on par with a custom model and thus showing that de-identification of some datasets can be accomplished without a customized system.Table For the large labeled dataset “A” we use i2b2-2014; for the partially labeled dataset “B” we run experiments using physionet, mimic-radiology, and mimic-discharge.Figure The larger mimic-discharge shows that over ~ 500 labels, one can train solely on “B”. Around 1000 labels, performance reaches that of fully customized (97.1%); around this number, one sees diminishing returns from further labeling effort. At ~ 80 labels, the “A then B” model is already better than the off-the-shelf system, again demonstrating the usefulness of even small labeled datasets.We train using i2b2-2014 as dataset “A” with different token embeddings, and select test dataset “B” from physionet, mimic-radiology, mimic-discharge, and mimic-echo. For each choice of “B”, we evaluate three different token embeddings: the generic GloVe embedding, embed-mimic, and the embed-mimic-* that matches “B”. . For the mimic datasets, we give recall/precision/F1 on Name PHI; for physionet, whose labeling matches i2b2-2014’s, we also report on all types.Table In our work we follow the same datasets through various levels of system customization, thereby creating a robust picture of the performance a health organization can expect from a de-identification system under different scenarios. This is while obtaining results on par with the available literature at comparable data points , 20, 37.Automated de-identification systems can be used to add an extra layer of security while working with trusted research collaborators, or to minimize exposure of PHI to human annotators who will complete the de-identification task. Automated de-identification can also play a role in a HIPAA-compliant data release, with the additional step of “Expert Determination,” wherein a human expert in the field determines if the de-identification process has ensured that “the risk is very small that the information could be used ... to identify an individual.”  SelectinBased on our results, we present broad guidelines to inform an organization’s approach to de-identification using machine learning.Organizations able to label on the order of 10 K PHI examples can expect their fully customized system to have a recall of 97–99%. Organizations also have the control to fine-tune the balance between recall and precision.Organizations should try using an off-the-shelf system before committing to customization. Although performance varied widely, our experiments showed that recall can be dramatically improved (to 89–99%) with simple heuristics gleaned from manual error analysis.Organizations with the resources to provide a small amount of labeled data will benefit from partial customization. Labeling even a small amount of PHI, ~ 20 to ~ 80 examples, will raise system performance over an off-the-shelf solution. Labeling ~ 1000 PHI will give results on par with full customization.Organizations can avoid the cost and privacy concerns of labeling data, yet still gain in performance over off-the-shelf-systems, by creating a custom embedding using a large set of their unlabeled data.These guidelines generalize from results on available datasets, and thus cannot provide performance guarantees. In practice one can ensure better baseline performance with additional de-identification techniques, such as adding organization-specific or generic heuristics, or enhancing a pure machine learning system with a human in the loop.Our results highlight the need for additional medical corpora with identical labeling schemes. Contributions of notes from a variety of healthcare systems, large and small, encompassing different jargon and distributions of identifiers, would go a long way towards the goal of building a truly generic, off-the-shelf system requiring no customization. Such a collection could also form a universally recognized benchmark for evaluating commercial offerings.


The next generation sequencing technology allows us to obtain a large amount of short DNA sequence (DNA-seq) reads at a genome-wide level. DNA-seq data have been increasingly collected during the recent years. Count-type data analysis is a widely used approach for DNA-seq data. However, the related data pre-processing is based on the moving window method, in which a window size need to be defined in order to obtain count-type data. Furthermore, useful information can be reduced after data pre-processing for count-type data.z-values for a follow-up analysis. For an illustration, an application to the DNA-seq data from a pair of normal and tumor cell lines is presented with a change-point analysis of z-values to detect DNA copy number alterations.In this study, we propose to analyze DNA-seq data based on the related distance-type measure. Distances are measured in base pairs (bps) between two adjacent alignments of short reads mapped to a reference genome. Our experimental data based simulation study confirms the advantages of distance-type measure approach in both detection power and detection accuracy. Furthermore, we propose artificial censoring for the distance data so that distances larger than a given value are considered potential outliers. Our purpose is to simplify the pre-processing of DNA-seq data. Statistically, we consider a mixture of right censored geometric distributions to model the distance data. Additionally, to reduce the GC-content bias, we extend the mixture model to a mixture of generalized linear models (GLMs). The estimation of model can be achieved by the Newton-Raphson algorithm as well as the Expectation-Maximization (E-M) algorithm. We have conducted simulations to evaluate the performance of our approach. Based on the rank based inverse normal transformation of distance data, we can obtain the related Our distance-type measure approach is novel. It does not require either a fixed or a sliding window procedure for generating count-type data. Its advantages have been demonstrated by our simulation studies and its practical usefulness has been illustrated by an experimental data application.The online version of this article (10.1186/s12864-019-5491-x) contains supplementary material, which is available to authorized users. The next generation sequencing technology has advanced significantly during the recent decade. It allows us to obtain a large amount of short DNA sequence (DNA-seq) reads at a genome-wide level . DNA-seqDNA-seq has been widely used for the detection of copy number variation/alteration , 4. BothDue to the limitations of current sequencing techniques, the sequencing data cannot be obtained for certain genomic regions. Then, the related count-type measure is simply zero. For a DNA sequence containing such a genomic region, a large gap without any short reads is observed. Then, the related distance-type measure can be very large  and an artificial censoring can be considered.In this study, we consider distance-type measure. Distances are measured in base pairs (bps) between two adjacent alignments of short reads mapped to a reference genome. Our distance-type measure approach is novel. It does not require either a fixed or a sliding window procedure for generating count-type data. Furthermore, we propose artificial censoring for the distance data so that distances larger than a given value are considered potential outliers. Our purpose is to simplify the pre-processing of DNA-seq data.Statistically, we consider a mixture of right censored geometric distributions to model the distance data. Additionally, to reduce the GC-content bias, we extend the mixture model to a mixture of generalized linear models (GLMs). Our approach can be considered as an alternative to the method proposed by Shen and Zhang . The estIn the following, we first demonstrate how to conduct experimental data based simulation study to confirm the advantages of distance-type measure approach in both detection power and detection accuracy. Then, we introduce our statistical models and the related estimation procedure. The model performance can be demonstrated by simulation studies. The practical usefulness of our approach can be illustrated by an application to the DNA-seq data from a pair of normal and tumor cell lines  as well We consider a novel distance-type measure approach to the analysis of DNA-seq data. Distances are measured in base pairs (bps) between two adjacent alignments of short reads mapped to a reference genome. An illustration is shown in Fig. We first introduce how to compare the distance-type vs. count-type data so that more advantages of distance-type data can be confirmed. Then, we introduce a finite mixture , 14 of gThe widely used coverage depth (or count-type) based approach can be briefly described as follows. For a genomic region, it is divided into non-overlapping smaller regions (windows or bins) and the number of sequencing reads is counted for each bin. For example, if a genomic region is divided into 100 bins, then 100 count-type data can be obtained from these bins. The original sequencing read data are then simplified to count-type data by this approach. In this study, the DNA sequencing data were collected for the detection of copy number variations. A certain type of data (distance or count) is generated from the original sequencing read data . Between these two types of data, our choice is decided based on the statistical power for detecting changes. To demonstrate the advantage of our approach, we perform a simulation study to evaluate whether the distance based approach can be more statistically powerful than the coverage depth based approach. We consider a relatively simple scenario: if two genomic regions are known to be different in copy numbers, then which approach is more likely to detect the change? Furthermore, can accurate detection of change location be achieved when the distance-type data are used in the copy number variation/alteration analysis?Application to Sequencing Data, but we used the original sequencing read data). In this region, we randomly selected two bins with length L for each bin. For each of the original sequencing reads from the second bin, we removed it randomly with 50% probability. (The purpose was to simulate a deletion of 50% copy number.) We kept all the original sequencing reads from the first bin. In this way, we simulated sequencing data for two genomic regions with different copy numbers. The distance-type data can be obtained as described at the beginning of Methods section. To obtain count-type data, we divided each region into bins with length W for each bin (and then counted the number of sequencing reads in each bin). To achieve an unbiased comparison between the distance-type data and the count-type data, we used the non-parametric Komogorov-Smirnov two-sample test to compare the empirical distributions of simulated data for two genomic regions. Notice that either the distance based approach or the coverage depth based approach is to generate a certain type of data . It is necessary to compare which type of data can be more statistically powerful in the detection of changes. Our evaluation strategy does not depend on data type, as it does not require any parametric or statistical distribution assumptions.To address which type of data is more likely to detect the change (detection power), we consider the following simulation. Our simulation data were generated based on some real experimental sequencing data . We generated data based on the region with position 10-30Mbps from the normal sample  For each setting, we repeated our simulation and analysis for 5000 times.p-value as the identified change location. In this way, from each simulation repetition, we obtained the identified change location and its related p-value. Although the lowest p-value was considered, some cutoff values would still be usually considered in practice. As all the possible change locations were screened, a cutoff value lower than the conventional cutoff value 0.05 would be usually considered. Therefore, we considered the following cutoff values for the lowest p-values: 0.01, 0.001, 0.0001, 0.00001, 0.000001 and 0.0000001. Based on the results for detection power comparison , we obtained more analysis results from the above simulation setting. In each simulation repetition, we considered the data for two genomic regions were connected. We evaluated how accurate the boundary (change) between two regions could be detected. As the data for two regions were ordered, the change location could be between any two consecutive data (for both distance-type and count-type). Therefore, we screened all the possible locations and considered the location with the lowest p-value was lower than the cutoff value, then we calculated the difference between the identified and true change location; otherwise, we considered the difference as no-detection (ND). Then, we would evaluate whether the overall difference based on the distance-type data was lower than the overall difference based on the count-type data. (Notice that a lower difference means a more accurate detection.) Statistically, the traditional Wilcoxon-Mann-Whitney rank sum test would be an appropriate choice (for testing “distance-type based difference” < “count-type based difference”). In the test, for these ND differences, we changed them to be a value larger than any observed differences as no-detection (ND) would be the worst scenario.For each simulation repetition, if the returned Y is a random variable (e.g. distance-type measure) with a right censored geometric distribution with p as the probability of success. Notice that a geometric distribution can be considered as a model for the number of failures until first success. Let T be a given constant for the censoring value . The indicator function δ = 1 if y ≥ T and δ = 0 if y < T. The probability distribution function of right censored geometric distribution is given by:Suppose Mixture of Right Censored Geometric Distribution.Y1, … , Yn denote a random sample of size n. Each mixture component is a right censored geometric distribution given by Eq.  censored at the same value T.g-component GLMs. Let y1, … , yn be n independent observations of the response variable. For each yi, there is a covariate xi. π1, … , πg are the component proportions as defined above. The mixture is given by:fj = (pj(x)(1 − pj(x))y)δ((1 − pj(x))T + 1)δ)(1 − . Notice that each pj is a function of x. We first define μj = (1 − pj)/pj and then we use a link function ηj = log(μj) = βj0 + β1x for j = 1, … , g. Here, β1 is common for all different components since it represents the GC-content effect. Furthermore, the component proportions π1, … , πg do not depend on the covariate since they are considered as global parameters . Therefore, the vector Ψ of parameters is given by Ψ = . The intercepts  are different for different distribution components.Based on the above mixture of right censored geometric distribution, a mixture of right censored geometric distribution based GLMs can be described. Eq.  can be eThe log-likelihood based on Eq.  is givenΨ. For each yi, missing component information can be considered. Accordingly, the vector {zij} is introduced: zij = 1 if yi belongs to the j-th component of the mixture model; zij = 0 otherwise . Then, the log-likelihood of “complete data” is given by:The EM algorithm  can be uFor the E-step, we calculate:For the M-step, we first calculate:β10,  … , βg0, β1) requires a numerical optimization procedure, which is used to solve the following equation system.The calculation for :Based on the Newton-Ralphson method, we have the following iterative equation for The details for this iterative equation as well as the EM algorithm are given in the Additional file g0 < g1. We test the null hypothesis H0 : g = g0 against H1 : g = g1. We first use the observed data to fit a mixture model with g0 components and a mixture model with g1 components. A LRT score can be calculated for the observed data. Then, we can simulate data based on the fitted model with g0 components (parametric bootstrap). For each set of simulated data, we can calculate the related bootstrapped LRT score. After B rounds of parametric bootstrap repetitions, the p-value of observed LRT score can be approximately calculated based on B bootstrapped LRT scores. The number of components in the mixture model is determined by a likelihood ratio test (LRT) based approach. Consider two integers p-values less than the cutoff value). When the regions are as long as 200 kbps, the advantage of distance based approach (distance-type data) can be clearly observed. It is more difficult to detect changes as the regions become shorter. However, the advantage of distance based approach (distance-type data) becomes even clearer when the regions are as short as 50 kbps. When the regions are as long as 1Mbps, two approaches perform overall similarly. From Table For the detection power based comparison, our evaluation results are summarized in Table p-value less than a given cutoff value is presented for each type of data. The proportion based on the distance-type data is always higher. Then, the one-sided p-value for each pair of comparison test is presented in Table p-values for Wilcoxon-Mann-Whitney rank sum test is actually simple.) When L is relatively short as 150,000 bps, the detection accuracy based on the distance-type data is always better than the detection of accuracy based on the count-type data. When L is increased to 300,000 bps, the detection accuracy based on the distance-type data is still clearly better when the cutoff value is 0.001 or lower. When L is relatively long as 450,000 bps, the detection accuracy based on the count-type data is better, but the detection accuracy based on the distance-type data is still better when the cutoff value is 0.0001 or lower. For the detection accuracy based comparison, our evaluation results are summarized in Table 1 = 0.3, π2 = 0.5, π3 = 0.2 and p1 = 0.002, p2 = 0.02, p3 = 0.2  to generate the empirical distribution of p-values. The empirical distribution plots for these p-values show uniform distribution like patterns (results not shown). Therefore, the hypothesis testing p-values can be appropriately calculated by the parametric bootstrap procedure.1 = 0.3, π2 = 0.5,  π3 = 0.2 and β10 = 6.0,  β20 = 3.6, β30 = 1, 1, β1 = 2.0. The choice of the component proportions are the same as above and the choice of β10, β20, β30 was based on setting β1 = 2.0 and using p1 = 0.002, p2 = 0.02, p3 = 0.2 as above (consistent with a previous simulation). For estimation performance evaluation, we simulated 1000 times with 10,000 samples each with no censoring (NC), censoring at 570, 937, and 1309 respectively . For estimation performance evaluation, we simulated 1000 times with 10,000 samples each with no censoring (NC), censoring at 1241, 895 and 549  decreases with the increasing censoring value. The RMSE also decreases as the artificial censoring value for observations increases. The median gets closer to the true value with the increasing censoring value of the observations. The inter-quartile range gets narrower with increasing sample size and the censoring value of the observations as well. Overall, both simulation studies support the use of right censoring in mixture GLM.p-values.  Additional file Furthermore, we have evaluated the effect of censoring on the testing performance. A data set can be simulated as above and we can test the hypothesis of 3 components vs. 4 based on 100 parametric bootstrap p-values. These steps can be repeated 200 times for the first simulation scenario or 100 times for the second simulation scenario  to generate the empirical distribution of the We considered an application to the DNA-seq data from Chiang et al. . The dathttp://bioinfo-out.curie.fr/projects/freec .) Notice that, as the GC-content data were based on a reference genome, the model coefficient for GC-content was first estimated based on the normal sample data and then fixed in the model for tumor sample data . This approach (pseudo-maximum likelihood estimation) was based on the work by Gong and Samaniego [A 4-component mixture of right censored geometric distribution based generalized linear models was estimated for the normal sample and then for the tumor sample Table . GC-contamaniego .  was calculated by:Fj was the CDF based on fj given in Eq.  have been widely considered in practice. The purpose is to make the (transformed) data more similar to normal distributions. Both rank-based INTs and non-rank based INTs can be considered –26. We cn in Eq. . Then, aThe comparison between the distance data (log2 of distance+ 1) and the related INT values  is shown in Fig. p-value calculation [Then, for the INT values related to these artificially right censored distance data, we simply assign a value 10 /2), where m is the number of bins .For a change-point analysis of the above selected region (as an illustration), we considered a recursive combination approach . For eachttp://www.genome.ucsc.edu/cgi-bin/hgTracks) shows that the cytobands p13.2, p13.3, p21.1, p21.2, p21.3 and p22.1 are related to this detected region. To understand biological significance of this detected deletion, a literature search shows that this region has been investigated by many studies on the relationship between these cytobands and breast cancer [The change-point analysis results are shown in Fig. t cancer , 31.Fig.Count-type data analysis is a widely used approach for DNA-seq data. It requires either a fixed or a sliding window procedure for generating count-type data, and useful information can be reduced after this data pre-processing. In this study, we proposed to analyze DNA-seq data based on the related distance-type measure. Our experimental data based simulation study confirmed the advantages of distance-type measure approach in both detection power and detection accuracy. Furthermore, we proposed a mixture of generalized linear models for analyzing the recent DNA-seq data. Our model was based on the right censored geometric distribution. According to the mathematical details provided in the Additional file Similar as the exponential distribution (for continuous measurements), which is widely used to model waiting time processes, the geometric distribution (for discrete measurements) is also a memoryless probability distribution. The artificial right-censoring has been proposed to overcome the difficult in modeling extreme distance values.  Due to different biological properties in different genomic regions, which results in different distance distributions, different geometric components are necessary to model the data. As one geometric distribution does not well fit the distance data, a mixture of right-censored geometric distributions has been proposed. To understand the connection between a probability distribution component and its related underling biological properties, we need further statistical and biological investigations, which will be an interesting topic for our future research.In our application, interesting change-points were observed from the tumor cell line data. The impact from GC-content was considered in our analysis. In this study, we chose a set of relatively simple DNA-seq data for the purpose of illustration of our approach. Our approach can also be applied to paired-end DNA-seq data. In practice, either paired-end reads or single-end reads can be considered in sequencing experiments. If only the starting position of each paired-end read is considered, then the analysis scenario is equivalent to the analysis of single-end reads. However, this strategy may not well utilize the advantage of paired-end reads.  To consider the distance based approach and to utilize the advantage of paired-end reads, we will need to investigate which modifications or extensions are necessary to our method proposed in this study. This will be an interesting topic for our future research.With the next generation sequencing technology, a large amount of short DNA sequence reads can be obtained at a genome-wide level. DNA-seq data have been increasingly used in different areas of biomedical studies. Although the count-type based approaches have been widely used for DNA-seq data analysis, a moving window based data pre-processing (or similar) is required with a pre-specified window size. Useful information can be reduced after the data pre-processing. In this study, we have developed a distance-type measure based method. Distances are measured in base pairs (bps) between two adjacent alignments of short reads mapped to a reference genome. This is a novel approach. Its advantages have been demonstrated by our simulation studies and its practical usefulness has been illustrated by an experimental data application.Additional file 1:Mathematical details. (PDF 271 kb)Additional file 2:Supplemental figures. (PDF 1697 kb)


The convergence results are illustrated with specific examples of expansions in q-Fourier–Bessel series.We study Fourier–Bessel series on a  In the following we will use the standard notation for the q-calculus . The third Jackson q-Bessel function or, for some authors, the Hahn–Exton q-Bessel function, is defined asq-analogues of the Bessel function defined by Jackson. We will often drop the superscript for notational convenience and simply write q-integral.However, for a certain l follow  for basil follow  for q-caulus and  for q-Be,3, from  is used own fact , 42 thatq-linear grid was developed, using a q-analogue of the exponential function and the corresponding q-trigonometric functions introduced by Exton  to denote the greatest integer which does not exceed x will be adopted.Now we will consider the sumFor a given sequence m it can be proved thatUsing induction on q-linear grid associated with f is defined as the sumThe Fourier–Bessel series on a q-Fourier–Bessel series but also other Fourier systems based on discrete orthogonality relations, as in [With a view to studying pointwise convergence of the series  when \domentclasspt{minimas, as in –24. TherLet Let Let given in , satisfiq-integral  in \docfined in  convergection of .q-Sturm–Liouville systems was developed. In particular, it was shown that the set of all normalized eigenfunctions forms an orthonormal basis for q-Sturm–Liouville systems are pointwise convergent.In  a rigoroBy  and 4.24.2 one mgiven by ,6.1\docun in the variable x andFor n. Consider the basic difference relation (2.12) of , p. 693\n and j are nonnegative integers, relation .For C, independent of k and n, such thatk and n. Using Theorem A we may write, for k large enough,One must show that there exists (i) When ary 3 of ,\documen (12) in , p. 1205ary 2 of , the fun(ii) Now, let us consider q-shifted factorial q-Bessel function and its derivative.The asymptotic behavior of stigated , Lemma 1q)∞ from  with theq)∞ from . We provk,For large values of p, one can resort to (We will present only the main steps of the proof. Computing the derivative of the function esort to –2.8) to to\documo Jacobi , p. 500 Notice that the above lemma implies that q-integral  upper bound for the q-Fourier–Bessel series.We are now in conditions to prove our main result. The beef of the proof consists of finding an q-Hölder type inequality [q-integrals appearing on the right side of ,,6.22),  and 6.2 into ine term in  can be b2),using . To deal;q).From  it follok→∞ and –4.3) gi gi\documudied in , §18.22 q-binomial Theorem [g is a q-analogue of g in uniform convergent q-Fourier–Bessel series. Setting g is q-linear Hölder of order q-Fourier series Consider mentclass2pt{minimerefore, –4.3) en en\docum


Triacylglycerol (TAG) from photosynthetic microalgae is a sustainable feedstock for biodiesel production. Physiological stress triggers microalgal TAG accumulation. However excessive physiological stress will impair the photosynthesis system seriously thus decreasing TAG productivity because of the low biomass production. Hence, it is critical to quantitatively and timely monitor the degree of the stress while the microalgal cells growing so that the optimal TAG productivity can be obtained.F/Fm′, which is the chlorophyll fluorescence parameter that reflects the effective capability of PS II, was identified to be a critical factor to indicate the degree of stress of the cells. In addition, the concept of a nitrogen stress index has been defined to quantify the degree of stress. Based on this index and by monitoring ΔF/Fm′ and guiding the supply of nitrogen in culture medium to maintain a stable degree of stress, a stable and efficient semi-continuous process for TAG production has been established.The lack of an on-line monitored indicator has limited our ability to gain knowledge of cellular “health status” information regarding high TAG productivity. Therefore, to monitor the degree of nitrogen stress of the cells, we investigated the correlation between the photosynthetic system II (PS II) quantum yield and the degree of stress based on the high relevancy between photosynthetic reduction and nitrogen stress-induced TAG accumulation in microalgal cells. ΔF/Fm′ indicator will have a significant impact on microalgal TAG production, especially for the outdoor controllable cultivation of microalgae on a large scale.The results indicate that the semi-continuous cultivation process with a controlled degree of stress by monitoring the Δ Microiodiesel . Physioliodiesel .2 fixation by degradation of ribulose-1,5-bisphosphate (RuBP) carboxylase/oxygenase (Rubisco) for the recycling of nitrogen. Limitation of CO2 fixation then decreases the consumption of ATP and NADPH and leads to an excess of NADPH and electrons [N-stress is the most effective means to trigger TAG accumulation , 6, 7. Wlectrons . The suflectrons , whereaslectrons –13. In alectrons , 15. ThePhotosynthesis is a coordinated physiological process that exclusively provides both energy and the material foundation for photoautotrophic microalgae . TherefoTetraselmis subcordiformis and Isochrysis zhangjiangensis, critical values of chlorophyll fluorescence existed where the maximum productivities of energy storage substances, such as carbohydrates, were obtained [Nannochloropsis oceanica IMET1, we established a reproducible TAG production process by monitoring photosynthetic activity to maintain a constant degree of N-stress under semi-continuous cultivation.Chlorophyll fluorescence analysis is a powerful tool for the study of photosynthesis in both plants and algae –19. It aobtained , 23. TheNannochloropsis oceanica IMET1 from the University of Maryland Biotechnology Institute was cultivated in seawater with modified F/2 medium at the ambient temperature of 25 ± 1 °C. In the present study, a 500-mL bubble column bioreactor (5 cm in diameter) was used with the air flow at 100 mL/min  filtered by a 0.22-μm membrane as described by Pan et al. [2 s cool white fluorescent lights were provided from one side of the bioreactor under a 14-h/10-h light/dark cycle. After pre-culture in the bioreactor with sufficient nutrient elements until the cells reached exponential growth phase, the cells were inoculated into new bioreactors at an initial biomass concentration of ~ 0.18 mg/mL .n et al. . AdditioF/Fm′ was monitored. During the semi-continuous, ΔF/Fm′ was used as an indicator of the degree of N-stress. When ΔF/Fm′ reached the predetermined level, which was determined in batch cultivation as the monitoring point, harvest and dilution were conducted to start a new cycle of cultivation. During the dilution, the relative amount of fresh medium replenished to the system was defined as the “renewal rate”. For each cycle during the same semi-continuous cultivation process, the initial biomass concentration and supplementary N amount were the same.For the batch culture, the initial N amount was 15 mg/L in the medium, and no additional N was added into the medium until harvest. For the semi-continuous cultivation, the inoculum was the N-stressed cells from the batch. During cultivation, the fluorescence parameter Δ4HCO3 and were then dried at 60 °C to achieve a constant weight. The DW was the difference between the final weight and the weight of the empty tube as described by Zhu et al. [F/Fm′ was measured using a chlorophyll fluorometer  by the method described by Yao et al. [Fm′ and F′ were obtained, and Fm′ refers to the maximum fluorescence at the light-adapted state. The effective photosynthesis capability of PS II, ΔF/Fm′, was calculated according to Eq. . The fatty acid composition was determined by gas chromatography after transesterification and according to Liu et al. [i and CiTAG− represent the DW and TAG content at the end of cycle i, and the symbols with the subscript of cycle 0 represent the corresponding data of the initial state of cycle i.The TAG content was calculated using the method developed by Wang and Shen and based on the concept of characteristic fatty acid, which provided a way to quantify TAGs in microalgae precisely by immediate transesterification from the wet biomass and required smaller samples than conventional methods , 28. Sheu et al. . TAG proF/Fm′. As shown in Fig. F/Fm′, which reflects the effective photosynthesis capability of PS II, decreased consistently with changes in intracellular N from 0.56 to 0.42. As shown in Fig. The N-stress indicator should be automatically and rapidly measured. It should be sensitive to N-stress, including the content of intracellular N and other parameters of physiology or biochemistry, such as the metabolite content, elemental content, photosynthesis activity, and respiration rate. Therefore, during cell growth and TAG accumulation under N-stress, the physiological characteristics and cellular biochemical properties were investigated including the DW, TAG content, intracellular N content and content of N in the medium and the chlorophyll fluorescence parameter ΔF/Fm′ is a promising indicator of N-stress in microalgae due to the high correlation between intracellular N and ΔF/Fm′. By investigating the growth of N. oceanica IMET1 under N-stress, it was found that during the period of day 3 to day 6 when the highest TAG productivity under N-stress was obtained, there was a high correlation between intracellular N and ΔF/Fm′. The correlation coefficient between intracellular N and ΔF/Fm′ reached 0.97 . In the other species of microalgae, a correlation also exists between intracellular N and ΔF/Fm′. For T. subcordiformis and I. zhangjiangensis, the correlations were y = 0.0741x + 0.151 (R2 = 0.9779) and y = 0.168x + 0.1597 (R2 = 0.9302), respectively (unpublished data). In short, it is not a special case where the ΔF/Fm′ and intracellular N synchronously change while cells are under the N-stress condition. The great advantages of ΔF/Fm′ are its characteristic of in situ and rapid measurement. In addition, the photosystem (whose activity ΔF/Fm′ reflects) plays the role of the hub in microalgal metabolism. Therefore, we propose monitoring the degree of N-stress using ΔF/Fm′ as an on-line indicator and then making the unsteady status of high TAG productivity reproducible to realize continuous TAG production.When comparing all of the parameters, intracellular N is a critical parameter to trigger the initiation of TAG accumulation. There were studies that made efforts on the determining the proper N-stress and emphasized the importance of accurate control of intracellular N when to optimize the lipid productivity , 31. HowF/Fm′ stayed at a relatively high level, above 63 mg/mL day and 0.45, which was 78% of the maximum value of 0.58, respectively, as shown in Fig. Nmax is the maximum intracellular N content of the cells under N-replete conditions, Nmin refers to the minimum intracellular N for the cells to only keep alive subjected to severe N-stress for a long time, and Ni is the intracellular N of the cells at day i. Therefore, NSI ranges from 0 to 1 and a greater value of NSI indicates a higher degree of N-stress of the cells. Then, the NSIs of the day3-cell, day4-cell and day5-cell were 0.62, 0.72 and 0.81, respectively, and represent aggravated N-stress of the cells.F/Fm′ was used as an N-stress indicator in situ. The cultivation of the day4-cell is used as an example to describe in detail the actual operation. The ΔF/Fm′ of the day4-cell was 0.503 at the inoculation, and this value was set as the critical point of monitoring. After supplying N into the medium and culturing for a few days, when the ΔF/Fm′ reached the same level as the preset critical point at inoculation, the cultivation was ended. Specifically, as shown in Fig. F/Fm′ reached 0.507, which was the same level as the initial ΔF/Fm′, that is, 0.503 for the day4-cell, the cultivation was ended. The operations of the day3-cell and day5-cell were the same as those of the day4-cell, and actual changes in ΔF/Fm′ can be seen in Fig. F/Fm′. In addition, from the results of the TAG productivity obtained by cultivation of the 3 types of N-stressed cells shown in Fig. F/Fm′ was higher and indicated a faster growth rate than the other two kinds of cells, the TAG accumulation was lower because the cells encountered a lower degree of N-stress, which ultimately resulted in less TAG productivity than the day4-cell. The result of the day3-cell indicated that the NSI of 0.62 was not large enough. Different from the day3-cell, a lower growth rate of day5-cells led to a lower TAG productivity. In contrast to the day3-cell, day5-cells encountered excessive N-stress with a large NSI of 0.81, which resulted in too much impairment of the cell growth. Therefore, according to the above results, the day4-cell experienced the proper degree of N-stress with an NSI of 0.72, and the corresponding ΔF/Fm′ of 0.503 could be considered the critical monitoring point.High TAG productivity is highly related to the degree of N-stress, and determination of the proper degree of N-stress of microalgal cells is required. The proper degree of N-stress could not only induce TAG accumulation but also maintain the cells at a relatively high photosynthesis capability for growth and thus ensure the balance of the contradiction between TAG accumulation and fast growth. From day 3 to day 5, as shown in Fig. F/Fm′ as an N-stress indicator, a semi-continuous cultivation of the proper N-stressed day4-cell was established by controlling the N supplying strategy. The initial N concentration in the medium and the renewal rate were optimized as the key operating parameters to improve the preliminary established semi-continuous process since the two operating parameters were reported to greatly affect the specific growth rate of the cells and productivity of the target products in a semi-continuous cultivation [F/Fm′ was monitored daily to guide semi-continuous recycling, and the critical point of monitoring for ΔF/Fm′ was 0.503. When the ΔF/Fm′ reached the preset critical point of monitoring, a new cycle was restarted. The harvest and additional proportion of medium were the same and were operated according to the renewal rate in Table The proper degree of N-stress has been disclosed for TAG stable production. Using Δtivation –34. The To keep the culture conditions consistent with that of the batch cultivation in “F/Fm' recovered to the peak of 0.55 and 0.52 during each cycle in experiment b and c after N replenishment and recovered to 0.58 in experiment a. Obviously, resupplying the N amount has a large effect on the level of photosynthetic activity recovery. In experiments b and c, the cells did not recover to the health status as in experiment a, leading to the lower cell growth and lower TAG productivity.Experiments b and c were performed to investigate the effects of the initial N concentration in medium on TAG productivity. As shown in Fig. Another critical operating factor was the renewal rate of the semi-continuous cultivation mode . In thisCompared to the healthy cells with sufficient nutrition, there was an obvious reduction in the photosynthesis activity of the proper N-stressed cells as shown in Fig. F/Fm′ is a rapid and in situ N-stress indicator and could precisely guide the control of the degree of N-stress during the cultivation of microalgae for efficient and stable TAG production.Taken together, the data indicate that ΔF/Fm', was identified as a perfect online indicator for the degree of N-stress of the cells. Based on NSI, a novel semi-continuous cultivation strategy of precise N-stress control by the accurate monitoring of ΔF/Fm' as an N-stress indicator was established for stable and efficient TAG production. The renewal rate and initial N concentration were optimized as 0.6 and 24 mg/mL, respectively. This new cultivation strategy provides significant guidance for outdoor microaglal cultivation in industrial applications and controlled indoor cultivation.It is vital to quantify the degree of N-stress when it is applied to obtain maximum TAG productivity during microalgal cultivation. The concept of the nitrogen stress index, NSI, was defined. The photosynthetic activity parameter of microalgae, Δ


Scientific Reports 10.1038/s41598-018-37641-0, published online 06 February 2019Correction to: The original version of this Article contained an error in the title of the paper, where “catechinopyranocyanidins” was incorrectly given as “catechiopyranocyanidins”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supplementary Information file.


Nature Communications; 10.1038/s41467-018-05327-w; published online 06 August 2018Correction to: y-axis labels ‘tDNA’ and ‘ctDNA’ were inadvertently inverted. This has been corrected in the PDF and HTML versions of the Article.The original version of this Article contained an error in Fig. 2, in which the left


Statistical Inference of Large-scale Gaussian Graphical Model) that includes four main approaches in statistical inference of high-dimensional GGM. Unlike the existing tools, SILGGM provides statistically efficient inference on both individual gene pair and whole-scale gene pairs. It has a novel and consistent false discovery rate (FDR) procedure in all four methodologies. Based on the user-friendly design, it provides outputs compatible with multiple platforms for interactive network visualization. Furthermore, comparisons in simulation illustrate that SILGGM can accelerate the existing MATLAB implementation to several orders of magnitudes and further improve the speed of the already very efficient R package FastGGM. Testing results from the simulated data confirm the validity of all the approaches in SILGGM even in a very large-scale setting with the number of variables or genes to a ten thousand level. We have also applied our package to a novel single-cell RNA-seq data set with pan T cells. The results show that the approaches in SILGGM significantly outperform the conventional ones in a biological sense. The package is freely available via CRAN at https://cran.r-project.org/package=SILGGM.Gene co-expression network analysis is extremely useful in interpreting a complex biological process. The recent droplet-based single-cell technology is able to generate much larger gene expression data routinely with thousands of samples and tens of thousands of genes. To analyze such a large-scale gene-gene network, remarkable progress has been made in rigorous statistical inference of high-dimensional Gaussian graphical model (GGM). These approaches provide a formal confidence interval or a p-value rather than only a single point estimator for conditional dependence of a gene pair and are more desirable for identifying reliable gene networks. To promote their widespread use, we herein introduce an extensive and efficient R package named SILGGM ( PLOS Computational Biology Software paper.This is a Gene co-expression network is an undirected graph, where each node represents a gene and each edge between two genes shows a significant co-expression relationship . It has Previous high-throughput sequencing technologies like microarray and bulk RNA-seq have generated many high-dimensional gene expression data sets with a huge number of genes, but these data sets usually have a small number of subjects or samples. Recently, the emergence of the droplet-based single-cell RNA-seq , 7 has mDue to the assumption of the intrinsically sparse structure of a gene network, two main streams of approaches have been developed in estimating conditional dependence of genes using high-dimensional GGM: (i.) the graphical Lasso, which is a penalized-likelihood approach for precision matrix of GGM –10 and  that includes B_NW_SL, D-S_NW_SL, D-S_GL and GFC_SL or GFC_L. SILGGM has significantly increased the efficiency of each approach using fast algorithms, the “Rcpp” library [To enhance the influence of these cutting-edge statistical inference works in practical usage and address the computational challenge in high-dimensional settings even with large sample sizes, we develop a more comprehensive package called SILGGM ′ follows a multivariate normal distribution with mean μ  and covariance matrix Σ. The conditional dependence between each pair of variables is reflected in a precision matrix Ω = (ωij)p×p = Σ−1, the inverse of Σ. For instance, if Xi and Xj are conditionally dependent, then equivalently, the corresponding element in Ω is ωij ≠ 0 [Xi as the ith gene. Therefore, the inference between gene i and j is equivalent to the inference of an individual ωij, and the global inference of whole-scale gene pairs is based on a multiple testing procedure with all ωij’s.In GGM, a set of  ωij ≠ 0 . In the p (the number of genes) allowed to be far larger than n (the number of subjects). The SILGGM package has one main function SILGGM with various arguments and its workflow is described in We focus on the high-dimensional settings with n by p gene expression data matrix as an input. The gene names can be specified in each column by users. Without loss of generality, the data matrix is further centralized by subtracting its mean or standardized by subtracting the mean and adjusting the variance to one before the formal statistical inference, but the final results are returned in an original scale.The setup of the SILGGM function is very simple. It only takes an ωij, while the last one is proposed particularly for simultaneous inference of all ωij’s. All of the four methods (see more details in O(p) or O(sp) runs of nodewise Lasso-type regressions are conducted among the other three methods. The second step is to obtain (p2 − p)/2 test statistics: (i.) the estimators The method argument in the function SILGGM supports four approaches in rigorous statistical inference: B_NW_SL, D-S_NW_SL, D-S_GL, GFC_SL or GFC_L. In the original four papers, the first three methods are developed for inference of each individual ωij’s, so the implementations of the other three methods can also be extended to global inference under its FDR framework [α-level sequence with α = 0.05,0.1 is pre-specified by the alpha argument in the function and it can be customized by users with different values.As it can be seen, GFC_SL or GFC_L essentially relies on asymptotically normal test statistics for testing on ramework  that hasi and j, SILGGM not only provides the estimator i and j is encoded in the th element of a p by p symmetric matrix with diagonal elements equal to 0. For global inference with a pre-specified α-level sequence, SILGGM further returns the estimated FDR sequence based on p by p adjacency matrices with each off-diagonal element value of 1 = conditionally dependent or 0 = conditionally independent). If the true structure of a gene network is available , SILGGM also includes the estimated power sequence with respect to the estimated FDR sequence. Users can input the true structure in a matrix format via the true_graph argument in the SILGGM function.Outputs are shown with the different type of inference. For individual inference of gene n = 258 asthmatic children [p = 1953 genes with the largest inter-sample variance  and obtain the corresponding network visualization shown in In addition to present the above outputs from both individual and global inference in a matrix format, the function SILGGM provides the cytoscape_format argument as an alternative to show them in a table format that can be saved as a .csv file by using the csv_save argument to a directory specified by the directory argument. The .csv file is compatible with multiple popular platforms for network visualization. In order to show the validity of this alternative, we have applied SILGGM to a public gene expression microarray data set on the lymphoblastoid cells of children , 30 and o asthma . The firComputational efficiency is a prominent advantage of SILGGM. The core algorithms in the package are developed with the “Rcpp” library  which hiIn the first step, based on the same optimization in FastGGM , we pre-ested in ) are appested in  which isested in . To condested in  due to tested in . In addiested in  to boostO(p3) with naïve calculation (see (7) and (10) in O(sp2), and s is usually much smaller than p in high-dimensional settings.In the second step, we facilitate inner product operations to derive each de-sparsified test statistic. To be more specific, we consider the sparsity of Lasso-type estimators from the first step and make inner product calculations only on the non-zero elements. For D-S_NW_SL or D-S_GL, to obtain In addition to the aforementioned optimizations, we optimize the inner product operations between the whole data matrix and regression coefficients by considering the sparsity of estimated coefficients when solving each scaled Lasso problem. The idea behind the optimization is same as the one used in the second step, and it reduces the redundant steps and enables B_NW_SL to perform even faster in SILGGM than in FastGGM.We illustrate the efficiency of SILGGM through simulation studies and a real data analysis. In simulation, we consider four popular graph structures for gene network studies: Band graph, Hub graph, Erdös–Rényi (E-R) random graph and Scale-free random graph, as shown in http://math.sjtu.edu.cn/faculty/weidongl/Publication/code.rar in n = 100 and simulate three types of graph settings: Band, Hub and E-R  over 100 replications on a single CPU are recorded for GFC_L with FDR control at the 0.1 and the 0.2 levels using SILGGM and the MATLAB code, as shown in To the best of our knowledge, the online MATLAB code  times p, is around 4 or 5. The first column of Then, we evaluate the timing performance of B_NW_SL using SILGGM compared to the current package FastGGM. As shown in n = 800, p = 5000 and n = 800, p = 10000).In addition to the time evaluation, we validate the accuracy of estimation results from all the approaches for both individual and global inference in the very large-scale settings with relatively small sample sizes th gene pair (H0:ωij = 0 vs. H1:ωij ≠ 0) in terms of the estimation for an entire graph. The empirical Type I error  under a pre-specified level of 0.05 for p-values and the corresponding Type II error  are measured for Band graph  and the “zero partial correlation” set  respectively. Since GFC_SL or GFC_L provides no confidence intervals, we involve the other three approaches here. According to the results from the three graph settings, the overall performance of the confidence intervals among B_NW_SL, D-S_NW_SL and D-S_GL are good in terms of the entire graph structure. But in terms of the confidence intervals for the non-zero partial correlations, B_NW_SL and D-S_NW_SL outperform D-S_GL. Moreover, the performance of B_NW_SL is more stable than that of D-S_NW_SL in the different settings , E-R grribed in ) and ScaH0:ωij = 0 vs. H1:ωij ≠ 0 with 1 ≤ i < j ≤ p simultaneously. Therefore, to make global inference of all gene pairs in a large graph, we always recommend controlling FDR to avoid the inflation of false positives. The testing results from the three graph settings indicate that the FDRs of all the methods are effectively controlled below the desired level for both p = 5000 and p = 10000. The corresponding power values  and the Matthews correlation coefficients (MCCs) also demonstrate comparably good performance of all the methods  used in individual inference. Unlike individual inference, global inference generally requires a multiple testing procedure for tests on all n = 3555 cells. After filtering out the unexpressed genes, we consider p = 2000 genes with the largest inter-sample variance  counts , we needp(m) ∝ mλ− for some positive λ. Here, m refers to the node degree and p(m) denotes the probability of the m-degree nodes. Many studies have indicated that biological networks are scale-free, and the node degrees possess a power-law distribution [n > p in this data set, the values of n and p share the same order such that MLE becomes unstable and increases bias of estimation. Thus, all the inferred network structures by SILGGM are biologically meaningful and much more reliable. Furthermore, we can see that the performance of the other three methods based on the nodewise Lasso-type regressions in SILGGM is even better than that of D-S_GL since the plot of D-S_GL shows some noise in the tail.Motivated by , we applribution –45. The https://cran.r-project.org/package=SILGGM. The details of package installation are described in The source code of the package and a complete reference manual including dependencies, usage of all package functions and associated examples are freely available via CRAN at The package SILGGM is computationally efficient compared to the MATLAB implementation of GFC_L and the R package FastGGM. Since R is a publicly free platform and has been more widely used in biological research compared to MATLAB which is a piece of commercially licensed software and has less accessibility to biologists, the R platform-based SILGGM will play a more important role in accelerating the biological gene network studies. SILGGM is also statistically efficient with both individual and global inference due to the theoretical justification of the four approaches and the validation of estimation accuracy in simulation studies. The analytical results from the single-cell data with Pan T cells further reflect the statistical efficiency of SILGGM since inferred gene networks are more reliable. Moreover, the comprehensiveness of SILGGM allows users to have more flexible choices of methods depending on the specific purpose of their study. Due to its computational feasibility, analytical reliability in results and methodological comprehensiveness, SILGGM can become a valuable and powerful tool to a wide range of biological researchers for high-dimensional or even whole genome-wide co-expression network analysis.n is small (e.g. n = 100) because the tuning selection in GFC_L is beneficial for FDR control. When n becomes larger (e.g. n = 800) but may be still relatively small to p, users can choose any of the four approaches due to their similar performance among the different settings Click here for additional data file.S2 Appendix(RAR)Click here for additional data file.S3 Appendix(PDF)Click here for additional data file.S1 File(PDF)Click here for additional data file.S2 File(PDF)Click here for additional data file.S3 File(PDF)Click here for additional data file.S1 Table(PDF)Click here for additional data file.


A. In the classical case, this leads to a bibasic extension of Heine’s formula, which was implicit in a paper of Andrews which he wrote in 1966. As special cases, we recover extensions of many of Ramanujan’s q-Lauricella function, and show how to obtain very general transformation formulas of this type. The results obtained include transformations of an n-fold sum into an m-fold sum.We apply Heine’s method—the key idea Heine used in 1846 to derive his famous transformation formula for  Heine’s method, so named by Andrews and Berndt , was whaented in , Ch. 1. q-rising factorials from Gasper and Rahman [q-rising factorial is defined as k is a positive integer,q is called the ‘base.’ The infinite q-rising factorial is defined, for q-rising factorials when k is an arbitrary complex number. The most fundamental of the q-summation theorems is the q-binomial theorem [q-Binomial theorem, and is as follows.h and t are complex numbers. This leads to a transformation formula that follows from a very general identity of Andrews [q-rising factorials to be defined, and w, b) is replaced by .It is useful to recall Heine’s method. We need the notation of d Rahman . The q-r.Heine’s  proof ofentclass1pt{minimaq-binomial theorem, due to Milne [n is called the dimension of the sum. When classical. We use the notation A, namely,n and m are 1:n-dimensional sum into an m-dimensional sum. This is followed by some examples of multiple series extensions of Ramanujan’s transformations in Sect. The second variation of Heine’s idea comes from using multiple series extensions of the to Milne , Milne ato Milne , Gustafsto Milne , Kajiharto Milne , and oneto Milne . The ser case of  below. Centclass1pt{minimaq-binomial theorem. Our intention is to illustrate the approach, not to provide a comprehensive list. There are six multiple q-binomial theorems of this kind that can be combined together to give 21 such results.In this section, we will give four multiple series extensions of , using dz are indeterminate, and are such that the terms in the sum are well defined. We require We begin by using a theorem of Milne and Lilly , Th. 4.7 and  and is vq-binomial theorems in the literature. The following are the product sides of q-binomial theorems due to Milne [q-binomial theorems. Unfortunately, the q-binomial theorem due to the author and Schlosser [We will not provide any further chlosser  is not ah and t are complex numbers, and we have the usual convergence conditions, namely, In this section, we find multiple series extensions of many of Ramanujan’s  done in , Ch. 1. alent to , as we cIn the special cases we consider, many of the products appearing in the sum simplify. Our first set of examples is obtained as special cases of Theorem  [q-binomial theorem. Using the q-analogue of Euler’s transformation formula, we obtain the following transformation formula of double sums.In Heine’s method, instead of the Aside from the usual convergence conditions  with base d, e, f, w) with base The proof is a straightforward extension of the proof of . We use Note that when =e, then  reduces =e, then .There are many sions of . They folas. See , Th. 1.1 The symbols L, P,  and R are code for Left, Products, and Right, respectively. We will use two copies of this result. Let us represent the summands of the corresponding left- hand sides by The proof is similar to the proof to that of Theorem k and j and The first copy of  is a \doWe now see that the left-hand side can be written asq-Lauricella function. Andrews’ formula was generalized to a multibasic formula by Agarwal, Jain, and Choi [We now present another variation, where Heine’s method is iterated, to obtain a transformation formula that transforms a multiple sum to a single sum. This was first done by Andrews [and Choi , and indand Choi , which iq-Lauricella functions [We use the symbol for the dot productmentclass2pt{minimtions in  to whichp-fold sum and the right-hand side is a single sum. The identity has Note that the left-hand side of  is a p-fduces to .p times. The first step of the proof is as follows:q-binomial theorem asq-binomial theorems, such as . , , \documeProperty H as follows. A q-binomial theorem of the form H ifp multi-sums corresponding to the left-hand side of  Let The proof is a straightforward iteration of Heine’s method, and is left to the reader. Remark Recall the results of Sect. H. So we can intermix q-binomial theorems with transformation formulas too.Theorem q-binomial theorems appearing in the work of Milne and Lilly, Gustafson and Krattenthaler, and one implicit in the author’s work with Schlosser.First we take We take  as our bP1) from , and \doP2) from . Now Theq-binomial theorem (nq-binomial theorems to deal with a product of the formIn our second example of an extension of , we take theorem . We take case of  with basentclass1pt{minimaA. It is a continuation of our work [The objective of this paper is to study Heine’s method as applied to series of type our work , which aour work , Ch.  1.q-binomial theorems in [n-dimensional series to a multiple of an m-dimensional series. Early examples of such identities were given by Gessel and Krattenthaler [In Sect. orems in , 23, 24.enthaler , Kajiharenthaler , and Kajenthaler .Previously, Gustafson and Krattenthaler , 13 gaveSection d Berndt , Ch. 1. d Berndt  and Kratd Berndt  (who proentclass1pt{minimaq-Lauricella function given by Agarwal, Jain, and Choi [In Sect. and Choi , and indand Choi . Previouand Choi , §7 gaveq-binomial theorem of Macdonald [q-binomial theorems. The Property H follows from the homogeneity of the Macdonald polynomials. If we apply Heine’s method to Macdonald’s q-binomial theorem, we obtain bibasic Heine transformation formulas different in character from, but perhaps not as deep as, the Heine transformation formula given by Baker and Forrester [We have already noted that the master theorem applies to using transformation formulas as in Sect. acdonald  (see Kanacdonald , Th. 3.5orrester , Th. 2.3