dataesr/scientific-paragraphs-categorization huggingface dataset:
2 BIC. The R package plsdof, based on the work of  Krämer and Sugiyama (2011) , was used to compute this criterion. It works as follows: BIC = RSS /n + log(n)(γ /n) σ 2 , where γ represents the DoF of model (  8 ) and σ 2 is defined by  Krämer and Sugiyama (2011). The selected model is the one which represents the first local minimum of this adapted BIC criterion; related results are referred to by BICdof. Results linked to models obtaining the global minimum are also returned under the acronym BICglob. -In PLSGLR: 1 AIC. The AIC criterion  (Akaike 1974 ) can be computed whatever the distribution involved. However, no corrected DoF have yet been suggested for the PLSGLR framework. 2 BIC. As in the case of AIC, the BIC  (Schwarz 1978)  is calculable without correcting the DoF. 3 CV -MClassed.
Simulation plan To compare these criteria, simulations were performed by adapting the simul_data_UniYX function, available in the R package plsRglm  (Bertrand et al. 2014). First, four orthonormal vectors of size p are built. Let T ∈ M 4× p (R) be the matrix containing them. Then, rows of X are successively obtained using X i• = R i T + i , where R i = (r 1i ,. , r 4i ) ∈ R 4 is a vector of random draws from N 0, σ j , j = 1,. , 4 respectively, with (σ 1 , σ 2 , σ 3 , σ 4 ) = (10, 8, 6, σ 4 ), and i is drawn from N (0, 10 -2 ). Only the first three orthonormal vectors are linked to the simulated response, so that σ 4 varies during the simulations in order to understand the impact of increasing variability of this uninformative fourth component on the various criteria. Different processes were used to simulate response vectors, depending on the desired distribution.
As a constant in all simulation schemes, the first three orthonormal vectors are involved, so that whatever the framework, simulations are performed to obtain a relevant subspace of dimension 3. Also, a noise parameter in y helps us to determine the robustness of the criteria being examined with respect to increasing values of σ 5 , which characterizes its standard deviation. Fixed sets of values for σ 4 and σ 5 are given, depending on the framework, and described in the corresponding sections. For more details about these simulation processes, see Supplementary Materials   [25, 50] ]. For each fixed pair (σ 4 , σ 5 ), which represents the standard deviation of the uninformative fourth component in X and the additional random noise standard deviation in y, respectively, we simulated 100 datasets.
Each dataset is based on p l covariates, 1 l 100. The p l numbers are obtained by sampling with replacement in Ω n. Testing these criteria on 100 different datasets allows us to calculate a mean value of the number of components for each fixed couple (σ 4 , σ 5 ) as well as an estimated variance that represents the inherent stability of the various criteria. Lastly, the number of bootstrap replicates was fixed at R = 500 and CI were constructed by setting α = 0. 05. More indepth details of the data simulation framework is available in Supplementary Materials 3. The aim of the simulations is twofold. First, a comparison of the chosen criteria, both through their results for the number of components and their robustness against different random noise variances. Second, predictive abilities are compared using predictive normalized mean squared errors (PNMSE), calculated on 80 additional simulated samples per dataset in the n < p framework, used as test sets.
Normalization is performed by dividing the predictive mean squared errors (PMSE) related to the obtained model by the PMSE linked to the trivial one (constant model equal to the mean of the training data). Furthermore, as mentioned in  Krämer and Sugiyama (2011, p. 702) , "the large test sample size ensures a reliable estimation of the test error. " Then, for each pair of values (σ 4 , σ 5 ), asymptotic t-tests with Welch-Satterthwaite DoF approximation  (Welch 1947 ) are performed to compare the PNMSE averages over the 100 simulated datasets related to each criterion. All tests have been run at level α = 0. 05.
Initial selection To select the best method, both between the Q2lv1o and the Q2K5 criteria, and between the BICdof and BICglob ones, results related to datasets with n > p, for the following sets of values for noise standard deviations (NSD), are considered: (A) : σ 4 ∈ {0. 01, 0. 21,. , 5. 81} σ 5 ∈ {0. 01, 0. 51,. , 20. 01}. The averages of the selected numbers of components over the 100 simulated datasets per couple are calculated. These averages, denoted by nb_comp and related to the BIC and Q 2 criteria, are presented graphically in Figs. 1  and 2  respectively as functions of σ 4 and σ 5 , respectively denoted sigma4 and sigma5. Based on results shown in Fig. 1 , BICglob has stability issues. We observe that this is mainly due to the adapted DoF not necessarily increasing as the number of components rises.
2009, p. 243) , the Q2K5 criterion is retained here for further comparisons. The means of number of components over the 100 simulated datasets per pair (σ 4 , σ 5 ) are displayed for the three criteria in Fig. 3. Variances of these numbers of components over the 100 simulated datasets per pair were also estimated, and are shown using boxplots in Fig. 4. Note that these variances approximate the inter-dataset variability for fixed values of σ 4 and σ 5 , not the intra-dataset one. In these results, we see that the Q2K5 criterion is the least robust against increasing noise variability in y, characterized by increasing values of σ 5 (sigma5). This lack of robustness leads it to globally underestimate the number of components. BICdof has a low computational requirements and is also the most robust against increasing values of σ 5.
86. 37 % of all its selected numbers of components are equal to three or four. However, as seen in Fig. 4 , the BICdof features the highest global variability in number of components selected over the 100 datasets involved per pair (σ 4 , σ 5 ). This is even more acute for datasets characterized by a fourth component standard deviation that is higher than that involved in the relevant subspace, i. e. , σ 4 > 3 i=1 σ 2 i = √ 200 14. 14. ( 10 ) In this particular case, our new bootstrap-based criterion retains stability, while the median of the BICdof results, for instance, more than triples (0. 25 to 0. 79) compared to that of the whole data. Moreover, BootYT is the most robust against increasing variability of the uninformative fourth component in X. As a preliminary conclusion based on these initial results, advising the use of a certain choice among the BICdof or BootYT criteria is not relevant in the n > p case.
PLSR: the n < p case As suggested by  Krämer and Sugiyama (2011) , a small training sample size allows us to consider high-dimensional settings. Averages of numbers of components over the 100 datasets per pair (σ 4 , σ 5 ) are displayed in Fig. 5. Graphical representations of variances are also shown in Fig. 6. In Fig. 5 , we see that the BICdof appears to suffer from overfitting issues. Moreover, based on the results in Fig. 6 , it returns results linked to out-of-range values of the variance, compared with the other two criteria. These two issues are mainly due to the extraction of 1678 (5. 8 %) results equal to 19 components, whereas 26184 (91. 2 %) trials give four or less components. By more carefully analyzing this phenomenon, it appears that the rate of 19 components is a globally decreasing function of σ 5.
PLSGLR results In this section, results related to the comparison between the bootstrap-based criterion and four other criteria (AIC, BIC, CV-MClassed and p_val, see Sect. 3. 1) are presented. Note that, in this framework, due to specific distributions linked to y and the link-functions g used, an increase in σ 5 does not lead to a linear increase of noise variance in y as it does for PLSR simulated datasets. However, the bijectivity of these link functions ensures that a spanned subspace of dimension three is extracted.
PLS-LR: the n > p case The following sets of values for NSD are considered for criteria comparison: (D) : σ 4 ∈ {0. 01, 0. 51,. , 9. 51} σ 5 ∈ {0. 01, 0. 51,. , 15. 51}. Both the means and variances of numbers of components over the 100 datasets per paire (σ 4 , σ 5 ) are displayed in Fig. 8. Based on these, CV-MClassed performs well in estimating the optimal number of components, on average. However, the downside is the higher variances related to its results than those of the others. Therefore, this criterion should be used with caution. The BootYT and p_val criteria return similar results in the n > p case. Both of them slightly underestimate the optimal number of components, but show stability in their results. The uncorrected DoF lead the AIC and BIC criteria to globally overestimate the number of components (Supplementary Material 4).
PLS-LR: the n < p case Here, the following sets of values for NSD are considered for criteria comparison: (E) : σ 4 ∈ {0. 01, 0. 51,. , 9. 51} σ 5 ∈ {0. 01, 0. 51,. , 9. 51}. Both the averages and variances of numbers of components over the 100 datasets per pair (σ 4 , σ 5 ) are displayed in Fig. 9. The CV-MClassed criterion retains both the same property of well estimating, on average, the number of components, and still has the variability issue. Concerning the two other criteria, we observe a greater underestimation issue linked to the p_val criterion than for BootYT. Furthermore, they both feature low variability.
PLS-PR: Row mean analysis In the PLS-PR case, the following sets of values for NSD are considered for respectively the n > p (F) and n < p (G) cases: Averages of number of components over the 100 datasets per pair (σ 4 , σ 5 ), related to the four criteria considered (AIC, BIC, p_val and BootYT), are shown in Fig. 11  for both the n > p and n < p frameworks. Apart from the bootstrap-based criterion, all criteria return an increasing number of components as σ 5 increases. These results lead us to conclude that our new bootstrap-based stopping criterion is the only one which is relevant for Poisson distributions, in that it selects, on average, a decreasing number of components as σ 5 increases. Based on these plots, no additional analyses of the numbers of components was done. Only the two criteria that give results, on average, closest to the expected result, are retained for further comparisons related to MSE, namely the p_val and BootYT criteria.
PLS-PR: MSE analysis First, training log(MSE) were computed using the n < p framework, and their means over all datasets related to each value of σ 5 are shown in Fig. 12. The global decrease in log(MSE) linked to p_val confirms that, as expected by the increasing number of extracted components observed in Sect. 5. 2. 1, this criterion models the random noise in y. In contrast, the bootstrap-based criterion shows a systematic increase in log(MSE), which empirically suggests that it better succeeds in separating the real information from the noise. Variances of PNMSE results over datasets related to each pair (σ 4 , σ 5 ) were computed. Means of these variances, related to fixed values of σ 5 , are displayed in Fig. 13. While results obtained by the bootstrap-based criterion are linked to acceptable variances when σ 5 1.
Illustration of CV issues: first applications on real datasets As mentioned by  Boulesteix (2014) , important issues concerning the stability of the q-fold CV procedure for the choice Fig. 16  Extracted number of components using q-fold CV-MClassed (left) and BootYT (right) criteria of tuning parameters, here the number of components, have been observed. These issues are directly induced by the value of q and by the random character of this resampling-based procedure while splitting the original dataset into two distinct sets, a training one and a test one. To illustrate consequences on the tuning parameter, we treated two real datasets. The first dataset was collected from patients with colon adenocarcinoma. It has 104 observations on 33 binary qualitative explanatory variables, and one binary response variable representing the cancer stage according to the Astler-Coller (AB vs.
CD) classification  (Astler and Coller 1954). This binary response leads us to perform PLS-logistic regressions. This dataset, named aze_compl, is available in the R package plsRglm  (Bertrand et al. 2014). We ran 100 times the selection process for the number of components using the CV-MClassed criterion, with q ∈ {3, 5, 10, 15, 30}. Then, we performed the same process using our new criterion. Results are shown in Fig. 16. Results obtained through q-fold CV, with q = n, are displayed in Fig. 16 , and typical examples for these types of issue. Depending on the choice of q and the way the different folds are split, the extracted number of components can be dramatically different. In addition, obtaining a complete distribution of the number of components is essentially impossible, due to the high number of different possibilities for splitting the original datasets into q groups.
Proposition 2 Let n = pq + r, 0 r q -1 be the Euclidean division of n by q. Then, the number of distinct partitions of the original dataset into r ( p + 1)-elements subsets and (qr ) p-elements subsets for a CV does not depend on the order of their placement, and is equal to: f (n, q) = n! r ! (q -r )! × 1 ( p + 1)! r × 1 p! q-r. (11) Leave-one-out CV, which is the only complete CV ( f (n, n) = 1, i. e. , there is only one way to choose n folds out of n observations), selects one component. However, it suffers from variance issues concerning the bias-variance tradeoff on the estimation of the prediction error  (Hastie et al. 2009; Kohavi 1995). Our new criterion is more stable on this dataset and leads the user to choose the number of components, in this case three, via a more accurate process.
The second example is a benchmark dataset, called "Processionnaire du Pin", which is treated in depth by  Tenenhaus (1998). It has 33 observations each with 10 explanatory variables, and is also available in the R package plsRglm under the name pine. More details on this dataset are available in  Tenenhaus (1998). The same process was applied to this second example, with the usual PLS regressions. Thus, we can compare the Q 2 criterion obtained through q-fold CV, q ∈ {2, 3, 5, 10, 15}, and our new criterion. The Q 2 criterion obtained through leave-one-out CV chooses one significant component. All results are shown in Fig. 17. Here, q-fold CV does not suffer from stability issues as seen before, since the Q 2 criterion is much more stable than the minimization of the number of misclassified values.
However, extracting one component is not recommended. Tenenhaus (1998) , after a complete analysis of this dataset, showed that four components is the best decision. This under-estimating issue linked to the Q 2 criterion confirms the simulation results obtained in Sect. 4. 2. Thus, while the Q 2 criterion under-estimates this optimal number of components, our new bootstrap-based criterion selects four components more than 80 % of the time.
Application on an allelotyping dataset In this section, we focus on an allelotyping study. Our method is applied to a dataset that concerns 267 subjects with colon cancer. Measures were made on 33 microsatellites, in search of an allelic imbalance that indicates an abnormal number of allele copies of a nearby gene of interest. The aim of the study was to find the microsatellite subsets that would best discriminate left and right colon tumors. Thus, the univariate response corresponds to the original location of a colon tumor, leading to a binary response y, taking the value 0 (resp. 1) if it was on the right colon (resp. left). This dataset is available in Supplementary Material 6 and more details are available in  Weber et al. (2007). This dataset contains missing values, so a preprocessing step was performed in order to complete it, using the R package mice.
Thus, performing a single CV to find the number of components using this criterion, must be avoided. As expected, the BootYT criterion returns stable results and selects, in almost 80 % of cases, 6 components. We also tested the robustness of these three criteria through a bootstrap re-sampling process with 100 bootstrap iterations, as well as a jackknife method. These two resam- These results confirm the high resampling robustness of our new criterion compared to CV-MClassed. The p_val criterion has comparable robustness but, based on our simulation results, higher bias. Based on these results and our simulations, we can reasonably conclude that for this dataset, the optimal number of components is 6.
Discussion A new bootstrap-based stopping criterion for PLS component construction was developed, characterized by a high level of stability and robustness with respect to noise level compared to other well-known criteria. Its implementation requires a function integrating the bootstrap process while building the PLS components. Though not yet available online, an equivalent form can be coded using existing R functions, notably from the plsRglm and boot packages. It requires recomputing, for each increment of k, the entire set of PLS components. While this style of implementation performs more operations than needed, it does not take much more time, since the main part of the computational cost comes from the bootstrap process. Indeed, our bootstrap-based criterion has a shortcoming with respect to computational time, which is greater than the other methods since it requires, in the PLSR framework, [(k max + 1) × p l + (K + 1)]× R least squares regressions per dataset.
1b )  11, 19. Labels were propagated along the cell lineage (Supplementary Video 1). This data was suitable to investigate the parameters characterising cell behaviour, including cell displacements, cell divisions, cell volume, cell shape and cell contact changes. Temporal and spatial rescaling of embryo-level dynamics. The quantitative inter-individual comparison of morphogenetic features was first measured at the global embryo level at each time step via the total number of cells N(t), total cellular volume W(t) and total cellular surface area Z(t). Because embryos appeared to grow at different speeds, a spatiotemporal rescaling of the corresponding curves was necessary to establish a baseline for inter-individual comparison and the assessment of variability (Fig. 3a-c ), i. e. to provide comparable charts while preserving the overall shape of the functions.
2b ). Shaded areas: incomplete cycles, for which an accurate estimation was not possible. (j to l) Same as (a to c) with the prototype curves added in black and grey. Due to limits on the observation windows, however (Fig. 2d ), some cell cycles were incompletely or not recorded, hence some groups g were missing cells. In total, there were 252 exploitable distributions, corresponding to 4 cell types across 5 cell generations or less (depending on the dataset) and 4-6 cell features (depending on the cell cycle and the availability of A g and B g ) in each one of the 5 embryos (Fig. 3d-f  and Supplementary Figs  3  and 4 ). Graphical assessment of these histograms, supported by a statistical test (Supplementary Figs  5  and 6 ), led to their classification into normal distributions for cell cycle lengths and mitosis times, and log-normal distributions for average volumes and surfaces.
This allowed us to calculate the mean μ and standard deviation σ of each random variable in each cell group g and provide an idealised parametric representation of the whole cohort's dataset in the form of 252 (μ, σ) pairs. From these statistical measures, we concluded that random variables were largely independent from each other along one cell lineage: we detected no significant correlations among single features (temporal or spatial) of mother, daughter and sister cells (Supplementary Fig. 9  and Supplementary Table  3 ). Moreover, while the average cell cycle lengthened through generations with increasing fluctuations, the cell cycle length of a daughter did not depend on the cell cycle length of its mother. At the spatial level, we observed a variation in the range of ± 20% around 0. 5 for the average daughter/mother volume ratio, which mitigated the common assumption of average cell volume conservation across mitosis through consecutive generations at cleavage stages.
The statistical approach elaborated in this paper to characterise the cell lineage of various individual organisms is general enough to be applied to other species. The cell branching process that gives rise to the cell lineage is observed in many different organisms. Our mathematical formulation is directly applicable to organisms undergoing periods of cleavage during their development, where cells' main activity is to divide. Our MATLAB library is readily available to analyse digital cell lineage trees and produce the corresponding prototypical stochastic cell lineages annotated with cell volume and cell surface measurements.
Digital embryo reconstruction. Reconstructed embryos were obtained by running the raw image datasets through the automated image processing workflow of the BioEmergences platform. A detailed description of the reconstruction methods, including the mathematical algorithms used, was published previously  6  (Supplementary Note 1). The three main components described in the next sections are: nucleus centre detection, cell trajectory tracking, and cell membrane segmentation. Nucleus centre detection and tracking. Cell positions were placed at the local intensity maxima of cell centre images filtered by a difference of Gaussians (Supplementary Note 1. 1). Tracking of the cell nuclei was obtained by first constructing tree branches connecting nearest cells on consecutive images when that proximity criterion was reciprocal.
Then, mitoses were detected by linking the cells left over without a predecessor to their closest neighbour at the previous time step. Finally, the resulting lineage tree was refined via the minimisation of a functional defined by an energy term based on the assumption that the embryo essentially behaves like an elastic mass (i. e. that neighbour cells at time t remain close at time t + 1). This functional was minimised using a simulated annealing heuristic, searching for the lowest-energy configuration by randomly changing a link, removing/adding a link or removing/adding a cell. To validate the tracking, all cell positions and trajectories were checked by at least two experts through visual inspection, comparing the digital reconstruction with the raw data using our Mov-IT visualisation software  6.
All detectable errors (false positive and false negative nuclei, false links, missing links, false positive and false negative mitosis events) were corrected to build error-free lineages. All five lineage trees were fully validated (Supplementary Note 1. 2 and 1. 3). Cell shape segmentation. To remove the noise and smooth the images while faithfully preserving edge information, the data was first filtered by geodesic mean curvature flow (GMCF)  30. Cell shapes were then obtained by applying a generalised version of the subjective surface (SubSurf) method  31, 32  for its ability to reconstruct missing boundaries, making it particularly suitable for this type of data (Supplementary Note 1. 4). Temporal and spatial rescaling. The measured number of cells, total cellular volume and total cellular surface area presented similar patterns of evolution without overlapping.
Acknowledgements This work was supported by a  DIM 2011 Région Paris Ile-de-France scholarship  to P. V. , a  DGA scholarship  to J. D. ,  EC NEST "Adventure" 012916, EC NEST "Measuring the Impossible" 28892, ZF-Health EC  project  HEALTH-F4-2010-242048 ,  France BioImaging infrastructure   ANR-10-INBS-04 ,  ANR BioSys Morphoscale, ANR Regenr and InterDIM 2011 Région Paris Ile-de-France.