 Research Article
 Open Access
Feature selection for highdimensional temporal data
 Michail Tsagris^{1}Email authorView ORCID ID profile,
 Vincenzo Lagani^{1} and
 Ioannis Tsamardinos^{1}
https://doi.org/10.1186/s1285901820237
© The Author(s) 2018
 Received: 17 July 2017
 Accepted: 11 January 2018
 Published: 23 January 2018
Abstract
Background
Feature selection is commonly employed for identifying collectivelypredictive biomarkers and biosignatures; it facilitates the construction of small statistical models that are easier to verify, visualize, and comprehend while providing insight to the human expert. In this work we extend established constrainedbased, featureselection methods to highdimensional “omics” temporal data, where the number of measurements is orders of magnitude larger than the sample size. The extension required the development of conditional independence tests for temporal and/or static variables conditioned on a set of temporal variables.
Results
The algorithm is able to return multiple, equivalent solution subsets of variables, scale to tens of thousands of features, and outperform or be on par with existing methods depending on the analysis task specifics.
Conclusions
The use of this algorithm is suggested for variable selection with highdimensional temporal data.
Keywords
 Time course data
 Longitudinal data
 Regression
 Variable selection
 Multiple solutions
Background
Temporal data measure a set of timevarying quantities over time on a population. They are often employed to understand the dynamics of evolution of a system, the effects of a perturbation (interventional studies), or the differences in dynamics between two groups (such as in casecontrol studies). Such data arise in many fields, namely bioinformatics, medicine, agriculture and econometrics, just to name a few.
Two broad categories of temporal data can be defined, depending on the sampling procedure: longitudinal data arise when the same samples are repeatedly measured at different times points, while time–course (a.k.a repeated crosssectional) data are produced when distinct samples (from the same population) are measured at each time point (e.g., in case of destructive testing). In contrast, timeseries data that often arise in econometrics, measure samples at regular time intervals and are often of a much larger temporal extent than temporal data in biology.
The correlation structure of temporal data, which includes autocorrelation of the same quantity over time or over the same sample requires special analysis techniques. For example, longitudinal data are often modeled with mixed models, which allow to properly account for withinsubject correlations.
Feature selection (a.k.a. variable selection) in predictive modeling can be defined as the task of selecting one or more minimalsize and (collectively) optimally predictive feature subsets for a target outcome. Reducing the number of features results in smaller, easiertoverify, understand, visualize, and apply predictive models; most importantly perhaps, it provides important insight to the data generating mechanism. This is no accident, as feature selection has been theoretically connected to causal discovery and the causal data generating model [1]. A typical example of a feature selection task is the identification of the genes whose expression allows the early diagnosis of a given disease. In the context of temporal data, each feature has a temporal extent and a time trajectory that can be employed for prediction.
To the best of our knowledge, most variable selection methods proposed so far for temporal data are devised for studies where the number of samples is larger than the number of predictors, i.e., p<n. This limits the applicability of these algorithms to “omics” types of data such as transcriptomics, epigenomics and genomics, where p is usually order of magnitudes larger than n.
Constraintbased, MarkovBlanket variableselection methods form a class of algorithms that are inspired by the theory of (Causal) Bayesian Networks [2] and include HITON, MMMB, MMPC, SES and others [3–5]. The Markov Blanket of the target outcome T is defined as a minimalsize set that renders all other variables conditionally independent of T. Under certain broad conditions it has been shown to be the solution to the feature selection problem [1]. If the data distribution can be represented with a faithful Bayesian Network (BN) [6] then the Markov Blanket of T is unique and has an interesting graphical interpretation: it comprises of the neighbors of T (i.e., the parents and children of T) and the spouses (parents or common children) of T in any such (unknown) faithful BN graph.

The target variable is timevarying. In this scenario the task consists of identifying the predictors that are associated with the outcome of interest in the course of time. An example is modeling how a gene expression progresses over time on the basis of other gene expressions. Missing values can occur, or not all subjects may have measurements for all time points (unbalanced design). This case can be further subdivided in two subscenarios: the Temporallongitudinal scenario, the same samples are being studied at all time points (longitudinal data), and the Temporaldistinct scenario, where different samples are being studied for each time points. The latter typically arises when it is impossible to repeat the measurements on the same sample: prototypical examples are animal studies where specimens are killed for collecting internal organs at different time points.

The target variable is a static (nontemporal) variable. In some studies the predictors are measured over time, however the dependent variable is static. An example is the study of geneexpressions differences between two mice groups (target). The task in this case is to identify the minimal set of genes whose trajectories, considered together, allow to best discriminate between the two groups. Also for this scenario we can identify two subcases, namely the Staticlongitudinal scenario, where the same samples are measure over time, and the Staticdistinct scenario, where different samples are considered at each time point.
These scenarios represent the most common designs for biological studies involving temporal data, and are widely applied in other fields as well. Other scenarios/study designs are of course possible (for example measurements might be repeatedly taken for each sample at each time point), however we consider them less relevant and out of the scope of the present paper.
In this paper we use the Statisticaly Equvialent Signatures (SES) algorithm [5, 10] as a prototype for the class of constraintbased algorithms. The predictors selected by SES (signature) are the neighbors of T in any BN (faithfully) representing the data distribution. This is a subset of the full Markov Blanket but it has been shown to be a good approximation for predictive purposes in extensive empirical studies [11]. Some algorithms (HITON, MMMB) do continue in trying to identify the full Markov Blanket which also includes the spouses of T at the expense of computational time. SES can successfully scale up to cases where p>>n, preserving excellent predictive capabilities [5]. We measure the time complexity of the algorithm in terms of the number of performed conditional independence tests. Each variable must be contrasted against each subset of the selected signature before being eliminated. This would require a number of tests in the order of O(p·2^{ s }), where p is the number of variables and s the number of selected variables. However, we only allow conditioning upon maximum k variables at the time, decreasing the complexity of the algorithm to O(p·s^{ k }). This means that the algorithm can still require an exponential number of tests with respect to the size of the selected signatures; however, in our experience the actual computational requirements of the algorithm are much lower, also due to the parsimonious signature often retrieved.
A desired feature of SES is the fact that it heuristically and efficiently attempts to identify statistically, equivalent solutions, i.e., minimalsized feature subsets with the same optimal predictive performance. As mentioned before, when the distribution is faithful to a BN the solution is unique; however, in practice whether due to finite sample or deviations from assumptions there are multiple (empirically) equivalent solutions. Identifying all equivalent solutions is important when feature selection is employed for knowledge discovery and getting insight to the domain under study. Returning an arbitrarilychosen single solution S may mislead the domain expert into thinking that all other variables are either redundant or irrelevant, when the situation can be reversed if selecting some other feature subset S^{′}.
In our empirical study, we compare SES against the stateoftheart feature selection algorithms for the above 4 scenarios on geneexpression data. SES successfully scales up to tens of thousands of gene trajectories. In terms of selection quality and predictive performance, SES outperforms other methods in the Temporallongitudinal scenario, is on par or better in the Staticlongitudinal and Staticdistinct scenarios while selecting many fewer variables, while it is outperformed in the Temporaldistinct scenario.
The rest of the paper is organized as follows. The “Methods” section introduces conditional independence testing for temporal data, as well as the SES algorithm. A comparative evaluation of the proposed approaches against LASSOinspired algorithms is then performed on real, high dimensional omics data. Discussion and conclusions end the paper.
Related work
In general, variable selection algorithms can be classified into two main categories, filter based and wrappers [12]. Methods of the first class select a subset of relevant features independently of the modeling algorithm that will be subsequently applied. On the other hand, wrapper methods try to select the set of features that optimize the performance of a specific classifier. A large bulk of literature has been published on the subject, with methods using several different approaches [13–23].
Finally, embedded methods are modeling algorithms whose operation automatically lead to the selection of the most relevant features (e.g., classification and regression trees [24]).
Many variable selection methods for classification of high dimensional biological data (particularly gene expression) have been proposed in the last decades [25]. For a recent review and open problems with regard to variable selection in high dimensional data the reader is addressed to BolónCanedo et al. [26].
In this work, we have carefully reviewed the current literature for identifying the most related and recent variable selection methods suitable for the four scenarios depicted above. Particularly, we have sought methods both applicable on temporal data and scalable to highdimensional problems (i.e, thousands of candidate predictors).
In brief, the glmmLasso algorithm seems to be the most wellperforming method for studies that belong to the Temporallongitudinal scenario, according to the comparison performed in [27]. This algorithm combines mixedmodels representation of complex variance structures with the sparsity of LASSO solutions; as a drawback, the resulting model is nonconvex and difficult to optimize. In the Temporaldistinct and staticdistinct scenarios there is no withinsample variance, and these two cases can be addressed with variable selection algorithms designed for nontemporal data. The Staticlongitudinal scenario corresponds to discriminant analysis in longitudinal data, and not much research has been performed in the context of variable selection, see for example [28–30].
Available approaches for the Temporallongitudinal scenario
Several approaches for variables selection were proposed in the last 15 years for studies where both the outcome and the predictors are measured over time on the same samples. Most of these approaches use either Generalized Linear Mixed Models (GLMM) or Generalized Estimating Equations (GEE).
On GLMM, Ni et al. [31] proposed a doublepenalized likelihood approach in semiparametric mixed models. Bondell and coauthors [32] proposed an algorithm that performs simultaneous selection of the fixed and random factors using a modified Cholesky decomposition and maximum penalized likelihood estimation, along with the smoothly clipped absolute deviation (SCAD). A similar approach, using adaptive LASSO penalty functions instead of SCAD, was presented as well [33]. Zhao et al. [34] suggested using a basis function approximations and a partial group SCAD penalty for semiparametric varying coefficient partially linear mixed models, while Tang et al. [35] focused on quantile varying coefficient models via penalizing the L_{ γ } norm. Schelldorfer et al. [36] proposed an L_{1}penalty term for linear mixed models, and this work was later extended to include Poisson and binary logistic regression [37]. A method quite similar to the one of [37] was proposed in [27]; however the latter uses a gradient ascent algorithm whereas the former uses a coordinate gradient descent method based on a quadratic approximation of the penalized loglikelihood. Finally, a comparison of model selection methods for linear mixed models based on four major approaches is presented in [38]: information criteria such as AIC or BIC, shrinkage methods based on penalized loss functions such as LASSO, fence (adhoc procedures) and Bayesian techniques.
The literature is less extensive when it comes to GEE. The use of a modified AIC, termed quasilikelihood information criterion (QIC), was proposed in [39]. Cantoni and coauthors [40] first used a generalised Mallow’s criterion, and subsequently [41] used a Markov chain Monte Carlo (MCMC) procedure for variable selection without visiting all possible candidate models. The case of missingatrandom data was addressed in [42] by using a missing longitudinal information criterion selecting the optimal model and the correlation structure. Finally, a penalized GEE method that is consistent even when the working correlation structure is misspecified was presented in [43].
Some Bayesian techniques include [44–46] among others. The first used a Cholesky decomposition of the random effects covariance matrix and introduced a further decomposition of the Cholesky decomposed lower triangular matrix. The elements of the resulting diagonal matrix are assigned zeroinflated truncatedGaussian priors and MCMC methods are applied. However, these types of approaches are discouraged [47], as they are computationally heavy and are prior dependent. Han and coauthors [45] compared a number of methods for comparing two linear mixed models using Bayes factors. They also mentioned that these kinds of methods require substantial human intervention and high computational power.
A common drawback of all the procedures presented so far is that they are applicable only on a small number of candidate predictors. The only exceptions are presented in [35–37], [43], that were tested on 100, 200, 500 and 1000 candidate predictors in their respective simulation studies. To note, these studies do not report information about the computational time required by the algorithms. Moreover, authors do not usually provide implementations of the methods they propose. The only methodologies available as R packages are the one presented by [36], under the name GLMMLasso, and the glmmLasso package by [27], which offers linear, Poisson and binary logistic mixed models.
Available approaches for the staticlongitudinal scenario
This scenario refers to the task of discriminant analysis in longitudinal data. According to the concise review presented in [48], variable selection is somewhat not heavily researched in this context. More recently, L_{1} type constrains such as LASSO and SCAD allowing for grouped variables [28] were suggested. Matsui et al. [49] extended previous work to include multinomial logistic regression where the variables are selected in a grouped way. Finally, approaches based on functional regression also exist in the literature, see for example [50].
Available approaches for the Temporaldistinct and Staticdistinct scenarios
Both the Temporaldistinct and Staticdistinct scenarios are defined over timecourse data measured at each time point on different samples. Thus, the withinsample variance cannot be modeled for these scenarios. This allows variable selection methods devised for nontemporal data, as the widely used LASSO [51], to be applied in this context.
The LASSO algorithm started gaining popularity after the work in [52] who suggested the least angle regression as a better and faster way to solve its underlying optimization problem. A coordinate descent algorithm, which allows using the LASSO penalty in the context of generalized linear models was then suggested [53]. This latter approach is implemented in the R package glmnet [54].
Grouped Lasso (gLASSO, [55]) was developed to handle categorical predictors, which are often encoded in linear modeling as groups of binary variables (dummy variables). For the sake of consistency, the dummy variables corresponding to a single categorical predictor should be either included or excluded altogether (“as a group”) in the final LASSO solution. More recently, a quite efficient gLASSO implementation was proposed by [56], with their code made available in the R package gglasso [57].
Methods
In this section we discuss in detail how to adapt constraintbased method for temporal data analysis. First, we will briefly present Generalized Linear Mixed Models (GLMM) and Generalized Estimating Equations (GEE). Both techniques are suitable for devising conditional independence tests for temporal data with (un)balanced study designs. For a thorough comparison between GLMM and GEE see [58, 59].
Generalised linear mixed models
The vector β is the (p+1)dimensional vector of coefficients for the n_{ i }×(p+1) fixed effects design matrix X_{ i }, which contains the predictor variables. The vector b_{ i }∼N_{ q }(0,Σ) is the qdimensional vector of coefficients for the n_{ i }×q random effects matrix W_{ i }, while Σ is the randomeffects covariance matrix. The vector \(\mathbf {e}_{i} \sim N_{n_{i}}\left (\mathbf {0}, \sigma ^{2}\mathbf {I}_{n_{i}}\right)\) is the n_{ i } dimensional withingroup error vector which follows a spherical normal distribution with zero mean vector and fixed variance σ^{2}.
We used the exchangeable or compound symmetry (CS) structure on the covariance matrix Σ. We decided not to use a first order autoregressive covariance (AR(1)) structure as a hyperparameter of the GLMM method, since this type of structure did not improve the performance of generalised estimating equations (presented below) and would add a high computational burden to the fitting of GLMM.
K stands for the number of subjects and the total sample size (number of measurements) is equal to \(N=\sum _{i=1}^{K}n_{i}\). The link function g connects the linear predictors on the right hand side of (1) with the distribution of the target variable. Common link functions are the identity, for normally distributed target variables, and the logit function for binomial responses.
The possibility of specifying random effects allows mixed models to adequately represent between and withinsubject variability, and to model the deviates of each subject from the average behavior of the whole population. These characteristics make GLMMs particularly suitable for temporal and longitudinal data [9].
Generalised estimating equations
Generalised Estimating Equations (GEE), developed by [60, 61], are an alternative to mixed models for modeling data with complex correlation structures. In contrast to GLMM which are subject specific, GEE contain only fixed effects and thus are population specific.
CS assumes that correlations of measurements for the same subject at different timepoints are always the same, regardless of the temporal distance between them. Depending on the specific application, this might be not very realistic. In contrast, the AR(1) structure assumes that the correlation between measurements at different time points for the same subject decreases exponentially as the temporal gap between them increases.
A precise numerical estimation of α is critical in GEE modeling; we use the jackknife variance estimator suggested by [62], which is quite suitable for cases when the number of subjects is small (K≤30), as in many biological studies. The simulation studies conducted by [63] and [64] showed that the approximate jackknife estimates are in many cases in good agreement with the fully iterated ones.
Conditional independence tests for the Temporallongitudinal scenario
We devise two independence tests based on GLMMs (Eq. 1) and GEEs (Eq. 2) respectively. This scenario assumes the predictors and the target variable are measured at a fixed set of timepoints τ={τ_{1},…,τ_{ m }} in the same set of subjects. For balanced designs, all subjects are measured at all timepoints, i.e, n_{ i }=n,∀i. The target variable is often a geneexpression trajectory and thus, in the rest of the paper and for this scenario we assume a continuous target.
where 1 is a vector of 1s, a is the global intercept, b_{ i } stands for the random intercept of the ith subject, γ, δ and β are the coefficient of the predictors, and the generic link function g(.) (Eq. 1) has been substituted with the identity one.
This formulation stems from two specific modeling choices: (a) we use the vector of actual time points τ as a covariate, in order to model the baseline effect of the time on the trajectory of the target variable. Time becomes a linear predictor of the target. Other choices are possible, but would require more timepoints that are typically not available in geneexpression data. (b) We include random intercepts, meaning we allow a different starting point for the estimated trajectory of each subject. This choice leads to W_{ i }=1_{ n }_{ i },∀i, where 1_{ n } is a vector of ones of size n. However, we do not allow random slopes, thus assuming all subjects have the same dynamics. This choice was dictated by the need of avoiding model overspecification, especially considering the small sample size of the datasets used in the experimentation.
Pinheiro and Bates [9] suggests the use of the Ftest for comparing the two models, where only the model, the full, under the alternative is fitted and the significance of the coefficient β is tested. Another possible choice would be the loglikelihood ratio test, however the Ftest is preferable for small samples, since the type I error is better controlled with the F distribution.
GEE fitting does not compute a likelihood [59] and thus, no loglikelihood ratio test can be computed. A Wald test is used instead here again and the significance of the coefficient β is tested. Because of the lack of likelihood computation, its effectiveness in assessing conditional independence is questionable [65]. Despite these theoretical considerations, the experimental results proved the test to be quite effective in our context.
Conditional independence tests for the Staticlongitudinal scenario
The Staticlongitudinal scenario assumes longitudinal data with continuous predictors and a static target variable T that is either binary or multicategory. The goal is to discriminate between two or more groups on the basis of timedepending covariates. As in the Temporallongitudinal scenario, the presence of longitudinal data requires to take into account the withinsubject correlations.
where Γ_{ Z } are the coefficients corresponding to the set of conditioning variables Z and Γ_{ X } are the coefficients corresponding to the variable X. A logit function g(.) is used for linking the linear predictors to the binomial (or multinomial) outcome. The loglikelihood ratio test (calibrated with a χ^{2} distribution) is used to decide which of the two models is to be preferred.
Conditional independence tests for the Temporaldistinct and Staticdistinct scenarios
In these two scenarios different subjects are sampled at each time point (timecourse data), and subjectspecific correlation structures cannot be modeled. For the Temporaldistinct scenario, where the target variable is continuous, it is thus possible to use models (5) for assessing conditional independence. In absence of subjectspecific correlation structures the GEE models reduce to standard linear models that can be compared with the standard Ftest. A similar approach can be used for the Staticdistinct scenario, where the outcome is binary or multinomial, by using a logit link function instead of the identity.
The SES algorithm
First introduced in [10], the SES algorithm attempts to identify the set(s) of predictors (signatures) that are minimal in size and provide optimal predictive performances for a target variable T. The basic idea is that if ∃Z, s.t., Ind(X;TZ), then X is superfluous for predicting T. Thus, SES repetitively applies a test of conditional independence until it identifies the predictors that are associated with T regardless of the conditioning set used. Under certain conditions, these variables are the neighbors of T in a Bayesian Network representing the data at hand [2]. An interesting characteristic of SES is that it can return multiple, statistically indistinguishable predictive signatures. As discussed in [68], limited sample size, high collinearity or intrinsic characteristics of the data may produce several signatures with the same size and predictive power. From a biological perspective, multiple equivalent signatures may arise from redundant mechanisms, for example genes performing identical tasks within the cell machinery. The SES algorithm is further explained in the Additional file 1 and in [5].
Equipping constraintbased methods with conditional independence test for temporal data
SES belongs to the class of constraintbased featureselection methods [4]. This type of algorithm processes the data exclusively through tests of conditional independence that assess Ind(X;TZ). This means that in order to extend any constraintbased methods to temporal data it is sufficient to equip an appropriate test, such as the ones defined in Eqs.(4)(7).
Experimentation on real data
The experimental evaluation aims at assessing the capabilities of the proposed conditional independence tests in real setting. For each scenario we identified several geneexpression datasets over which we applied the SES algorithm equipped with the conditional independence test most suitable for the data at hand. The feature subsets identified by SES were then fed to modeling methods for obtaining testable predictions.
Furthermore, in each scenario we contrasted SES against a feature selection algorithm belonging to the family of LASSO methods. This class of algorithms has proven to be wellperforming in several applications, including variable selection in temporal data (see the Section regarding the literature review). Particularly, we compare against glmmLasso [27] for the Temporallongitudinal scenario, with standard LASSO regression [51] for the Temporaldistinct scenario, and the grouped LASSO (GLASSO) for classification [54, 56] in the Staticlongitudinal and Staticdistinct scenarios.
We excluded from this comparative analysis approaches that a) do not scaleup to thousands of variables (e.g., Bayesian procedures), b) require a number of time points much larger than the applications taken into consideration in this work (as for functional regression, [69]), and c) in general do not have available implementations.
The configuration settings of all algorithms involved in the experimentation were optimized by following an experimentation protocol specifically devised for estimating and removing any bias in performance estimation due to overfitting.
Datasets
We thoroughly searched the Gene Expression Omnibus database (GEO, http://www.ncbi.nlm.nih.gov/) for datasets with temporal measurements. Keywords “longitudinal”, “time course”, “time series” and “temporal” returned nearly 1000 datasets. We only kept datasets having at least 15 measurement and at least three time points, and complete information about the design of the study generating the data. This resulted in at least 6 datasets for each scenario, except for the Staticlongitudinal scenario, where we identified 4 datasets with at least 8 measurements. Detailed information on the selected datasets are available in the (Additional file 1: Tables S5 and S6).
Modeling approaches
For the Staticlongitudinal scenario, logistic or multinomial regression was applied on the columns of the matrix Γ selected by SES, depending on the outcome at hand. The grouped Lasso (GLASSO, [56]) algorithm was used for comparison. GLASSO allows to specify groups of variables that can enter the final model only altogether. Particularly, the GLASSO was applied on the whole matrix Γ, forcing the algorithm to either select or discard predictors in pairs, following the way columns in Γ correspond to the original predictors.
For Temporaldistinct and Staticdistinct scenarios SES was always coupled with standard linear, logistic or multinomial regression (depending on the specific outcome), while the standard LASSO algorithm (binary outcome) and GLASSO (multinomial outcome) were used for comparison (see Additional file 1 for further details).
In all analyses SES’ hyperparameters maximum conditioning variables size k and significance level a varied between {3,4,5} and {0.05,0.1}, respectively. The λ penalty values generated by the Least Angle Square (LARS) algorithm [52] were used for the LASSO models of all scenarios, apart from the temporallongitudinal. LARS cannot be adapted to this latter scenario, and thus the range of values was separately determined for each dataset, by using all integer values between λ_{ min }, the smallest value guarantying the invertibility of the Hessian matrix in each fold, and λ_{ max }, the highest value after which no variable was selected.
Experimentation protocol
We used the mfold crossvalidation procedure with the TibshiraniTibshirani (TT) bias correction [70] for model selection and performance evaluation. In the standard crossvalidation protocol the available samples are partitioned in m folds, with approximately an equal number of samples each. Each fold is in turn heldout for testing, while the remaining data form the training set. The current modeling approach is applied several times on the training set, once for each predetermined configuration setting, and the predictive performances of the corresponding models are evaluated on the holdout fold. The configuration with the best average performance is then used for training a final model on the whole dataset. In all experimentation m was set to either 4 or 5, so that to have at least two measurements in each fold. Particularly, folds correspond to one or more subjects in the Staticlongitudinal scenario, and to one or more time points in the other scenarios.
where e_{ i } is the performance on fold i, while \(\hat {\pmb {\theta }}\) and \(\hat {\pmb {\theta }}_{i}\) are the configurations corresponding to the best average performance and to the best performance of the ith fold, respectively. Signs in (9) should be interchanged if the performance metric assigns higher scores to better models.
The statistical significance of the difference between average performances is computed through permutationbased ttests, where single performances are randomly permuted for approximating the null distribution.
All of the simulations, computations and time measurements were performed on a desktop with Intel Core i53470 CPU @ 3.2 processor, 4 GB RAM memory using a 64bit R version 3.2.2.
Results and discussion
Coupling SES with GLMM and GEE
Temporallongitudinal scenario: comparison between SES equipped with GLMM (SESglmm) and SES equipped with GEE
Dataset  MSPE  Average time (in seconds)  

SESglmm  SESgee(CS)  SESgee(AR(1))  SESglmm  SESgee(CS)  SESgee(AR(1))  
GDS5088  0.131 (0.000)  0.189 (0.1)  0.289 (0.018)  1562.51 (230.53)  1022.45 (217.99)  933.14 (180.34) 
GDS4395  0.116 (0.007)  0.156 (0.019)  0.298 (0.028)  21167.21 (26089.48)  4862.15 (1724.89)  5577.80 (1890.15) 
GDS4822  0.066 (0.000)  0.055 (0.001)  0.045 (0.004)  1785.66 (321.92)  2103.96 (490.74)  1492.30 (205.03) 
GDS3326  0.062 (0.001)  0.052 (0.000)  0.063 (0.002)  6617.09 (472.16)  3167.78 (795.74)  2348.69 (390.10) 
GDS3181  0.805 (0.096)  0.458 (0.000)  0.458 (0.00)  1684.90 (206.26)  1011.44 (152.59)  748.18 (105.32) 
GDS4258  0.074 (0.000)  0.149 (0.003)  0.152 (0.002)  4135.76 (506.15)  2818.024 (418.97)  2078.52 (462.30) 
GDS3915  0.527 (0.038)  0.553 (0.01)  0.439 (0.000)  669.18 (63.93)  511.82 (84.22)  491.91 (108.64) 
GDS3432  0.057 (0.001)  0.060 (0.008)  0.038 (0.003)  3275.22 (474.06)  2213.11 (371.68)  2104.05 (546.76) 
Average  0.230 (0.280)  0.209 (0.192)  0.223 (0.172)  5112.2 (6756.04)  2378.56 (1566.36)  1971.82 (1611.13) 
For each dataset the crossvalidated, TTcorrected Mean Squared Prediction Error (MSPE) is reported (standard deviation in parenthesis), along with the respective computational time in Table 1. Average performances are reported at the bottom line. Methods are indicated as SESglmm, SESgee(CS)) and SESgee(AR(1)), corresponding to SES coupled with GLMM and GEE, the latter using either the CS or AR(1) covariance structure. All methods obtain statistically equivalent results in terms of MSPE (all paired permutationbased ttest pvalues are above 0.37). The average computational time largely varies, with SESgee(AR(1)) being the fastest of the three methods (all paired permutationbased ttest pvalues are below 0.002). For all methods, computational times strongly depend upon the number of variables of each dataset, in a loglinear way (see Additional file 1).
Since the three versions produced equally predictive results, in the remaining of the analysis we use only SESglmm, in order to ensure a comparison as fair as possible with the GLMM based method glmmLasso.
glmmLasso scalability in highdimensional data
The same analysis was repeated on all datasets selected for the Temporallongitudinal scenario, consistently achieving similar results (Additional file 1). Consequently, for each dataset related to the Temporallongitudinal scenario only 2000 randomly selected predictors were retained in all subsequent analyses, so that the experimentation could be performed in a reasonable time and to allow a fair comparison between SESglmm and glmmLasso (see Additional file 1: Table S9 for the values of the penalty parameter used in glmmLasso).
Results on the four scenarios
Crossvalidated, TTcorrected performances of SES and LASSOtype methods on the four scenarios
Temporallongitudinal scenario  Temporaldistinct scenario  

MSPE  Selected vars  MSPE  Selected vars  
Dataset  SESglmm  glmmLasso  SESglmm  glmmLasso  Dataset  SES  LASSO  SES  LASSO 
GDS5088  0.081 (0.026)  0.160 (0.042)  5.25 (0.85)  5.15 (8.65)  GDS3859  0.068 (0.006)  0.019 (0.002)  3.5 (0.51)  11.81 (4.66) 
GDS4395  0.104 (0.041)  0.640 (0.568)  5.37 (0.56)  12.35 (13.61)  GDS972  0.022 (0.000)  0.001 (0.000)  5.83 (0.92)  22.2 (9.85) 
GDS4822  0.115 (0.484)  0.765 (0.436)  4.75 (0.85)  3.16 (5.77)  GDS947  0.056 (0.000)  0.054 (0.026)  5.92 (0.65)  12.40 (5.40) 
GDS3326  0.135 (0.021)  0.234 (0.139)  5.42 (0.78)  2.42 (7.45)  GDS964  0.033 (0.000)  0.003 (0.000)  5.73 (0.69)  25.69 (11.86) 
GDS3181  0.971 (0.484)  0.684 (0.257)  4.17 (0.87)  0.35 (2.15)  GDS2688  0.184 (0.006)  0.005 (0.001)  5.79 (1.06)  20.64 (10.93) 
GDS4258  0.234 (0.096)  9.882 (4.518)  3.83 (0.51)  1.48 (4.06)  GDS2135  0.053 (0.002)  0.014 (0.003)  3.80 (0.76)  10 (5.72) 
GDS3432  0.357 (0.017)  2.283 (1.572)  1.67 (3.51)  0.08 (0.55)  Av. diff.  0.053 ^{ a }  12.03 ^{ a }  
GDS3915  0.059 (0.002)  0.150 (0.055)  5.12 (0.80)  1.66 (4.62)  
Av. diff.  1.59 ^{ b }  1.12 ^{ b }  
Staticdistinct scenario  Staticlongitudinal scenario  
PCC  Selected vars  PCC  Selected vars  
Dataset  SES  LASSO  SES  LASSO  Dataset  SES  GLASSO  SES  GLASSO 
GDS4319  0.873 (0.000)  0.995 (0.000)  2.1 (0.31)  8 (0.00)  GDS4146  1.000 (0.000)  0.858 (0.142)  1.00 (0.00)  0.42 (1.38) 
GDS3924  0.729 (0.000)  0.528 (0.104)  2.75 (0.44)  53.56 (28.55)  GDS4518  0.750 (0.000)  0.417 (0.333)  1.75 (0.44)  3.04 (2.15) 
GDS3184  0.556 (0.067)  0.578 (0.111)  3.00 (0.00)  10.62 (5.16)  GDS4820  0.500 (0.000)  0.667 (0.167)  2.00 (0.00)  5.14 (3.19) 
GDS3145  0.953 (0.000)  0.594 (0.125)  1.5 (0.88)  0.6 (0.55)  GDS1840  0.625 (0.000)  0.500 (0.250)  1.5 (0.51)  2.67 (2.03) 
GDS2882  0.800 (0.000)  0.750 (0.000)  1.5 (0.88)  0.25 (0.50)  Av. diff.  0.108  1.23  
GDS2851  0.722 (0.000)  0.694 (0.000)  2.25 (0.44)  0.75 (0.50)  
GDS1784  0.861 (0.000)  0.694 (0.000)  1.75 (0.85)  0.5 (0.58)  
GDS2456  1.000 (0.000)  0.739 (0.000)  1.2 (0.41)  0.44 (0.53)  
Av. diff.  0.115 ^{ b }  6.52 ^{ b } 
On average, SES equipped with conditional independence tests for temporal data outperforms the corresponding LASSO algorithms, in terms of predictive performance, in all scenarios, except for the Temporaldistinct scenario. We also note that LASSO methods did not select any variable in at least one fold of cross validation for several datasets, as indicated by an average number of selected variables < 1 (baseline predictive models are produced in these cases). When LASSO methods select at least one variable in each fold, their variability in number of selected variables is considerably higher than the one of SES. Particularly, for the Temporallongitudinal scenario SESglmm largely outperforms, in terms of predictive performance, glmmLasso in all datasets except one (GDS3181), where glmmLasso is only marginally superior (See Additional file 1: Table S10). For the Temporaldistinct scenario the results are quite turned around, with LASSO having better predictive performances than SES, although at the cost of identifying larger and unstable sets of variables. Finally, SES generally outperforms LASSO in the Staticlongitudinal and Staticdistinct scenarios, both in terms of average PCC and number of selected features. No variables were selected for dataset GDS3944 by neither method, and thus we excluded this dataset from the results.
Temporallongitudinal scenario: comparison between SESglmm and glmmLasso based on 20 replications with different target variable (gene) and independently randomly selected 2000 genes as predictor variables
Dataset  GDS5088  GDS4395  GDS4822  GDS3326  GDS3181  GDS4258  GDS3432  GDS3915 

Average difference  3.560(4.118)  0.188(0.516)  0.003(0.134)  0.180(0.506)  0.020(0.04)  0.139(0.288)  0.000(0.355)  0.093(0.455) 
Proportion  19/20  7/20  9/20  13/20  15/20  10/20  10/20  8/20 
pvalue  0.0001 ^{ a }  0.128  0.938  0.0015 ^{ a }  0.0312 ^{ b }  0.024 ^{ b }  0.9946  0.3842 
where i=1,2,3,4 represents the 4 time points, j=1,⋯,11 the 11 subjects, and NC stands for the genomic region chr1:232358622−232358886. The variance of the random intercepts is equal to 0.0018, corresponding to a 7.8% of the total variability. All coefficients are significant (maximum Wald test pvalue 0.02), with positive coefficients indicating predictors whose trajectories over time agree with the one of the target genes, while the opposite holds for negative coefficients. The coefficient of the time effect has a negative sign; speculations on Fig. 2c suggest that a quadratic effect would perhaps perform better, but adding more parameters would easily lead to overfitting, due to the limited number of time points and subjects. Figure 2e and f show the temporal trajectories of gene TSIX and Ppp1r42, respectively, which were included in the optimal predictive signatures of dataset GDS4146 (Staticlongitudinal) and GDS2882 (Staticdistinct). In both cases the trajectories of the two genes markedly differ between the two classes.
Conclusions
In this work we described how constraintbased, feature selection methods can be extended for the analysis of (high dimensional) temporal data, by equipping them with suitable conditional independence tests. The main contribution of this work is thus indicating how a whole class of stateoftheart, provably wellperforming feature selection methods [4] can be easily extended to data characterized by (a) measurements taken over time and (b) high dimensionality, settings frequently encountered in biological studies as well as in other fields. Furthermore, conditional independence tests are the cornerstone of any constrainedbased method for (causal) network reconstruction [71]; under this respect, this work also paves the path for extending this type of algorithms to temporal data.
We assessed the performances of the proposed approach by evaluating a prototypical constraintbased method, the SES algorithm, on several realworld gene expression datasets. Each dataset belongs to one out of four different scenarios, which represent common study designs for temporal data. The Temporallongitudinal and Staticlongitudinal scenarios represent longitudinal studies with timedependent or static target variable, respectively, while the Staticlongitudinal and Staticdistinct scenarios refer to the case of different samples measured at each time point. The Temporallongitudinal and Temporaldistinct scenarios required devising conditional independence tests able to take into account the idiosyncrasies of their respective data. The conditional independence test devised for the Temporallongitudinal scenario addresses the within sample variation by employing the GLMM and GEE modeling techniques; the tests devised for the Staticlongitudinal scenario uses a twostep regression strategy for addressing the problem of discriminant analysis in longitudinal data.
In the context of our experimentation, SES outperformed stateoftheart methods belonging to the class of LASSO algorithms in three out of four scenarios. Particularly, in the Temporallongitudinal scenario, SESglmm clearly superseded the glmmLasso algorithm [27]. Moreover, SESglmm easily scales to tens of thousands of variables, while glmmLasso computation requirements become rapidly prohibitive.
A key feature of the SES algorithm is its ability to produce multiple solutions, signatures, i.e. more than one set of predictor variables, which are statistically equivalent, as demonstrated in a recent publication [5]. Many times in biological studies, and not only, the outcome of the study, the final model, or the selected variables, is not what expected by the expert in the field. This could be justified by the fact that the chosen model is statistically equivalent to the expected model. And hence, a degree of missinformation has been delivered. Multicollinearity among the predictor variables should not be treated as a disease, but rather as a means of extracting extra information about the data.
We assessed the equivalence of the signatures produced by SES also in the context of this experimentation. For every signature computed by SES we fitted a predictive model and calculated the corresponding MSPE. The distribution of these MSPE values confirmed that SES produces signatures whose predictive value is close to each other (see Additional file 1: Tables S7, and S8 and Figures S7, S8 and S9 for details).
The results showed in “Results on the four scenarios” section and Table 2 purposely do not report information regarding the computational time. In the Temporallongitudinal scenario SES’ computational requirements can be one or two orders of magnitude smaller than the ones of the glmmLasso. The opposite though is true for the other scenarios. A prototypical SES run on the GDS3859 dataset (45,100 variables and 23 samples, Temporaldistinct scenario) requires on average 278 s, while LASSO returns an answer in less than half a second. This difference is mainly due to implementation issues: the code of SES is written in R, while LASSO is based on a fast FORTRAN implementation. In addition, the SES algorithm needs to perform some additional computation for identifying multiple signatures. Other constraintbased methods, e.g., MMPC, that return only one signature are expected to be faster, in case the computational cost is an important parameter to consider.
Finally, we performed overrepresentation analysis on the pathways provided by the Kyoto Encyclopedia of Genes and Genomes (KEGG [72]) using the hyper geometric test. To ensure an adequate statistical power, we performed this analysis only for the 12 datasets where the probesets selected by SES correspond to five or more genes. For each dataset we used the pathways of the proper species. We found that on average 10 pathways are significant at FDR level 0.1 for each signature (the lists of enriched pathways are in the Additional file 1), meaning that the selected genes are significantly overrepresented in known biological mechanisms. This indicates that performing feature selection with constraintbased methods coupled with conditional independence tests for temporal data can also provide biological insights, along with wellperforming predictive models.
The main limitation of the present study is the relatively low sample size of several datasets, that makes difficult to precisely estimate performances. However, we note that the work presented by [73, 74] showed that the TT protocol is able to provide precise estimation even with relatively small sample size, computationally more efficiently than the more complex nestedcross validation protocol.
Future work will focus on several directions. Computationally, we have addressed the computational cost of the linear mixed models by using our implementations. We plan to add the other scenarios in this direction. Parallel computations for the first step of the algorithm will be made available for the other methods as well. More intriguing, the application of the tests introduced in this work on constraintbased, causaldiscovery methods is currently under investigation.
Declarations
Acknowledgements
The first author would like to express his acknowledgments to Dimitris Rizopoulos and Janice Scealy for answering some of his questions.
Funding
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/20072013) / ERC Grant Agreement n. 617393.
No funding body played any role in the design or conclusion of the present study.
Availability of data and materials
All data are publicly available from the Gene Expression Omnibus database (GEO, http://www.ncbi.nlm.nih.gov/). The R codes for the LongitudinalTemporal Scenario are available from the R package MXM https://cran.rproject.org/web/packages/MXM/index.html.
Authors’ contributions
MT has participated in the design of the study, wrote the code, performed the experiments and drafted the manuscript. VL has contributed to the manuscript and to the design and supervision of the study. IT contributed the initial idea, and to the design and supervision of the study. All authors have read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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