Feature selection for high-dimensional temporal data

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 Bolón-Canedo 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 high-dimensional problems (i.e, thousands of candidate predictors).

In brief, the glmmLasso algorithm seems to be the most well-performing method for studies that belong to the Temporal-longitudinal scenario, according to the comparison performed in [27]. This algorithm combines mixed-models representation of complex variance structures with the sparsity of LASSO solutions; as a drawback, the resulting model is non-convex and difficult to optimize. In the Temporal-distinct and static-distinct scenarios there is no within-sample variance, and these two cases can be addressed with variable selection algorithms designed for non-temporal data. The Static-longitudinal 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 Temporal-longitudinal 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 double-penalized likelihood approach in semi-parametric mixed models. Bondell and co-authors [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 semi-parametric 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 L₁-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 log-likelihood. 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 (ad-hoc procedures) and Bayesian techniques.

The literature is less extensive when it comes to GEE. The use of a modified AIC, termed quasi-likelihood information criterion (QIC), was proposed in [39]. Cantoni and co-authors [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 missing-at-random 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 zero-inflated truncated-Gaussian 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 co-authors [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 static-longitudinal 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, L₁ 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 Temporal-distinct and Static-distinct scenarios

Both the Temporal-distinct and Static-distinct scenarios are defined over time-course data measured at each time point on different samples. Thus, the within-sample variance cannot be modeled for these scenarios. This allows variable selection methods devised for non-temporal 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].

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 q-dimensional vector of coefficients for the n _i ×q random effects matrix W _i , while Σ is the random-effects 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 within-group error vector which follows a spherical normal distribution with zero mean vector and fixed variance σ².

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 hyper-parameter 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 within-subject 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 time-points 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 Temporal-longitudinal 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 time-points τ={τ₁,…,τ _m } in the same set of subjects. For balanced designs, all subjects are measured at all time-points, i.e, n _i =n,∀i. The target variable is often a gene-expression 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 i-th 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 time-points that are typically not available in gene-expression 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 over-specification, especially considering the small sample size of the datasets used in the experimentation.

Pinheiro and Bates [9] suggests the use of the F-test 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 log-likelihood ratio test, however the F-test 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 log-likelihood 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 Static-longitudinal scenario

The Static-longitudinal scenario assumes longitudinal data with continuous predictors and a static target variable T that is either binary or multi-category. The goal is to discriminate between two or more groups on the basis of time-depending covariates. As in the Temporal-longitudinal scenario, the presence of longitudinal data requires to take into account the within-subject 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 log-likelihood ratio test (calibrated with a χ² distribution) is used to decide which of the two models is to be preferred.

Conditional independence tests for the Temporal-distinct and Static-distinct scenarios

In these two scenarios different subjects are sampled at each time point (time-course data), and subject-specific correlation structures cannot be modeled. For the Temporal-distinct scenario, where the target variable is continuous, it is thus possible to use models (5) for assessing conditional independence. In absence of subject-specific correlation structures the GEE models reduce to standard linear models that can be compared with the standard F-test. A similar approach can be used for the Static-distinct 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;T|Z), 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].

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 gene-expression 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 well-performing in several applications, including variable selection in temporal data (see the Section regarding the literature review). Particularly, we compare against glmmLasso [27] for the Temporal-longitudinal scenario, with standard LASSO regression [51] for the Temporal-distinct scenario, and the grouped LASSO (GLASSO) for classification [54, 56] in the Static-longitudinal and Static-distinct scenarios.

We excluded from this comparative analysis approaches that a) do not scale-up 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 over-fitting.

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 Static-longitudinal 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 Static-longitudinal 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 Temporal-distinct and Static-distinct 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’ hyper-parameters 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 temporal-longitudinal. 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 m-fold cross-validation procedure with the Tibshirani-Tibshirani (TT) bias correction [70] for model selection and performance evaluation. In the standard cross-validation protocol the available samples are partitioned in m folds, with approximately an equal number of samples each. Each fold is in turn held-out 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 hold-out 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 Static-longitudinal 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 i-th 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 permutation-based t-tests, 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 i5-3470 CPU @ 3.2 processor, 4 GB RAM memory using a 64-bit R version 3.2.2.

Coupling SES with GLMM and GEE

For each dataset the cross-validated, TT-corrected 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 permutation-based t-test p-values are above 0.37). The average computational time largely varies, with SESgee(AR(1)) being the fastest of the three methods (all paired permutation-based t-test p-values are below 0.002). For all methods, computational times strongly depend upon the number of variables of each dataset, in a log-linear 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 high-dimensional data

Preliminary analyses pointed out glmmLasso’s limited ability of efficiently (in computational terms) scaling up to a few thousands of predictors (glmmLasso’s implementation is limited to 17,000 variables). We characterized glmmLasso scalability by running the algorithm on increasingly larger numbers of randomly selected variables. Figure 2a reports the results obtained from dataset GSD5088. Different lines report time performances of glmmLasso, and SES equipped with different conditional independence tests. glmmLasso requirements in terms of computational time increase in a super-linear way with the number of predictors (see Additional file 1: Figures S1 and S2 for time comparisons with all datasets). An interesting feature of the SES implementation that is worthy to mention is the fact that information about the univariate associations (test statistics and associated p-values) is stored. Hence, when the hyper-parameters change values, the algorithm begins from the second step. For the 6 pairs of configurations (pairs of a and _k ) used in our experimental analysis this results in a significant amount of computational savings.

The same analysis was repeated on all datasets selected for the Temporal-longitudinal scenario, consistently achieving similar results (Additional file 1). Consequently, for each dataset related to the Temporal-longitudinal 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

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 Temporal-distinct 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 Temporal-longitudinal 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 Temporal-distinct 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 Static-longitudinal and Static-distinct 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.

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 p-value 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 over-fitting, 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 (Static-longitudinal) and GDS2882 (Static-distinct). In both cases the trajectories of the two genes markedly differ between the two classes.