1 Introduction
High dimensionality is increasingly ubiquitous in numerous scientific fields including genetics, economics and physics. Reducing the dimensionality is a challenge that most statistical methodologies must meet not only to remain interpretable but also to achieve reliable predictions. In linear regression models, dimension reduction techniques often correspond to variable selection methods. Approaches for variable selection are already implemented in publicly available, open-source software, e.g., the well-known R packages glmnet (Friedman et al. 2010) and spikeslab (Ishwaran et al. 2013). The R package glmnet implements the Elastic net methodology (Zou and Hastie 2005), which is a generalization of both the LASSO (Tibshirani 1996) and the ridge regression [RR; Hoerl and Kennard (1970)]. The R package spikeslab in turn, implements the Spike and Slab methodology (Ishwaran and Rao 2005), which is a Bayesian approach for variable selection.
Dimension reduction cannot, however, be restricted to variable selection. Indeed, the field can be extended to include approaches which aim at creating surrogate covariates that summarize the information contained in initial covariates. Since the emblematic principal component regression [PCR; Jolliffe (1982)], many other methods spread in the recent literature. As specific examples, we may refer to the OSCAR methodology (Bondell and Reich 2008), or the PACS methodology (Sharma et al. 2013) which is a generalization of the latter approach. Those methods mainly proposed variable clustering within a regression model as a way to reduce the dimensionality. Despite their theoretical and practical appeal, implementations of those methods were often proposed only through MATLAB (The MathWorks Inc. 2014) or R scripts, limiting thus the flexibility and the computational efficiency of their use. The CLusterwise Effect REgression (CLERE) methodology (Yengo et al. 2014), was recently introduced as a novel methodology for simultaneous variable clustering and regression. The CLERE methodology is based on the assumption that each regression coefficient is an unobserved random variable sampled from a mixture of Gaussian distributions with an arbitrary number \(g\) of components. In addition, all components in the mixture are assumed to have different means (\(b_1,\ldots,b_g\)) and equal variances \(\gamma^2\).
In this paper, we propose two new features for the CLERE model. First, the stochastic EM (SEM) algorithm is proposed as a more computationally efficient alternative to the Monte Carlo EM (MCEM) algorithm previously introduced in Yengo et al. (2014). Secondly, the CLERE model is enhanced with the possibility of constraining the first component to have its mean equal to 0, i.e. \(b_1 = 0\). This enhancement is mainly aimed at facilitating the interpretation of the model. Indeed when \(b_1\) is set to \(0\), variables assigned to the cluster associated with \(b_1\) might be considered less relevant than other variables provided \(\gamma^2\) is small enough. Those two new features were implemented in the R package clere (Yengo and Canouil 2015). The core of the package is a C++ program interfaced with R using the R packages Rcpp (Eddelbuettel and François 2011) and RcppEigen (Bates and Eddelbuettel 2013). The R package clere can be downloaded from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=clere.
The outline of the present paper is the following. In the next section the definition of the model is recalled and the strategy to estimate the model parameters is explained. Afterwards, the main functionalities of the R package clere are presented. Real data analyses are then provided, aiming at illustrating the good predictive performances of CLERE, with noticeable parsimony ability, compared to standard dimension reduction methods. Finally, perspectives and further potential improvements of the package are discussed in the last section.
2 Model definition and notation
Our model is defined by the following hierarchical relationships:
\[\label{eq:clere0} \left\{ \begin{matrix}{l} y_i \sim\mathcal{N}\left(\beta_0+\sum_{j=1}^{p}{\beta_j x_{ij}},\sigma^2\right),\\ \beta_j |\mathbf{z}_j \sim \mathcal{N}\left(\sum^{g}_{k=1}{ b_k z_{jk} },\gamma^2 \right),\\ \mathbf{z}_j = \left(z_{j1},\ldots,z_{jg} \right)\sim \mathcal{M}\left(1,\pi_1,\ldots,\pi_g\right)\text{,} \end{matrix} \right. \tag{1} \]
where \(\mathcal{N}(\mu,\sigma^2)\) is the normal distribution with mean \(\mu\) and variance \(\sigma^2\), \(\mathcal{M}\left(1,\pi_1,\ldots,\pi_g\right)\) the one-order multinomial distribution with parameters \(\mathbf \pi=\left(\pi_1,\ldots,\pi_g \right)\), where, \(\forall\) \(k=1,\ldots,g\) \(\pi_k>0\) and \(\sum^g_{k=1}{\pi_k}=1\), and \(\beta_0\) is a constant term. For an individual \(i=1,\ldots,n\), \(y_i\) is the response and \(x_{ij}\) is an observed value for the \(j\)-th covariate. \(\beta_j\) is the regression coefficient associated with the \(j\)-th covariate (\(j=1,\ldots,p\)), which is assumed to follow a mixture of \(g\) Gaussians. The variable \(\mathbf{z}_j\) indicates from which mixture component \(\beta_j\) is drawn (\(z_{jk}=1\) if \(\beta_j\) comes from component \(k\) of the mixture, \(z_{jk}=0\) otherwise). Let’s note that model ((1)) can be considered as a variable selection-like model by constraining the model parameter \(b_1\) to be equal to 0. Indeed, assuming that one of the components is centered in zero means that a cluster of regression coefficients have null expectation, and thus that the corresponding variables are not significant for explaining the response variable. This functionality is available in the package.
Let \(\mathbf \beta = \left(\beta_1,\ldots,\beta_p \right)\), \(\mathbf y = (y_1,\ldots ,y_n)'\), \(\mathbf{X} = (x_{ij})\), \(\mathbf Z = (z_{jk})\), \(\mathbf b = (b_1,\ldots, b_g)'\) and \(\mathbf \pi= (\pi_1,\ldots, \pi_g)'\). Moreover, \(\log p(\mathbf y|\mathbf X;\mathbf \theta)\) denotes the log-likelihood of model ((1)) assessed for the parameter vector \(\mathbf \theta = \left(\beta_0,\mathbf b,\mathbf\pi,\sigma^2,\gamma^2\right)\). Model ((1)) can be interpreted as a Bayesian approach. However, to be fully Bayesian a prior distribution for parameter \(\mathbf \theta\) would have been necessary. Instead, we proposed to estimate \(\mathbf \theta\) by maximizing the (marginal) log-likelihood, \(\log p(\mathbf{y}|\mathbf{X};\mathbf \theta)\). This partially Bayesian approach is referred to as Empirical Bayes [EB; Casella (1985)]. Let \(\mathcal{Z}\) be the set of \(p\times g\)-matrices partitioning \(p\) covariates into \(g\) groups. Those matrices are defined as \[\mathbf Z = \left(z_{jk}\right)_{1\leq j \leq p, 1\leq k \leq g} \in \mathcal{Z} \Leftrightarrow \forall j=1,\ldots,p\text{ } \begin{cases} \exists!\text{ } k\text{ such as }z_{jk} = 1\\ \text{For all }k'\neq k\text{ }z_{jk'}=0. \end{cases}\] The log-likelihood \(\log p(\mathbf y|\mathbf X;\mathbf \theta)\) is defined as
\[\label{eq:likelihood} \log p(\mathbf y|\mathbf X;\mathbf \theta) = \log\left[ \sum_{\mathbf Z\in\mathcal{Z}} {\int_{\mathbb{R}^p}{p(\mathbf y,\mathbf \beta,\mathbf Z|\mathbf X;\mathbf \theta)}\mathrm{d}\mathbf \beta }\right]\text{.} \tag{2} \]
Since it requires integrating over \(\mathcal{Z}\) with cardinality \(g^p\), evaluating the likelihood becomes rapidly computationally unaffordable.
Nonetheless, maximum likelihood estimation is still achievable using the expectation maximization (EM) algorithm (Dempster et al. 1977). The latter algorithm is an iterative method which starts with an initial estimate of the parameter and updates this estimate until convergence. Each iteration of the algorithm consists of two steps, denoted as the E and the M steps. At each iteration \(d\) of the algorithm, the E step consists in calculating the expectation of the log-likelihood of the complete data (observed + unobserved) with respect to \(p(\mathbf \beta,\mathbf Z|\mathbf y,\mathbf X;\mathbf \theta^{(d)})\), the conditional distribution of the unobserved data given the observed data, and the value of the parameter at the current iteration, \(\mathbf \theta^{(d)}\). This expectation, often denoted as \(Q(\mathbf \theta|\mathbf \theta^{(d)})\) is then maximized with respect to \(\mathbf \theta\) at the M step.
In model ((1)), the E step is analytically intractable. A broad literature devoted to intractable E steps recommends the use of a stochastic approximation of \(Q(\mathbf \theta|\mathbf \theta^{(d)})\) through Monte Carlo (MC) simulations (Wei and Tanner 1990; Levine and Casella 2001). This approach is referred to as the MCEM algorithm. Besides, mean-field-type approximations are also proposed (Govaert and Nadif 2008; Mariadassou et al. 2010). Despite their computational appeal, the latter approximations do not generally ensure convergence to the maximum likelihood (Gunawardana and Byrne 2005). Alternatively, the SEM algorithm (Celeux et al. 1996) was introduced as a stochastic version of the EM algorithm. In this algorithm, the E step is replaced with a simulation step (S step) that consists in generating a complete sample by simulating the unobserved data using \(p(\mathbf \beta,\mathbf Z|\mathbf y,\mathbf X;\mathbf \theta^{(d)})\) providing thus a sample \((\mathbf \beta^{(d)},\mathbf Z^{(d)})\). Note that the Monte Carlo algorithm we use to perform this simulation is the Gibbs sampler. After the S step follows the M step which consists in maximizing \(p(\mathbf \beta^{(d)},\mathbf Z^{(d)}|\mathbf y,\mathbf X;\mathbf \theta)\) with respect to \(\mathbf \theta\). Alternating those two steps generates a sequence \(\left(\mathbf \theta^{(d)}\right)\), which is a Markov chain whose stationary distribution (when it exists) concentrates around the local maxima of the likelihood.
3 Estimation and model selection
In this section, two algorithms for model inference are presented: the Monte-Carlo Expectation Maximization (MCEM) algorithm and the Stochastic Expectation Maximization (SEM) algorithm. The section starts with the initialization strategy common to both algorithms and continues with the detailed description of each algorithm. Then, model selection (for choosing \(g\)) and variable selection are discussed.
Initialization
The two algorithms presented in this section are initialized using a primary estimate \({\beta_j}^{(0)}\) of each \(\beta_j\). The latter can be chosen either at random, or obtained from univariate regression coefficients or penalized approaches like LASSO and ridge regression. For large SEM or MCEM chains, initialization is not a critical issue. The choice of the initialization strategy is therefore made to speed up the convergence of the chains. A Gaussian mixture model with \(g\) component(s) is then fitted using \({\mathbf \beta}^{(0)} = \left(\beta_1^{(0)},\ldots,{\beta}^{(0)}_p\right)\) as observed data to produce starting values \(\mathbf{b}^{(0)}\), \(\mathbf \pi^{(0)}\) and \({\gamma^2}^{(0)}\) respectively for parameters \(\mathbf{b}\), \(\boldsymbol{\pi}\) and \({\gamma^2}\). Using maximum a posteriori (MAP) clustering, an initial partition \(\mathbf{Z}^{(0)} = \left(z^{(0)}_{jk}\right)\in\mathcal{Z}\) is obtained as \[\forall j\in\{1,\ldots,p\},\text{ }z^{(0)}_{jk} = \begin{cases} 1 & \text{ if }k = {argmin}_{k'\in\{1,\ldots,g\}}{\left({\beta_j}^{(0)}-b^{(0)}_{k'}\right)^2},\\ 0 & \text{otherwise.} \end{cases}\] \(\beta_0\) and \(\sigma^2\) are initialized using \({\mathbf \beta}^{(0)}\) as follows: \[\beta_0^{(0)} = \frac{1}{n}\sum^{n}_{i=1}{\left(y_i-\sum^{p}_{j=1}{\beta^{(0)}_j x_{ij}}\right)} \text{ and } {\sigma^2}^{(0)} = \frac{1}{n}\sum^{n}_{i=1}{\left(y_i-\beta_0^{(0)} -\sum^{p}_{j=1}{\beta^{(0)}_j x_{ij}}\right)^2} .\]
MCEM algorithm
The stochastic approximation of the E step
Suppose at iteration \(d\) of the algorithm that we have \(\left\{\left(\mathbf \beta^{(1,d)},\mathbf Z^{(1,d)}\right),\ldots, \left(\mathbf \beta^{(M,d)},\mathbf Z^{(M,d)}\right)\right\}\), \(M\) samples from \(p\left(\mathbf \beta,\mathbf Z|\mathbf y,\mathbf X;\mathbf \theta^{(d)}\right)\). Then the MC approximation of the E step can be written as
\[\label{eq:approxEstep} Q\left(\mathbf \theta|\mathbf \theta^{(d)}\right) = \mathbb{E}\left[\log p(\mathbf{y},\mathbf \beta,\mathbf{Z}|\mathbf{X};\mathbf \theta^{(d)}) |\mathbf{y},\mathbf{X};\mathbf \theta^{(d)}\right] \approx \frac{1}{M}\sum^{M}_{m=1}{\log p(\mathbf{y},\mathbf \beta^{(m,d)},\mathbf{Z}^{(m,d)}|\mathbf{X};\mathbf \theta^{(d)})}\text{.} \tag{3} \]
Sampling from \(p\left(\mathbf \beta,\mathbf Z|\mathbf y,\mathbf X;\mathbf \theta^{(d)}\right)\) is not straightforward. However, we can use a Gibbs sampling scheme to simulate unobserved data, taking advantage of \(p\left(\mathbf \beta|\mathbf Z,\mathbf y,\mathbf X;\mathbf \theta^{(d)}\right)\) and \(p\left(\mathbf Z|\mathbf \beta,\mathbf y,\mathbf X;\mathbf \theta^{(d)}\right)\) from which it is easy to simulate. These distributions, i.e., Gaussian and multinomial, respectively, are described below in Equations ((4)) and ((5)).
\[\label{eq:gibbs1} \left\{ \begin{matrix} \mathbf \beta|\mathbf{Z,y,X};\mathbf \theta^{(d)} \sim \mathcal{N}\left(\mathbf \mu^{(d)},\mathbf \Sigma^{(d)} \right),\\ \mathbf \mu^{(d)} = \left[ \mathbf{X'X} + \frac{{\sigma^2}^{(d)}}{{\gamma^2}^{(d)}}\text{I}_{p} \right]^{-1}\mathbf{X}'\left(\mathbf{y}-\beta^{(d)}_0\mathbf 1_p\right) + \frac{{\sigma^2}^{(d)}}{{\gamma^2}^{(d)}}\left[ \mathbf{X'X} + \frac{{\sigma^2}^{(d)}}{{\gamma^2}^{(d)}}\text{I}_{p} \right]^{-1}\mathbf{Zb}^{(d)},\\ \mathbf \Sigma^{(d)} = {\sigma^2}^{(d)}\left[ \mathbf{X'X} + \frac{{\sigma^2}^{(d)}}{{\gamma^2}^{(d)}}\text{I}_{p} \right]^{-1}, \end{matrix} \right. \tag{4} \]
and, noting that \(p\left(\mathbf Z|\mathbf \beta,\mathbf y,\mathbf X;\mathbf \theta^{(d)}\right)\) does not depend on \(\mathbf{X}\) nor \(\mathbf{y}\),
\[\label{eq:gibbs2} p\left(z_{jk}=1|\mathbf \beta; \mathbf \theta^{(d)}\right) \propto \pi^{(d)}_k\exp\left(-\frac{\left(\beta_j - b^{(d)}_k\right)^2}{2{\gamma^2}^{(d)}} \right). \tag{5} \]
In Equation ((4)), \(\text{I}_{p}\) and \(\mathbf 1_p\) stand for the identity matrix with dimension \(p\) and the vector of \(\mathbb{R}^p\) where all elements are equal to 1. To efficiently sample from \(p\left(\mathbf \beta|\mathbf Z,\mathbf y,\mathbf X;\mathbf \theta^{(d)}\right)\) a preliminary singular vector decomposition of matrix \(\mathbf X\) is necessary. Once this decomposition is performed the overall complexity of the approximate E step is \(\mathcal{O}\left[M(p^2+pg)\right]\).
The M step
Using the \(M\) draws obtained by Gibbs sampling at iteration \(d\), the M step is straightforward as detailed in Equations ((6)) to ((10)). The overall computational complexity of that step is \(\mathcal{O}\left( Mpg\right)\).
\[ \begin{aligned} \label{eq:mstep_pi} \pi^{(d+1)}_k &= \frac{1}{Mp}\sum^{M}_{m=1}{ \sum^{p}_{j=1}{z^{(m,d)}_{jk} } }\text{,} \end{aligned} \tag{6} \]
\[\begin{aligned} \label{eq:mstep_b} b^{(d+1)}_k &= \frac{1}{Mp\pi^{(d+1)}_k}\sum^{M}_{m=1}{\sum^{p}_{j=1}{ z^{(m,d)}_{jk} \beta_{j}^{(m,d)} } }\text{,} \end{aligned} \tag{7} \]
\[\begin{aligned} \label{eq:mstep_gamma} {\gamma^2}^{(d+1)} &= \frac{1}{Mp}\sum^{M}_{m=1}{\sum^{p}_{j=1}{ \sum^{g}_{k=1}{z^{(m,d)}_{jk} \left(\beta_{j}^{(m,d)}-b^{(d+1)}_k\right)^2} } }\text{,} \end{aligned} \tag{8} \]
\[\begin{aligned} \label{eq:mstep_beta0} \beta^{(d+1)}_0 &= \frac{1}{n}\sum^{n}_{i=1}{ \left[y_i-\sum^{p}_{j=1}{\left(\frac{1}{M}\sum^{M}_{m=1}{\beta^{(m,d)}_j} \right)x_{ij}} \right]}\text{,} \end{aligned} \tag{9} \]
\[\begin{aligned} \label{eq:mstep_sigma} {\sigma^2}^{(d+1)} &= \frac{1}{nM}\sum^{M}_{m=1}{\sum^{n}_{i=1}{ \left(y_i - \beta^{(d+1)}_0-\sum^{p}_{j=1}{ \beta^{(m,d)}_j x_{ij} }\right)^2 } }\text{.} \end{aligned} \tag{10} \]
SEM algorithm
In most situations, the SEM algorithm can be considered as a special case of the MCEM algorithm (Celeux et al. 1996), obtained by setting \(M=1\). In model ((1)), such a direct derivation leads to an algorithm where the computational complexity remains quadratic with respect to \(p\). To reduce that complexity, we propose a SEM algorithm based on the integrated complete data likelihood \(p(\mathbf y, \mathbf Z|\mathbf X;\mathbf \theta)\) rather than \(p(\mathbf y,\mathbf \beta,\mathbf Z|\mathbf X;\mathbf \theta)\). A closed form of \(p(\mathbf y, \mathbf Z|\mathbf X;\mathbf \theta)\) is available and given in the following.
Closed form of the integrated complete data likelihood
Let the SVD decomposition of matrix \(\mathbf X\) be \(\mathbf {USV}'\), where \(\mathbf U\) and \(\mathbf V\) are respectively \(n \times n\) and \(p \times p\) orthogonal matrices, and \(\mathbf S\) is a \(n \times p\) rectangular diagonal matrix where the diagonal terms are the eigenvalues \(\left(\lambda^2_1,\ldots,\lambda^2_n\right)\) of matrix \(\mathbf {XX}'\). We now define \(\mathbf X^u = \mathbf {U}'\mathbf{X}\) and \(\mathbf y^u = \mathbf {U}'\mathbf{y}\). Let \(\mathbf M\) be the \(n\times (g+1)\) matrix where the first column is made of 1’s and where the additional columns are those of matrix \(\mathbf X^u \mathbf Z\). Let also \(\mathbf t=\left(\beta_0,\mathbf b\right)\in \mathbb{R}^{(g+1)}\) and \(\mathbf R\) be a \(n\times n\) diagonal matrix where the \(i\)-th diagonal term equals \(\sigma^2 + \gamma^2\lambda^2_i\). With these notations we can express the complete data likelihood integrated over \(\mathbf \beta\) as
\[\begin{aligned} \label{eq:ICDLL} \log p\left(\mathbf{y,Z}|\mathbf{X};\mathbf \theta\right) = -\frac{n}{2}\log\left( 2\pi\right)-\frac{1}{2}\sum^{n}_{i=1}{\log\left( \sigma^2 + \gamma^2\lambda^2_i\right)}-\frac{1}{2}\left(\mathbf{y}^u-\mathbf {Mt}\right)'\mathbf R^{-1}\left(\mathbf{y}^u-\mathbf {Mt}\right) \\ + \sum^{p}_{j=1}{\sum^{g}_{k=1}{z_{jk}\log \pi_k}}\text{.} \end{aligned} \tag{11} \]
Simulation step
To sample from \(p\left(\mathbf{Z}|\mathbf{y,X};\mathbf \theta\right)\) we use a Gibbs sampling strategy based on the conditional distributions \(p\left(\mathbf{z}_j|\mathbf y,\mathbf Z^{-j}, \mathbf X;\mathbf \theta\right)\), \(\mathbf Z^{-j}\) denoting the set of cluster membership indicators for all covariates but the \(j\)-th. Let \(\mathbf w^{-j} = \left(w^{-j}_1,\ldots,w^{-j}_n\right)'\), where \(w_i^{-j} = y^u_i-\beta_0 - \sum_{l\neq j}{\sum^{g}_{k=1}{z_{lk} x^u_{il} b_k}}\). The conditional distribution \(p(z_{jk} = 1|\mathbf{Z}^{-j},\mathbf{y},\mathbf{X};\mathbf \theta)\) can be written as
\[\label{eq:Gibbs} p(z_{jk} = 1|\mathbf{Z}^{-j},\mathbf{y},\mathbf{X};\mathbf \theta) \propto \pi_k\exp\left[-\frac{b^2_k}{2} \left(\mathbf{x}^{u}_j\right)'\mathbf{R}^{-1} \mathbf{x}^{u}_j+b_k \left(\mathbf{w}^{-j}\right)'\mathbf{R}^{-1}\mathbf x^{u}_j\right]\text{,} \tag{12} \]
where \(\mathbf{x}^{u}_j\) is the \(j\)-th column of \(\mathbf X^u\). In the classical SEM algorithm, convergence to \(p\left(\mathbf{Z}|\mathbf{y,X};\mathbf \theta\right)\) should be reached before updating \(\mathbf \theta\). However, a valid inference can still be ensured in settings when \(\mathbf \theta\) is updated only after one or few Gibbs iterations. These approaches are referred to as SEM-Gibbs algorithm (Biernacki and Jacques 2013). The overall computational complexity of the simulation step is \(\mathcal{O}\left( npg\right)\), i.e., it is linear in \(p\) and not quadratic any more, in contrast to the previous MCEM.
To improve the mixing of the generated Markov chain, we start the simulation step at each iteration by creating a random permutation of \(\left\{1,\ldots,p\right\}\). Then, according to the order defined by that permutation, we update each \(z_{jk}\) using \(p(z_{jk} = 1|\mathbf{Z}^{-j},\mathbf{y},\mathbf{X};\mathbf \theta)\).
Maximization step
\(\log p\left(\mathbf{y,Z}|\mathbf{X};\mathbf \theta\right)\) corresponds to the marginal log-likelihood of a linear mixed model (Searle et al. 1992), which can be written as
\[\label{eq:mixedmodel0} \mathbf{y}^u = \mathbf M \mathbf t + \mathbf {\lambda v} + \mathbf \varepsilon \tag{13} \]
where \(\mathbf v\) is an unobserved random vector such as \(\mathbf v \sim \mathcal{N}\left(0,\gamma^2\text{I}_{n}\right)\), \(\mathbf \varepsilon \sim \mathcal{N}\left(0,\sigma^2\text{I}_{n}\right)\) and \(\mathbf \lambda = \text{diag}\left(\lambda_1,\ldots,\lambda_n\right)\). The estimation of the parameters of model ((13)) can be performed using the EM algorithm, as in Searle et al. (1992). We adapt below the EM equations defined in Searle et al. (1992), using our notations. At iteration \(s\) of the internal EM algorithm, we define \(\mathbf R^{(s)} = {\sigma^2}^{(s)}\mathbf I_{n} + {\gamma^2}^{(s)}\mathbf \lambda'\mathbf \lambda\). The detailed internal E and M steps are given below.
Internal E step
\[\begin{align} v^{(s)}_\sigma & = \mathbb{E}\left[\left(\mathbf{y}^u - \mathbf{Mt}^{(s)} - \mathbf {\lambda v}\right)'\left(\mathbf{y}^u - \mathbf{Mt}^{(s)} - \mathbf {\lambda v}\right)|\mathbf{y}^u\right] \\ & = {\sigma^4}^{(s)}\left(\mathbf{y}^u - \mathbf{Mt}^{(s)}\right)'\mathbf R^{(s)}\mathbf R^{(s)}\left(\mathbf{y}^u - \mathbf{Mt}^{(s)}\right) + n\times {\sigma^2}^{(s)} - {\sigma^4}^{(s)} \sum^{n}_{i=1}{\frac{1}{{\sigma^2}^{(s)}+{\gamma^2}^{(s)}\lambda^2_i}}\text{.}\\ v^{(s)}_\gamma & = \mathbb{E}\left[\mathbf v'\mathbf v|\mathbf{y}^u\right] \\ & = {\gamma^4}^{(s)}\left(\mathbf{y}^u - \mathbf{Mt}^{(s)} \right)'\mathbf R^{(s)}\mathbf \lambda'\mathbf \lambda \mathbf R^{(s)}\left(\mathbf{y}^u - \mathbf{Mt}^{(s)}\right) + n \times{\gamma^2}^{(s)} - {\gamma^4}^{(s)} \sum^{n}_{i=1}{\frac{\lambda^2_i}{{\sigma^2}^{(s)}+{\gamma^2}^{(s)}\lambda^2_i}}\text{.}\\ \mathbf h^{(s)} & = \mathbb{E}\left[\mathbf{y}^u - \mathbf {\lambda v} |\mathbf{y}^u\right] = \mathbf{Mt}^{(s)} + {\sigma^2}^{(s)} \{R^{(s)}\}^{-1} \left(\mathbf{y}^u - \mathbf{Mt}^{(s)}\right)\text{.} \end{align} \]
Internal M step
\[\begin{aligned} {\sigma^2}^{(s+1)} &= v^{(s)}_\sigma / n\text{,}\\ {\gamma^2}^{(s+1)} &= v^{(s)}_\gamma / n\text{,}\\ \mathbf{t}^{(s+1)} &= \left[\mathbf M'\mathbf M\right]^{-1}\mathbf M'\mathbf h^{(s)}\text{.} \end{aligned}\]
Given a non-negative user-specified threshold \(\delta\) and a maximum number \(N_{\max}\) of iterations, Internal E and M steps are alternated until \[|\log p\left(\mathbf{y,Z}|\mathbf{X};\mathbf \theta^{(s)}\right)- \log p\left(\mathbf{y,Z}|\mathbf{X};\mathbf \theta^{(s+1)}\right)|<\delta\text{ or } s = N_{\max}\text{.}\] The computational complexity of the M step is \(\mathcal{O}\left( g^3 + ngN_{max}\right)\), thus not involving \(p\).
Attracting and absorbing states
Absorbing states. The SEM algorithm described above defines a Markov chain where the stationary distribution is concentrated around values of \(\mathbf \theta\) corresponding to local maxima of the likelihood function. This chain has absorbing states in values of \(\mathbf \theta\) such as \(\sigma^2=0\) or \(\gamma^2=0\). In fact, the internal M step reveals that updated values for \(\sigma^2\) and \(\gamma^2\) are proportional to previous values of those parameters.
Attracting states. We empirically observed that attraction around \(\sigma^2=0\) was quite frequent when using the MCEM algorithm, especially when \(p>n\) and when the number \(M\) of draws was small. We therefore advocate to use at least 5 draws (\(M \geq 5\) using option
nsampin the functionfitClere).
Model selection
Once the MLE \(\widehat{\mathbf \theta}\) is calculated (using one of the algorithms), the maximum log-likelihood and the posterior clustering matrix \(\mathbb{E}\left[\mathbf Z|\mathbf{y, X};\widehat{\mathbf \theta} \right]\) are approximated using MC simulations based on Equations ((11)) and ((12)). The approximate maximum log-likelihood \(\widehat{l}\), is then utilized to calculate AIC (Akaike 1974) and BIC (Schwarz 1978) for model selection. In model ((1)), those criteria can be written as
\[\begin{aligned} \text{BIC} &= -2\widehat{l} + 2(g+1)\log (n),\\ \text{AIC} &= -2\widehat{l} + 4(g+1)\text{.} \end{aligned}\]
An additional criterion for model selection, namely the ICL criterion (Biernacki et al. 2000) is also implemented in the R package clere. The latter criterion can be written as \[\text{ICL} = \text{BIC} - \sum^{p}_{j=1}{\sum^{g}_{k=1}{\pi_{jk} \log ( \pi_{jk} ) }}\text{,}\] where \(\pi_{jk} = \mathbb{E}\left[z_{jk}|\mathbf{y, X};\widehat{\mathbf \theta} \right]\).
Interpretation of the special group of variables associated with \(b_1=0\)
The constraint \(b_1=0\) is mainly driven by an interpretation purpose. The meaning of this group depends on both the total number \(g\) of groups and the estimated value of parameter \(\gamma^2\). In fact, when \(g>1\) and \(\gamma^2\) is small, covariates assigned to that group are likely less relevant to explain the response. Determining whether \(\gamma^2\) is small enough is not straightforward. However, when this property holds, we may expect the groups of covariates to be separated. This would for example translate in the posterior probabilities \(\pi_{j1}\) being larger than 0.7. In addition to the benefit in interpretation, the constraint \(b_1=0\), reduces the number of parameters to be estimated and consequently the variance of the predictions performed using the model.
4 Package functionalities
The R package clere mainly implements a function for parameter
estimation and model selection: the function fitClere(). Four
additional methods are also implemented in the package: for graphical
representation, plot(); summarizing the results, summary(); for
getting the predicted clusters of variables, clusters(); and for
making predictions from new design matrices, predict(). Examples of
calls to the functions presented in this section are given in the next
section.
The main function fitClere()
The main function fitClere() has only three mandatory arguments: the
vector of response y (size \(n\)), the matrix of explanatory variables
x (size \(n\times p\)) and the number \(g\) of groups of regression
coefficients which is expected. The optional parameter analysis, when
it takes the value "aic", "bic" or "icl", allows to test all the
possible number of groups between \(1\) and \(g\). The choice between the
two proposed algorithms is possible thanks to the parameter algorithm,
but we encourage the users to use the default value, the SEM algorithm,
which generally over-performs the MCEM algorithm (see the first
experiment of the next section).
Several other parameters allow to tune the different numbers of iterations of the estimation algorithm. In general, the higher are these parameter values, the better is the quality of the estimation but the heavier is also the computing time. What we advice is to use the default values, and to graphically check the quality of the estimation with plots as in Figure 1: If the values of the model parameters are quite stable for a sufficient large part of the iterations, this indicates that the results are ok. If the stability is not reached sufficiently early before the end of the iterations, a higher number of iterations should be chosen.
Finally, among the remaining parameters (note that the complete list can
be obtained with help("fitClere")), two are especially important:
parallel allows to run parallel computations (if compatible with the
user’s computer) and sparse allows to impose that one of the
regression parameters is equal to 0 and thus to introduce a cluster of
not significant explanatory variables.
Methods summary(), plot(), clusters() and predict()
The summary() method for an object returned by fitClere() prints an
overview of the estimated parameters and returns the estimated
likelihood and information based model selection criteria (AIC, BIC and
ICL). The corresponding plot() method produces graphs such as ones
presented in Figure 1.
nBurn of iterations
discarded before calculating maximum likelihood
estimates.The clusters() method takes one argument of class “Clere” as returned
by fitClere() and a threshold argument. This function assigns each
variable to the group where associated conditional probability of
membership is larger than the given threshold. If conditional
probabilities of membership are larger than the specified threshold for
more than one group, then the group having the largest probability is
returned and a warning is printed. If, moreover, there are several ex
aequo on that largest probability, then the group with the smallest
index is returned. When threshold = NULL, the
maximum a posteriori (MAP) strategy is used to infer the clusters.
The predict() method has two arguments: a “Clere” object and a design
matrix \(\mathbf X_{new}\). Using that new design matrix, the predict()
method returns an approximation of
\(\mathbb{E}\left[\mathbf X_{new}\mathbf \beta|\mathbf y, \mathbf X; \hat{\mathbf \theta}\right]\).
5 Numerical experiments
This section presents two sets of numerical experiments. The first set of experiments aims at comparing the MCEM and SEM algorithms in terms of computational time and estimation or prediction accuracy. The second set of experiments is aimed at comparing CLERE to standard dimension reduction techniques. The latter comparison is performed on both simulated and real data.
SEM algorithm versus MCEM algorithm
Description of the simulation study
In this section, a comparison between the SEM algorithm and the MCEM algorithm is performed. This comparison is performed using the four following performance indicators:
Computational time (CT) to run a pre-defined number of SEM/MCEM iterations. This number was set to 2,000 in this simulation study.
Mean squared estimation error (MSEE) defined as \[\text{MSEE}_a = \mathbb{E}\left[(\mathbf{\theta}-\widehat{\mathbf{\theta}}_a)'(\mathbf{\theta}-\widehat{\mathbf{\theta}}_a)\right]\text{,}\] where \(a\in\left\{\text{\texttt{"SEM","MCEM"}}\right\}\) and \(\widehat{\mathbf{\theta}}_a\) is an estimated value for parameter \(\mathbf \theta\) obtained with algorithm \(a\). Since \(\mathbf \theta\) is only known up to a permutation of the group labels, we chose the permutation leading to the smallest MSEE value.
Mean squared prediction error (MSPE) defined as \[\text{MSPE}_a = \mathbb{E}\left[(\mathbf{y^v-X^v\widehat{\theta}_a})'(\mathbf{y^v-X^v\widehat{\theta}_a})\right]\text{,}\] where \(\mathbf y^v\) and \(\mathbf X^v\) are respectively a vector of responses and a design matrix from a validation data set.
Maximum log-likelihood (ML) reached. This quantity was approximated using 1,000 samples from \(p(\mathbf Z|\mathbf y;\widehat{\mathbf \theta})\).
Three versions of the MCEM algorithm were proposed for comparison with
the SEM algorithm, depending on the number \(M\) (or nsamp) of Gibbs
iterations used to approximate the E step. That number was varied
between 5, 25 and 125. We chose these iterations numbers so as to cover
different situations, from a situation in which the number of iterations
is too small to a situation in which that number seems sufficient to
expect having reached convergence of the simulated Markov
chain. Those versions were respectively denoted
MCEM\(_5\), MCEM\(_{25}\) and MCEM\(_{125}\). The comparison was performed
using 200 simulated data sets. In order to consider high-dimensional
situations with sizes allowing to reproduce multiple simulations in a
reasonable time,each training data set consisted
of \(n=25\) individuals and \(p=50\) variables. Validation data sets used to
calculate MSPE consisted of 1,000 individuals each. All covariates were
simulated independently according to the standard Gaussian distribution:
\[\forall(i,j)\text{
}x_{ij}\sim\mathcal{N}(0,1)\text{.}\] Both training and validation
data sets were simulated according to model ((1)) using
\(\beta_0 = 0\), \(\mathbf b = (0,3,15)'\),
\(\mathbf \pi = (0.64,0.20,0.16)'\), \(\sigma^2=1\) and \(\gamma^2=0\). This
is equivalent to simulate data according to the standard linear
regression model defined by: \[y_i \sim
\mathcal{N}\left(\sum^{32}_{j=1}{0 \times x_{ij}} +
\sum^{42}_{j=33}{3 \times x_{ij}} + \sum^{50}_{j=43}{15 \times
x_{ij}},1\right).\]
All algorithms were run using 10 different random starting points. Estimates yielding the largest likelihood were then used for the comparison.
Results of the comparison
Table 1 summarizes the results of the comparison between the algorithms. The MCEM\(_{5}\) algorithm was 1.3 times faster than the SEM algorithm however the latter algorithm poorly performed regarding all other performance criteria (estimation quality, prediction error, likelihood maximization). This observation illustrates the importance of setting a large number \(M\) of draws to improve the estimation. Indeed, when increasing this number to 25 or 125, we observed an improvement in the estimation accuracy but no noticeable improvement in the likelihood. In turn, the SEM algorithm was quite efficient compared to the MCEM\(_{25}\) and MCEM\(_{125}\) algorithms. This algorithm not only ran faster (between 3 and 13-fold faster than MCEM\(_{25}\) and MCEM\(_{125}\) – see Table 1), but also reached predictive performances close to the oracle (i.e., using the true parameter). These good performances are mainly explained by the fact that the SEM algorithm most of the time (66.5% of the time) reached a better likelihood than the other algorithms.
The results of this simulation study were made available as an internal
data set named algoComp in the R package clere. More details can be
obtained using the command help("algoComp").
| % of times | Median | ||
| Performance indicators | Algorithms | the algorithm was best | (Std. Err.) |
| CT (seconds) | SEM | 0 | 2.5 ( 0.053 ) |
| MCEM\(_5\) | 100 | 1.9 ( 0.016 ) | |
| MCEM\(_{25}\) | 0 | 7.1 ( 0.027 ) | |
| MCEM\(_{125}\) | 0 | 32.8 ( 0.121 ) | |
| MSEE | SEM | 58 | 0.31 ( 10.4 ) |
| MCEM\(_5\) | 12 | 20.14 ( 2843.3 ) | |
| MCEM\(_{25}\) | 16.5 | 8.86 ( 3107.5 ) | |
| MCEM\(_{125}\) | 13.5 | 4.02 ( 5664.2 ) | |
| MSPE | SEM | 51.5 | 1.3 ( 46.1 ) |
| MCEM\(_5\) | 12 | 75.7 ( 64.3 ) | |
| MCEM\(_{25}\) | 15.5 | 58.7 ( 55.2 ) | |
| MCEM\(_{125}\) | 21 | 51.6 ( 51.1 ) | |
| True parameter | — | 1.1 ( 0.013 ) | |
| ML | SEM | 66.5 | \(-79.3\) ( 1.2 ) |
| MCEM\(_5\) | 11.5 | \(-110.7\) ( 2.0 ) | |
| MCEM\(_{25}\) | 14.5 | \(-111.6\) ( 1.9 ) | |
| MCEM\(_{125}\) | 7.5 | \(-116.2\) ( 1.7 ) | |
| True parameter | — | \(-77.6\) ( 0.34 ) |
Comparison with other methods
Description of the methods
In this section, we compare our model to standard dimension reduction
approaches in terms of MSPE. Although a more detailed comparison was
suggested in Yengo et al. (2014), we propose here a quick illustration of the
relative predictive performance of our model. The comparison is achieved
using data simulated according to the scenario described above in
Section SEM algorithm versus MCEM algorithm . The methods selected for comparison are the Ridge regression
(Hoerl and Kennard 1970), the Elastic net (Zou and Hastie 2005), the LASSO (Tibshirani 1996),
PACS (Sharma et al. 2013), the method of Park and colleagues (Park et al. 2007 subsequently denoted AVG) and the Spike and Slab model (Ishwaran and Rao 2005 subsequently denoted SS). The first three methods are implemented in the
freely available R package glmnet. With the latter package, the tuning
parameter lambda was selected using the function cv.glm (with 5
folds) aiming at minimizing the mean squared error (option
type = "mse"). In particular for the Elastic net, the second tuning
parameter alpha (measuring the amount of mixture between the \(L^1\) and
\(L^2\) penalty) was jointly selected with lambda to minimize the mean
squared error. The R package glmnet proposes a procedure for
automatically selecting values for lambda. We therefore used this
default procedure while we selected alpha among
\(\{0, 0.1,0.2,\ldots,0.9,1\}\). The PACS methodology proposes to estimate
the regression coefficients by solving a penalized least squares
problem. It imposes a constraint on \(\mathbf \beta\) that is a weighted
combination of the \(L^1\) norm and the pairwise \(L^\infty\) norm.
Upper-bounding the pairwise \(L^\infty\) norm enforces the covariates to
have close coefficients. When the constraint is strong enough, closeness
translates into equality achieving thus a grouping property. For PACS,
no software was available. Only an R script was released on Bondell’s
web page1. Since this R script was running very slowly, we decided to
reimplement it in C++ and observed a 30-fold speed-up of computational
time. Similarly to Bondell’s R script, our implementation uses two
parameters lambda and betawt. Our reimplementation of Bondell’s
script was included in the R package clere in the function
fitPacs(). In Sharma et al. (2013), the authors suggest assigning betawt with
the coefficients obtained from a ridge regression model after the tuning
parameter was selected using AIC. In this simulation study we used the
same strategy; however the ridge parameter was selected via 5-fold cross
validation. 5-fold CV was preferred to AIC since selecting the ridge
parameter using AIC always led to estimated coefficients equal to zero.
Once betawt was selected, lambda was chosen via 5-fold cross
validation among the following values: 0.01, 0.02, 0.05, 0.1, 0.2, 0.5,
1, 2, 5, 10, 20, 50, 100, 200 and 500. All other default parameters of
their script were unchanged. The AVG method is a two-step approach. The
first step uses hierarchical clustering of covariates to create
surrogate covariates by averaging the variables within each group. Those
new predictors are afterwards included in a linear regression model,
replacing the primary variables. A variable selection algorithm is then
applied to select the most predictive groups of covariates. To implement
this method, we followed the algorithm described in Park et al. (2007) and
programmed it in R. The Spike and Slab model is a Bayesian approach for
variable selection. It is based on the assumption that the regression
coefficients are distributed according to a mixture of two centered
Gaussian distributions with different variances. One component of the
mixture (the spike) is chosen to have a small variance, while the other
component (the slab) is allowed to have a large variance. Variables
assigned to the spike are dropped from the model. We used the R package
spikeslab to run the Spike and Slab models. Especially, we used the
function spikeslab from that package to detect influential variables.
The number of iterations used to run the function spikeslab was 2,000
(1,000 discarded).
When running fitClere(), the number nItEM of SEM iterations was set
to 2,000. The number g of groups for CLERE was chosen between 1 and 5
using AIC (option analysis = "aic"). Two versions of CLERE were
considered: the one with all parameters estimated and the one with \(b_1\)
set to 0. The latter approach is subsequently denoted CLERE\(_0\) (option
sparse = TRUE).
Results of the comparison
Figure 2 summarizes the comparison between the methods. In this simulated scenario, CLERE outperformed the other methods in terms of prediction error. These good performances were improved when parameter \(b_1\) was set to 0. CLERE was also the most parsimonious approach with an averaged number of estimated parameters equal to 7.7 (6.9 when \(b_1=0\)). The second best approach was PACS which also led to parsimonious models. The superiority of such methods could be expected since in the simulation model the regression coefficients are gathered in three groups. Overall variable selection approaches yielded the largest prediction error in this simulation. CLERE, PACS and Spike and Slab had the largest computational times (CT). For CLERE and PACS this loss in CT was compensated by a a strong improvement in prediction error as explained above, while Spike and Slab yielded the worst prediction error in addition to being the slowest approach in this example.
The results of this simulation study were made available as an internal
data set in the R package clere. The object is called numExpSimData
and more details can be obtained using the command
help("numExpSimData").
Real data sets analysis
Description of the data sets
We used in this section the real data sets Prostate and eyedata from
the R packages lasso2
(Lokhorst et al. 2014) and flare
(Li et al. 2014) respectively. The Prostate data set comes from a study that
examined the correlation between the level of prostate specific antigen
and a number of clinical measures in \(n=97\) men who were about to
receive a radical prostatectomy. This data set is a benchmark data set
used in multiple publications about high-dimensional regression model,
including Tibshirani (1996; Hastie et al. 2001), and was chosen here in
order to illustrate the performance of CLERE in comparison to the
competing methods. We used the prostate specific
antigen (variable lpsa) as response variable and the \(p=8\) other
measurements as covariates.
The eyedata data set is extracted from the published study of
Scheetz et al. (2006). This data set consists of gene expression levels
measured at \(p=200\) probes in \(n=120\) rats. The response variable
utilized was the expression of the TRIM32 gene which is a biomarker of
the Bardet-Bidel Syndrome (BBS). We chose this data set to illustrate
the performances of CLERE on a (manageable) high-dimensional problem
which is the actual context for which this method was developped
(Yengo et al. 2014).
Those two data sets were utilized to compare CLERE to the same methods used in the previous section where the simulation study was presented. All methods were compared in terms of out-of-sample prediction error estimated using 5-fold cross validation (CV). Those CV statistics were then averaged and compared across the methods in Table 2.
Running the analysis
Before presenting the results of the comparison between CLERE and its
competitors, we illustrate the commands to run the analysis of the
Prostate data set. The data set is loaded by typing:
R> data("Prostate", package = "lasso2")
R> y <- Prostate[, "lpsa"]
R> x <- as.matrix(Prostate[, -which(colnames(Prostate) == "lpsa")]) Possible training (xt and yt) and validation (xv and yv) sets
are generated as follows:
R> itraining <- 1:(0.8*nrow(x))
R> xt <- x[ itraining,]; yt <- y[ itraining]
R> xv <- x[-itraining,]; yv <- y[-itraining]The fitClere() function is run using the AIC to select the number of
groups between 1 and 5. To lessen the impact of local minima in the
model selection, 5 random starting points are used. This can be
implemented by:
R> Seed <- 1234
R> mod <- fitClere(y = yt, x = xt, g = 5, analysis = "aic", parallel = TRUE,
+ nstart = 5, sparse = TRUE, nItEM = 2000, nBurn = 1000,
+ nItMC = 10, dp = 5, nsamp = 1000, seed = Seed)
R> summary(mod)
-------------------------------
| CLERE | Yengo et al. (2013) |
-------------------------------
Model object 2 groups of variables ( Selected using AIC criterion )
---
Estimated parameters using SEM algorithm are
intercept = -0.1339
b = 0.0000 0.4722
pi = 0.7153 0.2848
sigma2 = 0.395
gamma2 = 4.065e-08
---
Log-likelihood = -78.31
Entropy = 0.5464
AIC = 168.63
BIC = 182.69
ICL = 183.23
R> plot(mod)Running the command plot(mod) generates the plot given in
Figure 1. We can also access the cluster memberships by
running the command clusters(). For example, running the command
clusters(mod, threshold = 0.7) yields
R> clusters(mod, thresold = 0.7)
lcavol lweight age lbph svi lcp gleason pgg45
2 2 1 1 1 1 1 1 In the example above 2 variables, being the cancer volume (lcavol) and
the prostate weight (lweight), were assigned to group 2
(\(b_2=0.4737\)). The other 6 variables were assigned to group 1
(\(b_1=0\)). Posterior probabilities of membership are available through
the slot P in the object of class “Clere”.
R> mod@P
Group 1 Group 2
lcavol 0.000 1.000
lweight 0.000 1.000
age 1.000 0.000
lbph 1.000 0.000
svi 0.764 0.236
lcp 1.000 0.000
gleason 1.000 0.000
pgg45 1.000 0.000 The covariates were respectively assigned to their group with a
probability larger than 0.7. Moreover, given that parameter \(\gamma^2\)
had a very small value (\(\widehat{\gamma^2} = 4.065\times10^{-8}\)), we
can argue that cancer volume and prostate weight are the only relevant
explanatory covariates. To assess the prediction error associated with
the model we can run the command predict() as follows:
R> error <- mean((yv - predict(mod, xv))^2)
R> error
[1] 1.543122 Results of the analysis
Table 2 summarizes the prediction errors and the number
of parameters obtained for all the methods. CLERE\(_0\) had the lowest
prediction error in the analysis of the Prostate data set and the
second best performance for the eyedata data set. The AVG method was
also very competitive compared to the variable selection approaches
stressing thus the relevance of creating groups of variables to reduce
the dimensionality (especially in the eyedata data set). It is worth
noting that in both data sets, imposing the constraint \(b_1=0\) improved
the predictive performance of CLERE.
In the Prostate data set, CLERE robustly identified two groups of
variables representing influential (\(b_2>0)\) and not relevant variables
(\(b_1=0\)). In the eyedata data set in turn, AIC led to selecting only
one group of variables. However, this did not lessen the predictive
performance of the model since CLERE\(_0\) was second best after AVG,
while needing significantly less parameters. PACS performed badly in
both data sets. The Elastic net showed good predictive performances
compared to the variable selection methods like LASSO or the Spike and
Slab model. Ridge regression and Elastic net had comparable results in
both data sets. In line with the results of the simulation study, we
observed that despite a larger computational time (CT), CLERE and
CLERE\(_0\) had a reduced mean squared error compared to the fastest
methods. However, this improvement was less substantial than observed in
the simulation study given the differences in CT. This increased CT may
be explained by the fact that no simple stopping rule is proposed when
fitting CLERE. We may therefore contemplate that a smaller number of SEM
iterations could have been used to yield a similar prediction error.
Indeed, when looking at Figure 1, we see that convergence
was achieved almost from the first 10 iterations. Still, the observed CT
for CLERE being around 22s for the eyedata data set and around 3s for
the Prostate data set remains affordable in these examples.
The results of this analysis on real data were made available as an
internal data set named numExpRealData in the R package clere. Using
the command help("numExpRealData") more details can be obtained.
| 100\(\times\)Averaged CV statistic | Number of parameters | CT (seconds) | |
| (Std. Error) | (Std. Error) | (Std. Error) | |
Prostate data set |
|||
| LASSO | 90.2 ( 29 ) | 5.6 ( 0.7 ) | 0.064 ( 0.007 ) |
| RIDGE | 86.8 ( 24 ) | 8.0 ( 0 ) | 0.065 ( 0.002 ) |
| Elastic net | 90.3 ( 24 ) | 8.0 ( 0 ) | 0.065 ( 0.002 ) |
| STEP | 442.4 ( 137 ) | 8.0 ( 0 ) | 0.004 ( 0.001 ) |
| CLERE | 82.4 ( 25 ) | 6.0 ( 0 ) | 1.1 ( 0.1 ) |
| CLERE\(_0\) | 74.5 ( 26 ) | 5.0 ( 0 ) | 2.7 ( 0.8 ) |
| Spike and Slab | 85.6 ( 26 ) | 5.6 ( 0.7 ) | 4.2 ( 0.03 ) |
| AVG | 90.2 ( 27 ) | 6.2 ( 0.4 ) | 0.44 ( 0.06 ) |
| PACS | 90.6 ( 34 ) | 5.6 ( 0.4 ) | 0.053 ( 0.002 ) |
eyedata |
|||
| LASSO | 0.73 ( 0.1 ) | 21.2 ( 2 ) | 0.18 ( 0.01 ) |
| RIDGE | 0.74 ( 0.1 ) | 200.0 ( 0 ) | 0.24 ( 0.004 ) |
| Elastic net | 0.74 ( 0.1 ) | 200.0 ( 0 ) | 0.23 ( 0.003 ) |
| STEP | 1142.6 ( 736 ) | 95.0 ( 0 ) | 0.083 ( 0.002 ) |
| CLERE | 0.73 ( 0.1 ) | 4.0 ( 0 ) | 21.5 ( 0.2 ) |
| CLERE\(_0\) | 0.72 ( 0.1 ) | 3.0 ( 0 ) | 21.1 ( 0.1 ) |
| Spike and Slab | 0.81 ( 0.2 ) | 12.4 ( 0.9 ) | 10.3 ( 0.1 ) |
| AVG | 0.70 ( 0.04 ) | 15.6 ( 2 ) | 10.6 ( 0.4 ) |
| PACS | 2.0 ( 0.9 ) | 3.0 ( 0.3 ) | 108.9 ( 28 ) |
6 Conclusions
We presented in this paper the R package clere. This package implements two efficient algorithms for fitting the CLusterwise Effect REgression model: the MCEM and the SEM algorithms. The MCEM algorithm is to be preferred when \(p<n\); the SEM algorithm is more efficient for high-dimensional data sets (\(n<p\)). The good performance of SEM over MCEM could have been expected regarding the computational complexities of the two algorithms that are \(\mathcal{O}\left( npg + g^3 + N_{max}ng\right)\) and \(\mathcal{O}\left( M(p^2 + pg)\right)\) respectively. In fact, as long as \(p>n\), the SEM algorithm has a lower complexity. However, the computational time to run our SEM algorithm is more variable compared to MCEM as its does not have a closed form. We finally advocate the use of the MCEM algorithm only when \(p\ll n\). Another important feature for model interpretation is proposed by constraining the model parameter \(b_1\) to equal 0, which leads to variable selection. Such a constraint may also lead to a reduced prediction error. We illustrated on a real data set, how to run an analysis, based on a detailed R script. Although our numerical experiments showed that the CLERE method tended to be slower than variable selection methods, it still provided better or competitive predictive performance. In addition, the CLERE model was often more parsimonious than other models and was easily interpretable since groups of regression coefficients/variables could be summarized using a single parameter.
Our model can easily be extended to the analysis of binary responses. This extension will be made available in a forthcoming version of the package. Another direction for future research will be to develop an efficient stopping rule for the proposed SEM algorithm, specific to our context. Such a criterion is expected to improve the computational performance of our estimation algorithm.
CRAN packages used
glmnet, spikeslab, clere, Rcpp, RcppEigen, lasso2, flare
CRAN Task Views implied by cited packages
Bayesian, HighPerformanceComputing, MachineLearning, NumericalMathematics, Survival
Note
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