1. Introduction
For every soccer team, the coach is responsible for directing the team to victory. The primary aim is to score as many goals as possible and, at the same time, thwart the other team from doing the same. The coach may choose some players over others. Even among team members, some players attack, others defend, whereas some are good as half-back players. Moreover, the core players that form the playing team do not remain the same throughout the game. Keeping in mind the dynamics of the game, the coach may direct some players to replace others, ensuring the primary aim of the game, to win, remains intact. In close comparison with this framework, the cell does not operate very differently. The coach of the cell, the DNA within the nucleus, directs different team members, TF and genes, to execute cellular functions and complex biological processes, which help the cell to adapt to varying dynamics, including external stimuli, as well as internal changes. The team members, TFs and genes, work together to express or suppress different metabolic pathways at different instances of the cell’s life. Particularly, these TFs contain DNA binding domains that allow them to bind to specific regions of DNA, called promoters [1]. By binding to these promoters, TFs initiate the process of converting genes into proteins. Transcription factor activities (TFAs) refer not only to the connectivity of any particular TF, but also to its level of activity. The connectivity of a particular TF informs its team members to collaborate in order to regulate RNA polymerase, which in its turn controls in terms of expressing or suppressing genes. TFAs cannot be measured directly; rather, they can be inferred from gene expression data. Furthermore, TRN represent interactions between genes and TFAs within a cell and offer a global perspective in the cellular behavior. Understanding the structure of TRNs and estimating TFAs provide insight into the cellular dynamics present in healthy and diseased tissues and organs and hold the potential to help in diagnosing, characterizing and determining cures for various diseases [2].
In the literature, several computational frameworks have been proposed to analyze regulatory interactions, which are briefly summarized below. The first class models the TRN as a dynamic system. Particularly, [3] and [4] describe gene expression as a linear and continuous time first-order differential equation. On the other hand, Boolean network models [5,6] quantize gene expressions by only two discrete levels: ON and OFF. The expression level of each gene is the Boolean function of the expression levels of other genes. These methods are generally performed using a small number of time series data and, thus, lead to an under-determined problem [7]. Another approach for TRN reconstruction is referred to as the co-expression (or relevance) networks, in which two genes are connected if the similarity between them exceeds a predefined threshold. Examples of similarity measures used in constructing relevance networks include correlation [8] and mutual information [9,10]. Relevance networks are helpful to understand the fundamental topological features of biological networks, but they do not infer causal relations among genes. The algorithms falling into the third category are commonly described as probabilistic graphical models [11,12,13], which include Gaussian graphical models (GGMs) and Bayesian networks (BNs). In GGMs, the network or graph is constructed based on the notion of conditional independence, and two genes are connected if and only if they are independent given the expression levels of all other genes. GGMs are formulated using undirected graphs and represent an example of full conditional models, since the conditional dependency is considered with respect to all other genes. On the other hand, BNs entail directed acyclic graphs, and the conditional dependency is measured with respect to all subsets of the other genes [11,14]. One limitation of probabilistic graphic models is that they have strong assumptions on the joint distribution that prevent representing or interpreting some biological relationships. For example, cyclic graphs are not allowed in the BN framework. In this way, it ignores self-feedback loops among genes that are natural features in genetic networks. Additionally, the applications of probabilistic graphic models are generally limited to the network with the number of experimental measurements significantly larger than the number of genes, since analyzing the structure of large-scaled genetic networks using probabilistic graphic models is highly complex. Besides dynamic models, co-expression networks and probabilistic graphical models, structural equation modeling (SEM) also represents a widely-used technique for TRN inference [15,16]. Generally, an SEM consists of a structural model and a measurement model. The structural model describes the causal relations between the latent variables, while the measurement model depicts the relations between latent variables and observed measurements.
Recently, studies dedicated to TRN inference using the network component analysis (NCA) technique have begun to emerge in the literature [17]. NCA establishes a parameter estimation problem and reconstructs TRNs following a statistical signal processing viewpoint. Since NCA-based algorithms do not require time series data, they can collect the experimental data from different time intervals and combine them to increase the samples size and prevent the under-determination problem. Even with a limited number of experiments, NCA-based algorithms are still able to reconstruct TRNs with a large number of TFs and genes (See Section 3.3 for more details). Moreover, NCA-based algorithms take advantage of some prior knowledge about the connectivity patterns of the genetic network, which is becoming available via high-throughput experiments [18] or data mining of interaction information [19,20,21]. The assumed mathematical model for NCA is represented by the following system of linear equations [17]:(1) X=AS+Γ where X∈RN×K represents the log ratios of expression values of N genes at K time points of the microarray dataset, A∈RN×M denotes the connectivity strength between N genes and M TFs, S∈RM×K stands for the activities of M TFs at K time points and Γ∈RN×K represents the measurement noise. Examples of two TRNs with six genes and four TFs, but different connectivity topologies, are shown in Figure 1.
Generally, in Equation (1), X cannot be uniquely decomposed as the product of two matrices A and S, unless further constraints are imposed. Principal component analysis (PCA) [22] and independent component analysis (ICA) [23] represent two conventional statistical algorithms that can provide valid solutions provided that the input signals present in S are independent and/or orthogonal. However, such an assumption generally does not hold for biological signals in practice. Accounting for this fact, Liao et al. [17] proposes NCA, which incorporates the prior information about TF-gene regulation, to infer TRNs. As will be discussed in detail in Section 2, NCA is an iterative computational algorithm that ensures the uniqueness of decomposition solutions. Due to some drawbacks of NCA, such as the stringent conditions required to apply NCA, several alternative NCA-based algorithms have been proposed in the literature to improve NCA from different perspectives, such as less restrictive assumptions, lower computational complexity and higher robustness against noise, outliers and modeling errors.
Figure 1  Examples of two transcription regulatory networks (TRNs) with six genes and four transcription factors (TFs), but different connectivity topologies.   The rest of the paper, which proposes to provide a review of the major algorithms reported for NCA, is organized as follows. Section 2 introduces the NCA framework and the mathematical details of the NCA algorithm. Extensions of NCA are presented in Section 3. These extensions still rely on the NCA algorithm, but improve the applicability range of NCA by requiring less stringent assumptions. In Section 4, alternative NCA-based algorithms proposed in the literature for TRN inference are surveyed. A few illustrative computer simulation results highlighting the performance of major NCA algorithms are presented in Section 5. In addition, the comparison of these algorithms and some recommendations on how to choose the appropriate algorithm are discussed in Section 6 based on the simulation results in Section 5. Finally, Section 7 summarizes the content of this paper.