Consider a -dimensional, zero-mean, stationary, Gaussian random process , . We are interested in learning the conditional independence graph (CIG) [1, 2, 3, 4] of from the finite-length observation of a single process realization. We consider the high-dimensional regime, where the number of observed process samples is (much) smaller than [5, 6, 7, 8, 9, 10, 11]
. In this case, consistent estimation of the CIG is possible only if structural assumptions on the vector process are made. Specifically, we will consider CIGs that aresparse in the sense of containing relatively few edges. This problem is relevant, e.g., in the analysis of the time evolution of air pollutant concentrations [1, 2] and in medical diagnostic data analysis .
, specifically on vector autoregressive (VAR) models. In this paper, we develop and analyze a nonparametric approach, which only requires the vector process to be spectrally smooth. The smoothness notion we use is quantified in terms of moments of the matrix-valued auto-covariance function (ACF) of the process. Compared to[10, 8, 9, 12, 13], our approach applies to a considerably more general class of processes including VAR processes as a special case.
Our main conceptual contribution resides in recognizing that the problem of inferring the sparse CIG of a Gaussian vector process is a special case of a block-sparse signal recovery problem [14, 15, 16], i.e., a multitask learning problem [17, 18]. While for the special case of a VAR process with sparse CIG, a block-sparse structure was already identified in , we show that a (different) block-sparse structure exists for general stationary time series. This stems from the fact that the CIG of a general stationary time series is encoded in the continuous ensemble of values of the spectral density matrix , . Based on this insight, we develop a multitask LASSO [17, 19] formulation of the sparse CIG estimation problem. Our main analytical contribution is Theorem 4.1, which provides conditions for our scheme to correctly identify the CIG with high probability.
2 Problem Formulation
Consider a -dimensional, zero-mean, stationary, real, Gaussian random process with (matrix-valued) ACF . The ACF is assumed summable, i.e., for some matrix norm . The spectral density matrix (SDM) of the process is defined as , and we assume that
for all , where and
denote the smallest and largest eigenvalue of, respectively. In particular, (1) implies that the matrix is nonsingular for all . We will furthermore require the vector process to be such that satisfies certain smoothness properties which are expressed in terms of moments of the ACF defined as
Here, is a nonnegative weight function that typically increases with .
The CIG of the process is the graph with node set representing the scalar component processes and edge set , where if and only if the component processes and are conditionally independent given all remaining component processes . The neighborhood of node is defined as . We restrict ourselves to processes with sparse CIG in the sense that
The graphical model selection problem we consider can now be stated as the problem of inferring the CIG , or more precisely its edge set , from the observation , where is the sample size. Since is Gaussian with nonsingular for all , it follows from [1, 2, 20] that if and only if for all . The edge set therefore corresponds to the locations of the nonzero entries of , and our graphical model selection problem amounts to determining these locations. We are interested in estimating the CIG from regularly spaced samples , with , , and large enough for the following to hold:
The implication from left to right in (4) follows trivially from what was said above. The implication from right to left is satisfied, e.g., for processes with all entries of being rational functions in , provided that is larger than the maximum degree of the numerator polynomials of . Another sufficient condition for the implication from right to left to hold is the following.
The restriction to the finite set of frequencies is made for expositional convenience. The general theory developed in this paper goes through for , with our inference procedure becoming a multitask learning problem with a continuum instead of a finite number, , of tasks.
3 Graphical Model Selection
Our method for inferring the CIG is inspired by the approach employed in [10, 7]. We first estimate the SDM , , by means of a multivariate spectral estimator. Then we use this estimate to perform neighborhood regression, which yields an estimate of the support (i.e., the locations of the nonzero entries) of and, via (4), the CIG. Neighborhood regression is performed by solving a multitask learning problem using multitask LASSO (mLASSO) .
With regards to the first step, it is natural to estimate using the multivariate Blackman-Tukey estimator :
Here, for and, by symmetry of the ACF, for . Furthermore, the window function is chosen such that
is positive semidefinite. All window functions with nonnegative discrete-time Fourier transform are admissible[21, Sec. 2.5.2]. In the high-dimensional regime, where the number of observations is smaller than the number of nodes, the matrices in (5) will be rank-deficient (to see this, note that each column of is a linear combination of ). Simply inverting , for , and inferring the edge set via (4) is therefore not possible.
To cope with this issue, we reduce the problem of finding the support of the matrices to multitask learning problems (one for each node). This can be done as follows. First note that, because of (4), the union of the supports of the th rows of the matrices , determines the neighborhood . The , as shown next, can then be obtained by solving multitask learning problems. For simplicity of exposition and without loss of generality we assume in the following. Given , , we define and via
where is the positive definite square root of . We next decompose into its orthogonal projection onto and the orthogonal complement thereof according to
with and , where is the pseudo-inverse of . The significance of this construction is expressed by the following proposition.
The neighborhood of node is determined by the joint support of the , according to
where the addition in (8) is elementwise.
We first note that (4) implies
where is the vector containing the entries . Next, we show that , which will finalize the proof. By the construction of and in (6), we have
The essence of Proposition 3.1 is that it reduces the problem of determining the neighborhood to that of finding the joint support of the . Recovering the based on the observations (7) is now recognized as a multitask learning or generalized multiple measurement vector problem [17, 18, 23], which in turn is a special case (with additional structure) of a block-sparse signal recovery problem [14, 15, 16]. Specifically, a multi-task learning problem can be cast as a block-sparse signal recovery problem by stacking the individual linear models in (7) into a single linear model; the resulting system matrix is block-diagonal. The approach described in  for VAR processes, albeit leading to a block-sparse recovery problem, does not result in a block-diagonal system matrix.
An efficient method for solving the multi-task learning problem at hand is the mLASSO, which can be formulated as follows (e.g., ):
where is the LASSO parameter, and with given by . To compute the estimate , one does not need to compute and by taking the square root of as in (6). To see this, we note that (LABEL:equ_multitask_lasso_graph_sel_statistic) is equivalent to
and, by (9), and are submatrices of . Therefore, working with (3) instead of (LABEL:equ_multitask_lasso_graph_sel_statistic) has the advantage that the square root does not need to be computed in order to determine .
In summary, we have shown that the neighborhood can be found via the support of the mLASSO estimate (3). Recognizing that this estimate depends on , which is unknown, motivates the following inference algorithm (for general ), which simply uses instead of in (3).
Given the observation , the parameter , the threshold parameter , and the mLASSO parameter (the choice of and will be discussed in Section 4), perform the following steps:
Step 1: For each , compute the SDM estimate according to (5).
Step 2: Compute the mLASSO estimate for each as
where is the submatrix of obtained by deleting its th column and th row, and is obtained by deleting the th entry in the th column of .
Step 3: Estimate the neighborhood of node as the index set
with given by .
Our algorithm can be regarded as a generalization of the algorithm proposed in  for i.i.d. random processes to general stationary random processes. The new element here is that since we consider general Gaussian vector processes, we have multiple measurements available to determine the CIG. This is exploited through the use of mLASSO instead of plain LASSO as employed in .
4 Performance Guarantees
We now present conditions for our CIG selection scheme to correctly identify, with high probability, the neighborhoods , and in turn the edge set , of the underlying CIG. Our analysis yields allowed growth rates for the problem dimensions, i.e., the number of scalar process components and the maximum node degree , as functions of the sample size . Moreover, we provide concrete choices for the threshold parameter in (13) and the mLASSO parameter in (1).
A necessary and sufficient condition for mLASSO to correctly identify the joint support of the underlying parameter vector is the incoherence condition [24, Eqs. (4)–(5)]. This condition is a worst-case (in our case, over frequency ) condition  in that it needs the system matrices in (7) (again, we consider node ) to be “well-conditioned” for all . In other words, the incoherence condition does not predict any performance improvement owing to the availability of measurements (7) instead of just one. We will therefore base our performance analysis on the multitask compatibility constant , which is defined, for a given index set of size , as
with . Here, where is the restriction of the vector to the entries in . Invoking the concept of the multitask compatibility constant will be seen below to yield an average (across frequency ) requirement on the SDM for Algorithm 1 to correctly identify the CIG.
We start by defining the class of -dimensional, zero-mean, stationary, Gaussian processes with CIG of maximum node degree (cf. (3)) and SDM satisfying (1) and (4). The remaining parameters characterizing this class are defined as follows:
Minimum partial coherence : This parameter quantifies the minimum partial correlation between the spectral components of the process. In particular, we require that, for every , ,
Consider a process belonging to the class . Let be the estimate of given by (13), based on sample size and with the choices (in (1)) and (in (13)). Then, if for some , the sample size and the ACF moment satisfy
with , the probability of Algorithm 1 delivering the correct edge set is at least , i.e., .
Theorem 4.1 shows that success is guaranteed with high probability if the sample size scales logarithmically in and polynomially in , and if the process is sufficiently smooth, i.e., is sufficiently small.
Let us particularize Theorem 4.1 to the special case of a VAR(1) process as considered in [9, 13, 12], i.e., with i.i.d. noise . As in , we take to be the adjacency matrix of a dependency graph , which is related to—but in general different from—the CIG . We assume that is a simple graph of maximum node degree and the nonzero entries of the adjacency matrix are all equal to a single positive number . The VAR(1) process we consider satisfies (1) and (3) by its definition and belongs to the class with , , , , and . Moreover, condition (4) is satisfied as soon as since the entries of are rational functions in with numerator degree . The threshold in (LABEL:eq:condonN) becomes , with a constant that is independent of , , , and . In contrast, the corresponding threshold for the method in  is , with a constant that is independent of , , , and . The difference between the growth rates of these thresholds with respect to may be explained by the fact that the method in  is tailored to VAR processes whereas our approach applies to general (spectrally smooth) stationary processes.
5 Numerical Results
We generated222Matlab code to reproduce the results in this section is available at http://www.nt.tuwien.ac.at/about-us/staff/alexander-jung/. a Gaussian process of dimension by applying a finite impulse response (FIR) filter of length to a zero-mean, stationary, white, Gaussian noise process . The covariance matrix was chosen such that the resulting CIG satisfies (3) with . The filter coefficients are such that the magnitude of the associated transfer function is uniformly bounded from above and below by positive constants, thereby ensuring that conditions (1) and (4) (for arbitrary ) are satisfied. We then computed the estimates using Algorithm 1 with window function and . We set and , where , , and was varied in the range .
In Fig. 1, we show receiver operating characteristic (ROC) curves with the average fraction of false alarms and the average fraction of correct decisions for varying mLASSO parameter . Here, denotes the neighborhood estimate obtained from Algorithm 1 in the -th simulation run. We averaged over independent simulation runs.
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