Order Selection of the Linear Mixing Model for Complex-Valued FMRI Data

Abstract

Functional magnetic resonance imaging (fMRI) data are originally acquired as complex-valued images, which motivates the use of complex-valued data analysis methods. Due to the high dimension and high noise level of fMRI data, order selection and dimension reduction are important procedures for multivariate analysis methods such as independent component analysis (ICA). In this work, we develop a complex-valued order selection method to estimate the dimension of signal subspace using information-theoretic criteria. To correct the effect of sample dependence to information-theoretic criteria, we develop a general entropy rate measure for complex Gaussian random process to calibrate the independent and identically distributed (i.i.d.) sampling scheme in the complex domain. We show the effectiveness of the approach for order selection on both simulated and actual fMRI data. A comparison between the results of order selection and ICA on real-valued and complex-valued fMRI data demonstrates that a fully complex analysis extracts more meaningful components about brain activation.

Appendix

We present the entropy rate of a complex-valued second-order stationary Gaussian random process using a widely linear model following an approach similar to the one given in [27].

Given a second-order stationary and zero mean random process Z _k , the covariance function is defined by $R(m) = E\{Z_{k+m}Z^\ast_k\}$ and the pseudo covariance function [25], also called the relation function [31], as $\tilde{R}(m) = E\{Z_{k+m}Z_k\}$. Without loss of generality, the random processes and vectors discussed in this paper are assumed to be zero mean. A random process is called second-order stationary if it is wide sense stationary and its pseudo covariance function only depends on the index difference. The Fourier transform of the covariance function yields the power spectrum (or spectral density) function S(ω). Similarly, we define the Fourier transform of the pseudo covariance function as the pseudo power spectrum function $\tilde{S}(\omega)$.

Entropy rate is a measure of average information in a random sequence, which can be written for a complex random process Z _k as

$$ h_c(Z) = \lim\limits_{n \rightarrow \infty} \frac{H(Z_1,Z_2,\ldots,Z_n)}{n} $$

(5)

when the limit exists. As in the real case, $H(Z_1,Z_2,\ldots,Z_n) \leq \sum_{k=1}^n H(Z_k)$, with equality if and only if the random variables Z _k are independent. Therefore the entropy rate can be used to measure the sample dependence and it reaches the upper bound when all samples of the process are independent.

The widely linear filter is introduced in [32], and any second-order stationary complex signal can be modeled as the output of a widely linear system driven by a circular white noise, which cannot be achieved by a strictly linear system [31]. Given the input and output vectors x, y ∈ ℂ^N, a widely linear system is expressed as

$$ \mathbf{y} = \mathbf{F}\mathbf{x}+\mathbf{G}\mathbf{x}^\ast $$

where F and G are complex-valued impulse responses in matrix form. The system function of a widely linear system is the pair of functions [F(ω), G(ω)]. By using entropy rate analysis on multivariate Guassian random process [5, 39] and [29], the relation of entropy rate for input and output of a widely linear system is given by

$$ \begin{array}{rll} h_c(Y) &=& h_c(X) \\ &&+\,\frac{1}{4\pi} \int_{-\pi}^{\pi} \log \left\{ \left[ |F(\omega)|^2 - |G(\omega)|^2 \right]\right.\\&&\times\,\left. \left[ |F(-\omega)|^2 - |G(-\omega)|^2 \right] \right\} d\omega \end{array} $$

Theorem 1

If z(n) is a complex second-order stationary Gaussian random process with power spectrum function S(ω) and pseudo power spectrum function $\tilde{S}(\omega)$ , its entropy rate h _c is given by

$$ h_c = \log(\pi e) + \frac{1}{4\pi} \int_{-\pi}^{\pi} \log \left[ S(\omega)S(-\omega)-|\tilde{S}(\omega)|^2 \right] d\omega. $$

The proof of Theorem 1 is given in [39].

For a second-order circular process, we have $\tilde{S}(\omega)=0$, thus yielding the entropy rate of a second-order circular Gaussian random process as

$$ h_{{\rm circ}} = \log(\pi e) + \frac{1}{4\pi} \int_{-\pi}^{\pi} \log \left[ S(\omega)S(-\omega) \right] d\omega. $$

For the general case in Theorem 1, $|\tilde{S}(\omega)|^2 \geq 0$. Hence, for the second-order circular and noncircular Gaussian random sequences with the same covariance function R(m), we have

$$ h_{{\rm noncirc}} \leq h_{{\rm circ}}, $$

which can be also verified using the result for complex entropy [25, 36] and the definition of entropy rate given in Eq. 5.