Sparse Multivariate Autoregressive Modeling for Mild Cognitive Impairment Classification

Abstract

Brain connectivity network derived from functional magnetic resonance imaging (fMRI) is becoming increasingly prevalent in the researches related to cognitive and perceptual processes. The capability to detect causal or effective connectivity is highly desirable for understanding the cooperative nature of brain network, particularly when the ultimate goal is to obtain good performance of control-patient classification with biological meaningful interpretations. Understanding directed functional interactions between brain regions via brain connectivity network is a challenging task. Since many genetic and biomedical networks are intrinsically sparse, incorporating sparsity property into connectivity modeling can make the derived models more biologically plausible. Accordingly, we propose an effective connectivity modeling of resting-state fMRI data based on the multivariate autoregressive (MAR) modeling technique, which is widely used to characterize temporal information of dynamic systems. This MAR modeling technique allows for the identification of effective connectivity using the Granger causality concept and reducing the spurious causality connectivity in assessment of directed functional interaction from fMRI data. A forward orthogonal least squares (OLS) regression algorithm is further used to construct a sparse MAR model. By applying the proposed modeling to mild cognitive impairment (MCI) classification, we identify several most discriminative regions, including middle cingulate gyrus, posterior cingulate gyrus, lingual gyrus and caudate regions, in line with results reported in previous findings. A relatively high classification accuracy of 91.89 % is also achieved, with an increment of 5.4 % compared to the fully-connected, non-directional Pearson-correlation-based functional connectivity approach.

Appendix

The forward Orthogonal Least Squares algorithm and Error Reduction Ratios

Here, a brief introduction of the OLS algorithm is given as follows. Due to the use of the linear polynomial model in Eq. (3), matrix X is often referred to as the regression matrix. This regression matrix X can be orthogonally decomposed as

$$ X= UV, $$

(7)

where V is an q × q unit upper triangular matrix and

$$ \mathrm{U}=\left[{u}_1,{u}_2,\cdots, {u}_n\right], $$

(8)

is a (N − q) × q matrix with orthogonal columns that satisfy

$$ {\mathrm{U}}^{\mathrm{T}}U=P, $$

(9)

and P is a positive diagonal matrix P = diag[p ₁, p ₂, ⋯, p _d ] with p _i  = u _i , u _i , where the symbol 〈⋅, ⋅ 〉 denotes the inner product of two vectors, i.e.,

$$ \left\langle {u}_i,{u}_i\right\rangle ={\displaystyle \sum_{t=1}^N}{u}_i(t){u}_i(t). $$

(10)

Equation (3) can now be expressed as

$$ Y=\left(X{V}^{-1}\right)\left( V\theta \right)+E=(U)\left( V\theta \right)+E= UG+E, $$

(11)

where G = [g ₁, g ₂, ⋯, g _q ]^T is an auxiliary parameter vector, which can be calculated directly from Y and U by means of the property of orthogonality as ${g}_i=\frac{\left\langle Y,{u}_i\right\rangle }{\left\langle {u}_i,{u}_i\right\rangle },i=1,2,\cdots, q$.

Rewrite Eq. (11) as

$$ Y={\displaystyle \sum_i^q{u}_i{g}_i+E,} $$

(12)

and calculate the inner product u _i , Y, by substituting Y by Eq. (12), as

$$ \left\langle {u}_i,Y\right\rangle =\left\langle {u}_i,{\displaystyle \sum_{i=1}^q}{u}_i{g}_i\right\rangle ={\displaystyle \sum_{i=1}^q}\left\langle {u}_i,{u}_i{g}_i\right\rangle ={g}_i\left\langle {u}_i,{u}_i\right\rangle, $$

(13)

Calculate the inner product Y, Y from the Eq. (12)

$$ \left\langle Y,Y\right\rangle =\left\langle {\displaystyle \sum_{i=1}^q}{u}_i{g}_i,Y\right\rangle ={\displaystyle \sum_{i=1}^q}\left\langle {u}_i{g}_i,Y\right\rangle ={\displaystyle \sum_{i=1}^q}{g}_i^2\left\langle {u}_i,{u}_i\right\rangle, $$

(14)

Dividing both sides of Eq. (14) by Y, Y, then yields

$$ 1={\displaystyle \sum_{i=1}^q}\frac{g_i^2\left\langle {u}_i,{u}_i\right\rangle }{\left\langle Y,Y\right\rangle }, $$

(15)

The error reduction ratio ERR _i due to u _i can be presented by

$$ ER{R}_i=\frac{g_i^2\left\langle {u}_i,{u}_i\right\rangle }{\left\langle Y,Y\right\rangle }, $$

(16)

The error reduction ratio often provides a simple and effective means for selecting a subset of significant terms from a large number of candidate terms in a forward regression manner. A term can be selected if it produces the largest value of ERR _i among the rest of the candidate terms. The selection procedure will be terminated when

$$ 1-{\displaystyle \sum_{i=1}^{q_0}} ER{R}_i<\varepsilon, $$

(17)

where ε is a desired tolerance, and this leads only to a subset model of q ₀ terms (q ₀ ≪ q). The detailed procedure for derivation can be seen from Billings et al. (1989).