A method for accurate group difference detection by constraining the mixing coefficients in an ICA framework

ICA and Infomax

ICA is a technique that transforms a given set of observations into components that are mutually independent. It is based on a simple generative model given as

(1)

The observed data X consists of measurements such as one or more stacked features with dimensions of N × L (subjects by voxels), and S with dimensions of M × L, contain M independent sources such as brain activation maps. Here, A is the mixing matrix (N × M) containing the subject‐specific loading parameters and can be divided into two parts, A _h (controls) and A _p (patients) for group analysis (see Fig. 1). In joint ICA case, S is the joint component and each task (e.g., a sensorimotor task) shares the same A. The aim of ICA is to find the unmixing matrix W = A ⁻¹ so that the source estimation U = W · X is as close as possible to the true source S. The back‐reconstructed sources for each group, U _h and U _p can be calculated as U _h = A ${}_{\text{h}}^{\text{−1}}$ · X _h = W _h X _h, and U _p = A ${}_{\text{p}}^{\text{−1}}$ · X _p = W _p X _p. If data is noisy and the joint component is allowed to show different manifestations on different populations, U _h, and U _p are different from each other.

Among the ICA algorithms (Infomax, fastICA, JADE, EVD, SIMBEC, and AMUSE) that have been applied to brain imaging data, Infomax has proven to be a reliable choice, (see, for example Correa et al., 2007). For this reason, we introduce CC‐ICA within the Infomax framework as an example in this article. Recently, several efficient ICA algorithms, e.g., EFICA, EWASOBI, COMBI, and new JADE methods (Koldovský and Tichavský, 2007, 2008; Li and Zhang, 2007; Tregellas et al., 2004) have been developed and implemented in ICALAB3 (Cichocki et al., 2007). To the best of our knowledge, so far they have not been applied to fMRI data, which is typically higher in dimension and noisier than multimedia data. We also compare the estimation performance of seven ICA algorithms, including Infomax, CC‐Infomax, fastICA, EFICA, COMOBI, EWASOBI, and one of the new JADE algorithms, FJADE, for further validation.

Infomax (Bell and Sejnowski, 1995) can be shown to be equivalent to maximum likelihood (ML) estimation and is closely related to most other ICA algorithms (e.g., FastICA) under certain conditions (Adali et al., 2008; Cardoso, 1997). For example, if the demixing matrix W in Infomax is constrained to be orthogonal and the nonlinearity in both is matched to the source density, Infomax and maximization of negentropy (the FastICA type approach) can be shown to be equivalent in terms of cost functions they use. More importantly, both Infomax/ML and algorithms based on maximization of negentropy are optimal (again if the source distribution at the output of the unmixing matrix is matched to the hypothesized source distribution) in that they achieve desirable large sample properties of maximum likelihood.

Statistical Assumption

To account for group inferences, the ICA model can be extended as

(2)

Considering a fixed number of subjects, including n _h healthy controls and n _p patients, n _h + n _p = N. A _h, A _p are mixing matrices such that each column represents loading parameters for one shared component. For the ith component, i = 1,2…M, assume scalar A _h,i, A _p,i (the ith column of matrix A _h, A _p) are its corresponding subject‐specific loading parameters for two groups. The mean of A _h,i and A _p,i (i.e., and ) represent the contribution of the ith component to two groups. By incorporating group difference related to A _h, A _p, CC‐ICA is trying to increase the extracted ICs' accuracy and their sensitivity to group‐effects. The sensitivity to group‐effects built into the CC‐Infomax algorithm results in the components reflecting the most group difference being ranked at the top of the sort order.

We initially assume that distributions of A _h,i, A _p,i have the same variance. The T statistic to test whether their means are different can be calculated as follows:

(3)

where s ² is the unbiased estimator of the scalar's variance, and N−2 is the number of degrees of freedom for two‐tailed significance testing. The larger the absolute value of T _i, the higher the probability that and are significantly different.

If this assumption is violated, namely, scalars A _h,i,A _p,i have different variances, suppose n _h < n _p, then the unequal variance t‐test can be used alternatively as

(4)

(5)

where A _h,i(n), A _p,i(n) denotes the coefficient of the nth subject in A _h,i and A _p,i, respectively. If n _h = n _p = N/2, the above equation can be rewritten as

(6)

(7)

where z _i is an intermediate scalar used to calculate T _i; s ${}_{\mathit{\text{i}}}^{\text{2}}$ is the biased variance estimator of z _i.

The cost function C of CC‐ICA is constructed such that in addition to the given objective function H, which should be maximized for achieving independence, the sum of the squared statistic T _i between A _h,i and A _p,i is added for joint maximization;

(8)

where λ is the constraint strength, the suffix i represents the constrained component(s). In the optimization section, we explain how to determine the constrained components.

CC‐ICA Learning Process

Before the ICA processing, a typical principal component analysis (PCA) is used for dimensionality reduction and data whitening. Assume Λ and B are, respectively, the diagonal eigenvalue and eigenvector matrix of the covariance matrix E{X·X ^T}, Λ = diag{λ₁, λ₂…λ_N} where X is zero mean. The top M eigenvectors (M < N) with the largest λ_i are selected, thus Λ is changed to Λ′, an M × M diagonal matrix, and B is reduced to a corresponding M × N matrix B′. Therefore, the reduced data X′ is given by:

(9)

where the M × N matrix D is called whitening matrix, which ensures that the covariance matrix of X′ is given by the identity matrix. D is used in transformation of mixing matrix A when the constraint takes effect.

The main purpose of CC‐ICA is to estimate an unmixing matrix W by maximizing the augmented cost function C, which will optimize the independence criterion while simultaneously improving the estimation of group difference. The learning process is as follows:

For a selected iterative ICA algorithm, we assume the unmixing matrix obtained at the kth iteration is W _k,sICA, thus W _k,sICA will iteratively converge with the maximization of H. If the weight change ΔW _k,sICA does not stabilize (measured by a tolerance factor), the constraint will not take effect, namely, λ = 0, W _k = W _k,sICA. Otherwise, λ > 0, W _k,sICA is updated to W _k via maximization of the constraint ΣT ${}_{\mathit{\text{i}}}^{\text{2}}$ at each iteration (more details are shown below), W _k is then updated to W _k+1,sICA according to the selected iterative ICA algorithm [in case of Infomax, the natural gradient learning rule (Amari, 1998) is adopted]. The above learning process is repeated until convergence of W _k achieved, which is to some degree similar to that used in set‐membership normalized least mean square (SM‐NLMS) algorithm (Gollamudi et al., 1998).

If the constraint is incorporated, ΣT ${}_{\mathit{\text{i}}}^{\text{2}}$ is maximized by the steepest ascent learning rule using partial derivatives of A _h,i and A _p,i, where A = [W _k,sICA D]⁻¹. For the equal variance two sample t‐test:

(10)

(11)

For the unequal variance two sample t‐test, the learning process of A _h,i and A _p,i is similar to above, here we only show the condition when n _h = n _p = N/2.

(12)

(13)

Thus, the unmixing matrix of CC‐ICA, W _k, is given by integrating the updated A _h and A _p,

(14)

In this article, we use Infomax as an example of the iterative ICA algorithm, hence the first term in (8) is given by

(15)

(16)

where H is the differential entropy, f(·) is the probability density function, and E[·] is the expectation operator. Here, x′ denotes the random vector for the whitened data X′ and u is the random vector of source estimates. The nonlinear function g(·) is chosen based on the form of the expected underlying source distribution. The sigmoid function has been shown to be quite robust to violations of the underlying model for a wide variety of data types (Lee et al., 1999); thus, it is an attractive choice as the nonlinearity, especially since most sources of interest in fMRI analysis are super‐Gaussian (Calhoun and Adali, 2006). Therefore, the M × M matrix W _k,sICA is updated using the natural gradient learning rule (Amari, 1998) as shown in Eq. (17), which alleviates the need to calculate the inverse of W _k−1 in the gradient updates, where k is the iteration index.

(17)

Dynamic CC‐ICA Optimization

Appropriate optimization strategies are adopted to enable CC‐ICA to adaptively adjust the learning process with the dataset and to avoid generating spurious group difference. We incorporate the following strategies: (1) dynamic selection of the constrained components, (2) an adaptive constraint strength λ, and (3) a stopping rule (switch) on the constraint. With these optimizations, CC‐ICA works reliably and shows an improvement over regular ICA in both accuracy and component sort order, as we show in the simulations and hybrid data.

First, the specific components that are constrained are not fixed; instead, they are allowed to vary during the optimization. After the weight change ΔW _k,sICA stabilizes (i.e., for the first l steps run only iterative ICA algorithms until W _k = W _k,sICA start to converge, k = 1,2…l, in case of Infomax, l ∈ [10,15]), T ² values of all components are calculated and sorted at the current and the following steps; half or more number of components with the highest T ² values are selected to be constrained. In summary, the fewer components we constrain, the more they fluctuate during the initial learning process, but as W _k converges, the constrained components become stable.

Second, we adaptively adjust the constraint strength λ. Specifically, after the constrained components are determined at the lth step, the constraint ΣT ${}_{\mathit{\text{i}}}^{\text{2}}$ is incorporated with a strength λ₀ (initial learning rate) which is estimated by a testing loop at the (l + 1)th step. Starting with a heuristic learning rate λ_h, W _l+1,sICA is updated to W _l+1 according to Eq. (14), we then calculate the matrix correlation between W _l+1,sICA and W _l+1. If the absolute correlation value is lower than a certain threshold th_corr (e.g., th_corr = 0.99), then λ_h is scaled down by a typical annealing rate 0.9 and the above matrix correlation testing is repeated, until convergence results in the final λ₀. Furthermore, to achieve mutual independence between the components, the slope of ΔW _k (or entropy) during the last s steps (e.g., s = 10), i.e., Slope_ΔW,10 or Slope_entropy,10, can be used to supervise the learning process so that the entropy maximization dominates the W _k update process. Our simulations show these two options have similar results, except the former hasa smaller computational load, and so we use Slope_ΔW,10 to adjust the constraint strength. When Slope_ΔW,10 > 0, λ₀ is scaled down by 0.9.

Finally, we have put an adaptive stopping rule in place, which enables CC‐ICA to converge to regular ICA if no group differences exist. When there are few or no components with a true group difference, the Slope_ΔW,10 can also act as a switch for the constraint. In this case, entropy will fall suddenly and consequently ΔW _k will increase abruptly, so that when Slope_ΔW,10 exceeds, the threshold th_stop (e.g., th_stop = 0.001), λ is set to be zero and the learning process converges to regular ICA. Hence, the supervision by Slope_ΔW,10 prevents the constraint from producing spurious group difference.

We have rigorously tested our algorithm to ensurerobustness and stability. Assume the mixtures are whitened in each case, all the parameters mentioned above (i.e., l ∈ [10,15], th_corr ∈ [0.95,0.99], s ∈ [5,10], th_stop ∈ [0.0005,0.001]) and their ranges are tested via multiple datasets including simulations and real data. Convergence is achieved in every case without exception, and in general, the best value of the optimization parameters increased slightly with the increase of the source dimension L.

Performance Evaluation Metrics

CC‐ICA is designed to enhance group differences detection. Five metrics are selected to evaluate performance of CC‐ICA on different data sets:

• 1

  The P value that determines the probability of two groups having significantly different mean in the mixing coefficients. Since our constraint works directly on coefficients and intends to decrease the P value, the constrained ICs are theoretically expected to have smaller P values than the corresponding ones in standard ICA.

• 2

  The accuracy of the recovered ICs. If we have access to the ground truth (as in a simulation), the estimated components should be the same as the “true” sources; the correlation between the true source and the extracted component are used to indicate the component accuracy.

• 3

  The accuracy of the mixing coefficients. We compute the correlation of constrained columns of the mixing matrix A with the real loading parameters in synthetic and hybrid data set.

• 4

  The J ‐divergence (the symmetric Kullback‐Leibler divergence) between the back‐reconstructed sources for two groups. It is a measure reflecting the mutual information between the estimated source distributions of two groups (Calhoun et al., 2004a). For real fMRI data, J‐divergence quantifies group differences in the spatial distributions of the back‐reconstructed sources U _h and U _p; the higher J‐divergence value a component has, the larger difference of source distribution exists between controls and patients.

• 5

  Sort order of component by P value or J ‐divergence. Basically the ability of an algorithm to correctly rank in order the components that show significant group differences. The components are sorted by P value from low to high or by J‐divergence from high to low. The sort order can be considered as the sensitivity of the algorithm to group difference. We also quantify the error rate of the estimated sort order of components compared with the ground truth in simulation section.

Data Sets

Synthetic data

Infomax when used with the sigmoid nonlinearity performs well with super‐Gaussian signals (Lee et al., 1999), and so five zero‐mean super‐Gaussian signals with a length of 2000 samples, denoted as S _h, S _p for each group, were randomly mixed to obtain the pure signal matrix R with 20 rows. , every row of R is denoted as r _j, j = 1,2, …, 20. A _h and A _p are simulated via a normal distribution with a range of [0–1]. The mixing coefficients of the 2nd, 3rd, and 5th sources are set to have a significantly different mean of between groups (reflected in corresponding columns of A _h and A _p, P < 0.05). Sources S _h and S _p have similar shapes, as Figure 2(a) shown, yet exhibit some variations between the two groups such as different frequency and width of the signal periods. Specifically, the 4th and 5th sources have the largest group difference, which is measured by the mean square error (MSE) defined in Eq. (18),

(18)

where S _h,i and S _p,i represent the ith true source for each group. K = 2000 is the length of source. Zero‐mean Gaussian noise with different variance is then added to sequence r _j to generate the observed data sequence x _j under two SNR conditions. The SNR for the whole data sets is the mean of the SNRs of total artificial noised mixtures. It is defined as Eq. (19) in units of dB:

(19)

where std(·) is the standard derivation function

Figure 2.

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We also perform a comparison of multiple existing ICA algorithms by generating five sources with both supergaussian and subgaussian signals (not shown) without group differences. The source length is set to be 50,000 samples in order to approximately match the number of voxels in our fMRI data.

Hybrid fMRI data

We created hybrid data by superimposing the known sources on actual SPM contrast images generated from auditory oddball (target) task (AOD) and Sternberg tasks, using data from our earlier work, collected at the Olin Neuropsychiatry Research Center (Calhoun et al., 2006b), to examine the “coupled” activation across multiple tasks.

We consider three conditions for the added known sources: (1) a different spatial pattern and different loading parameters between groups; (2) the same spatial pattern and different loading parameters between groups; (3) the same spatial pattern and the same loading parameters between groups. For the different tasks, the artificial sources are added to different positions; for the same task, the sources simulating the two groups (u _h and u _p) are either different or same. The hybrid data generation of the first condition is illustrated in Figure 4(a), where each source is generated into a 21 × 21 grid pattern, as defined in Eqs. (20) and (21). In other two conditions, u _p is set the same with u _h.

(20)

(21)

Figure 4.

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The mixing coefficients of u _p and u _h, denoted as a _h and a _p, are random numbers drawn from a normal distribution. In the first and second condition, the ranges used were [0.5–1.5] and [0–1] for patients a _p and controls a _h, respectively, providing a difference in the mean of the loading parameters. In the third condition, a _p is set the same with a _h. The hybrid activation images of AOD and SB task are then flattened to vectors and stacked side by side to obtain the observed data matrix (subjects by voxels). Each task shared the same mixing matrix as Figure 1 shown before.

Experiments on Synthetic Data

In the first simulation, we compare CC‐ICA with regular ICA and choose appropriate assumptions for the former by testing. There are two statistical assumptions for the constraint, namely, an equal or unequal variance two sample t‐test. Moreover, two choices for the learning rate adjustment (Slope_ΔW,10 or Slope_entropy,10) can be selected. So for both high and low SNR condition, we run CC‐Infomax five times to analyze the mixed noisy data X under four combinations of the above options and compare their results with Infomax using the three performance metrics (the 2nd, 3rd, and 5th metric) listed before. Here all components are constrained in CC‐Infomax since there are only five components. Results show that CC‐Infomax can identify the expected independent components (ICs) successfully with similar performance for all four combinations. The constraint with an equal variance t‐test assumption slightly outperforms the unequal variance t‐test, while the two options for supervising the learning rate provide almost identical results. Since ΔW is already calculated at every step while the entropy needs to be computed additionally, using the slope of ΔW for supervision has a smaller computational load. Therefore, in the following applications of CC‐Infomax, we use Slope_ΔW,10 to adjust the constraint strength in optimization process.

The back‐reconstructed sources U _h = A ${}_{\text{h}}^{\text{−1}}$ · X _h and U _p = A ${}_{\text{p}}^{\text{−1}}$ · X _p under a low SNR condition (SNR = 0.5 dB) are displayed in Figure 2(b,c). Compared with the ground truth S _h and S _p in Figure 2(a), CC‐Infomax generates results with a relatively higher accuracy than Infomax and it clearly identifies the small signal variation between groups, especially the 4th and the 5th sources which show the largest group difference on patterns. Figure 3 illustrates the unmixed sources U = A ⁻¹ X by Infomax and CC‐Infomax when SNR = 0.5 dB, which demonstrate the similarity between U _h and U _p.

Table I listed all performance metrics evaluated under both SNR conditions. The accuracy of the mixing coefficients and the unmixed ICs are evaluated via correlation values with the ground truth, where the “|mean|” denotes the mean of the absolute correlation values across the five components.

In either high or low SNR conditions, CC‐Infomax show high accuracy on both the mixing coefficients and sources, whereas Infomax performs less well in noisy condition, especially for identifying the IC with larger group difference (the 4th and 5th IC). Specifically, for the 4th component, the accuracy of the back‐reconstructed sources for the two groups is increased from 0.656, 0.708 (Infomax) to 0.977, 0.950 (CC‐Infomax). And for the 5th component, the absolute correlation of its loading parameters with ground truth is increased from 0.39 (Infomax) to 0.96 (CC‐Infomax). For the component with few group differences such as the 1st IC, the two algorithms perform equivalently and both attain high accuracy.

The improved estimation by CC‐Infomax directly results in the correct identification of the three components with group‐biased coefficients (top three ICs sorted by P values from low to high); in contrast, Infomax does not correctly identify them (shown in bold italic in Table I) in the low SNR case. The error rate of the estimated sort order of components when compared with the ground truth is quantified as . CC‐Infomax is 86.7% correct, whereas Infomax is only 60% correct; thus, CC‐Infomax shows much more consistency with the ground truth in component ranking.

In the second simulation, we compare performance of seven aforementioned ICA algorithms. The five sources with group differences are mixed together with noise, generating 20 mixtures with an average SNR of 5 dB. We run 50 Monte Carlo runs and randomly create mixing coefficient matrix at every time. Testing results of all seven algorithms are listed in Table II. CC‐ICA was found to show the highest estimation accuracy.

In the third simulation, we use another five sources to generate five mixed signals at a SNR of 10 dB. No group difference was introduced into the simulation. Twenty Monte Carlo runs are adopted for each algorithm and the mixing matrix is randomly created at every run. Performance evaluation is given in Table III. If a given source was not identified at all, its correlation is set to zero.

In Tables II and III, “Corr” denotes the mean correlation between inference and ground truth across all runs and all components; the bold number indicates the best performance on this metric. We can see that Infomax is almost always similar to or better than other ICA algorithms and its performance is quite stable. On the other hand, CC‐Infomax increases the accuracy of extracted component beyond infomax and the other algorithms when group differences are present; otherwise, it performs almost identically to infomax, without generating spurious group difference. We also show this in our hybrid data application.

Results From Hybrid Multitask fMRI Data

Since the superimposed sources have a known pattern, it is straightforward to recognize them (partially composed of actual data) from the unmixed data. Such an approach enables us to evaluate the performance of CC‐Infomax and Infomax on extracting maximally spatially independent maps under a low contrast to noise ratio (CNR) of 0.5. CNR is computed as the maximum value of the known source (difference in signal between region of interest) divided by the standard deviation of the fMRI data at the same source added region across the subjects.

For every condition of the added known sources, the hybrid data are unmixed by three algorithms: Infomax and CC‐Infomax with equal or unequal variance t‐test assumption. According to a modified minimum description length (MDL) criterion (Li et al., 2007), all three algorithms were run five times using eight components. The final decomposition result is the average of five runs. For CC‐Infomax, four components are constrained during the optimization. Before they are constrained, a Lilliefors test was performed to test the normality of their mixing coefficients. All the constrained components pass the test at a significance level of 1%, thus their loading parameters obey the normal distribution.

The joint ICA applications of all three algorithms successfully select out the coupled sources into a separate component and enable us to visualize where in each data set the coupling occurs. As shown in Figure 4(b), the target IC indicates the added sources on right positions for both AOD and SB features. Two sample t‐test are performed between groups on loading parameters of the target IC, the resulting P values and its sort order among eight ICs (from low to high) are listed in Table IV. The J‐divergence of the target IC are also calculated and sorted from high to low, its corresponding sort order is showed in Table IV as well.

For condition 1, the target IC should have the smallest P value and the largest J divergence among all the extracted components, since the added sources were designed so. CC‐Infomax demonstrates the correct sort order of the two measures, whereas Infomax shows a poor estimation on component sort order. In condition 2, the three algorithms have similar results, whereas CC‐Infomax with equal variance t‐test assumption is most sensitive to group difference (see component sort order); this is also reflected in condition 1. In condition 3, there is no group difference existing for the added source. All algorithms generate very reasonable results for the target IC and the performance of CC‐Infomax with equal variance t‐test assumption is almost identical with Infomax. All above results support the claim that CC‐Infomax is more sensitive in detecting group difference than Infomax, and also does not appear to generate spurious results when there is no real group difference. Further, the CC‐Infomax with equal variance t‐test assumption slightly outperforms the unequal variance t‐test assumption. So the CC‐Infomax results shown in Figure 4 were generated based on the former assumption.

The correlation of loading parameters of the target IC versus the ground truth are calculated under all three conditions and listed in Table IV. The first condition is plotted in Figure 4(c), where patients are coded in yellow squares and controls are coded in blue circles. As expected, the controls showed a lower mean than the patients did and the target IC shows the highest correlation among 8 extracted ICs. However, CC‐Infomax (equal variance t‐test) shows an increased accuracy versus Infomax (0.8151 vs. 0.7392). Hence, by incorporating prior statistical group information, more accurate components are estimated (both the mixing matrix and the source itself) in CC‐Infomax and the sort order of components by P value or J‐divergence correctly identifies the components with group differences.

Results of Real fMRI Data Analysis

We can also examine more directly the relationship between the activation regions of SM and SB tasks by applying CC‐Infomax and Infomax to a real fMRI data set. Since there is no ground truth as reference for real fMRI data, we focus on the sort order of IC by P value and J‐divergence. Sixteen components were estimated according to the MDL criteria. Before being added the constraint, the mixing coefficients of all components were subjected to a Lilliefors test. The null hypothesis of normal distribution was not rejected at a significance level of 1%. A two sample F test is also performed on A _h and A _p in coupling, results show the loading parameters of most of components have the same variance between groups at the significance level of 1%. All three algorithms, Infomax, CC‐Infomax with equal or unequal variance assumption, were run five times and then averaged to get the result. We were interested in evaluating the performance of constraining different number of components initially, (e.g., constraining 4, 8, 12, and 16 ICs, respectively) and found that the results were basically the same when allowing half or more of the components to be constrained. This ensures all possible components capturing group differences have the possibility to be constrained.

One of the major advantages of ICA is the filtering of artifacts. For example, we have tested in simulation/real fMRI data and found that ICA can decompose the cerebrospinal fluid (CSF) artifact into an independent component. The extracted components are then inspected visually and the components revealing obvious artifacts (e.g., CSF artifacts) are eliminated from the group analysis. When we use the different variance assumption, the extracted top eight components sorted by P values are the same. So we only list CC‐Infomax results derived from equal variance assumption as well as the results from Infomax in Table V, where the components' sort order and its corresponding J‐divergence are indicated. The most similar component derived from Infomax is shown in the same column with CC‐Infomax.

The most significant difference identified by Infomax showed P < 0.036 for the mixing coefficient, with a small J‐divergence sorted 14th, meaning this component did not show group difference in the joint histogram distribution of the identified source. The one most similar to it in the CC‐Infomax was ranked third. According to our analysis in simulation and hybrid data sets, the component with the smallest P value in CC‐Infomax is the most group‐distinguishing IC. Further investigation on J‐divergence show that, the target IC also has the largest J‐divergence among all components, confirming that it is most sensitive to group difference (on both mixing coefficients and source distribution). The J‐divergence of the most similar IC from Infomax is only ranked 10th among 16 ICs and its P value (P = 0. 202) does not identify a significant group difference in the loadings. Thus the use of CC‐Infomax resulted in a change in sort order of the sources from Infomax, a result which may prove especially important for the identification of relevant disease biomarkers.

Figure 5 illustrates the thresholded Z maps (|Z| > 2.5) of the IC with the smallest P value derived from two algorithms. The SM feature is shown in the middle and the SB feature is shown on the right. The ICA loadings of each group are illustrated on the left, where the healthy controls have a significantly larger mean (P < 0.05) than the schizophrenia patients for both components. For CC‐infomax, see Figure 5(a), strong activations are demonstrated in temporal lobe for SM task and in parietal lobe and frontal lobe (dorsolateral prefrontal cortex) for the SB task; On the other hand, for Infomax, parietal lobe, primary somatosensory cortex, and much smaller regions in temporal lobe are activated for SM task, while primary motor cortex is greatest for the SB task. The Talairach coordinates of the target IC displayed in Figure 5(a) are given in Table VI.

To examine the joint task activity in more detail, the joint histogram images (Calhoun et al., 2006b) were computed for each participant and then averaged within each group as follows: voxels surviving the threshold for the SB part of the joint source were sorted in descending order by the component voxel values (the same was done for voxels in the SM part of the joint source). This procedure resulted in two sets of voxel coordinates. Histograms were then generated by pairing these two voxel sets. These pairings were used to generate single subject 2D histograms of SB vs. SM fMRI signal (as estimated by the SPM contrast image). For the voxels included, the 2D histogram can be considered an estimate of the joint distribution function for the two tasks (e.g., P(f _sb, f _sm), where f _sb, f _sm indicates the fMRI signal amplitude for the SB or SM tasks respectively).

Figure 6 illustrates the joint histogram by subtracting of the patient group average from the control group average with orange areas larger in controls and blue areas larger in patients, where (a) is the first sorted IC of CC‐Infomax by J‐divergence (also with the smallest P value) and (b) is the most similar IC to (a) got from the Infomax. In general, more SM task voxels were active for CC‐Infomax and the SB task showed an increased kurtosis for the patients (the number of voxels that were increased or decreased was greater). The activated voxels in (a) have larger absolute values between the two groups than those in (b), thus a higher J‐divergence value (2.790 vs. 0.358).