Group ICA, Dual Regression and IVA组 ICA、双回归与独立向量分析

15.1 From one subject to many从单个被试到多个被试

Most fMRI studies compare networks between people: patients and controls, young and old, before and after treatment. ICA of each subject separately returns components in arbitrary order, sign and scale (Section 1.3第 1.3 节), and sometimes with different splits of the same network, so the components must be matched across subjects before anything can be compared. Matching by spatial correlation or by clustering is possible but error-prone, especially at higher model orders. Group methods build the correspondence into the model instead. They differ in what they assume to be shared: the maps (temporal concatenation, Section 15.2第 15.2 节); the time courses (spatial concatenation, Svensén, Kruggel & Benali, 2002, which suits experiments in which all subjects follow the same stimulus timing, but not rest); both maps and time courses up to subject-specific loadings (tensor PICA; Beckmann & Smith, 2005); or neither, with corresponding components linked only by statistical dependence (independent vector analysis, Section 15.6第 15.6 节). Calhoun, Liu and Adalı (2009) review these approaches. This chapter develops the ones most used in practice, then turns to what is computed from the components: connectivity between networks and links between imaging modalities.

大多数 fMRI 研究要在人与人之间比较网络：患者与对照、年轻人与老年人、治疗前与治疗后。对每个被试单独做 ICA，得到的成分顺序、符号与尺度都是任意的（Section 1.3第 1.3 节），同一个网络有时还会以不同的方式被拆分，因此在做任何比较之前，必须先把成分在被试之间对应起来。按空间相关或聚类来匹配是可行的，但容易出错，模型阶数较高时尤其如此。组方法则把对应关系直接写进模型。它们的区别在于假设哪些东西是共享的：共享空间图（时间拼接，Section 15.2第 15.2 节）；共享时间过程（空间拼接，Svensén, Kruggel & Benali, 2002，适合所有被试遵循相同刺激时序的实验，不适合静息态）；空间图和时间过程都共享、只差被试特有的载荷（张量 PICA；Beckmann & Smith, 2005）；或者都不共享，相互对应的成分只靠统计依赖联系起来（独立向量分析，Section 15.6第 15.6 节）。Calhoun、Liu 与 Adalı（2009）综述了这些方法。本章先讨论实践中最常用的几种，再转向由成分计算的量：网络之间的连接，以及不同成像模态之间的联系。

15.2 Temporal concatenation时间拼接

Let $\bX_i \in \R^{T\times V}$, $i = 1, \dots, M$, be the preprocessed data of $M$ subjects, registered to a common template and centered over voxels. Group ICA by temporal concatenation (Calhoun, Adalı, Pearlson & Pekar, 2001) assumes that the subjects share the same maps but have their own time courses, $\bX_i = \bA_i\bS$. Stacking the subjects in time gives an ordinary spatial ICA model ((14.1)) with $MT$ "sensors":

设 $\bX_i \in \R^{T\times V}$（$i = 1, \dots, M$）是 $M$ 个被试经过预处理、配准到同一模板并在体素方向中心化后的数据。时间拼接的组 ICA（Calhoun, Adalı, Pearlson & Pekar, 2001）假设各被试共享相同的空间图，但各有自己的时间过程，即 $\bX_i = \bA_i\bS$。把各被试在时间方向上堆叠起来，就得到一个普通的空间 ICA 模型（(14.1)），其“传感器”个数为 $MT$：

The stacked matrix is large, and its PCA reduction is computed in two stages. Each subject's data are first reduced to $q$ dimensions, $\bY_i = \bF_i\bX_i \in \R^{q\times V}$, where the rows of $\bF_i \in \R^{q\times T}$ are the $q$ leading eigenvectors of $\frac1V\bX_i\bX_i^\top$. The reduced data are stacked into $\bY \in \R^{Mq\times V}$ and reduced again to $n$ dimensions, where the rows of $\bG \in \R^{n\times Mq}$ are the $n$ leading eigenvectors of $\frac1V\bY\bY^\top$. Partition $\bG = [\bG_1, \dots, \bG_M]$ into blocks $\bG_i \in \R^{n\times q}$, so that $\bG\bY = \sum_i \bG_i\bY_i$. Spatial ICA of the $n\times V$ matrix $\bG\bY$ ((14.2)) gives the group maps $\hat\bS$ and an invertible $n\times n$ mixing matrix $\bB$ with

堆叠后的矩阵很大，它的 PCA 降维分两步进行。先把每个被试的数据降到 $q$ 维：$\bY_i = \bF_i\bX_i \in \R^{q\times V}$，其中 $\bF_i \in \R^{q\times T}$ 的各行是 $\frac1V\bX_i\bX_i^\top$ 的前 $q$ 个特征向量。再把降维后的数据堆叠成 $\bY \in \R^{Mq\times V}$，并降到 $n$ 维，其中 $\bG \in \R^{n\times Mq}$ 的各行是 $\frac1V\bY\bY^\top$ 的前 $n$ 个特征向量。把 $\bG$ 分块为 $\bG = [\bG_1, \dots, \bG_M]$，$\bG_i \in \R^{n\times q}$，于是 $\bG\bY = \sum_i \bG_i\bY_i$。对 $n\times V$ 矩阵 $\bG\bY$ 做空间 ICA（(14.2)），得到组空间图 $\hat\bS$ 与可逆的 $n\times n$ 混合矩阵 $\bB$，满足

The subject-level step removes noise specific to each subject and keeps memory requirements manageable; the group-level step keeps the dimensions that the subjects share. Here the reducing matrices have orthonormal rows, $\bF_i\bF_i^\top = \bI$ and $\bG\bG^\top = \bI$, as in the derivation of Erhardt et al. (2011). Implementations such as GIFT also whiten at each stage, dividing the rows of $\bF_i$ and of $\bG$ by the square roots of the corresponding eigenvalues. At the group level this only moves a diagonal factor into $\bB$; at the subject level it changes how the back-reconstructed maps relate to the time courses (Remark 15.3注 15.3). With $T = 100$, $q = 5$, $M = 6$ and $n = 3$, as in the example of Section 15.4第 15.4 节, $\bY$ is $30\times V$ and $\bG\bY$ is $3\times V$.

被试水平的降维去除每个被试特有的噪声，并使内存需求保持在可控范围内；组水平的降维保留各被试共有的维度。这里降维矩阵的行是单位正交的：$\bF_i\bF_i^\top = \bI$，$\bG\bG^\top = \bI$，与 Erhardt 等（2011）的推导一致。GIFT 等软件还在每一步做白化，即把 $\bF_i$ 与 $\bG$ 的各行除以相应特征值的平方根。组水平的白化只是把一个对角因子并入 $\bB$；被试水平的白化则会改变反重建得到的空间图与时间过程之间的关系（Remark 15.3注 15.3）。取 $T = 100$，$q = 5$，$M = 6$，$n = 3$（与Section 15.4第 15.4 节的例子相同），则 $\bY$ 为 $30\times V$，$\bG\bY$ 为 $3\times V$。

15.3 Back-reconstruction反重建

Group maps are only half the answer: comparisons between subjects need each subject's own time courses and maps. Back-reconstruction obtains them by running the reductions backwards. If the group PCA loses little, $\bY \approx \bG^\top\bG\bY = \bG^\top\bB\hat\bS$, and the rows that belong to subject $i$ give $\bY_i \approx \bG_i^\top\bB\hat\bS$; if the subject PCA loses little, $\bX_i \approx \bF_i^\top\bY_i$. This suggests the subject time courses ($T\times n$) and maps ($n\times V$)

组空间图只是答案的一半：被试之间的比较需要每个被试自己的时间过程与空间图。反重建通过把降维步骤倒推回去来得到它们。若组水平 PCA 损失很少，则 $\bY \approx \bG^\top\bG\bY = \bG^\top\bB\hat\bS$，取出属于被试 $i$ 的那些行，得到 $\bY_i \approx \bG_i^\top\bB\hat\bS$；若被试水平 PCA 损失很少，则 $\bX_i \approx \bF_i^\top\bY_i$。由此得到被试的时间过程（$T\times n$）与空间图（$n\times V$）：

This is the back-reconstruction of Calhoun et al. (2001), written in the present notation; Erhardt et al. (2011) call it GICA1. Because $\bF_i^\top$ has orthonormal columns, $\check\bA_i^+ = (\bG_i^\top\bB)^+\bF_i$, so $\check\bS_i = \check\bA_i^+\bX_i$: the subject maps are the least-squares regression of the subject's data on the subject's time courses. This identity relies on the orthonormal rows of $\bF_i$; Remark 15.3注 15.3 describes what changes when the subject-level reduction is whitened. Erhardt et al. (2011) compared several back-reconstruction schemes on simulated and task data. One of them, GICA3, which they introduced, produces subject maps that add up to the group maps, with subject time courses and maps that reproduce each subject's data about as accurately as the PCA reduction allows; they recommended GICA3 together with subject-specific PCA, and GICA3 is the back-reconstruction recommended in the GIFT toolbox. They also observed that GICA1 approximates dual regression, the method of the next section. Proposition 15.2命题 15.2 makes this exact.

这就是 Calhoun 等（2001）的反重建方法，这里用本章的记号写出；Erhardt 等（2011）称之为 GICA1。由于 $\bF_i^\top$ 的列单位正交，$\check\bA_i^+ = (\bG_i^\top\bB)^+\bF_i$，所以 $\check\bS_i = \check\bA_i^+\bX_i$：被试的空间图就是该被试数据对其时间过程的最小二乘回归。这一恒等式依赖于 $\bF_i$ 的行单位正交；被试水平的降维做白化时有何变化，见Remark 15.3注 15.3。Erhardt 等（2011）在仿真数据和任务数据上比较了几种反重建方案。其中由他们提出的 GICA3 给出的被试空间图之和等于组空间图，而且被试的时间过程与空间图重现被试数据的精度，大致达到 PCA 降维所允许的程度；他们推荐将 GICA3 与被试特异的 PCA 配合使用，GIFT 工具箱也把 GICA3 作为推荐的反重建方法。他们还指出 GICA1 近似于下一节的双回归。Proposition 15.2命题 15.2把这一近似关系精确化。

15.4 Dual regression双回归

Dual regression (Beckmann et al., 2009; Filippini et al., 2009), called spatio-temporal regression (STR) by Erhardt et al. (2011), skips the PCA bookkeeping and uses only a set of group maps $\hat\bS$, from group ICA or from elsewhere, together with the subject's own data. Stage 1 is a spatial regression: each image of subject $i$ is regressed on the group maps, which gives the subject's time courses. Stage 2 is a temporal regression: each voxel's time series is regressed on these time courses, which gives the subject's maps. Stage 1 solves $\min_{\bA}\|\bX_i - \bA\hat\bS\|_F$ and stage 2 solves $\min_{\bS}\|\bX_i - \hat\bA_i\bS\|_F$; in closed form,

双回归（Beckmann et al., 2009; Filippini et al., 2009），Erhardt 等（2011）称之为时空回归（spatio-temporal regression, STR），它不需要 PCA 的记账，只用一组组空间图 $\hat\bS$（来自组 ICA 或其他途径）和被试自己的数据。第一步是空间回归：把被试 $i$ 的每一幅图像对组空间图回归，得到该被试的时间过程。第二步是时间回归：把每个体素的时间序列对这些时间过程回归，得到该被试的空间图。第一步求解 $\min_{\bA}\|\bX_i - \bA\hat\bS\|_F$，第二步求解 $\min_{\bS}\|\bX_i - \hat\bA_i\bS\|_F$，其闭式解为

In the FSL implementation the time courses can be variance-normalized before stage 2. Replacing $\hat\bA_i$ by $\hat\bA_i\bD_i^{-1}$, with $\bD_i$ the diagonal matrix of their standard deviations, multiplies the rows of $\hat\bS_i$ by these standard deviations (Exercise 15.2习题 15.2), so the maps then reflect both the shape and the amplitude of each network (Nickerson et al., 2017). The subject maps are finally compared across subjects voxel by voxel, for instance with permutation tests. The closed forms show what the subject maps contain.

在 FSL 的实现中，可以在第二步之前把时间过程方差归一化。把 $\hat\bA_i$ 换成 $\hat\bA_i\bD_i^{-1}$（$\bD_i$ 是由各时间过程标准差构成的对角矩阵），相当于把 $\hat\bS_i$ 的各行乘以这些标准差（Exercise 15.2习题 15.2），于是空间图同时反映每个网络的形状和幅度（Nickerson et al., 2017）。最后在体素水平上比较各被试的空间图，例如用置换检验。闭式解揭示了被试空间图里到底包含什么。

Without variance normalization, then, every subject map equals the group map plus a correction built only from the part of the subject's data that lies outside the span of the group maps, regressed on time courses that were themselves estimated with the group maps. The two stages rest on different assumptions, common maps in stage 1 and subject-specific maps in stage 2, a point stressed by Erhardt et al. (2011). For group maps from spatial ICA, $\hat\bS\hat\bS^\top = V\bI$ (Exercise 14.5习题 14.5), so stage 1 reduces to $\hat\bA_i = \frac1V\bX_i\hat\bS^\top$ and each time course is computed independently of the other maps; stage 2 is different, since the time courses are correlated and every map depends on all of them. The relation to back-reconstruction is the following.

可见，不做方差归一化时，每个被试的空间图都等于组空间图加上一个修正项；修正项只由被试数据中落在组空间图张成空间之外的部分构成，并且是对用组空间图估计出的时间过程回归得到的。两步回归依据的假设不同：第一步假设空间图是共同的，第二步假设空间图因被试而异，Erhardt 等（2011）强调过这一点。若组空间图来自空间 ICA，则 $\hat\bS\hat\bS^\top = V\bI$（Exercise 14.5习题 14.5），第一步化为 $\hat\bA_i = \frac1V\bX_i\hat\bS^\top$，每条时间过程的计算与其他空间图无关；第二步则不同，因为各时间过程相关，每张空间图都依赖于全部时间过程。与反重建的关系如下。

The difference is the part of the subject's data that its own PCA discarded, regressed on the group maps: dual regression uses all of the subject's data, GICA1 only what survived the subject-level reduction. The group-level reduction causes no difference, because what it discards is orthogonal, across voxels, to the group maps. The following script runs the whole pipeline on six synthetic subjects whose three maps are the same blobs shifted at random by up to four voxels, with subject-specific random-walk time courses.

两者之差是被试自身 PCA 舍弃的那部分数据对组空间图的回归：双回归使用了被试的全部数据，GICA1 只用了经被试水平降维后保留下来的部分。组水平的降维不造成差别，因为它舍弃的部分在体素方向上与组空间图正交。下面的脚本在六个合成被试上运行整个流程：每个被试的三张空间图是同样的斑块，只是随机平移了至多四个体素，时间过程是各被试特有的随机游走。

import numpy as np

def fastica(Z, rng, iters=200):                       # symmetric FastICA, g = tanh
    U = np.linalg.qr(rng.standard_normal((len(Z), len(Z))))[0]
    for _ in range(iters):
        Y = U @ Z
        U = np.tanh(Y) @ Z.T / Z.shape[1] - np.diag((1 - np.tanh(Y)**2).mean(1)) @ U
        u, _, vt = np.linalg.svd(U); U = u @ vt
    return U

def pca_rows(X, k):                                   # k leading eigenvectors, as rows
    d, E = np.linalg.eigh(X @ X.T / X.shape[1])
    return E[:, ::-1][:, :k].T, d[::-1][:k]

rng = np.random.default_rng(0)
side, T, M, q, n = 30, 100, 6, 5, 3                   # grid, scans, subjects, PCA dims
yy, xx = np.mgrid[0:side, 0:side]
blob = lambda cy, cx, r: np.exp(-((yy - cy)**2 + (xx - cx)**2) / (2 * r**2)).ravel()
Xs, Ss = [], []
for i in range(M):                                    # blobs shifted at random per subject
    S = np.vstack([blob(cy + rng.integers(-4, 5), cx + rng.integers(-4, 5), r)
                   for cy, cx, r in [(8, 8, 3), (20, 22, 4), (8, 22, 3)]])
    A = np.cumsum(rng.standard_normal((T, n)), 0)    # subject-specific time courses
    X = (A - A.mean(0)) / A.std(0) @ S + 0.3 * rng.standard_normal((T, side**2))
    Xs.append(X - X.mean(1, keepdims=True)); Ss.append(S)

Fs = [pca_rows(X, q)[0] for X in Xs]                  # subject PCA: q x T each
Y = np.vstack([F @ X for F, X in zip(Fs, Xs)])        # temporal concatenation: Mq x V
G, lam = pca_rows(Y, n)                               # group PCA: n x Mq
Z = np.diag(lam ** -0.5) @ G @ Y
U = fastica(Z, rng)
S_g, B = U @ Z, np.diag(lam ** 0.5) @ U.T             # group maps; G Y = B S_g

def match(S_est, S_true):                             # mean |corr| of best-matching rows
    return np.abs(np.corrcoef(S_est, S_true)[:n, n:]).max(0).mean()
r0, r1, r2 = [], [], []
for X, F, Gi, S in zip(Xs, Fs, np.split(G, M, axis=1), Ss):
    S_1 = np.linalg.pinv(Gi.T @ B) @ F @ X            # GICA1 back-reconstruction
    A_dr = X @ np.linalg.pinv(S_g)                    # dual regression, stage 1
    S_dr = np.linalg.pinv(A_dr) @ X                   # dual regression, stage 2
    r0.append(match(S_g, S)); r1.append(match(S_1, S)); r2.append(match(S_dr, S))
print(Y.shape, S_g.shape)
print(np.round([np.mean(r0), np.mean(r1), np.mean(r2)], 3))

The first line, (30, 900) (3, 900), gives the sizes of the concatenated data $\bY$ ($Mq\times V$) and of the group maps. The second gives mean correlations between estimated and true subject maps: $0.773$ if the group maps themselves are used for every subject, $0.928$ for GICA1 and $0.930$ for dual regression. Both recover much of the spatial variability between subjects, and they nearly agree, as Proposition 15.2命题 15.2 leads one to expect when the subject PCA keeps most of the signal.

第一行 (30, 900) (3, 900) 给出拼接后数据 $\bY$（$Mq\times V$）与组空间图的尺寸。第二行是估计的被试空间图与真实被试空间图之间的平均相关系数：直接把组空间图用于每个被试时为 $0.773$，GICA1 为 $0.928$，双回归为 $0.930$。两种方法都恢复了被试之间空间变异的大部分，而且结果几乎一致；当被试水平的 PCA 保留了大部分信号时，这正是Proposition 15.2命题 15.2所预期的。

15.5 Spatially constrained ICA and NeuroMark空间约束 ICA 与 NeuroMark

Back-reconstruction and dual regression obtain subject components by regression and never ask them to be independent. Spatially constrained ICA instead runs ICA on each subject's data and uses reference maps to guide it. In the semi-blind spatial ICA of Lin et al. (2010), each component maximizes a non-Gaussianity contrast subject to a constraint that keeps it close to a spatial reference, such as a mask defined by an atlas. In group-information-guided ICA (GIG-ICA; Du & Fan, 2013), the references are the group maps, and each subject component is found by a multi-objective optimization that balances its independence against its similarity to the corresponding group component. Schematically, with $\bz$ the whitened data of subject $i$ at a random voxel and $\bar s_l$ the value of reference map $l$ there, the unit vector $\bw_l$ that extracts component $l$ solves

反重建与双回归都通过回归得到被试的成分，从不要求它们相互独立。空间约束 ICA 则对每个被试的数据做 ICA，并用参考空间图来引导它。在 Lin 等（2010）的半盲空间 ICA 中，每个成分在保持与某个空间参考（例如由图谱定义的掩模）足够接近的约束下，最大化一个非高斯性对比函数。在组信息引导 ICA（GIG-ICA；Du & Fan, 2013）中，参考就是组空间图，每个被试成分通过一个多目标优化求得，在独立性与它和对应组成分的相似性之间取得平衡。示意地说，记 $\bz$ 为被试 $i$ 在随机体素处的白化数据，$\bar s_l$ 为参考图 $l$ 在该处的取值，则提取成分 $l$ 的单位向量 $\bw_l$ 求解

two objectives combined, for example, by a weighted sum, where $J$ is a negentropy approximation (Section 6.7第 6.7 节) and the reference is standardized. Because every subject is decomposed on its own, the order and meaning of the components are fixed by the references, and new subjects can be processed one at a time without rerunning a group analysis.

两个目标可以通过加权和等方式合并，其中 $J$ 是负熵的近似（Section 6.7第 6.7 节），参考图已标准化。由于每个被试单独分解，成分的顺序与含义由参考图固定，新被试可以逐个处理，无需重新做组分析。

NeuroMark (Du et al., 2020) turns this into a pipeline for comparisons across studies. Group ICA of two large healthy-control data sets, from the Human Connectome Project and the Genomics Superstruct Project with 1,828 subjects in total, yielded 53 intrinsic networks that replicated across the two data sets; they are grouped into seven functional domains: subcortical, auditory, sensorimotor, visual, cognitive control, default mode and cerebellar. The networks of each new subject are estimated by spatially constrained ICA with these templates as references, which keeps components and connectivity measures comparable across data sets and disorders. Du et al. applied the pipeline to several brain disorders, including schizophrenia, autism, mild cognitive impairment and Alzheimer's disease, bipolar disorder and major depression.

NeuroMark（Du et al., 2020）把这一思路做成了可用于跨研究比较的流程。对两个大型健康对照数据集（来自人类连接组计划与基因组超结构计划，共 1,828 名被试）做组 ICA，得到 53 个在两个数据集之间可复现的内在网络，它们分属七个功能域：皮层下、听觉、感觉运动、视觉、认知控制、默认模式与小脑。每个新被试的网络都以这些模板为参考，用空间约束 ICA 估计，从而使成分与连接指标在不同数据集和不同疾病之间保持可比。Du 等把这一流程用于多种脑疾病，包括精神分裂症、自闭症、轻度认知障碍与阿尔茨海默病、双相障碍和重度抑郁。

15.6 Independent vector analysis独立向量分析

Temporal concatenation assumes common maps; dual regression and constrained ICA estimate subject maps around group references. Independent vector analysis (IVA; Kim, Eltoft & Lee, 2006) treats the subjects symmetrically and links them only through statistical dependence. There are $K$ data sets, here the subjects after their own PCA reduction, each with its own mixing matrix:

时间拼接假设空间图是共同的；双回归与约束 ICA 围绕组参考来估计被试的空间图。独立向量分析（independent vector analysis, IVA；Kim, Eltoft & Lee, 2006）则对称地对待各被试，只通过统计依赖把它们联系起来。设有 $K$ 个数据集（这里是经过各自 PCA 降维的被试），每个数据集有自己的混合矩阵：

The vector $\bs_i$ collects the $i$-th source of every data set and is called the $i$-th source component vector (SCV). The model assumes that the SCVs $\bs_1, \dots, \bs_n$ are mutually independent, while the components within an SCV may be, and are expected to be, dependent: the default-mode map of one subject resembles that of another. For spatial ICA of fMRI the samples are again voxels, so $s_i^{[k]}$ is the value of map $i$ of subject $k$ at a random voxel.

向量 $\bs_i$ 汇集了各数据集的第 $i$ 个源，称为第 $i$ 个源成分向量（source component vector, SCV）。模型假设各 SCV $\bs_1, \dots, \bs_n$ 相互独立，而同一 SCV 内部的分量可以相互依赖，而且通常预期是相互依赖的：一个被试的默认模式图与另一个被试的相似。对 fMRI 的空间 ICA 而言，样本仍然是体素，所以 $s_i^{[k]}$ 是被试 $k$ 的第 $i$ 张空间图在随机体素处的取值。

IVA estimates unmixing matrices $\bW^{[1]}, \dots, \bW^{[K]}$, with outputs $\by^{[k]} = \bW^{[k]}\bx^{[k]}$ and estimated SCVs $\by_i = (y_i^{[1]}, \dots, y_i^{[K]})^\top$, by minimizing the mutual information among the estimated SCVs,

IVA 估计解混矩阵 $\bW^{[1]}, \dots, \bW^{[K]}$，输出为 $\by^{[k]} = \bW^{[k]}\bx^{[k]}$，估计的 SCV 为 $\by_i = (y_i^{[1]}, \dots, y_i^{[K]})^\top$，所用的准则是最小化各估计 SCV 之间的互信息

The transformation is block diagonal, which gives the first identity below; the second splits the entropy of an estimated SCV into its marginal entropies minus the mutual information $I(\by_i)$ among its components:

变换是分块对角的，由此得到下面第一个等式；第二个等式把一个估计 SCV 的熵分解为各边缘熵之和减去其分量之间的互信息 $I(\by_i)$：

Hence the cost is

因此代价函数为

(Anderson, Adalı & Li, 2012; Adalı, Anderson & Fu, 2014). Without the terms $I(\by_i)$, the cost would be, up to a constant, the sum of $K$ separate ICA costs (6.4). These terms reward dependence within each estimated SCV, and they are what aligns the components across data sets. With two data sets separated correctly but in different orders, an estimated SCV would pair sources from two different SCVs, which are independent, so its $I(\by_i)$ would vanish; with a common order $I(\by_i) = I(\bs_i) > 0$, which lowers the cost (Exercise 15.3习题 15.3). IVA thus solves the correspondence problem as part of the estimation.

（Anderson, Adalı & Li, 2012; Adalı, Anderson & Fu, 2014）。若去掉 $I(\by_i)$ 项，代价函数就是（相差一个常数）$K$ 个各自独立的 ICA 代价 (6.4) 之和。这些项奖励每个估计 SCV 内部的依赖，正是它们把各数据集的成分对齐。若两个数据集都已正确分离但顺序不同，则估计的 SCV 会把两个不同 SCV 中的源配在一起，而它们是独立的，所以其 $I(\by_i)$ 为零；顺序一致时 $I(\by_i) = I(\bs_i) > 0$，代价更低（Exercise 15.3习题 15.3）。这样，IVA 在估计过程中顺带解决了对应问题。

Algorithms differ in the density they assume for the SCVs. The first IVA algorithms (Kim, Eltoft & Lee, 2006) used a super-Gaussian multivariate density of Laplacian type. IVA-G (Anderson, Adalı & Li, 2012) models each SCV as multivariate Gaussian and so uses only second-order statistics, the correlations of corresponding components across data sets. Even Gaussian sources can then be separated, provided their correlation structures across data sets differ. Two Gaussian SCVs whose covariance matrices satisfy $\bSigma_m = \bD\bSigma_l\bD$ for a diagonal $\bD$ cannot be separated (Exercise 15.5习题 15.5), and Anderson, Fu, Phlypo and Adalı (2014) showed that obstructions of this kind are essentially the only ones. The situation parallels SOBI (Theorem 9.6定理 9.6), with data sets in place of time lags.

各种算法的区别在于对 SCV 假设的密度。最早的 IVA 算法（Kim, Eltoft & Lee, 2006）采用拉普拉斯型的超高斯多元密度。IVA-G（Anderson, Adalı & Li, 2012）把每个 SCV 建模为多元高斯，因而只利用二阶统计量，即对应成分在各数据集之间的相关。这样一来，即使是高斯源也能分离，只要它们在各数据集之间的相关结构不同。若两个高斯 SCV 的协方差矩阵满足 $\bSigma_m = \bD\bSigma_l\bD$（$\bD$ 为对角矩阵），它们就无法分离（Exercise 15.5习题 15.5）；Anderson、Fu、Phlypo 与 Adalı（2014）证明，这类障碍本质上是唯一的障碍。这与 SOBI 的情形（Theorem 9.6定理 9.6）相似，只是用数据集代替了时间延迟。

Lee et al. (2008) applied IVA to group fMRI. Michael et al. (2014) compared group ICA with back-reconstruction and IVA on simulated data with spatially varying networks. As the variability of the maps across subjects grew, IVA stayed close to the ground truth, whereas group ICA produced subject maps that were partly composites of other maps. They also noted that group ICA remains preferable when group-level maps are the goal, and that IVA is more demanding in memory and computation.

Lee 等（2008）把 IVA 用于组 fMRI 分析。Michael 等（2014）在空间网络存在变异的仿真数据上比较了带反重建的组 ICA 与 IVA：随着空间图在被试之间的变异增大，IVA 的估计仍接近真实值，而组 ICA 得到的被试空间图部分地混入了其他空间图。他们也指出，若目标是组水平的空间图，组 ICA 仍更合适，而且 IVA 对内存与计算的要求更高。

15.7 Functional network connectivity功能网络连接

Spatial ICA leaves the time courses unconstrained, so their correlations carry information about how networks interact. Jafri et al. (2008) introduced functional network connectivity (FNC), the correlation among the time courses of spatially independent components, and applied it to schizophrenia. For subject $i$, let $\tilde\bA_i$ be the time courses of (15.4), or of any back-reconstruction, after centering in time and usually detrending and low-pass filtering. The static FNC is the $n\times n$ correlation matrix

空间 ICA 对时间过程不加约束，所以时间过程之间的相关携带了网络之间相互作用的信息。Jafri 等（2008）提出了功能网络连接（functional network connectivity, FNC），即空间独立成分的时间过程之间的相关，并把它用于精神分裂症研究。对被试 $i$，记 $\tilde\bA_i$ 为 (15.4)（或任何一种反重建）得到的时间过程，经过时间上的中心化，通常还做了去趋势与低通滤波。静态 FNC 是 $n\times n$ 相关矩阵

where $\diag(\bC_i)$ is the diagonal matrix with the same diagonal as $\bC_i$. The entries of $\bR_i$ are usually Fisher-transformed, $z = \tanh^{-1}(r)$, before statistics across subjects, because the transformed values have approximately constant variance. Connectivity also changes within a scan. Dynamic FNC computes the correlation matrix (15.7) from the time points inside a window that slides along the scan, often with tapered weights, and studies how the matrices evolve. Allen et al. (2014) used tapered windows of about 44 seconds and clustered the windowed matrices of all subjects with k-means into a few recurring connectivity states, whose frequency and transitions can be compared between groups. Short windows give noisy estimates (Exercise 15.4习题 15.4): the window length trades temporal resolution against precision, and fluctuations in windowed correlations must be distinguished from the sampling variability that stationary data would also show.

其中 $\diag(\bC_i)$ 是与 $\bC_i$ 对角元相同的对角矩阵。在做跨被试统计之前，$\bR_i$ 的元素通常要做 Fisher 变换 $z = \tanh^{-1}(r)$，因为变换后的值方差近似恒定。连接在一次扫描之内也会变化。动态 FNC 用沿扫描滑动的窗口内的时间点计算相关矩阵 (15.7)（常用渐变加权的窗），研究这些矩阵如何演化。Allen 等（2014）使用约 44 秒的渐变窗，并用 k 均值把所有被试的窗口相关矩阵聚成少数几个反复出现的连接状态，进而在组间比较各状态出现的频率与相互转换。窗口越短，估计越嘈杂（Exercise 15.4习题 15.4）：窗长需要在时间分辨率与精度之间权衡，而且窗口相关的起伏必须与平稳数据同样会表现出的抽样波动区分开来。

15.8 Multimodal fusion多模态融合

ICA can also link different kinds of data from the same subjects. Suppose each of $M$ subjects contributes one feature map from each of two modalities, for instance a gray-matter map from structural MRI and a contrast map from a task fMRI experiment, and stack them into $\bX_1 \in \R^{M\times V_1}$ and $\bX_2 \in \R^{M\times V_2}$, one row per subject. Joint ICA (Calhoun et al., 2006) assumes that the modalities share the mixing matrix:

ICA 还能把同一批被试的不同类型数据联系起来。设 $M$ 个被试中每人在两种模态下各提供一张特征图，例如结构 MRI 的灰质图与任务 fMRI 实验的对比图，把它们按每个被试一行堆叠成 $\bX_1 \in \R^{M\times V_1}$ 与 $\bX_2 \in \R^{M\times V_2}$。联合 ICA（joint ICA；Calhoun et al., 2006）假设两种模态共享混合矩阵：

Each joint source, a row of $[\bS_1, \bS_2]$, is a pair of linked maps, one per modality, and the corresponding column of $\bA$ gives its expression in each subject, the subject loadings. The samples are the $V_1 + V_2$ features, and independence is imposed on the joint sources. Differences between groups in the loadings, such as between patients and controls, point to linked changes in both modalities. Joint ICA (15.8) forces the modalities to share their loadings exactly. Parallel ICA (Liu et al., 2009) relaxes this. It runs a separate ICA for each modality, $\bX_1 = \bA_1\bS_1$ and $\bX_2 = \bA_2\bS_2$, and adds to the combined Infomax objective a term that rewards correlation between the loadings of a component of one modality and those of a component of the other, so that linked components emerge where the data support them; Liu et al. used it to relate fMRI components to genetic (SNP) components. In both models the link between modalities runs through the subject dimension, the one thing the data sets have in common.

每个联合源（$[\bS_1, \bS_2]$ 的一行）是一对相互关联的图，每种模态一张；$\bA$ 中对应的列给出它在每个被试中的表达强度，称为被试载荷。样本是 $V_1 + V_2$ 个特征，独立性施加在联合源上。载荷在组间（例如患者与对照之间）的差异，意味着两种模态中存在相互关联的变化。联合 ICA (15.8) 强制两种模态的载荷完全相同。并行 ICA（parallel ICA；Liu et al., 2009）放宽了这一点：它对每种模态分别做 ICA，$\bX_1 = \bA_1\bS_1$，$\bX_2 = \bA_2\bS_2$，并在合并的 Infomax 目标函数上加一项，奖励一种模态某个成分的载荷与另一种模态某个成分的载荷之间的相关，使相互关联的成分只在数据支持时才出现；Liu 等用它把 fMRI 成分与遗传（SNP）成分联系起来。在两种模型中，模态之间的联系都通过被试这一维度建立，这是各数据集唯一的共同之处。

15.9 Exercises习题