Independent Component Analysis与fMRI

From https://pnrc.cchmc.org/files/documents/summer-training-course/01_GIFT_FNC_yw.pdf

ICA（独立成分分析） finds the independent components (also called factors, latent variables or sources) by maximizing the statistical independence of the estimated components

目的： 从线性组合中分离出信号源

前提假设：
1. 是线性组合起来的（Linear mixing）
2. 各成分之间相互统计独立（independence）
3. 各信号是非高斯分布的（Non-gaussian）

两种形式：（以定义独立（independence）的方式来划分）
1. 最小化互信息量（mutual information） — >infomax
2. 最大化非高斯性（non-Gaussianity）—> FastICA

ICA v.s. PCA:

• Principle component Analysis: find the components which can explain
  the maximum amount of variance possible presented in the observed
  data.
• Both are subspace projection techniques.
• Basic goal of PCA is to decorrelate the signal by projecting the data onto
  orthogonal axes, while the goal of ICA is to find statistically independent
  components from the signal.
• PCA: minimize the reprojection error from compressed data; ICA:
  minimize the statistical dependence between the ICs.
• PCs are orthogonal and ranked in order; ICs are statistical independent,
  but not orthogonal to each other and not ranked.
• There is no closed form expression to find M (iterative algorithms).
• Important property: the new sets of components are uncorrelated with
  each other (thus PCA is often used in the pre-processing procedure of
  ICA).

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ICA 在 fMRI 数据分析中的应用：

好处： no assumptions about the form of the hemodynamic response function (HRF) which is needed for generalized linear modeling (GLM).

目的： to identify a number of unknown sources of signal, ICA assumes that these sources are mutually and statistically independent in space (sICA) or time (tICA)

sICA 更prefer一点 because temporal
points (few hundreds, corresponding to each occurrence of a functional image acquisition) are small compared to spatial locations (more than millions voxels in a functional image).

ICA 软件包：

• GIFT (http://ictab.sourceforge.net)
  Matlab, multiple ICA algorithms (infomax, fastica, …etc.)
  Graphic User Interface (GUI) and sorting options
• AnalyzeFMRI (http://www.stats.ox.ac.uk/~marchinini/software.html)
  R, FastICA
• BrainVoyager (http://www.brainvoyager.com)
  Commercial (not free, but FDA approved), FastICA
• ICALAB
  Matlab, many ICA algorithms, no user-friendly fMRI data display
• FSL melodic (http://www.fmrib.ox.ac.uk/analysis/research/melodic)
  C, FastICA

IC的数量：

• 太多： over-splitting
• 太少： over-clumping
• 对一般 fMRI 数据来说，20 - 40个component都是合理的
• 有相关算法可以帮助估计数量（GIFT软件中含有）
  
  • Akaike’s information criterion (AIC)
  • Minimum description length (MDL)
  • Bayesian information criterion (BIC)

（See more in V.D. Calhoun, et al. (2001) HBM, vol.14:140-151）

预处理

• Centering (subtract the mean to create a zero mean signal)
  
  • Simplify the ICA analysis (zero mean variable – so ICs are zeromean
    too according to the equation [take expectations on both
    sides of the Y=MX equation).
• Whitening (transform Y linearly so that new Y is white)
• Dimension reduction can be achieved using Principle Component Analysis or Singular Value Decomposition (to make sure its components are uncorrelated and their
  variances equal unity).

Group ICA