MATLAB高级探索：数据分析与概率建模详解

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本资源是一份高级MATLAB探索性数据分析与计算统计教程，由Mark Steyvers编写，发布日期为2015年1月14日。这份详细指南涵盖了MATLAB在数据处理过程中的关键应用，旨在帮助用户深入理解并掌握该工具在数据分析和统计计算中的进阶技术。首先，章节一探讨了探索性数据分析（Exploratory Data Analysis，EDA）。这部分从基本的数据组织与总结（1.1节）开始，讲解如何整理和提炼数据信息，以便对数据有初步认识。视觉化数据（1.2节）是重要的环节，通过图表展示，可以直观地发现数据模式、异常值和趋势，这对于理解数据特性至关重要。接着，第二部分（2.1节）涉及维度降维技术，特别是独立成分分析（Independent Component Analysis，ICA），它用于识别数据中的潜在独立成分。这部分介绍了ICA的应用场景，并解释了其工作原理。还提供了MATLAB示例，展示了如何在实际项目中实施这些技术。进入第三部分，概率模型构建（Probabilistic Modeling），作者介绍了从随机变量中进行抽样（3.1节）的基础知识，包括标准分布的使用以及非标准分布的样本生成方法。其中，逆变换法是核心技巧，针对离散和连续变量分别讨论了具体策略，如反变换法和 rejection sampling。第四部分深入探讨了马尔科夫链蒙特卡洛（Markov Chain Monte Carlo，MCMC）方法。这部分首先概述了蒙特卡罗积分的概念（4.1节），随后详细讲解了马尔可夫链（4.2节）、MCMC方法的整合（4.3节）以及Metropolis算法，包括Metropolis-Hastings算法的介绍及其在多变量分布上的应用（4.6.1 和 4.6.2 节）。此外，Gibbs采样（4.7节）作为MCMC的一种常见策略，也被详细讲解。在整个教程中，读者将学到如何利用MATLAB进行复杂的数据预处理、可视化、概率建模和高级统计计算，这将极大提升他们在实际项目中运用MATLAB进行数据分析的能力。无论是初学者还是经验丰富的用户，这份指南都能提供有价值的知识和实用技巧。

CHAPTER 2. DIMENSIONALITY REDUCTION 13

Figure 2.2: Independent components extracted from natural images

Figure 2.3: Independent components extracted from resting state fMRI. (from Storti et al.

(2013). Frontiers in Neuroscience)

CHAPTER 2. DIMENSIONALITY REDUCTION 15

observed mixtures x without knowing the mixing parameters a and having very little knowl-

edge about the nature of the original signals. Independent component analysis (ICA) is one

technique to approach this problem.

Note that ICA is formulated as a linear model – the original source signals are combined

in a linear fashion to obtain the observed signals. We can rewrite the model such that the

observed signals are the “inputs” and we use weights w

, w

, and w

to create new

outputs y

(t) and y

(t):

(t) = w

(t) + w

(t)

(t) = w

(t) + w

(t)

(2.2)

We can also write the previous two equations in matrix form:

Y = W X

X = A S

(2.3)

Through linear algebra

, it can be shown that if we set the weight matrix W to the inverse

of the original weight matrix A, then Y=S. This means that there exists a set of weights W

that will transform the observed mixtures X to the source signals S. The question now is

how to ﬁnd the appropriate set of weights W. In the examples below, we will ﬁrst show some

demonstrations of ICA with a function that will automatically produce the independent

components. We will then give some example code that will illustrate the method behind

ICA.

2.1.3 Matlab examples

There are many diﬀerent toolboxes for ICA. We will use the fastica toolbox because it

is simple and fast. It is available at http://research.ics.aalto.fi/ica/fastica/code/

dlcode.shtml. Note that when you need to apply ICA to specialized tasks such as EEG

artifact removal or resting-state analysis, there are more suitable Matlab packages for these

tasks.

To get started, let’s give a few demonstrations of ICA in the context of blind source

separation. Listing 2.1 shows Matlab code that produces the visual output show in Figure

2.5. In this example, there are two input signals corresponding to sine waves of a diﬀerent

wave lengths. These two signals are weighted to produce two mixtures x

and x

. These are

provided as inputs to the function fastica which produces the independent components y

and y

. Note how the reconstructed signals are not quite the same as the origal signals. The

ﬁrst independent component (y

) is similar to the second source signal (s

) and the second

independent component (y

) is similar to the ﬁrst source signal (s

), but in both cases, the

amplitudes are diﬀerent. This highlights two properties of ICA: the original ordering and

variances of the source signals cannot be recovered.

Note that if W = A

−1

, we can write A

−1

X = A

−1

AS = S

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