├── main.pdf ├── main.docx ├── pic ├── cqu.jpg ├── cqu.eps └── graphic.eps ├── README.md ├── attachments └── images │ └── Principle-of-PCA │ ├── 01.png │ ├── 02.png │ ├── 03.png │ ├── 05.png │ ├── 06.png │ └── 07.png ├── reference.bib ├── cqu.latex ├── chinese-gb7714-2005-numeric.csl ├── main.md └── cqu.cls /main.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/main.pdf -------------------------------------------------------------------------------- /main.docx: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/main.docx -------------------------------------------------------------------------------- /pic/cqu.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/pic/cqu.jpg -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Markdown 学术写作例子 2 | 3 | 详细用法请查看[我的博客](http://zale.site/articles/2016/05/Academia-Writing-Using-Markdown-and-Pandoc.html) 4 | -------------------------------------------------------------------------------- /attachments/images/Principle-of-PCA/01.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/attachments/images/Principle-of-PCA/01.png -------------------------------------------------------------------------------- /attachments/images/Principle-of-PCA/02.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/attachments/images/Principle-of-PCA/02.png -------------------------------------------------------------------------------- /attachments/images/Principle-of-PCA/03.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/attachments/images/Principle-of-PCA/03.png -------------------------------------------------------------------------------- /attachments/images/Principle-of-PCA/05.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/attachments/images/Principle-of-PCA/05.png -------------------------------------------------------------------------------- /attachments/images/Principle-of-PCA/06.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/attachments/images/Principle-of-PCA/06.png -------------------------------------------------------------------------------- /attachments/images/Principle-of-PCA/07.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/zl810881283/Academia-Writing-with-Markdown-Using-Pandoc/HEAD/attachments/images/Principle-of-PCA/07.png -------------------------------------------------------------------------------- /reference.bib: -------------------------------------------------------------------------------- 1 | @article{马小虎2014基于鉴别稀疏保持嵌入的人脸识别算法, 2 | title={基于鉴别稀疏保持嵌入的人脸识别算法}, 3 | author={马小虎 and 谭延琪}, 4 | journal={自动化学报}, 5 | volume={40}, 6 | number={1}, 7 | pages={73--82}, 8 | year={2014} 9 | } 10 | @article{qiao2010sparsity, 11 | title={Sparsity preserving projections with applications to face recognition}, 12 | author={Qiao, Lishan and Chen, Songcan and Tan, Xiaoyang}, 13 | journal={Pattern Recognition}, 14 | volume={43}, 15 | number={1}, 16 | pages={331--341}, 17 | year={2010}, 18 | publisher={Elsevier} 19 | } 20 | @article{de2011embedding, 21 | title={An embedding dimension reduction algorithm based on sparse analysis}, 22 | author={De-Qin, Yan and Sheng-Lan, Liu and Yan-Yan, Li}, 23 | journal={Acta Automatica Sinica}, 24 | volume={37}, 25 | number={11}, 26 | pages={1306--1312}, 27 | year={2011} 28 | } 29 | @inproceedings{marcellin2000overview, 30 | title={An overview of JPEG-2000}, 31 | author={Marcellin, Michael W and Gormish, Michael J and Bilgin, Ali and Boliek, Martin P}, 32 | booktitle={Data Compression Conference, 2000. Proceedings. DCC 2000}, 33 | pages={523--541}, 34 | year={2000}, 35 | organization={IEEE} 36 | } 37 | @article{elad2006image, 38 | title={Image denoising via sparse and redundant representations over learned dictionaries}, 39 | author={Elad, Michael and Aharon, Michal}, 40 | journal={Image Processing, IEEE Transactions on}, 41 | volume={15}, 42 | number={12}, 43 | pages={3736--3745}, 44 | year={2006}, 45 | publisher={IEEE} 46 | } 47 | @inproceedings{yang2008image, 48 | title={Image super-resolution as sparse representation of raw image patches}, 49 | author={Yang, Jianchao and Wright, John and Huang, Thomas and Ma, Yi}, 50 | booktitle={Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on}, 51 | pages={1--8}, 52 | year={2008}, 53 | organization={IEEE} 54 | } 55 | @inproceedings{huang2006sparse, 56 | title={Sparse representation for signal classification}, 57 | author={Huang, Ke and Aviyente, Selin}, 58 | booktitle={Advances in neural information processing systems}, 59 | pages={609--616}, 60 | year={2006} 61 | } 62 | @inproceedings{davenport2007smashed, 63 | title={The smashed filter for compressive classification and target recognition}, 64 | author={Davenport, Mark A and Duarte, Marco F and Wakin, Michael B and Laska, Jason N and Takhar, Dharmpal and Kelly, Kevin F and Baraniuk, Richard G}, 65 | booktitle={Electronic Imaging 2007}, 66 | pages={64980H--64980H}, 67 | year={2007}, 68 | organization={International Society for Optics and Photonics} 69 | } 70 | @article{yang2007feature, 71 | title={Feature selection in face recognition: A sparse representation perspective}, 72 | author={Yang, Allen Y and Wright, John and Ma, Yi and Sastry, S Shankar}, 73 | journal={submitted to IEEE Transactions Pattern Analysis and Machine Intelligence}, 74 | year={2007}, 75 | publisher={Citeseer} 76 | } 77 | @article{殷俊2013核稀疏保持投影及生物特征识别应用, 78 | title={核稀疏保持投影及生物特征识别应用}, 79 | author={殷俊 and 杨万扣}, 80 | journal={电子学报}, 81 | volume={41}, 82 | number={4}, 83 | pages={639--645}, 84 | year={2013} 85 | } 86 | 87 | @inproceedings{roth1999nonlinear, 88 | title={Nonlinear discriminant analysis using kernel functions}, 89 | author={Roth, Volker and Steinhage, Volker}, 90 | booktitle={Advances in neural information processing systems}, 91 | year={1999}, 92 | organization={Citeseer} 93 | } 94 | @article{scholkopf1998nonlinear, 95 | title={Nonlinear component analysis as a kernel eigenvalue problem}, 96 | author={Sch{\"o}lkopf, Bernhard and Smola, Alexander and M{\"u}ller, Klaus-Robert}, 97 | journal={Neural computation}, 98 | volume={10}, 99 | number={5}, 100 | pages={1299--1319}, 101 | year={1998}, 102 | publisher={MIT Press} 103 | } 104 | -------------------------------------------------------------------------------- /cqu.latex: -------------------------------------------------------------------------------- 1 | \documentclass[bachelor,zhspacing]{cqu} %单面打印版本 2 | \usepackage{etex} 3 | 4 | 5 | %%在这增加你需要的其它包 6 | \definecolor{hellgelb}{rgb}{1,1,0.8} 7 | \definecolor{colKeys}{rgb}{0,0,1} 8 | \definecolor{colIdentifier}{rgb}{0,0,0} 9 | \definecolor{colComments}{rgb}{1,0,0} 10 | \definecolor{colString}{rgb}{0,0.5,0} 11 | \usepackage{listings} 12 | \lstset{% 13 | float=hbp,% 14 | basicstyle=\ttfamily\small, % 15 | identifierstyle=\color{colIdentifier}, % 16 | keywordstyle=\color{colKeys}, % 17 | stringstyle=\color{colString}, % 18 | commentstyle=\color{colComments}, % 19 | columns=flexible, % 20 | tabsize=4, % 21 | frame=single, % 22 | extendedchars=true, % 23 | showspaces=false, % 24 | showstringspaces=false, % 25 | numbers=left, % 26 | numberstyle=\tiny, % 27 | breaklines=true, % 28 | backgroundcolor=\color{hellgelb}, % 29 | breakautoindent=true, % 30 | captionpos=b,% 31 | xleftmargin=0pt% 32 | } 33 | 34 | \begin{document} 35 | 36 | %-----------------------------------论文题目------------------------------------------------- 37 | \xuehao{20121892} 38 | \cntitle{重庆大学\LaTeX 学位论文模板CQU使用说明} 39 | \cnauthor{张乐} 40 | \cnmajor{软件工程} 41 | \cnteacher{葛永新} 42 | \cnxueyuan{软件学院} 43 | \entitle{An Instruction of the \LaTeX\ Templet for Chongqing University Thesis} 44 | \enauthor{Le Zhang} 45 | \enmajor{Software Engineering} 46 | \enteacher{Prof. Yongxin Ge} 47 | \enxueyuan{College of Software} 48 | \cnkind{****} 49 | \enkind{****} 50 | %\cnzlteacher{ } %%助理教师，如果必要，还要将cqu.cls中的有关该项前的%号去掉 51 | %\enzlteacher{ } 52 | \cndate{二O一六年六月} 53 | \endate{June 2016} 54 | %%%%只需修改上面的相关信息%%%%%%%% 55 | \makecntitle 56 | \makeentitle 57 | %%%%%%%%%%%%%%%%%%%%%%%%%%% 58 | 59 | \pagenumbering{Roman} 60 | \setcounter{page}{0} 61 | %------------------------------------文章摘要------------------------------------------------------------ 62 | \cnkeywords{模板，摘要，论文，\LaTeX} 63 | \begin{cnabstract} 64 | 摘要是设计或论文内容不加注释和评论的简短陈述，应以第三人称陈述。它应具有独立性和自含性， 65 | 即不阅读设计或论文的全文，就能获得必要的信息，摘要的内容应包含与设计或论文同等量的主要信 66 | 息，供读者确定有无必要阅读全文，也供文摘等二次文献采用。\par 67 | 摘要一般应说明研究工作目的、实验研究方法、结果和最终结论等，而重点是结果和结论。摘要中一 68 | 般不用图、表、化学结构式、计算机程序，不用非公知公用的符号、术语和非法定的计量单位。\par 69 | 摘要页置于英文题名页后。 \par 70 | 中文摘要一般为400汉字左右，用小四号宋体。 \par 71 | 关键词是为了文献标引工作从设计（论文）中选取出来用以表示全文主题内容信息款目的单词或术 72 | 语。一般每篇设计（论文）应选取3~5个词作为关键词，关键词间用逗号隔开，最后一个词后不打标点 73 | 符号。以显著的字符排在同种语言摘要的下方。如有可能，尽量用《汉语主题词表》等词表提供的规 74 | 范词。\par 75 | 本文介绍重庆大学论文模板cqu的使用方法。本模板符合学校的本科论文格式基本要求，而硕博模板 76 | 有待完善。 77 | 本文的创新点主要有: 78 | \begin{itemize*} 79 | \item 用例子来解释模板的使用方法; 80 | \item 用废话来填充无关紧要的部分; 81 | \item 一边学习摸索一边编写新代码。 82 | \end{itemize*} 83 | (模板作者注:中文关键词定义cnkeywords应在使用中文摘要环境之前。英文关键词同理。) 84 | \end{cnabstract} 85 | \enkeywords{template, \LaTeX, abstract, paper} 86 | \begin{enabstract} 87 | An abstract of a dissertation is a summary and extraction of 88 | research work and contributions. Included in an abstract should be 89 | description of research topic and research objective, brief 90 | introduction to methodology and research process, and summarization 91 | of conclusion and contributions of the research. An abstract should be 92 | characterized by independence and clarity and carry identical 93 | information with the dissertation. It should be such that the general 94 | idea and major contributions of the dissertation are conveyed 95 | without reading the dissertation.\par 96 | An abstract should be concise and to the point. It is a 97 | misunderstanding to make an abstract an outline of the dissertation and 98 | words “the first chapter”, “the second chapter” and the like should be 99 | avoided in the abstract.\par 100 | Key words are terms used in a dissertation for indexing, 101 | reflecting core information of the dissertation. An abstract may 102 | contain a maximum of 5 key words, with semicolons used in between to 103 | separate one another. 104 | \end{enabstract} 105 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 106 | 107 | %--------------文章目录------------- 108 | \tableofcontents 109 | \listoffigures 110 | %\addcontentsline{toc}{section}{插图清单} 111 | \listoftables 112 | %\addcontentsline{toc}{section}{附表清单} 113 | 114 | 115 | %------------------------------------词汇------------------------------------------------------------ 116 | \begin{denotation}{2.5}{0} 117 | 118 | \item[cluster] 集群 119 | \item[Itanium] 安腾 120 | \item[SMP] 对称多处理 121 | \item[API] 应用程序编程接口 122 | \item[PI] 聚酰亚胺 123 | \item[劝 学] 君子曰：学不可以已。青，取之于蓝，而青于蓝；冰，水为之，而寒于水。 124 | 木直中绳。（车柔）以为轮，其曲中规。虽有槁暴，不复挺者，（车柔）使之然也。故木 125 | 受绳则直， 金就砺则利，君子博学而日参省乎己，则知明而行无过矣。吾尝终日而思 126 | 矣， 不如须臾之所学也；吾尝（足齐）而望矣，不如登高之博见也。登高而招，臂非加 127 | 长也， 而见者远； 顺风而呼， 声非加疾也，而闻者彰。假舆马者，非利足也，而致 128 | 千里；假舟楫者，非能水也，而绝江河， 君子生非异也，善假于物也。积土成山，风雨 129 | 兴焉；积水成渊，蛟龙生焉；积善成德，而神明自得，圣心备焉。故不积跬步，无以至千 130 | 里；不积小流，无以成江海。骐骥一跃，不能十步；驽马十驾，功在不舍。锲而舍之，朽 131 | 木不折； 锲而不舍，金石可镂。蚓无爪牙之利，筋骨之强，上食埃土，下饮黄泉，用心 132 | 一也。蟹六跪而二螯，非蛇鳝之穴无可寄托者，用心躁也。\pozhehao{} 荀况 133 | \end{denotation} 134 | 135 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 136 | 137 | \pagenumbering{arabic} 138 | 139 | $body$ 140 | 141 | %\include{chapters/summery} 142 | 143 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 144 | 145 | 146 | 147 | %\include{chapters/appendix} %%附录 148 | 149 | \end{document} 150 | -------------------------------------------------------------------------------- /chinese-gb7714-2005-numeric.csl: -------------------------------------------------------------------------------- 1 | 2 | 214 | -------------------------------------------------------------------------------- /main.md: -------------------------------------------------------------------------------- 1 | --- 2 | figureTitle: "图" 3 | figPrefix: "图" 4 | 5 | tableTitle: "表" 6 | tblPrefix: "表" 7 | 8 | eqnPrefix: "公式" 9 | 10 | lot: true 11 | lof: true 12 | 13 | natbib: true 14 | --- 15 | 16 | 17 | PCA（Principal Component Analysis）是一种常用的数据分析方法。PCA通过线性变换将原始数据变换为一组各维度线性无关的表示，可用于提取数据的主要特征分量，常用于高维数据的降维。网上关于PCA的文章有很多，但是大多数只描述了PCA的分析过程，而没有讲述其中的原理。这篇文章的目的是介绍PCA的基本数学原理，帮助读者了解PCA的工作机制是什么。如下图 @fig:test 和 @tbl:tab 18 | 19 | 当然我并不打算把文章写成纯数学文章，而是希望用直观和易懂的方式叙述PCA的数学原理，所以整个文章不会引入严格的数学推导。希望读者在看完这篇文章后能更好的明白PCA的工作原理。 20 | 21 | | Tables | Are | Cool | 22 | | ------------- |:-------------:| -----:| 23 | | col 3 is | right-aligned | $1600 | 24 | | col 2 is | centered | $12 | 25 | | zebra stripes | are neat | $1 | 26 | 27 | : Demonstration of simple table syntax. {#tbl:tab} 28 | 29 | # 数据的向量表示及降维问题 30 | 31 | 一般情况下[@马小虎2014基于鉴别稀疏保持嵌入的人脸识别算法]，在数据挖掘和机器学习中，数据被表示为向量。例如某个淘宝店2012年全年的流量及交易情况可以看成一组记录的集合，其中每一天的数据是一条记录，格式如下： 32 | 33 | (日期, 浏览量, 访客数, 下单数, 成交数, 成交金额) 34 | 35 | 其中“日期”是一个记录标志而非度量值，而数据挖掘关心的大多是度量值，因此如果我们忽略日期这个字段后，我们得到一组记录，每条记录可以被表示为一个五维向量，其中一条看起来大约是这个样子：@eq:ab 36 | 37 | $$(500,240,25,13,2312.15)^\mathsf{T}$${#eq:ab} 38 | 39 | 注意这里我用了转置，因为习惯上使用列向量表示一条记录（后面会看到原因），本文后面也会遵循这个准则。不过为了方便有时我会省略转置符号，但我们说到向量默认都是指列向量。 40 | 41 | 我们当然可以对这一组五维向量进行分析和挖掘，不过我们知道，很多机器学习算法的复杂度和数据的维数有着密切关系，甚至与维数呈指数级关联。当然，这里区区五维的数据，也许还无所谓，但是实际机器学习中处理成千上万甚至几十万维的情况也并不罕见，在这种情况下，机器学习的资源消耗是不可接受的，因此我们必须对数据进行降维。 42 | 43 | 降维当然意味着信息的丢失，不过鉴于实际数据本身常常存在的相关性，我们可以想办法在降维的同时将信息的损失尽量降低。 44 | 45 | 举个例子，假如某学籍数据有两列$M$和$F$，其中M列的取值是如何此学生为男性取值$1$，为女性取值$0$；而$F$列是学生为女性取值$1$，男性取值$0$。此时如果我们统计全部学籍数据，会发现对于任何一条记录来说，当$M$为$1$时$F$必定为$0$，反之当$M$为$0$时$F$必定为$1$。在这种情况下，我们将$M$或$F$去掉实际上没有任何信息的损失，因为只要保留一列就可以完全还原另一列。 46 | 47 | 当然上面是一个极端的情况，在现实中也许不会出现，不过类似的情况还是很常见的。例如上面淘宝店铺的数据，从经验我们可以知道，“浏览量”和“访客数”往往具有较强的相关关系，而“下单数”和“成交数”也具有较强的相关关系。这里我们非正式的使用“相关关系”这个词，可以直观理解为“当某一天这个店铺的浏览量较高（或较低）时，我们应该很大程度上认为这天的访客数也较高（或较低）”。后面的章节中我们会给出相关性的严格数学定义。 48 | 49 | 这种情况表明，如果我们删除浏览量或访客数其中一个指标，我们应该期待并不会丢失太多信息。因此我们可以删除一个，以降低机器学习算法的复杂度。 50 | 51 | 上面给出的是降维的朴素思想描述，可以有助于直观理解降维的动机和可行性，但并不具有操作指导意义。例如，我们到底删除哪一列损失的信息才最小？亦或根本不是单纯删除几列，而是通过某些变换将原始数据变为更少的列但又使得丢失的信息最小？到底如何度量丢失信息的多少？如何根据原始数据决定具体的降维操作步骤？ 52 | 53 | 要回答上面的问题，就要对降维问题进行数学化和形式化的讨论。而PCA是一种具有严格数学基础并且已被广泛采用的降维方法。下面我不会直接描述PCA，而是通过逐步分析问题，让我们一起重新“发明”一遍PCA。 54 | 55 | # 向量的表示及基变换 56 | 57 | 既然我们面对的数据被抽象为一组向量，那么下面有必要研究一些向量的数学性质。而这些数学性质将成为后续导出PCA的理论基础。 58 | 59 | ## 内积与投影 60 | 61 | 下面先来看一个高中就学过的向量运算：内积。两个维数相同的向量的内积被定义为： 62 | 63 | $$(a_1,a_2,\cdots,a_n)^\mathsf{T}\cdot (b_1,b_2,\cdots,b_n)^\mathsf{T}=a_1b_1+a_2b_2+\cdots+a_nb_n$$ 64 | 65 | 内积运算将两个向量映射为一个实数。其计算方式非常容易理解，但是其意义并不明显。下面我们分析内积的几何意义。假设$A$和$B$是两个$n$维向量，我们知道$n$维向量可以等价表示为$n$维空间中的一条从原点发射的有向线段，为了简单起见我们假设$A$和$B$均为二维向量，则$A=(x_1,y_1)$，$B=(x_2,y_2)$。则在二维平面上$A$和$B$可以用两条发自原点的有向线段表示，见下图： 66 | 67 | 68 | ![这是一张图片](./pic/graphic.eps){#fig:test} 69 | 70 | 好，现在我们从$A$点向$B$所在直线引一条垂线。我们知道垂线与$B$的交点叫做$A$在$B$上的投影，再设$A$与$B$的夹角是$a$，则投影的矢量长度为$\vert A\vert cos(a)$，其中$\vert A\vert =\sqrt{x_1^2+y_1^2}$是向量$A$的模，也就是$A$线段的标量长度。 71 | 72 | 注意这里我们专门区分了矢量长度和标量长度，标量长度总是大于等于$0$，值就是线段的长度；而矢量长度可能为负，其绝对值是线段长度，而符号取决于其方向与标准方向相同或相反。 73 | 74 | 到这里还是看不出内积和这东西有什么关系，不过如果我们将内积表示为另一种我们熟悉的形式： 75 | 76 | $$A\cdot B=|A||B|cos(a)$$ 77 | 78 | 现在事情似乎是有点眉目了：$A$与$B$的内积等于$A$到$B$的投影长度乘以$B$的模。再进一步，如果我们假设$B$的模为$1$，即让$\vert B\vert =1$，那么就变成了： 79 | 80 | $$A\cdot B=|A|cos(a)$$ 81 | 82 | 也就是说，**设向量$B$的模为$1$，则$A$与$B$的内积值等于$A$向$B$所在直线投影的矢量长度**！这就是内积的一种几何解释，也是我们得到的第一个重要结论。在后面的推导中，将反复使用这个结论。 83 | 84 | # 基 85 | 86 | 下面我们继续在二维空间内讨论向量。上文说过，一个二维向量可以对应二维笛卡尔直角坐标系中从原点出发的一个有向线段。例如下面这个向量： 87 | 88 | 89 | ![](./attachments/images/Principle-of-PCA/02.png) 90 | 91 | 在代数表示方面，我们经常用线段终点的点坐标表示向量，例如上面的向量可以表示为$(3,2)$，这是我们再熟悉不过的向量表示。 92 | 93 | 不过我们常常忽略，**只有一个$(3,2)$本身是不能够精确表示一个向量的**。我们仔细看一下，这里的$3$实际表示的是向量在$x$轴上的投影值是$3$，在$y$轴上的投影值是$2$。也就是说我们其实隐式引入了一个定义：以$x$轴和$y$轴上正方向长度为$1$的向量为标准。那么一个向量$(3,2)$实际是说在$x$轴投影为$3$而$y$轴的投影为$2$。注意投影是一个矢量，所以可以为负。 94 | 95 | 更正式的说，向量$(x,y)$实际上表示线性组合： 96 | 97 | $$x(1,0)^\mathsf{T}+y(0,1)^\mathsf{T}$$ 98 | 99 | 不难证明所有二维向量都可以表示为这样的线性组合。此处$(1,0)$和$(0,1)$叫做二维空间中的一组基。 100 | 101 | 102 | ![](./attachments/images/Principle-of-PCA/03.png) 103 | 104 | 所以，**要准确描述向量，首先要确定一组基，然后给出在基所在的各个直线上的投影值，就可以了**。只不过我们经常省略第一步，而默认以$(1,0)$和$(0,1)$为基。 105 | 106 | 我们之所以默认选择$(1,0)$和$(0,1)$为基，当然是比较方便，因为它们分别是$x$和$y$轴正方向上的单位向量，因此就使得二维平面上点坐标和向量一一对应，非常方便。但实际上任何两个线性无关的二维向量都可以成为一组基，所谓线性无关在二维平面内可以直观认为是两个不在一条直线上的向量。 107 | 108 | 例如，$(1,1)$和$(-1,1)$也可以成为一组基。一般来说，我们希望基的模是$1$，因为从内积的意义可以看到，如果基的模是$1$，那么就可以方便的用向量点乘基而直接获得其在新基上的坐标了！实际上，对应任何一个向量我们总可以找到其同方向上模为$1$的向量，只要让两个分量分别除以模就好了。例如，上面的基可以变为$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$和$(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$。 109 | 110 | 现在，我们想获得$(3,2)$在新基上的坐标，即在两个方向上的投影矢量值，那么根据内积的几何意义，我们只要分别计算$(3,2)$和两个基的内积，不难得到新的坐标为$(\frac{5}{\sqrt{2}},-\frac{1}{\sqrt{2}})$。下图给出了新的基以及$(3,2)$在新基上坐标值的示意图： 111 | 112 | 113 | ![](./attachments/images/Principle-of-PCA/05.png) 114 | 115 | 另外这里要注意的是，我们列举的例子中基是正交的（即内积为$0$，或直观说相互垂直），但可以成为一组基的唯一要求就是线性无关，非正交的基也是可以的。不过因为正交基有较好的性质，所以一般使用的基都是正交的。 116 | 117 | ## 基变换的矩阵表示 118 | 下面我们找一种简便的方式来表示基变换。还是拿上面的例子，想一下，将$(3,2)$变换为新基上的坐标，就是用$(3,2)$与第一个基做内积运算，作为第一个新的坐标分量，然后用$(3,2)$与第二个基做内积运算，作为第二个新坐标的分量。实际上，我们可以用矩阵相乘的形式简洁的表示这个变换： 119 | 120 | $$\begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 5/\sqrt{2} \\ -1/\sqrt{2} \end{pmatrix}$$ 121 | 122 | 太漂亮了！其中矩阵的两行分别为两个基，乘以原向量，其结果刚好为新基的坐标。可以稍微推广一下，如果我们有$m$个二维向量，只要将二维向量按列排成一个两行$m$列矩阵，然后用“基矩阵”乘以这个矩阵，就得到了所有这些向量在新基下的值。例如$(1,1)$，$(2,2)$，$(3,3)$，想变换到刚才那组基上，则可以这样表示： 123 | 124 | $$\begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 2/\sqrt{2} & 4/\sqrt{2} & 6/\sqrt{2} \\ 0 & 0 & 0 \end{pmatrix}$$ 125 | 126 | 于是一组向量的基变换被干净的表示为矩阵的相乘。 127 | 128 | **一般的，如果我们有$M$个$N$维向量，想将其变换为由$R$个$N$维向量表示的新空间中，那么首先将$R$个基按行组成矩阵$A$，然后将向量按列组成矩阵$B$，那么两矩阵的乘积$AB$就是变换结果，其中$AB$的第$m$列为$A$中第$m$列变换后的结果**。 129 | 130 | 数学表示为： 131 | 132 | $$\begin{pmatrix} p_1 \\ p_2 \\ \vdots \\ p_R \end{pmatrix} \begin{pmatrix} a_1 & a_2 & \cdots & a_M \end{pmatrix} = \begin{pmatrix} p_1a_1 & p_1a_2 & \cdots & p_1a_M \\ p_2a_1 & p_2a_2 & \cdots & p_2a_M \\ \vdots & \vdots & \ddots & \vdots \\ p_Ra_1 & p_Ra_2 & \cdots & p_Ra_M \end{pmatrix}$$ 133 | 134 | 其中$p_i$是一个行向量，表示第$i$个基，$a_j$是一个列向量，表示第$j$个原始数据记录。 135 | 136 | 特别要注意的是，这里$R$可以小于$N$，而$R$决定了变换后数据的维数。也就是说，我们可以将一$N$维数据变换到更低维度的空间中去，变换后的维度取决于基的数量。因此这种矩阵相乘的表示也可以表示降维变换。 137 | 138 | 最后，上述分析同时给矩阵相乘找到了一种物理解释：**两个矩阵相乘的意义是将右边矩阵中的每一列列向量变换到左边矩阵中每一行行向量为基所表示的空间中去**。更抽象的说，一个矩阵可以表示一种线性变换。很多同学在学线性代数时对矩阵相乘的方法感到奇怪，但是如果明白了矩阵相乘的物理意义，其合理性就一目了然了。 139 | 140 | # 协方差矩阵及优化目标 141 | 142 | 上面我们讨论了选择不同的基可以对同样一组数据给出不同的表示，而且如果基的数量少于向量本身的维数，则可以达到降维的效果。但是我们还没有回答一个最最关键的问题：如何选择基才是最优的。或者说，如果我们有一组$N$维向量，现在要将其降到$K$维（$K$小于$N$），那么我们应该如何选择$K$个基才能最大程度保留原有的信息？ 143 | 144 | 要完全数学化这个问题非常繁杂，这里我们用一种非形式化的直观方法来看这个问题。 145 | 146 | 为了避免过于抽象的讨论，我们仍以一个具体的例子展开。假设我们的数据由五条记录组成，将它们表示成矩阵形式： 147 | 148 | $$\begin{pmatrix} 1 & 1 & 2 & 4 & 2 \\ 1 & 3 & 3 & 4 & 4 \end{pmatrix}$$ 149 | 150 | 其中每一列为一条数据记录，而一行为一个字段。为了后续处理方便，我们首先将每个字段内所有值都减去字段均值，其结果是将每个字段都变为均值为$0$（这样做的道理和好处后面会看到）。 151 | 152 | 我们看上面的数据，第一个字段均值为$2$，第二个字段均值为$3$，所以变换后： 153 | 154 | $$\begin{pmatrix} -1 & -1 & 0 & 2 & 0 \\ -2 & 0 & 0 & 1 & 1 \end{pmatrix}$$ 155 | 156 | 我们可以看下五条数据在平面直角坐标系内的样子： 157 | 158 | 159 | ![](./attachments/images/Principle-of-PCA/06.png) 160 | 161 | 现在问题来了：如果我们必须使用一维来表示这些数据，又希望尽量保留原始的信息，你要如何选择？ 162 | 163 | 通过上一节对基变换的讨论我们知道，这个问题实际上是要在二维平面中选择一个方向，将所有数据都投影到这个方向所在直线上，用投影值表示原始记录。这是一个实际的二维降到一维的问题。 164 | 165 | 那么如何选择这个方向（或者说基）才能尽量保留最多的原始信息呢？一种直观的看法是：希望投影后的投影值尽可能分散。 166 | 167 | 以上图为例，可以看出如果向x轴投影，那么最左边的两个点会重叠在一起，中间的两个点也会重叠在一起，于是本身四个各不相同的二维点投影后只剩下两个不同的值了，这是一种严重的信息丢失，同理，如果向y轴投影最上面的两个点和分布在x轴上的两个点也会重叠。所以看来x和y轴都不是最好的投影选择。我们直观目测，如果向通过第一象限和第三象限的斜线投影，则五个点在投影后还是可以区分的。 168 | 169 | 下面，我们用数学方法表述这个问题。 170 | 171 | ## 方差 172 | 173 | 上文说到，我们希望投影后投影值尽可能分散，而这种分散程度，可以用数学上的方差来表述。此处，一个字段的方差可以看做是每个元素与字段均值的差的平方和的均值，即： 174 | 175 | $$Var(a)=\frac{1}{m}\sum_{i=1}^m{(a_i-\mu)^2}$$ 176 | 177 | 由于上面我们已经将每个字段的均值都化为0了，因此方差可以直接用每个元素的平方和除以元素个数表示： 178 | 179 | $$Var(a)=\frac{1}{m}\sum_{i=1}^m{a_i^2}$$ 180 | 181 | 于是上面的问题被形式化表述为：寻找一个一维基，使得所有数据变换为这个基上的坐标表示后，方差值最大。 182 | 183 | ## 协方差 184 | 185 | 对于上面二维降成一维的问题来说，找到那个使得方差最大的方向就可以了。不过对于更高维，还有一个问题需要解决。考虑三维降到二维问题。与之前相同，首先我们希望找到一个方向使得投影后方差最大，这样就完成了第一个方向的选择，继而我们选择第二个投影方向。 186 | 187 | 如果我们还是单纯只选择方差最大的方向，很明显，这个方向与第一个方向应该是“几乎重合在一起”，显然这样的维度是没有用的，因此，应该有其他约束条件。从直观上说，让两个字段尽可能表示更多的原始信息，我们是不希望它们之间存在（线性）相关性的，因为相关性意味着两个字段不是完全独立，必然存在重复表示的信息。 188 | 189 | 数学上可以用两个字段的协方差表示其相关性，由于已经让每个字段均值为$0$，则： 190 | 191 | $$Cov(a,b)=\frac{1}{m}\sum_{i=1}^m{a_ib_i}$$ 192 | 193 | 可以看到，在字段均值为$0$的情况下，两个字段的协方差简洁的表示为其内积除以元素数$m$。 194 | 195 | 当协方差为$0$时，表示两个字段完全独立。为了让协方差为$0$，我们选择第二个基时只能在与第一个基正交的方向上选择。因此最终选择的两个方向一定是正交的。 196 | 197 | 至此，我们得到了降维问题的优化目标：**将一组N维向量降为$K$维（$K$大于$0$，小于$N$），其目标是选择$K$个单位（模为$1$）正交基，使得原始数据变换到这组基上后，各字段两两间协方差为$0$，而字段的方差则尽可能大（在正交的约束下，取最大的$K$个方差）**。 198 | 199 | ## 协方差矩阵 200 | 201 | 上面我们导出了优化目标，但是这个目标似乎不能直接作为操作指南（或者说算法），因为它只说要什么，但根本没有说怎么做。所以我们要继续在数学上研究计算方案。 202 | 203 | 我们看到，最终要达到的目的与字段内方差及字段间协方差有密切关系。因此我们希望能将两者统一表示，仔细观察发现，两者均可以表示为内积的形式，而内积又与矩阵相乘密切相关。于是我们来了灵感： 204 | 205 | 假设我们只有a和b两个字段，那么我们将它们按行组成矩阵$X$： 206 | 207 | $$X=\begin{pmatrix} a_1 & a_2 & \cdots & a_m \\ b_1 & b_2 & \cdots & b_m \end{pmatrix}$$ 208 | 209 | 然后我们用$X$乘以$X$的转置，并乘上系数$\frac{1}{m}$： 210 | 211 | $$\frac{1}{m}XX^\mathsf{T}=\begin{pmatrix} \frac{1}{m}\sum_{i=1}^m{a_i^2} & \frac{1}{m}\sum_{i=1}^m{a_ib_i} \\ \frac{1}{m}\sum_{i=1}^m{a_ib_i} & \frac{1}{m}\sum_{i=1}^m{b_i^2} \end{pmatrix}$$ 212 | 213 | 奇迹出现了！这个矩阵对角线上的两个元素分别是两个字段的方差，而其它元素是$a$和$b$的协方差。两者被统一到了一个矩阵的。 214 | 215 | 根据矩阵相乘的运算法则，这个结论很容易被推广到一般情况： 216 | 217 | **设我们有$m$个$n$维数据记录，将其按列排成$n$乘$m$的矩阵$X$，设$C=\frac{1}{m}XX^\mathsf{T}$，则$C$是一个对称矩阵，其对角线分别个各个字段的方差，而第$i$行$j$列和$j$行$i$列元素相同，表示$i$和$j$两个字段的协方差**。 218 | 219 | ## 协方差矩阵对角化 220 | 221 | 根据上述推导，我们发现要达到优化目前，等价于将协方差矩阵对角化：即除对角线外的其它元素化为$0$，并且在对角线上将元素按大小从上到下排列，这样我们就达到了优化目的。这样说可能还不是很明晰，我们进一步看下原矩阵与基变换后矩阵协方差矩阵的关系： 222 | 223 | 设原始数据矩阵$X$对应的协方差矩阵为$C$，而$P$是一组基按行组成的矩阵，设$Y=PX$，则$Y$为$X$对$P$做基变换后的数据。设$Y$的协方差矩阵为$D$，我们推导一下$D$与$C$的关系： 224 | 225 | $$\begin{array}{l l l} D & = & \frac{1}{m}YY^\mathsf{T} \\ & = & \frac{1}{m}(PX)(PX)^\mathsf{T} \\ & = & \frac{1}{m}PXX^\mathsf{T}P^\mathsf{T} \\ & = & P(\frac{1}{m}XX^\mathsf{T})P^\mathsf{T} \\ & = & PCP^\mathsf{T} \end{array}$$ 226 | 227 | 现在事情很明白了！我们要找的$P$不是别的，而是能让原始协方差矩阵对角化的$P$。换句话说，优化目标变成了**寻找一个矩阵$P$，满足$PCP^\mathsf{T}$是一个对角矩阵，并且对角元素按从大到小依次排列，那么$P$的前$K$行就是要寻找的基，用$P$的前$K$行组成的矩阵乘以$X$就使得$X$从$N$维降到了$K$维并满足上述优化条件**。 228 | 229 | 至此，我们离“发明”PCA还有仅一步之遥！ 230 | 231 | 现在所有焦点都聚焦在了协方差矩阵对角化问题上，有时，我们真应该感谢数学家的先行，因为矩阵对角化在线性代数领域已经属于被玩烂了的东西，所以这在数学上根本不是问题。 232 | 233 | 由上文知道，协方差矩阵$C$是一个是对称矩阵，在线性代数上，实对称矩阵有一系列非常好的性质： 234 | 235 | 1. 实对称矩阵不同特征值对应的特征向量必然正交。 236 | 237 | 2. 设特征向量$\lambda$重数为$r$，则必然存在$r$个线性无关的特征向量对应于$\lambda$，因此可以将这$r$个特征向量单位正交化。 238 | 239 | 由上面两条可知，一个$n$行$n$列的实对称矩阵一定可以找到$n$个单位正交特征向量，设这$n$个特征向量为$e_1,e_2,\cdots,e_n$，我们将其按列组成矩阵： 240 | 241 | $$E=\begin{pmatrix} e_1 & e_2 & \cdots & e_n \end{pmatrix}$$ 242 | 243 | 则对协方差矩阵$C$有如下结论： 244 | 245 | $$E^\mathsf{T}CE=\Lambda=\begin{pmatrix} \lambda_1 & & & \\ & \lambda_2 & & \\ & & \ddots & \\ & & & \lambda_n \end{pmatrix}$$ 246 | 247 | 其中$\Lambda$为对角矩阵，其对角元素为各特征向量对应的特征值（可能有重复）。 248 | 249 | 以上结论不再给出严格的数学证明，对证明感兴趣的朋友可以参考线性代数书籍关于“实对称矩阵对角化”的内容。 250 | 251 | 到这里，我们发现我们已经找到了需要的矩阵$P$： 252 | 253 | $$P=E^\mathsf{T}$$ 254 | 255 | $P$是协方差矩阵的特征向量单位化后按行排列出的矩阵，其中每一行都是$C$的一个特征向量。如果设$P$按照$\Lambda$中特征值的从大到小，将特征向量从上到下排列，则用$P$的前$K$行组成的矩阵乘以原始数据矩阵$X$，就得到了我们需要的降维后的数据矩阵$Y$。 256 | 257 | 至此我们完成了整个PCA的数学原理讨论。在下面的一节，我们将给出PCA的一个实例。 258 | 259 | # 算法及实例 260 | 261 | 为了巩固上面的理论，我们在这一节给出一个具体的PCA实例。 262 | 263 | ## PCA算法 264 | 265 | 总结一下PCA的算法步骤： 266 | 267 | 设有$m$条$n$维数据。 268 | 269 | 1. 将原始数据按列组成$n$行$m$列矩阵$X$ 270 | 271 | 2. 将$X$的每一行（代表一个属性字段）进行零均值化，即减去这一行的均值 272 | 273 | 3. 求出协方差矩阵$C=\frac{1}{m}XX^\mathsf{T}$ 274 | 275 | 4. 求出协方差矩阵的特征值及对应的特征向量 276 | 277 | 5. 将特征向量按对应特征值大小从上到下按行排列成矩阵，取前$k$行组成矩阵$P$ 278 | 279 | 6. $Y=PX$即为降维到$k$维后的数据 280 | 281 | ## 实例 282 | 这里以上文提到的 283 | 284 | $$\begin{pmatrix} -1 & -1 & 0 & 2 & 0 \\ -2 & 0 & 0 & 1 & 1 \end{pmatrix}$$ 285 | 286 | 为例，我们用PCA方法将这组二维数据其降到一维。 287 | 288 | 因为这个矩阵的每行已经是零均值，这里我们直接求协方差矩阵： 289 | 290 | $$C=\frac{1}{5}\begin{pmatrix} -1 & -1 & 0 & 2 & 0 \\ -2 & 0 & 0 & 1 & 1 \end{pmatrix}\begin{pmatrix} -1 & -2 \\ -1 & 0 \\ 0 & 0 \\ 2 & 1 \\ 0 & 1 \end{pmatrix}=\begin{pmatrix} \frac{6}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{6}{5} \end{pmatrix}$$ 291 | 292 | 然后求其特征值和特征向量，具体求解方法不再详述，可以参考相关资料。求解后特征值为： 293 | 294 | $$\lambda_1=2,\lambda_2=\frac{2}{5}$$ 295 | 296 | 其对应的特征向量分别是： 297 | 298 | $$c_1\begin{pmatrix} 1 \\ 1 \end{pmatrix},c_2\begin{pmatrix} -1 \\ 1 \end{pmatrix}$$ 299 | 300 | 其中对应的特征向量分别是一个通解，$c_1$和$c_2$可取任意实数。那么标准化后的特征向量为： 301 | 302 | $$\begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{pmatrix},\begin{pmatrix} -1/\sqrt{2} \\ 1/\sqrt{2} \end{pmatrix}$$ 303 | 304 | 因此我们的矩阵$P$是： 305 | 306 | $$P=\begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}$$ 307 | 308 | 可以验证协方差矩阵$C$的对角化： 309 | 310 | $$PCP^\mathsf{T}=\begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}\begin{pmatrix} 6/5 & 4/5 \\ 4/5 & 6/5 \end{pmatrix}\begin{pmatrix} 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}=\begin{pmatrix} 2 & 0 \\ 0 & 2/5 \end{pmatrix}$$ 311 | 312 | 最后我们用$P$的第一行乘以数据矩阵，就得到了降维后的表示： 313 | 314 | $$Y=\begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}\begin{pmatrix} -1 & -1 & 0 & 2 & 0 \\ -2 & 0 & 0 & 1 & 1 \end{pmatrix}=\begin{pmatrix} -3/\sqrt{2} & -1/\sqrt{2} & 0 & 3/\sqrt{2} & -1/\sqrt{2} \end{pmatrix}$$ 315 | 316 | 降维投影结果如下图： 317 | 318 | 319 | ![](./attachments/images/Principle-of-PCA/07.png) 320 | 321 | # 进一步讨论 322 | 323 | 根据上面对PCA的数学原理的解释，我们可以了解到一些PCA的能力和限制。PCA本质上是将方差最大的方向作为主要特征，并且在各个正交方向上将数据“离相关”，也就是让它们在不同正交方向上没有相关性。 324 | 325 | 因此，PCA也存在一些限制，例如它可以很好的解除线性相关，但是对于高阶相关性就没有办法了，对于存在高阶相关性的数据，可以考虑Kernel PCA，通过Kernel函数将非线性相关转为线性相关，关于这点就不展开讨论了。另外，PCA假设数据各主特征是分布在正交方向上，如果在非正交方向上存在几个方差较大的方向，PCA的效果就大打折扣了。 326 | 327 | 最后需要说明的是，PCA是一种无参数技术，也就是说面对同样的数据，如果不考虑清洗，谁来做结果都一样，没有主观参数的介入，所以PCA便于通用实现，但是本身无法个性化的优化。 328 | 329 | 希望这篇文章能帮助朋友们了解PCA的数学理论基础和实现原理，借此了解PCA的适用场景和限制，从而更好的使用这个算法。 330 | 331 | 332 | \begin{thebibliography}{99} 333 | 334 | \end{thebibliography} 335 | -------------------------------------------------------------------------------- /cqu.cls: -------------------------------------------------------------------------------- 1 | %% This is file `cqu.cls',based on article styles and thuthesis styles( 清华大学模板). 2 | %% Copyright (C) 2009-2010 by G.C. 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\newcommand{\yihao}{\fontsize{26bp}{\baselineskip}\selectfont} 130 | \newcommand{\liuhao}{\fontsize{7.5bp}{\baselineskip}\selectfont} 131 | \newcommand{\xiaoliuhao}{\fontsize{6.5bp}{\baselineskip}\selectfont} 132 | \newcommand{\qihao}{\fontsize{5.5bp}{\baselineskip}\selectfont} 133 | \newcommand{\xiaoqihao}{\fontsize{5bp}{\baselineskip}\selectfont} 134 | %%%%%正文小四号，行距固定为20磅 135 | \renewcommand\normalsize{% 136 | \@setfontsize\normalsize{12bp}{20bp} 137 | \abovedisplayskip=10bp \@plus 2bp \@minus 2bp 138 | \abovedisplayshortskip=10bp \@plus 2bp \@minus 2bp 139 | \belowdisplayskip=\abovedisplayskip 140 | \belowdisplayshortskip=\abovedisplayshortskip} 141 | \newlength{\hangju} 142 | \setlength{\hangju}{20bp \@plus.2ex} 143 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%页面格式设置%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 144 | %%页面各边距根据学校论文要求的边距设置,可微调 145 | %%页眉：左侧显示“重庆大学本科学生毕业设计 （论文）”，右侧为该节（section）名称， 146 | %%页眉下有一宽度为0.8pt的直线 147 | %%页脚：居中显示该页页码 148 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 149 | \RequirePackage{geometry} 150 | \RequirePackage{fancyhdr} 151 | \if@bachelor 152 | \geometry{a4paper,left=3.5cm,right=2.5cm,% 153 | top=3.0cm,headheight=1.6cm,headsep=1cm,% 154 | bottom=2.5cm,footskip=1cm} 155 | \else 156 | \geometry{a4paper,left=3.5cm,right=2.5cm,% 157 | top=3.0cm,headheight=1.6cm,headsep=1cm,% 158 | bottom=2.5cm,footskip=1cm} 159 | \fi 160 | %%%%%页眉和页脚设置 161 | \pagestyle{fancy} 162 | \renewcommand{\sectionmark}[1]{\markboth{#1}{}} 163 | \renewcommand{\headrulewidth}{0.8pt} 164 | \lhead{\xiaowuhao 重庆大学本科学生毕业设计 （论文）} 165 | \rhead{\xiaowuhao \leftmark} 166 | \if@master 167 | \lhead{\wuhao 重庆大学硕士学位论文} 168 | \rhead{\wuhao \leftmark} 169 | \fi 170 | \if@doctor 171 | \lhead{\wuhao 重庆大学博士学位论文} 172 | \rhead{\wuhao \leftmark} 173 | \fi 174 | \if@twoside 175 | \lhead{}\rhead{} 176 | \fancyhead[EC]{\xiaowuhao \leftmark} 177 | \fancyhead[OC]{\xiaowuhao 重庆大学本科学生毕业设计 （论文）} 178 | \if@master 179 | \fancyhead[OC]{\wuhao 重庆大学硕士学位论文} 180 | \fancyhead[EC]{\wuhao \leftmark} 181 | \fi 182 | \if@doctor 183 | \fancyhead[OC]{\wuhao 重庆大学博士学位论文} 184 | \fancyhead[EC]{\wuhao \leftmark} 185 | \fi 186 | \fi 187 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%正文设置%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 188 | %%章节标题：最多三级，即X.X.X;第一级section，三号黑体居中，与上文间距2个中文字符，与下 189 | %%文间距1个中文字符;第二级subsection，小三号黑体居左;第三级subsubsection，四号黑体居左 190 | %%段落缩进：2个中文字符 191 | %%脚注：以带圈阿拉伯数字编号，按页编号 192 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 193 | %%%段落首行缩进2个中文字符 194 | \RequirePackage{indentfirst} 195 | \parindent 2.3em 196 | \setlength{\parskip}{0pt \@plus2pt \@minus0pt} %段落间距 197 | %%%脚注按页编号 198 | \RequirePackage{footnpag} 199 | \renewcommand{\thefootnote}{% 200 | \scriptsize\textcircled{\tiny\arabic{footnote}}} 201 | %%%章节标题设置 202 | \renewcommand\section{\@startsection {section}{1}{\z@}% 203 | {-3.5ex \@plus -1ex \@minus -.2ex}% 204 | {\hangju}% 205 | {\if@bachelor \vspace*{\hangju} 206 | \else 207 | \vspace*{0ex} 208 | \fi 209 | \centering\normalfont\sanhao\bfseries}} 210 | \renewcommand\subsection{\@startsection{subsection}{2}{\z@}% 211 | {-20bp \@plus -1ex \@minus -.2ex}% 212 | {1bp \@plus .2ex}% 213 | {\raggedright\normalfont\xiaosanhao\bfseries}} 214 | \renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% 215 | {-1bp \@plus -1ex \@minus -.2ex}% 216 | {1bp \@plus .2ex}% 217 | {\raggedright\normalfont\sihao\bfseries}} 218 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%目录设置%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 219 | %%利用\printcontent输出目录 220 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 221 | \def\@dotsep{1} % 目录点间的距离 222 | \renewcommand*\l@section[2]{% 223 | \ifnum 1>\c@tocdepth \else 224 | \vskip \z@ \@plus.2\p@ 225 | {\leftskip 0em\relax \rightskip \@tocrmarg \parfillskip -\rightskip 226 | \parindent 0em\relax\@afterindenttrue 227 | \interlinepenalty\@M 228 | \leavevmode 229 | \@tempdima 1em\relax 230 | \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip 231 | {\sihao \bf #1}\nobreak 232 | \leaders\hbox{$\m@th\mkern \@dotsep mu\hbox{.}\mkern \@dotsep mu$}\hfill 233 | \nobreak{\normalfont \normalcolor #2}% 234 | \par}% 235 | \fi} 236 | \renewcommand*\l@subsection[2]{% 237 | \ifnum 2>\c@tocdepth \else 238 | \vskip \z@ \@plus.2\p@ 239 | {\leftskip 1em\relax \rightskip \@tocrmarg \parfillskip -\rightskip 240 | \parindent 1em\relax\@afterindenttrue 241 | \interlinepenalty\@M 242 | \leavevmode 243 | \@tempdima 1.5em\relax 244 | \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip 245 | {\wuhao \bf #1}\nobreak 246 | \leaders\hbox{$\m@th\mkern \@dotsep mu\hbox{.}\mkern \@dotsep mu$}\hfill 247 | \nobreak{\normalfont \normalcolor #2}% 248 | \par}% 249 | \fi} 250 | \renewcommand*\l@subsubsection[2]{% 251 | \ifnum 3>\c@tocdepth \else 252 | \vskip \z@ \@plus.2\p@ 253 | {\leftskip 1.6em\relax \rightskip \@tocrmarg \parfillskip -\rightskip 254 | \parindent 1.6em\relax\@afterindenttrue 255 | \interlinepenalty\@M 256 | \leavevmode 257 | \@tempdima 2em\relax 258 | \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip 259 | {\wuhao #1}\nobreak 260 | \leaders\hbox{$\m@th\mkern \@dotsep mu\hbox{.}\mkern \@dotsep mu$}\hfill 261 | \nobreak{\normalfont \normalcolor #2}% 262 | \par}% 263 | \fi} 264 | %%重新定义生成目录 265 | \renewcommand\tableofcontents{\newpage% 266 | \if@bachelor \vspace*{-2\hangju} 267 | \else \vspace*{-\hangju} 268 | \fi % 269 | \section*{\contentsname\@mkboth{目录}{}}% 270 | \@starttoc{toc}% 271 | } 272 | %列表和图像目录 273 | \renewcommand\listoffigures{\newpage% 274 | \if@bachelor \vspace*{-2\hangju} 275 | \else \vspace*{-\hangju} 276 | \fi % 277 | \section*{\listfigurename}% 278 | \@starttoc{lof}} 279 | \renewcommand\listoftables{% 280 | \section*{\listtablename}% 281 | \@starttoc{lot}% 282 | } 283 | 284 | %%%%%%中文标题定义 285 | \renewcommand{\refname}{参\quad 考\quad 文\quad 献} 286 | \renewcommand{\contentsname}{\centerline{% 287 | \bf{\sanhao{目\qquad 录}}}} 288 | \renewcommand{\listfigurename}{\centerline{% 289 | \bf{\sanhao{插\quad 图\quad 清\quad 单}}}} 290 | \renewcommand{\listtablename}{\centerline{% 291 | \bf{\sanhao{附\quad 表\quad 清\quad 单}}}} 292 | \newcommand{\listequationname}{\bf{% 293 | \sanhao{公\quad 式\quad 索\quad 引}}} 294 | \newcommand{\equationname}{公式} 295 | \renewcommand{\indexname}{索引} 296 | \renewcommand{\figurename}{图} 297 | \renewcommand{\tablename}{表} 298 | \renewcommand{\appendixname}{附录} 299 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%浮动对象设置%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 300 | %%提供了排布一般图形和表格，含子图形的图形和并列图形或表格的各种浮动对象环境 301 | %%表格环境：mytable，标题在上，使用中英文标题的命令是\tabcaptionwithen 302 | %%图形环境：figure，标题在下，使用中英文标题的命令是\figcaptionwithen 303 | %%并列图形：twofigures，用来排布并列的两个图形 304 | %%子图形：subfigure，具体用法参见subfig宏包文档 305 | %%利用sidewaystable环境和\rotcaption命令，可用来旋转较宽的表格 306 | %%使用longtable环境来布跨页长表格 307 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 308 | \numberwithin{table}{section} 309 | \numberwithin{figure}{section} 310 | \newcommand{\entablename}{Table} 311 | \newcommand{\enfigurename}{Figure} 312 | 313 | 314 | %\newcounter{tabnote} 315 | %\stepcounter{tabnote} 316 | %\newcommand{\tabnote}{\textsuperscript{\thetabnote}} 317 | %\newcommand{\tabnote}[1]{\tabnote #1} 318 | 319 | \DeclareCaptionFont{wuhao}{\wuhao} 320 | \captionsetup{labelsep=quad,% 321 | justification=centering,% 322 | font=wuhao} %%一般的caption样式 323 | %%%%%表格环境名称 mytable 324 | \newenvironment{mytable}{% 325 | \floatstyle{plaintop}\restylefloat{table}% 326 | \begin{table}[H]\centering\wuhao}% 327 | {\end{table}} 328 | \DeclareCaptionLabelFormat{sectiontab}{% 329 | \tablename ~ \thetable} 330 | \captionsetup[table]{labelformat=sectiontab,% 331 | belowskip=6bp,% 332 | aboveskip={12bp-\intextsep}}%% 表格的caption样式 333 | \newcommand{\tabcaptionwithen}[2]{%\vspace*{-0.8em} % 334 | \addtocounter{table}{1}\centering\wuhao % 335 | \tablename ~ \thetable\quad #1% 336 | \\\entablename ~ \thetable ~ #2% 337 | \\} %%中英文表格标题 338 | 339 | %说明：排布表格中的脚注时，可以将脚注内容当作表格内容置于 340 | %表末处理，符号为\textsuperscript{*}，当然也可以使用threeparttable宏包 341 | %%%%%图片环境名称 figure 342 | \DeclareCaptionLabelFormat{sectionfig}{% 343 | \figurename ~ \thefigure} 344 | \captionsetup[figure]{labelformat=sectionfig,% 345 | belowskip={12bp-\intextsep},% 346 | aboveskip=6bp}%% 图片的caption样式 347 | \captionsetup[subfloat]{font=small,% 348 | labelformat=parens,% 349 | labelsep=space,% 350 | farskip=1ex}%% 子图的caption样式 351 | \newcommand{\figcaptionwithen}[2]{% 352 | \\\addtocounter{figure}{1}% 353 | \centering\wuhao% 354 | \figurename ~ \thefigure\quad #1% 355 | \\\enfigurename ~ \thefigure ~ #2% 356 | \vspace*{-0.7em}}%%中英文图像标题 357 | \newenvironment{twofigures}[2]{ %%两个并列图形环境 358 | \begin{figure} 359 | \begin{minipage}[t]{0.45\textwidth}% 360 | \centering #1% 361 | \end{minipage}% 362 | \qquad 363 | \begin{minipage}[t]{0.45\textwidth}% 364 | \centering #2% 365 | \end{minipage}% 366 | \end{figure}} 367 | 368 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%标题页设置%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 369 | %%有指导教师的将下面相关代码前的”%“号去掉 370 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 371 | \def\cntitlename{\bf{\xiaoerhao 重庆大学本科学生毕业设计（论文）}} 372 | \def\entitlename{\textbf{\xiaoerhao Graduation Design(Thesis) of Chongqing University}} 373 | %\def\entitlename{\textbf{\xiaoerhao } 374 | %\def\entitlename{\textbf{\xiaoerhao } 375 | \newtoks\cntitle %%论文题目 376 | \newtoks\entitle 377 | \newtoks\cnauthor %%作者 378 | \newtoks\enauthor 379 | \newtoks\xuehao %%学号 380 | \newtoks\cnmajor %%专业 381 | \newtoks\enmajor 382 | \newtoks\cnteacher %%指导教师 383 | \newtoks\enteacher 384 | \newtoks\cnxueyuan %%学院 385 | \newtoks\enxueyuan 386 | 387 | \newtoks\enzlteacher 388 | \newtoks\cndate %%日期 389 | \newtoks\endate 390 | \newtoks\cnkind %%学科门类 391 | \newtoks\enkind 392 | 393 | \if@bachelor 394 | \def\makecntitle{\null % 395 | \thispagestyle{empty} 396 | \begin{center}\vspace*{-3em}\bf\sanhao 397 | \renewcommand{\arraystretch}{1.5} 398 | \begin{tabular}{p{\linewidth}} 399 | \multicolumn{1}{c}{\xiaoerhao\cntitlename}\\ 400 | \tabularnewline \tabularnewline 401 | \begin{minipage}{\textwidth}\centering % 402 | \begin{spacing}{1.5} 403 | {\erhao\the\cntitle} 404 | \end{spacing} 405 | \end{minipage}\\ 406 | \tabularnewline \tabularnewline \tabularnewline 407 | \multicolumn{1}{c}{% 408 | \if@pdftex 409 | \includegraphics[scale=1.0]{pic/cqu.jpg} 410 | \else 411 | \includegraphics[scale=1.0]{pic/cqu.eps}\fi}\\ 412 | \tabularnewline \tabularnewline 413 | \multicolumn{1}{c}{% 414 | \begin{tabular}{l} 415 | 学\qquad 生：\the\cnauthor \\ 416 | 学\qquad 号：\the\xuehao \\ 417 | 指导教师：\the\cnteacher \\ 418 | % 助理指导教师：\the\cnzlteacher \\ 419 | 专\qquad 业：\the\cnmajor 420 | \end{tabular}} 421 | \end{tabular} \vfill 422 | {\xiaoerhao 重庆大学\the\cnxueyuan}\\[1em] 423 | {\the\cndate} 424 | \end{center}\newpage} 425 | \def\makeentitle{\null % 426 | \thispagestyle{empty} 427 | \begin{center}\vspace*{-3em}\bf\sanhao 428 | \renewcommand{\arraystretch}{1.5} 429 | \begin{tabular}{p{\linewidth}} 430 | \multicolumn{1}{c}{\entitlename}\\ 431 | \tabularnewline \tabularnewline 432 | \begin{minipage}{\textwidth}\centering % 433 | \begin{spacing}{1.5} 434 | {\erhao\the\entitle} 435 | \end{spacing} 436 | \end{minipage}\\ 437 | \tabularnewline \tabularnewline \tabularnewline 438 | \multicolumn{1}{c}{% 439 | \if@pdftex 440 | \includegraphics[scale=1.0]{pic/cqu.jpg} 441 | \else 442 | \includegraphics[scale=1.0]{pic/cqu.eps}\fi}\\ 443 | \tabularnewline \tabularnewline 444 | \multicolumn{1}{c}{% 445 | \begin{tabular}{l} 446 | Undergraduate：\the\enauthor \\ 447 | Supervisor：\the\enteacher \\ 448 | % 助理指导教师：\the\enzlteacher \\ 449 | Major：\the\enmajor 450 | \end{tabular}} 451 | \end{tabular} \vfill 452 | {\xiaoerhao\the\enxueyuan}\\[1em] 453 | {\xiaoerhao Chongqing University}\\[1em] 454 | {\the\endate} 455 | \end{center}} 456 | \else 457 | \def\makecntitle{\null % 458 | \thispagestyle{empty} 459 | \begin{center}\vspace*{-3em}\bf\sanhao 460 | \renewcommand{\arraystretch}{2} 461 | \begin{tabular}{p{\linewidth}} 462 | \begin{minipage}{\textwidth}\centering % 463 | \begin{spacing}{2} 464 | {\erhao\the\cntitle} 465 | \end{spacing} 466 | \end{minipage}\\ 467 | \tabularnewline \tabularnewline 468 | \multicolumn{1}{c}{% 469 | \if@pdftex 470 | \includegraphics[scale=1.0]{pic/cqu.jpg} 471 | \else 472 | \includegraphics[scale=1.0]{pic/cqu.eps}\fi}\\ 473 | \tabularnewline 474 | \multicolumn{1}{c}{\xiaoerhao% 475 | \if@master 重庆大学硕士学位论文 476 | \else 重庆大学博士学位论文 477 | \fi}\\ 478 | \tabularnewline 479 | \multicolumn{1}{c}{% 480 | \begin{tabular}{l} 481 | 学生姓名：\the\cnauthor \\ 482 | 指导教师：\the\cnteacher \\ 483 | % 助理指导教师：\the\cnzlteacher \\ 484 | 专\qquad 业：\the\cnmajor \\ 485 | 学科门类：\the\cnkind 486 | \end{tabular}} 487 | \end{tabular} \vfill 488 | {\xiaoerhao 重庆大学\the\cnxueyuan}\\[1em] 489 | {\the\cndate} 490 | \end{center}\newpage} 491 | \def\makeentitle{\null % 492 | \thispagestyle{empty} 493 | \begin{center}\vspace*{-3em}\bf\sanhao 494 | \renewcommand{\arraystretch}{1.5} 495 | \begin{tabular}{p{\linewidth}} 496 | \begin{minipage}{\textwidth}\centering % 497 | \begin{spacing}{2} 498 | {\erhao\the\entitle} 499 | \end{spacing} 500 | \end{minipage}\\ 501 | \tabularnewline 502 | \multicolumn{1}{c}{% 503 | \if@pdftex 504 | \includegraphics[scale=1.0]{pic/cqu.jpg} 505 | \else 506 | \includegraphics[scale=1.0]{pic/cqu.eps}\fi}\\ 507 | \tabularnewline 508 | \multicolumn{1}{c}{\xiaoerhao A Thesis Submitted to % 509 | Chongqing University}\\ 510 | \multicolumn{1}{c}{\xiaoerhao in Partial Fulfillment % 511 | of the Requirement for the}\\ 512 | \multicolumn{1}{c}{\xiaoerhao Degree of 513 | \if@master {Master} \else {Doctor}\fi of \the\enkind 514 | }\\ 515 | \multicolumn{1}{c}{by}\\ 516 | \multicolumn{1}{c}{\the\enauthor}\\ 517 | \tabularnewline 518 | \multicolumn{1}{c}{% 519 | \begin{tabular}{c} 520 | Supervisor：\the\enteacher \\ 521 | % 助理指导教师：\the\enzlteacher \\ 522 | Major：\the\enmajor 523 | \end{tabular}} 524 | \end{tabular} \vfill 525 | {\xiaoerhao\the\enxueyuan}\\[1em] 526 | {\xiaoerhao Chongqing University,~Chongqing,~China.}\\[1em] 527 | {\the\endate} 528 | \end{center}} 529 | \fi 530 | 531 | %%%%%%%%%%%%%%%%%%%%%%%%%定义摘要环境%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 532 | %%cnabstract:中文摘要环境，cnkeywords：中文关键词 533 | %%enabstract:英文摘要环境，enkeywords：英文关键词 534 | %%注意：\cnkeywords和\enkeywords必须放在各环境前面 535 | %%这两个环境页眉独立设置 536 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 537 | \newtoks\cnkeywords 538 | 539 | \newtoks\enkeywords 540 | 541 | \newenvironment{cnabstract}{\newpage % 542 | \if@bachelor 543 | \vspace*{-2\hangju} 544 | \else 545 | \vspace*{-\hangju} 546 | \fi 547 | \section*{摘\qquad 要\@mkboth{中文摘要}{}}}% 548 | {\par 549 | \if@bachelor 550 | \vskip 2\hangju \noindent 551 | \else 552 | \vskip \hangju \noindent 553 | \fi 554 | \hei{关键词：}\the\cnkeywords % 555 | \addcontentsline{toc}{section}{中文摘要} % 556 | } 557 | 558 | \newenvironment{enabstract}{\newpage % 559 | \if@bachelor 560 | \vspace*{-2\hangju} 561 | \else 562 | \vspace*{-\hangju} 563 | \fi 564 | \section*{ABSTRACT}% 565 | \if@bachelor\@mkboth{ABSTRACT}{} 566 | \else\@mkboth{英文摘要}{}\fi}% 567 | {\par 568 | \if@bachelor 569 | \vskip 2\hangju \noindent 570 | \else 571 | \vskip \hangju \noindent 572 | \fi 573 | \textbf{Keywords:}~~\the\enkeywords % 574 | \addcontentsline{toc}{section}{ABSTRACT}% 575 | } 576 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%致谢和附录设置%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 577 | %%致谢环境：ack，单独一页 578 | %%附录环境：cnappendix，apptitle定义附录名称，附录序号设置为大写字母，若有小节， 579 | %%从subsection开始 580 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 581 | %%%%%%定义致谢环境 582 | \newcommand{\@ackname}{致\qquad 谢} 583 | %\@mkboth{\refname}{\refname}% 584 | \newenvironment{ack}{\newpage % 585 | \if@bachelor 586 | \vspace*{-2\hangju} 587 | \else 588 | \vspace*{-\hangju} 589 | \fi 590 | \section*{\@ackname\@mkboth{致谢}{}}}{% 591 | \addcontentsline{toc}{section}{致谢} 592 | \newpage} 593 | 594 | %%%%%定义附录环境 595 | \newcounter{appsection} 596 | \renewcommand\theappsection{\Alph{appsection}} 597 | \stepcounter{appsection} 598 | \newcommand{\apptitle}[1]{% 599 | {\newpage 600 | \@mkboth{\appendixname\theappsection :~#1}{}% 601 | \centering \vspace*{-0.2ex} 602 | \interlinepenalty \@M 603 | \normalfont \sanhao \bfseries % 604 | \appendixname\theappsection :~~#1\par} 605 | \addcontentsline{toc}{section}{\appendixname\theappsection :~#1}% 606 | \vspace*{\hangju} 607 | \stepcounter{appsection}\stepcounter{section}} 608 | \newenvironment{cnappendix}{\par% 609 | \setcounter{section}{0}% 610 | \setcounter{subsection}{0}% 611 | \def\thesection{\@Alph\c@section}}{} 612 | %%简单的附录环境 613 | \let\simappendix\appendix 614 | \renewenvironment{simappendix}{% 615 | \if@bachelor 616 | \vspace*{-2\hangju} 617 | \else 618 | \vspace*{-\hangju} 619 | \fi 620 | \section*{附\qquad 录\@mkboth{\appendixname}{}% 621 | \appendix}}{} 622 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%参考文献%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 623 | %%重新定义参考文献环境 624 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 625 | \renewenvironment{thebibliography}[1]{\newpage% 626 | \setlength{\baselineskip}{20bp}\wuhao 627 | \if@bachelor 628 | \vspace*{-2\hangju} 629 | \else 630 | \vspace*{-\hangju} 631 | \fi 632 | \section*{\refname} 633 | \addcontentsline{toc}{section}{参考文献} 634 | \@mkboth{\refname}{\refname}% 635 | \list{\@biblabel{\@arabic\c@enumiv}}% 636 | {\renewcommand{\makelabel}[1]{##1\hfill} 637 | \settowidth\labelwidth{1em} 638 | \setlength{\labelsep}{0em} 639 | \setlength{\itemindent}{0pt} 640 | \setlength{\leftmargin}{\labelwidth+\labelsep} 641 | \addtolength{\itemsep}{-0.7em} 642 | \usecounter{enumiv}% 643 | \let\p@enumiv\@empty 644 | \renewcommand\theenumiv{\@arabic\c@enumiv}}% 645 | \sloppy\frenchspacing 646 | \clubpenalty4000 647 | \@clubpenalty \clubpenalty 648 | \widowpenalty4000% 649 | \interlinepenalty4000% 650 | \sfcode`\.\@m} 651 | {\def\@noitemerr 652 | {\@latex@warning{Empty `thebibliography' environment}}% 653 | \endlist\frenchspacing} 654 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%个人简历%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 655 | %% 定义个人简历 656 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 657 | %\newenvironment{resume}{% 658 | % \section*{\the\resumetitle}}{} 659 | %作者在攻读学位期间发表的论文目录： 660 | %作者在攻读学位期间取得的科研成果目录： 661 | %%%%%%%%%%%%%%%%%%%%%%数学环境%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 662 | %% 663 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 664 | \numberwithin{equation}{section} %%公式按节编号 665 | 666 | %允许太长的公式断行、分页等。 667 | \allowdisplaybreaks[4] 668 | \renewcommand\theequation{\ifnum \c@section>\z@ % 669 | \thesection-\fi\@arabic\c@equation} 670 | % 定理字样使用黑体，正文使用宋体，冒号隔开 671 | \renewcommand{\proofname}{\bf 证明} 672 | \renewenvironment{proof}[1][\proofname]{\par 673 | \pushQED{\qed}% 674 | \normalfont \topsep6\p@\@plus6\p@\relax 675 | \trivlist 676 | \item[\hskip\labelsep 677 | \itshape 678 | #1\@addpunct{:}]\ignorespaces 679 | }{% 680 | \popQED\endtrivlist\@endpefalse 681 | } 682 | 683 | \newtheorem{assumption}{假设}[section] 684 | \newtheorem{definition}{定义}[section] 685 | \newtheorem{proposition}{命题}[section] 686 | \newtheorem{lemma}{引理}[section] 687 | \newtheorem{theorem}{定理}[section] 688 | \newtheorem{axiom}{公理}[section] 689 | \newtheorem{corollary}{推论}[section] 690 | \newtheorem{example}{例}[section] 691 | \newtheorem{remark}{注释}[section] 692 | \newtheorem{problem}{问题}[section] 693 | \newtheorem{conjecture}{猜想}[section] 694 | %%%%%%%%%%%%%%%%%%%%%%主要符号对照表%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 695 | %%符号环境：denotation，有一个参数0和1,即，0表示不在目录中显示，1表示在目录中显示 696 | %%还有一个参数控制label的宽度，单位cm 697 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 698 | \newcommand{\@denotation@name}{主要符号对照表} 699 | \newenvironment{denotation}[2]{\newpage 700 | \ifnum1=#2 701 | \addcontentsline{toc}{section}{主要符号对照表} 702 | \fi 703 | \if@bachelor 704 | \vspace*{-2\hangju} 705 | \else 706 | \vspace*{-\hangju} 707 | \fi 708 | \@mkboth{\@denotation@name}{}% 709 | \section*{\@denotation@name} 710 | \noindent\begin{list}{}% 711 | {\vskip-30bp 712 | \renewcommand\makelabel[1]{##1\hfil} 713 | \setlength{\labelwidth}{#1cm} % 标签盒子宽度 714 | \setlength{\labelsep}{0.5cm} % 标签与列表文本距离 715 | \setlength{\itemindent}{0cm} % 标签缩进量 716 | \setlength{\leftmargin}{\labelwidth+\labelsep} % 左边界 717 | \setlength{\rightmargin}{0cm} 718 | \setlength{\parsep}{0cm} % 段落间距 719 | \setlength{\itemsep}{0cm} % 标签间距 720 | \setlength{\listparindent}{0cm} % 段落缩进量 721 | \setlength{\topsep}{0pt} % 标签与上文的间距 722 | }}{\end{list}\newpage} 723 | %%%%%%%%%%%%%%%%%%%%%%%%%%其它命令%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 724 | % 在导言区支持中文。 725 | \def\@active@cjk{ 726 | \count@=128 727 | \@whilenum\count@<254 \do{% 728 | \advance\count@ by 1 729 | \lccode`\~=\count@ 730 | \catcode\count@=\active 731 | \lowercase{\def~{\kern1ex}}}} 732 | \AtEndOfClass{\ifxetex\else\@active@cjk\fi} 733 | % \pkg{caption} 宏包会调用 \pkg{ragged2e}，\pkg{ragged2e} 又调用 734 | %\pkg{everysel}。而 \pkg{everysel} 宏包和 \pkg{CJK} 735 | % 宏包都要修改 \cs{selectfont}，一个是给 \cs{selectfont} 加 hook，一个是让 736 | % \cs{selectfont} 支持 \cs{CJKbold}。幸好两者本质上并不冲突，把他们源文件中对 737 | % \cs{selectfont} 重定义的部分合并一下就可以了。 738 | \def\cqu@fixselectfont{% 739 | \ifx\@EverySelectfont@EveryHook\@undefined\else 740 | \DeclareRobustCommand{\selectfont}{% 741 | \ifx\f@linespread\baselinestretch \else 742 | \set@fontsize\baselinestretch\f@size\f@baselineskip \fi 743 | \xdef\font@name{% 744 | \csname\curr@fontshape/\f@size\endcsname}% 745 | \pickup@font 746 | \font@name 747 | % CJK addition: 748 | \CJK@bold@false 749 | \csname \curr@fontshape\endcsname 750 | % everysel addition: 751 | \@EverySelectfont@EveryHook 752 | \@EverySelectfont@AtNextHook 753 | \gdef\@EverySelectfont@AtNextHook{}% 754 | % end additions 755 | \size@update 756 | \enc@update} 757 | \fi} 758 | % 把 CJK 环境放到合适的位置，以免导致其它宏包的命令位于 CJK 环境中而出现问题（比 759 | % 如 natbib 的``Multiple-defined labels''，同时自动开启 CJK。 760 | \ifxetex 761 | \AtBeginDocument{} 762 | \else 763 | \def\cqu@beginCJK{% 764 | \cqu@fixselectfont% 765 | \begin{CJK*}{UTF8}{song8}} 766 | \def\cqu@endCJK{\clearpage\end{CJK*}} 767 | \let\cqu@begindocumenthook\@begindocumenthook 768 | \let\cqu@enddocumenthook\@enddocumenthook 769 | \def\AtBeginDocument{\g@addto@macro\cqu@begindocumenthook} 770 | \def\AtEndDocument{\g@addto@macro\cqu@enddocumenthook} 771 | \def\@begindocumenthook{\cqu@begindocumenthook\cqu@beginCJK} 772 | \def\@enddocumenthook{\cqu@endCJK\cqu@enddocumenthook} 773 | \fi 774 | \AtEndOfClass{\sloppy} 775 | %%%%%%中文破折号 776 | \newcommand{\pozhehao}{\kern0.3ex\rule[0.8ex]{1.5em}{0.1ex}\kern0.3ex} 777 | \newcommand{\shengluehao}{\,……\,} 778 | 779 | % 设置一些 pdf 文档信息，依赖于 \pkg{hyperref} 宏包。 780 | %\hypersetup{% 781 | % pdftitle={\the\cntitle}, 782 | % pdfauthor={\the\cnauthor}, 783 | % %linktocpage,% 784 | % bookmarks,CJKbookmarks, 785 | % bookmarksopen=true, 786 | % bookmarksnumbered=true, 787 | % breaklinks, 788 | % pdfborder=1} 789 | 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