6. 统计学与线性代数预科

中国谚语

知彼知己，百战不殆；不知彼而知己，一胜一负；不知彼，不知己，每战必殆。 –《孙子·谋攻篇》

6.1. 符号

• M：样品数量

• n：功能的数量

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  第i个标签

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  ：第i个预测标签

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  : the mean of

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  .

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  ：标签向量。

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  ：预测的标签向量。

6.2. 线性代数预科

由于我已经在我的数字分析预科考试笔记中记录了线性代数的预备知识，所以有兴趣的读者可以参考。 [Feng2014] 更多详细信息（图 线性代数预科 ）

线性代数预科

6.3. 测量公式

6.3.1. 平均绝对误差

在统计学中， MAE (Mean absolute error ）是两个连续变量之间的差的度量。平均绝对误差如下：

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6.3.2. 均方误差

在统计学上， MSE (Mean Squared Error ）估计量的（估计未观测量的程序的）测量误差或偏差平方的平均值，即估计量和估计量之间的差。

System Message: WARNING/2 (text mse=frac 1 m sum i=1)

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6.3.3. 均方根误差

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6.3.4. 平方和

在统计数据分析中 TSS (Total Sum of Squares ）是作为标准方法的一部分出现的一个量，用于表示此类分析的结果。它被定义为每个观测值与总平均值的平方差之和。

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6.3.5. 解释平方和

在统计学上， ESS (Explained sum of squares ，或者称为回归的模型平方和或平方和。

ESS是预测值与响应变量平均值之差的平方和，由下式得出：

System Message: WARNING/2 (TEXT ESS=SUM I=1 ^MLEFT（HAT Y BAR Y RIGHT）^2)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 12 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mhsetup.sty)) ! LaTeX Error: File `dsfont.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.16 \def \Z{\mathbb{Z}}^^M No pages of output. Transcript written on math.log.

6.3.6. 残差平方和

在统计学中， RSS (Residual sum of squares ，也称为平方残差之和（SSR）或预测平方误差之和（SSE），是残差平方之和，由下式得出：

System Message: WARNING/2 (文本rss=sum i=1 ^m左（hat y u i-右）^2)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 12 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mhsetup.sty)) ! LaTeX Error: File `dsfont.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.16 \def \Z{\mathbb{Z}}^^M No pages of output. Transcript written on math.log.

6.3.7. 确定系数

System Message: WARNING/2 (R^2)

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System Message: WARNING/2 (R ^ 2：=frac ess tss=1-text rss overtext tss.，)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 12 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mhsetup.sty)) ! LaTeX Error: File `dsfont.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.16 \def \Z{\mathbb{Z}}^^M No pages of output. Transcript written on math.log.

注解

一般来说 (

System Message: WARNING/2 (\y^{{T}}{{\bar {{\y}}}}={{\hat {{\y}}}}^{{T}}{{\bar {{\y}}}})

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 12 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mhsetup.sty)) ! LaTeX Error: File `dsfont.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.16 \def \Z{\mathbb{Z}}^^M No pages of output. Transcript written on math.log.

，总平方和=解释平方和+剩余平方和，即：

System Message: WARNING/2 (TEXT TSS=TEXT ESS+TEXT RSS TEXT如果且仅当显示样式Y ^ T BAR Y Y BAR Y。)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 12 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mhsetup.sty)) ! LaTeX Error: File `dsfont.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.16 \def \Z{\mathbb{Z}}^^M No pages of output. Transcript written on math.log.

有关更多详细信息，请访问 Partitioning in the general ordinary least squares model .

6.4. 混淆矩阵

混淆矩阵

6.4.1. 回忆

System Message: WARNING/2 (文本回忆=frac 文本tp 文本tp+fn)

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6.4.2. 精密度

System Message: WARNING/2 (文本精度=frac 文本tp 文本tp+fp)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 12 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mhsetup.sty)) ! LaTeX Error: File `dsfont.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.16 \def \Z{\mathbb{Z}}^^M No pages of output. Transcript written on math.log.

6.4.3. 准确度

System Message: WARNING/2 (文本准确度=frac 文本tp+tn 文本总计)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 12 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mhsetup.sty)) ! LaTeX Error: File `dsfont.sty' not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.16 \def \Z{\mathbb{Z}}^^M No pages of output. Transcript written on math.log.

6.4.4.

System Message: WARNING/2 (F_1)

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得分

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6.5. 统计检验

6.5.1. 相关检验

• 皮尔逊相关：检验两个连续变量之间的关联强度。

• 斯皮尔曼相关性：检验两个序数变量之间的关联强度（不依赖于正态分布数据的假设）。

• 卡方：检验两个分类变量之间的关联强度。

6.5.2. 均值检验比较

• 配对t检验：检验两个相关变量之间的差异。

• 独立t检验：检验两个独立变量之间的差异。

• 方差分析：在计算结果变量的任何其他方差后，检验两组均值之间的差异。

6.5.3. 非参数试验

• Wilcoxon秩和检验：两个自变量之间的差异检验——考虑了差异的大小和方向。

• Wilcoxon符号秩检验：两个相关变量之间的差异检验——考虑了差异的大小和方向。

• 符号测试：测试两个相关变量是否不同-忽略变化幅度，只考虑方向。