主成分分析
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| - | == Introduction == | + | == Introduction 引言 == |
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【原文】：Principal Components Analysis (PCA) is a dimensionality reduction algorithm | 【原文】：Principal Components Analysis (PCA) is a dimensionality reduction algorithm | ||
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| - | == Example and Mathematical Background == | + | == Example and Mathematical Background 实例和数学背景 == |
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【原文】：For our running example, we will use a dataset | 【原文】：For our running example, we will use a dataset | ||
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[[File:PCA-rawdata.png|600px]] | [[File:PCA-rawdata.png|600px]] | ||
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【原文】：This data has already been pre-processed so that each of the features <math>\textstyle x_1</math> and <math>\textstyle x_2</math> | 【原文】：This data has already been pre-processed so that each of the features <math>\textstyle x_1</math> and <math>\textstyle x_2</math> | ||
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[[File:PCA-u1.png | 600px]] | [[File:PCA-u1.png | 600px]] | ||
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【原文】：I.e., the data varies much more in the direction <math>\textstyle u_1</math> than <math>\textstyle u_2</math>. | 【原文】：I.e., the data varies much more in the direction <math>\textstyle u_1</math> than <math>\textstyle u_2</math>. | ||
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\Sigma = \frac{1}{m} \sum_{i=1}^m (x^{(i)})(x^{(i)})^T. | \Sigma = \frac{1}{m} \sum_{i=1}^m (x^{(i)})(x^{(i)})^T. | ||
\end{align}</math> | \end{align}</math> | ||
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【原文】：If <math>\textstyle x</math> has zero mean, then <math>\textstyle \Sigma</math> is exactly the covariance matrix of <math>\textstyle x</math>. (The symbol "<math>\textstyle \Sigma</math>", pronounced "Sigma", is the standard notation for denoting the covariance matrix. Unfortunately it looks just like the summation symbol, as in <math>\sum_{i=1}^n i</math>; but these are two different things.) | 【原文】：If <math>\textstyle x</math> has zero mean, then <math>\textstyle \Sigma</math> is exactly the covariance matrix of <math>\textstyle x</math>. (The symbol "<math>\textstyle \Sigma</math>", pronounced "Sigma", is the standard notation for denoting the covariance matrix. Unfortunately it looks just like the summation symbol, as in <math>\sum_{i=1}^n i</math>; but these are two different things.) | ||
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the top (principal) eigenvector of <math>\textstyle \Sigma</math>, and <math>\textstyle u_2</math> is | the top (principal) eigenvector of <math>\textstyle \Sigma</math>, and <math>\textstyle u_2</math> is | ||
the second eigenvector. | the second eigenvector. | ||
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【初译】：假设<math>\textstyle x</math>的均值为零，那么<math>\textstyle \Sigma</math>就是<math>\textstyle x</math>的协方差矩阵。（符号<math>\textstyle \Sigma</math>，读"Sigma"，是协方差矩阵的表示符。虽然看起来与求和符号<math>\sum_{i=1}^n i</math>比较像，但他们是两个不同的概念。）由此可以得出，数据变化的主方向<math>\textstyle u_1</math>是协方差矩阵<math>\textstyle \Sigma</math>的主特征向量，而<math>\textstyle u_2</math>是次特征向量。 | 【初译】：假设<math>\textstyle x</math>的均值为零，那么<math>\textstyle \Sigma</math>就是<math>\textstyle x</math>的协方差矩阵。（符号<math>\textstyle \Sigma</math>，读"Sigma"，是协方差矩阵的表示符。虽然看起来与求和符号<math>\sum_{i=1}^n i</math>比较像，但他们是两个不同的概念。）由此可以得出，数据变化的主方向<math>\textstyle u_1</math>是协方差矩阵<math>\textstyle \Sigma</math>的主特征向量，而<math>\textstyle u_2</math>是次特征向量。 | ||
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\end{bmatrix} | \end{bmatrix} | ||
\end{align}</math> | \end{align}</math> | ||
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【原文】：Here, <math>\textstyle u_1</math> is the principal eigenvector (corresponding to the largest eigenvalue), | 【原文】：Here, <math>\textstyle u_1</math> is the principal eigenvector (corresponding to the largest eigenvalue), | ||
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| - | == Rotating the Data == | + | == Rotating the Data 旋转数据 == |
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【原文】：Thus, we can represent <math>\textstyle x</math> in the <math>\textstyle (u_1, u_2)</math>-basis by computing | 【原文】：Thus, we can represent <math>\textstyle x</math> in the <math>\textstyle (u_1, u_2)</math>-basis by computing | ||
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[[File:PCA-rotated.png|600px]] | [[File:PCA-rotated.png|600px]] | ||
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【原文】：This is the training set rotated into the <math>\textstyle u_1</math>,<math>\textstyle u_2</math> basis. In the general | 【原文】：This is the training set rotated into the <math>\textstyle u_1</math>,<math>\textstyle u_2</math> basis. In the general | ||
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| - | == Reducing the Data Dimension == | + | == Reducing the Data Dimension 数据降维 == |
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【原文】：We see that the principal direction of variation of the data is the first | 【原文】：We see that the principal direction of variation of the data is the first | ||
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\tilde{x}^{(i)} = x_{{\rm rot},1}^{(i)} = u_1^Tx^{(i)} \in \Re. | \tilde{x}^{(i)} = x_{{\rm rot},1}^{(i)} = u_1^Tx^{(i)} \in \Re. | ||
\end{align}</math> | \end{align}</math> | ||
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【原文】：More generally, if <math>\textstyle x \in \Re^n</math> and we want to reduce it to | 【原文】：More generally, if <math>\textstyle x \in \Re^n</math> and we want to reduce it to | ||
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| - | == Recovering an Approximation of the Data == | + | == Recovering an Approximation of the Data 数据还原 == |
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【原文】：Now, <math>\textstyle \tilde{x} \in \Re^k</math> is a lower-dimensional, "compressed" representation | 【原文】：Now, <math>\textstyle \tilde{x} \in \Re^k</math> is a lower-dimensional, "compressed" representation | ||
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= \sum_{i=1}^k u_i \tilde{x}_i. | = \sum_{i=1}^k u_i \tilde{x}_i. | ||
\end{align}</math> | \end{align}</math> | ||
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【原文】：The final equality above comes from the definition of <math>\textstyle U</math> [[#Example and Mathematical Background|given earlier]]. | 【原文】：The final equality above comes from the definition of <math>\textstyle U</math> [[#Example and Mathematical Background|given earlier]]. | ||
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| - | == Number of components to retain == | + | == Number of components to retain 选择主成分个数 == |
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【原文】：How do we set <math>\textstyle k</math>; i.e., how many PCA components should we retain? In our | 【原文】：How do we set <math>\textstyle k</math>; i.e., how many PCA components should we retain? In our | ||
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【二审】： | 【二审】： | ||
| - | == PCA on Images == | + | |
| + | == PCA on Images 对图像数据应用PCA算法 == | ||
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【原文】：For PCA to work, usually we want each of the features <math>\textstyle x_1, x_2, \ldots, x_n</math> | 【原文】：For PCA to work, usually we want each of the features <math>\textstyle x_1, x_2, \ldots, x_n</math> | ||
to have a similar range of values to the others (and to have a mean close to | to have a similar range of values to the others (and to have a mean close to | ||
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【二审】： | 【二审】： | ||
| - | == References == | + | == References 参考文献 == |
http://cs229.stanford.edu | http://cs229.stanford.edu | ||
{{PCA}} | {{PCA}} | ||
