Week8_2Principal Component Analysis

第 1 题

Consider the following 2D dataset:

Which of the following figures correspond to possible values that PCA may return for u(1)$u^{(1)}$ (the first eigenvector / first principal component)? Check all that apply (you may have to check more than one figure).

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*     答案: 1 2 *
* minimize the projection error:要找到投影距离最小的向量,是1和2,方向正还是负都是可以的. *

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第 2 题

Which of the following is a reasonable way to select the number of principal components k$k$ ?
(Recall that n$n$ is the dimensionality of the input data and m$m$ is the number of input examples.)

• Choose k to be the smallest value so that at least 99% of the variance is retained.
• Choose the value of k that minimizes the approximation error 1m∑mi=1||x(i)−x(i)approx||2$\frac{1}{m}\sum _{i=1}^m||x^{(i)}-x_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}}^{(i)}|{|}^2$.
• Choose k to be the smallest value so that at least 1% of the variance is retained.
• Choose k to be 99% of n (i.e., k=0.99∗n$k=0.99\ast n$, rounded to the nearest integer).

*     答案: 1 *
* 选项1: . 正确 *
* 选项1: . 正确 *
* 选项1: . 正确 *
* 选项1: . 正确 *

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第 3 题

Suppose someone tells you that they ran PCA in such a way that “95% of the variance was retained.” What is an equivalent statement to this?

• 1m∑mi=1||x(i)||21m∑mi=1||x(i)−x(i)approx||2≥0.05$\frac{\frac{1}{m}\sum _{i=1}^m||x^{(i)}|{|}^2}{\frac{1}{m}\sum _{i=1}^m||x^{(i)}-x_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}}^{(i)}|{|}^2}\ge 0.05$
• 1m∑mi=1||x(i)||21m∑mi=1||x(i)−x(i)approx||2≤0.05$\frac{\frac{1}{m}\sum _{i=1}^m||x^{(i)}|{|}^2}{\frac{1}{m}\sum _{i=1}^m||x^{(i)}-x_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}}^{(i)}|{|}^2}\le 0.05$
• 1m∑mi=1||x(i)−x(i)approx||21m∑mi=1||x(i)||2≤0.05$\frac{\frac{1}{m}\sum _{i=1}^m||x^{(i)}-x_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}}^{(i)}|{|}^2}{\frac{1}{m}\sum _{i=1}^m||x^{(i)}|{|}^2}\le 0.05$
• 1m∑mi=1||x(i)||21m∑mi=1||x(i)−x(i)approx||2≤0.95$\frac{\frac{1}{m}\sum _{i=1}^m||x^{(i)}|{|}^2}{\frac{1}{m}\sum _{i=1}^m||x^{(i)}-x_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}}^{(i)}|{|}^2}\le 0.95$

*     答案: 3 *
* 选项1: . 正确 *
* 选项1: . 正确 *
* 选项1: . 正确 *
* 选项1: . 正确 *

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第 4 题

Which of the following statements are true? Check all that apply.

• Given only z(i)$z^{(i)}$ and Ureduce$U_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}}$, there is no way to reconstruct any reasonable approximation to x(i)$x^{(i)}$.
• Even if all the input features are on very similar scales, we should still perform mean normalization (so that each feature has zero mean) before running PCA.
• PCA is susceptible to local optima; trying multiple random initializations may help.
• Given input data x∈Rn$x\in {\mathbb{R}}^n$, it makes sense to run PCA only with values of k that satisfy k≤n$k\le n$. (In particular, running it with k=n$k=n$ is possible but not helpful, and k>n$k>n$ does not make sense.)

*     答案: 2 4 *
* 选项1: . 正确 *

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第 5 题

Which of the following are recommended applications of PCA? Select all that apply.

• As a replacement for (or alternative to) linear regression: For most learning applications, PCA and linear regression give substantially similar results.
• Data compression: Reduce the dimension of your data, so that it takes up less memory / disk space.
• Data visualization: To take 2D data, and find a different way of plotting it in 2D (using k=2).
• Data compression: Reduce the dimension of your input data x(i)$x^{(i)}$, which will be used in a supervised learning algorithm (i.e., use PCA so that your supervised learning algorithm runs faster).

*     答案: 2 4 *
* 选项1: . 正确 *