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What are the differences between these two kinds of PCA?

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$ X = UDV^T $$$X = UDV^T$$

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1, there is also "Latent Variables and Factor Analysis" the authors write:

$ S = \sqrt(N)U $$$ S = \sqrt{N}U $$

$ A^T = DV^T/\sqrt(N) $$$ A^T = DV^T/\sqrt{N} $$

$ X = SA^T $$$ X = SA^T $$

Then:

$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $$$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $$

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt(N)$$\sqrt{N}$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

  7. It seems like the first is takes $UD$ and $V$ and the second one takes $U$ and $DV'$. Does it actually matter which one we take as long as the matrix is transpose accordingly?

What are the differences between two kinds of PCA?

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$ X = UDV^T $

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1, there is also "Latent Variables and Factor Analysis":

$ S = \sqrt(N)U $

$ A^T = DV^T/\sqrt(N) $

$ X = SA^T $

Then:

$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt(N)$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

  7. It seems like the first is takes $UD$ and $V$ and the second one takes $U$ and $DV'$. Does it actually matter which one we take as long as the matrix is transpose accordingly?

What are the differences between these two kinds of PCA?

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$$X = UDV^T$$

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1 "Latent Variables and Factor Analysis" the authors write:

$$ S = \sqrt{N}U $$

$$ A^T = DV^T/\sqrt{N} $$

$$ X = SA^T $$

Then:

$$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $$

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt{N}$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

  7. It seems like the first is takes $UD$ and $V$ and the second one takes $U$ and $DV'$. Does it actually matter which one we take as long as the matrix is transpose accordingly?

8 added 1 character in body
source | link

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$ X = UDV^T $

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1, there is also "Latent Variables and Factor Analysis":

$ S = \sqrt(N)U $

$ A^T = DV^T/\sqrt(N) $

$ X = SA^T $

Then:

$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt(N)$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

  7. It seems like the first is takes $UD$ and $V$ and the second one takes $U$ and $DV$$DV'$. Does it actually matter which one we take as long as the matrix is transpose accordingly?

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$ X = UDV^T $

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1, there is also "Latent Variables and Factor Analysis":

$ S = \sqrt(N)U $

$ A^T = DV^T/\sqrt(N) $

$ X = SA^T $

Then:

$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt(N)$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

  7. It seems like the first is takes $UD$ and $V$ and the second one takes $U$ and $DV$. Does it actually matter which one we take as long as the matrix is transpose accordingly?

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$ X = UDV^T $

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1, there is also "Latent Variables and Factor Analysis":

$ S = \sqrt(N)U $

$ A^T = DV^T/\sqrt(N) $

$ X = SA^T $

Then:

$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt(N)$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

  7. It seems like the first is takes $UD$ and $V$ and the second one takes $U$ and $DV'$. Does it actually matter which one we take as long as the matrix is transpose accordingly?

7 Added another question
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The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$ X = UDV^T $

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1, there is also "Latent Variables and Factor Analysis":

$ S = \sqrt(N)U $

$ A^T = DV^T/\sqrt(N) $

$ X = SA^T $

Then:

$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt(N)$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

  7. It seems like the first is takes $UD$ and $V$ and the second one takes $U$ and $DV$. Does it actually matter which one we take as long as the matrix is transpose accordingly?

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$ X = UDV^T $

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1, there is also "Latent Variables and Factor Analysis":

$ S = \sqrt(N)U $

$ A^T = DV^T/\sqrt(N) $

$ X = SA^T $

Then:

$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt(N)$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows:

$ X = UDV^T $

Then $ UD $ are the Principal Components and $ V $ are the directions.

However, later in the book, in section 14.7.1, there is also "Latent Variables and Factor Analysis":

$ S = \sqrt(N)U $

$ A^T = DV^T/\sqrt(N) $

$ X = SA^T $

Then:

$ X_p = a_{p1}S_1 + ... + a_{pp}S_p $

Questions:

  1. What can be done with the second one that can't be done with the first one?

  2. Is the first one for a "tall" matrix and the second one for a "wide" matrix? So then does the matrix as used in the first one need to be transposed before it is used in the second one?

  3. Why does the second one have $\sqrt(N)$ ?

  4. Any other mathematical differences that I have missed?

  5. If my matrix is set of functions/curves, what is the difference between having $V$ as the curves and $S$ as the curves?

  6. Is there any advantage to using full SVD over reduced SVD?

  7. It seems like the first is takes $UD$ and $V$ and the second one takes $U$ and $DV$. Does it actually matter which one we take as long as the matrix is transpose accordingly?

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4 principal not principle
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