9 fixed the tex

# 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

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?

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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