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Tom's Sept 5 edits of two lectures
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lectures/svd_intro.md

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**Properties:**
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* Where $U$ is constructed via a full SVD, $U^\top U = I_{p\times p}$ and $U U^\top = I_{m \times m}$
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* Where $\hat U$ is constructed via a reduced SVD, although $\hat U^\top \hat U = I_{p\times p}$ it happens that $\hat U \hat U^\top \neq I_{m \times m}$
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* Where $\hat U$ is constructed via a reduced SVD, although $\hat U^\top \hat U = I_{p\times p}$, it happens that $\hat U \hat U^\top \neq I_{m \times m}$
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We illustrate these properties for our example with the following code cells.
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* ${V_k}^{T}=\tilde{\epsilon_k}$ (the $k$th principal component)
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Because there are alternative algorithms for computing $P$ and $U$ for given a data matrix $X$, depending on algorithms used, we might have sign differences or different orders between eigenvectors.
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Because there are alternative algorithms for computing $P$ and $U$ for given a data matrix $X$, depending on algorithms used, we might have sign differences or different orders of eigenvectors.
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We can resolve such ambiguities about $U$ and $P$ by
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lectures/wald_friedman.md

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- Dynamic programming
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- Type I and type II statistical errors
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- a type I error occurs when you reject a null hypothesis that is true
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- a type II error is when you accept a null hypothesis that is false
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- a type II error occures when you accept a null hypothesis that is false
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- Abraham Wald's **sequential probability ratio test**
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- The **power** of a statistical test
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- The **critical region** of a statistical test

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