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private:writeups:svd [2020/10/18 19:20]
admin 		private:writeups:svd [2020/10/18 21:25] (current)
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 +====== Math ======
  
-$$X=\begin{bmatrix}+$$X_{(m,n)}=\begin{bmatrix}
 | & | &  & | \\  | & | &  & | \\ 
 x_1 & x_2 & \dots & x_m  \\  x_1 & x_2 & \dots & x_m  \\ 
Line 18: Line 19:
 \end{bmatrix} \end{bmatrix}
 \begin{bmatrix} \begin{bmatrix}
-\text{---}  & v_1^T & \text{---}  \\  +| & | &  & | \\  
-\text{---}  & v_2^T & \text{---}  \\  +v_1 & v_2 & \dots & v_m  \\  
-\text{---}  & \vdots & \text{---}  \\  +| & | &  & | 
-\text{---}  & v_m^T & \text{---}  +\end{bmatrix}^T 
-\end{bmatrix}$$+$$
  
 where $n$ is the number of points in each $x$, and $m$ is the number of $x$ values.  where $n$ is the number of points in each $x$, and $m$ is the number of $x$ values. 
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 $$ X = \sigma_1 u_1  v_1^T +  \sigma_2 u_2  v_2^T + \ldots + \sigma_m u_m  v_m^T$$ (assuming $n>>m$).   $$ X = \sigma_1 u_1  v_1^T +  \sigma_2 u_2  v_2^T + \ldots + \sigma_m u_m  v_m^T$$ (assuming $n>>m$).  
  
-We can therefore select the first "number" of these terms to +We can therefore select the first $r$ terms of this expansion to get a reduced-order approximation of X.  I.e. 
 +$$ X \approx \tilde{U}\tilde{\Sigma}\tilde{V}^T$$. 
 + 
 +====== Interpretation ====== 
 + 
 +Correlation between the rows and columns of X.   
 + 
 + 
 +U and V are the eigenvectors of the correlation matrices,$X X^T$ and $X^T X$, respectively.  Because calculating $U$ and $V$ this way is slow, more optimized methods are available.    Which is the inner product of $x_i ^T x_j$ where $i$ and $j$ are the $i$th and $j$th column of $X$.  If the inner product is large, it means that $x_i$ and $x_j$ are similar. If small, it means that  $x_i$ and $x_j$  are different.   
 + 
 + 
 +====== Principal component analysis (PCA) ====== 
 +This is the statistical interpretation of SVD. 
 + 
 +PCA finds elipsoids (depending on dimensionality) that account for the largest, second largest, etc. variance in your data.   
 + 
 +Note that X is defined differently as a matrix of horizontally aligned matrix (instead of vertical). 
 + 
 +https://www.youtube.com/watch?v=fkf4IBRSeEc 
 + 
 +Steps. 
 +1.  Compute mean row, X' 
 +2.  Subtract mean from X.  $B=X-X'$  
 +3.  Calc covariance matrix, $C=B^T B$ 
 +4.  Computer eigs of C.  Eigenvectors, V, and eigvenvalues, D. 
 +5.  Principal components, T=BV.  (V also called "loadings") 
 +If $B=U \Sigma V^T$, then $T=U \Sigma$.   
 + 
 +Takeaway, PCA can be found by saving the SVD of (X minus its row-mean).   
 + 
 + 
 +====== Least squares, regression, and pseudo-inverse ====== 
 + 
 +SVD allows us to generalized Ax=b to non-square A matrices.  (pseudo-inverse)
private/writeups/svd.1603048814.txt.gz · Last modified: 2020/10/18 19:20 by admin

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