update

2022-09-07 13:53:47 +02:00
parent eeca31435c
commit 70a0d9dc05
2 changed files with 353 additions and 389 deletions
@@ -580,22 +580,10 @@ which gives us, using the orthogonality of the matrices $\bm{U}$ and $\bm{V}$,,

 !bt
 \[
-\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{U}^T\bm{y}=\sum_{i=0}^{p-1}\bm{u}_i\bm{u}^T_j\bm{y},
+\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{U}^T\bm{y}=\sum_{i=0}^{p-1}\bm{u}_i\bm{u}^T_i\bm{y},
 \]
 !et

-
-
-
-
-Note here that when we perform the  multiplication of the various matrices, the orthogonal vectors of the matrix $\bm{U}$
-!bt
-\[
-\bm{U}=[\bm{u}_0,\bm{u}_1,\dots,\bm{u}_{n-1}],
-\]
-!et
-that belong to $i>p-1$, result in only  zeros when we perform the multiplications. This means that the sum above has non-zero elements only up to $i=p-1$. This corresponds also to the number of singular values (these are all non-zero).
-
 It means that the ordinary least square model (with the optimal parameters) $\bm{\tilde{y}}$, corresponds to an orthogonal transformation of the output (or target) vector $\bm{y}$ by the vectors of the matrix $\bm{U}$.