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mean squared norm のタイポ修正
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yuji96 committed Dec 15, 2023
1 parent 8095afd commit b7c9db1
Showing 1 changed file with 7 additions and 7 deletions.
14 changes: 7 additions & 7 deletions _posts/blog/2023-12-15-PCA-vs-PCoA.md
Original file line number Diff line number Diff line change
Expand Up @@ -131,25 +131,25 @@ $$
$$
d_{\bullet j}^2
= \frac{1}{n}\sum_i \left( \|x_i\|^2 + \|x_j\|^2 - 2x_i^Tx_j \right)
= \overline{\|x\|} + \|x_j\|^2 + 0
= \overline{\|x\|^2} + \|x_j\|^2 + 0
$$

ただし、$\overline{\|\|x\|\|} := \frac{1}{n}\sum_i \|\|x_i\|\|$.
ただし、$\overline{\|\|x\|\|^2} := \frac{1}{n}\sum_i \|\|x_i\|\|^2$.

3 項目 $d_{i \bullet}^2$:

$$
d_{i \bullet}^2
= \frac{1}{n}\sum_j \left( \|x_i\|^2 + \|x_j\|^2 - 2x_i^Tx_j \right)
= \|x_i\|^2 + \overline{\|x\|} + 0
= \|x_i\|^2 + \overline{\|x\|^2} + 0
$$

4 項目 $d_{\bullet \bullet}^2$:

$$
d_{i \bullet}^2
= \frac{1}{n^2}\sum_{i,j} \left( \|x_i\|^2 + \|x_j\|^2 - 2x_i^Tx_j \right)
= 2\overline{\|x\|} + 0
= 2\overline{\|x\|^2} + 0
$$

全部合わせると:
Expand All @@ -158,9 +158,9 @@ $$
\begin{aligned}
(CD^{(2)}C)_{ij}
&= (\|x_i\|^2 + \|x_j\|^2 - 2x_i^Tx_j) \\
&- (\overline{\|x\|} + \|x_j\|^2) \\
&- (\|x_i\|^2 + \overline{\|x\|}) \\
&+ 2\overline{\|x\|} \\
&- (\overline{\|x\|^2} + \|x_j\|^2) \\
&- (\|x_i\|^2 + \overline{\|x\|^2}) \\
&+ 2\overline{\|x\|^2} \\
&= - 2x_i^Tx_j
\end{aligned}
$$
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