Differences

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--- sand_box [2025/12/29 04:46] – hkimscil
+++ sand_box [2026/04/01 01:52] (current) – hkimscil
@@ Line 1: / Line 1: @@
-{{pasted:20251229-044558.png}}
+{{pasted:20260401-015229.png?300}}
+{{pasted:20251229-044558.png?300}}
+<tabbed>
+  * sand box:code01
+  * *sand box:output01
+</tabbed>
+----
+  graph TD
+    A(**mermaid**)-->B((__plugin__))
+    A-->C(((//for//)))
+    B-->D[["[[https://www.dokuwiki.org/dokuwiki|Dokuwiki]]"]]
+    C-->D
+\begin{eqnarray*}
+& & P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\\
+& & P(B \mid A) = \dfrac{P(B \cap A)}{P(A)}\\
+\\
+& & P(B \vert A) \;\; \text{  vs. } \;\; P(B \mid A) \\
+& & P(A \cap B) = P(A \mid B) * P(B) \\
+& & P(B \cap A) = P(B \mid A) * P(A) \\
+& & P(A \cap B) = P(A, B) \\
+\\
+& & \frac{3}{4 \pi} \sqrt{4 \cdot x^2 12} \\
+& & \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} \\
+& & {\it f}(x) = \frac{1}{\sqrt{x} x^2} \\
+& & e^{i \pi} + 1 = 0\;
+\end{eqnarray*}
+<WRAP tabs>
+  * [[:sand box/intro]]
+  * [[:sand box/body]]
+  * [[:sand box/conc]]
+</WRAP>
+{{tabinclude>sand_box:page1|Top page,sand_box:page2|Second page,*sand_box:page3}}
+<tabbed>
+  * *sand_box:page1
+  * sand_box:page2
+  * sand_box:page3
+</tabbed>
+[{{:r.regressionline3.png}}]
+\begin{align*}
+& \;\;\;\; \sum{(Y_i - \hat{Y_i})^2} \\
+&= \sum{(Y_i - (a + bX_i))^2}  \;\;\; \because \hat{Y_i} = a + bX_i \\
+&= \text{SSE or SS.residual} \;\;\; \text{(and this should be the least value.)} \\
+\end{align*}
+<WRAP box>
+\begin{align*}
+&\text{for a (constant)} \\ \\
+&\dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2} \\
+&=  \sum \dfrac{\text{d}}{\text{dv}} {(Y_i - (a + bX_i))^2} \\
+&= \sum{2 (Y_i - (a + bX_i))} * (-1) \;\;\;\; \\
+&\because \dfrac{\text{d}}{\text{dv for a}} (Y_i - (a+bX_i)) = -1 \\
+& = -2 \sum{(Y_i - (a + bX_i))} \\
+\\
+&\text{in order to have the least value, the above should be zero} \\
+\\
+&-2 \sum{(Y_i - (a + bX_i))} = 0 \\
+&\sum{(Y_i - (a + bX_i))} = 0 \\
+&\sum{Y_i} - \sum{a} - b \sum{X_i} = 0 \\
+&\sum{Y_i} - n*{a} - b \sum{X_i} = 0 \\
+&n*{a} = \sum{Y_i} - b \sum{X_i} \\
+&a = \dfrac{\sum{Y_i}}{n} - b \dfrac{\sum{X_i}}{n} \\
+&a = \overline{Y} - b \overline{X} \\
+\end{align*}
+</WRAP>
+<WRAP box>
+\begin{eqnarray*}
+\text{for b, (coefficient)} \\
+\\
+\dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2}  & = & \sum \dfrac{\text{d}}{\text{dv}} {(Y_i - (a + bX_i))^2} \\
+& = & \sum{2 (Y_i - (a + bX_i))} * (-X_i) \;\;\;\; \\
+& \because & \dfrac{\text{d}}{\text{dv for b}} (Y_i - (a+bX_i)) = -X_i \\
+& = & -2 \sum{X_i (Y_i - (a + bX_i))} \\
+\\
+\text{in order to have the least value, the above should be zero} \\
+\\
+-2 \sum{X_i (Y_i - (a + bX_i))} & = & 0 \\
+\sum{X_i (Y_i - (a + bX_i))} & = & 0 \\
+\sum{X_i (Y_i - ((\overline{Y} - b \overline{X}) + bX_i))} & = & 0 \\
+\sum{X_i ((Y_i - \overline{Y}) - b (X_i - \overline{X})) } & = & 0 \\
+\sum{X_i (Y_i - \overline{Y})} - \sum{b X_i (X_i - \overline{X}) } & = & 0 \\
+\sum{X_i (Y_i - \overline{Y})} & = &  b \sum{X_i (X_i - \overline{X})} \\
+b & = & \dfrac{\sum{X_i (Y_i - \overline{Y})}}{\sum{X_i (X_i - \overline{X})}} \\
+b & = & \dfrac{\sum{(Y_i - \overline{Y})}}{\sum{(X_i - \overline{X})}} \\
+b & = & \dfrac{ \sum{(Y_i - \overline{Y})(X_i - \overline{X})} } {\sum{(X_i - \overline{X})(X_i - \overline{X})}} \\
+b & = & \dfrac{ \text{SP} } {\text{SS}_\text{x}} \\
+\end{eqnarray*}
+</WRAP>
 <code>
 library(tidyverse)
@@ Line 295: / Line 390: @@
 </code>
-<tabbed>
-  * sand box:code01
-  * *sand box:output01
-</tabbed>
-{{clock}}
-----
-  graph TD
-    A(**mermaid**)-->B((__plugin__))
-    A-->C(((//for//)))
-    B-->D[["[[https://www.dokuwiki.org/dokuwiki|Dokuwiki]]"]]
-    C-->D
-\begin{eqnarray*}
-& & P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\\
-& & P(B \mid A) = \dfrac{P(B \cap A)}{P(A)}\\
-\\
-& & P(B \vert A) \;\; \text{  vs. } \;\; P(B \mid A) \\
-& & P(A \cap B) = P(A \mid B) * P(B) \\
-& & P(B \cap A) = P(B \mid A) * P(A) \\
-& & P(A \cap B) = P(A, B) \\
-\\
-& & \frac{3}{4 \pi} \sqrt{4 \cdot x^2 12} \\
-& & \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6} \\
-& & {\it f}(x) = \frac{1}{\sqrt{x} x^2} \\
-& & e^{i \pi} + 1 = 0\;
-\end{eqnarray*}
-<WRAP tabs>
-  * [[:sand box/intro]]
-  * [[:sand box/body]]
-  * [[:sand box/conc]]
-</WRAP>
-{{tabinclude>sand_box:page1|Top page,sand_box:page2|Second page,*sand_box:page3}}
-<tabbed>
-  * *sand_box:page1
-  * sand_box:page2
-  * sand_box:page3
-</tabbed>
-[{{:r.regressionline3.png}}]
-\begin{align*}
-& \;\;\;\; \sum{(Y_i - \hat{Y_i})^2} \\
-&= \sum{(Y_i - (a + bX_i))^2}  \;\;\; \because \hat{Y_i} = a + bX_i \\
-&= \text{SSE or SS.residual} \;\;\; \text{(and this should be the least value.)} \\
-\end{align*}
-<WRAP box>
-\begin{align*}
-&\text{for a (constant)} \\ \\
-&\dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2} \\
-&=  \sum \dfrac{\text{d}}{\text{dv}} {(Y_i - (a + bX_i))^2} \\
-&= \sum{2 (Y_i - (a + bX_i))} * (-1) \;\;\;\; \\
-&\because \dfrac{\text{d}}{\text{dv for a}} (Y_i - (a+bX_i)) = -1 \\
-& = -2 \sum{(Y_i - (a + bX_i))} \\
-\\
-&\text{in order to have the least value, the above should be zero} \\
-\\
-&-2 \sum{(Y_i - (a + bX_i))} = 0 \\
-&\sum{(Y_i - (a + bX_i))} = 0 \\
-&\sum{Y_i} - \sum{a} - b \sum{X_i} = 0 \\
-&\sum{Y_i} - n*{a} - b \sum{X_i} = 0 \\
-&n*{a} = \sum{Y_i} - b \sum{X_i} \\
-&a = \dfrac{\sum{Y_i}}{n} - b \dfrac{\sum{X_i}}{n} \\
-&a = \overline{Y} - b \overline{X} \\
-\end{align*}
-</WRAP>
-<WRAP box>
-\begin{eqnarray*}
-\text{for b, (coefficient)} \\
-\\
-\dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2}  & = & \sum \dfrac{\text{d}}{\text{dv}} {(Y_i - (a + bX_i))^2} \\
-& = & \sum{2 (Y_i - (a + bX_i))} * (-X_i) \;\;\;\; \\
-& \because & \dfrac{\text{d}}{\text{dv for b}} (Y_i - (a+bX_i)) = -X_i \\
-& = & -2 \sum{X_i (Y_i - (a + bX_i))} \\
-\\
-\text{in order to have the least value, the above should be zero} \\
-\\
--2 \sum{X_i (Y_i - (a + bX_i))} & = & 0 \\
-\sum{X_i (Y_i - (a + bX_i))} & = & 0 \\
-\sum{X_i (Y_i - ((\overline{Y} - b \overline{X}) + bX_i))} & = & 0 \\
-\sum{X_i ((Y_i - \overline{Y}) - b (X_i - \overline{X})) } & = & 0 \\
-\sum{X_i (Y_i - \overline{Y})} - \sum{b X_i (X_i - \overline{X}) } & = & 0 \\
-\sum{X_i (Y_i - \overline{Y})} & = &  b \sum{X_i (X_i - \overline{X})} \\
-b & = & \dfrac{\sum{X_i (Y_i - \overline{Y})}}{\sum{X_i (X_i - \overline{X})}} \\
-b & = & \dfrac{\sum{(Y_i - \overline{Y})}}{\sum{(X_i - \overline{X})}} \\
-b & = & \dfrac{ \sum{(Y_i - \overline{Y})(X_i - \overline{X})} } {\sum{(X_i - \overline{X})(X_i - \overline{X})}} \\
-b & = & \dfrac{ \text{SP} } {\text{SS}_\text{x}} \\
-\end{eqnarray*}
-</WRAP>