Differences

This shows you the differences between two versions of the page.

--- sand_box [2023/12/10 21:49] – hkimscil
+++ sand_box [2024/09/25 07:45] (current) – hkimscil
@@ Line 1: / Line 1: @@
 <WRAP tabs>
-  * [[start|Allgemein]]
+  * [[:sand box/intro]]
-  * [[personal:richtlinien:start|Richtlinien]]
+  * [[:sand box/body]]
-  * [[menue_themenbereiche|Firmenbereiche]]
+  * [[:sand box/conc]]
-  * [[menue_details|WIKI-Details]]
 </WRAP>
+{{tabinclude>page1|Top page,page2|Second page,*page3}}
+<tabbed>
+  * page1|top page
+  * page2
+  * *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>