deriviation_of_a_and_b_in_a_simple_regression
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| deriviation_of_a_and_b_in_a_simple_regression [2024/05/23 08:19] – hkimscil | deriviation_of_a_and_b_in_a_simple_regression [2024/05/23 08:31] (current) – hkimscil | ||
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| Line 6: | Line 6: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | <WRAP box> | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | \\ | ||
| \text{for a (constant)} \\ | \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} \\ | \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 \\ | + | & = & \sum{2 (Y_i - (a + bX_i))} * (-1) \; |
| + | & \because | ||
| & = & -2 \sum{(Y_i - (a + bX_i))} \\ | & = & -2 \sum{(Y_i - (a + bX_i))} \\ | ||
| \\ | \\ | ||
| Line 24: | Line 25: | ||
| a & = & \overline{Y} - b \overline{X} \\ | a & = & \overline{Y} - b \overline{X} \\ | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | </ | ||
| - | + | <WRAP box> | |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | \\ | ||
| \text{for b, (coefficient)} \\ | \text{for b, (coefficient)} \\ | ||
| \\ | \\ | ||
| - | \dfrac{\text{d}}{\text{dv}} \sum{(Y_i - (a + bX_i))^2} | + | \dfrac{\text{d}}{\text{dv}} \sum{(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 \\ | + | & = & \sum{2 (Y_i - (a + bX_i))} * (-X_i) \; |
| + | & \because | ||
| & = & -2 \sum{X_i (Y_i - (a + bX_i))} \\ | & = & -2 \sum{X_i (Y_i - (a + bX_i))} \\ | ||
| \\ | \\ | ||
| \text{in order to have the least value, the above should be zero} \\ | \text{in order to have the least value, the above should be zero} \\ | ||
| \\ | \\ | ||
| - | -2 \sum{X_i (Y_i - (a + bX_i))} & = & 0 \\ | + | -2 \sum{X_i (Y_i - (a + bX_i))} & = & 0 \\ |
| \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 \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}) - 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})} - \sum{b X_i (X_i - \overline{X}) } & = & 0 \\ | ||
| - | \sum{X_i (Y_i - \overline{Y})} & = & b \sum{X_i (X_i - \overline{X})} \\ | + | \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{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})}}{\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{ \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}} \\ | b & = & \dfrac{ \text{SP} } {\text{SS}_\text{x}} \\ | ||
| - | |||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | </ | ||
| - | |||
| - | |||
| - | {{: | ||
| - | {{: | ||
deriviation_of_a_and_b_in_a_simple_regression.1716419988.txt.gz · Last modified: 2024/05/23 08:19 by hkimscil