Least Squares/Rank Regression Equations - Revision history

Lisa Hacker at 19:15, 15 September 2023

2023-09-15T19:15:38Z

← Older revision		Revision as of 19:15, 15 September 2023
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	{{Template:~~LDABOOK_SUB~~\|Appendix A\|Least Squares/Rank Regression Equations}}		{{Template:LDABOOK\|Appendix A\|Least Squares/Rank Regression Equations}}

	==Rank Regression on Y==		==Rank Regression on Y==

Sharon Honecker: /* Example */ Corrected result for b for RRX

2016-07-28T00:10:20Z

Example: Corrected result for b for RRX

← Older revision		Revision as of 00:10, 28 July 2016
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	::<math>\widehat{b}=\frac{387.5-(56.5)(41.5)/8}{276.25-(41.5)^2/8}\,\!</math>		::<math>\widehat{b}=\frac{387.5-(56.5)(41.5)/8}{276.25-(41.5)^2/8}\,\!</math>
	::<math>\widehat{a}=-0.~~97002~~\,\!</math>		::<math>\widehat{b}=1.5484\,\!</math>

	:and:		:and:

Sharon Honecker: Fixed math formatting errors.

2015-10-12T19:24:28Z

Fixed math formatting errors.

← Older revision		Revision as of 19:24, 12 October 2015
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	==Rank Regression on Y==		==Rank Regression on Y==
	Assume that a set of data pairs (x1, y1), (x2, y2), ... , (xN, yN), were obtained and plotted. Then, according to the least squares principle, which minimizes the vertical distance between the data points and the straight line fitted to the data, the best fitting straight line to these data is the straight line y = + x such that:		Assume that a set of data pairs (x1, y1), (x2, y2), ... , (xN, yN), were obtained and plotted. Then, according to the least squares principle, which minimizes the vertical distance between the data points and the straight line fitted to the data, the best fitting straight line to these data is the straight line y = a + bx such that:

	::<math>\sum_{i=1}^N (\hat{a}+\hat{b} x_i - y_i)^2=min(a,b)\sum_{i=1}^N (a+b x_i-y_i)^2 \,\!</math>		::<math>\sum_{i=1}^N (\hat{a}+\hat{b} x_i - y_i)^2=min(a,b)\sum_{i=1}^N (a+b x_i-y_i)^2 \,\!</math>

	and where and are the least squares estimates of ''a'' and ''b'', and ''N'' is the number of data points.		and where <math>\hat{a}\,\!</math> and <math>\hat{b}\,\!</math> are the least squares estimates of ''a'' and ''b'', and ''N'' is the number of data points.

	To obtain <math>\hat{a}\,\!</math> and <math>\hat{b}\,\!</math>, let:		To obtain <math>\hat{a}\,\!</math> and <math>\hat{b}\,\!</math>, let:
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	Setting Eqns. (1) and (2) equal to zero yields:		Setting Eqns. (1) and (2) equal to zero yields:

	::<math>\sum_{i=1}^N (a+b x_i-y_i)~~x_i~~=\sum_{i=1}^N(\hat{y}_i-y_i)=-\sum_{i=1}^N(y_i-\hat{y}_i)=0 \,\!</math>		::<math>\sum_{i=1}^N (a+b x_i-y_i)=\sum_{i=1}^N(\hat{y}_i-y_i)=-\sum_{i=1}^N(y_i-\hat{y}_i)=0 \,\!</math>

	:and:		:and:
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	Solving the equations simultaneously yields:		Solving the equations simultaneously yields:

	::<math>\hat{a}=\frac{\displaystyle \sum_{i=1}^N y_i}{N}-\hat{b}\frac{\displaystyle \sum_{i=1}{N} x_i}{N}=\bar{y}-\hat{b}\bar{x} \,\!</math> (3)		::<math>\hat{a}=\frac{\displaystyle \sum_{i=1}^N y_i}{N}-\hat{b}\frac{\displaystyle \sum_{i=1}^N x_i}{N}=\bar{y}-\hat{b}\bar{x} \,\!</math> (3)

	:and:		:and:

	::<math>\hat{b}=\frac{\displaystyle \~~sum~~{i=1}^N x_i y_i-\frac{\displaystyle \sum_{i=1}^N x_i \sum_{i=1}^N y_i}{N}}{\displaystyle \sum_{i=1}^N x_i^2-\frac{\left(\displaystyle\sum_{i=1}^N x_i\right)^2}{N}}\,\!</math>(4)		::<math>\hat{b}=\frac{\displaystyle \sum_{i=1}^N x_i y_i-\frac{\displaystyle \sum_{i=1}^N x_i \sum_{i=1}^N y_i}{N}}{\displaystyle \sum_{i=1}^N x_i^2-\frac{\left(\displaystyle\sum_{i=1}^N x_i\right)^2}{N}}\,\!</math>(4)


	==Rank Regression on X==		==Rank Regression on X==
	Assume that a set of data pairs (x1, y1), (x2, y2), ... , (xN, yN) were obtained and plotted. Then, according to the least squares principle, which minimizes the horizontal distance between the data points and the straight line fitted to the data, the best fitting straight line to these data is the straight line x = + y such that:		Assume that a set of data pairs (x1, y1), (x2, y2), ... , (xN, yN) were obtained and plotted. Then, according to the least squares principle, which minimizes the horizontal distance between the data points and the straight line fitted to the data, the best fitting straight line to these data is the straight line x = a + by such that:

	::<math>\displaystyle\sum_{i=1}^N(\hat{a}+\hat{b}y_i-x_i)^2=min(a,b)\displaystyle\sum_{i=1}^N (a+by_i-x_i)^2 \,\!</math>		::<math>\displaystyle\sum_{i=1}^N(\hat{a}+\hat{b}y_i-x_i)^2=min(a,b)\displaystyle\sum_{i=1}^N (a+by_i-x_i)^2 \,\!</math>
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	Differentiating F with respect to a and b yields:		Differentiating F with respect to a and b yields:

	::<math>\frac{\partial F}{\partial a}=2\displaystyle\sum_{i=1}^N(a+by_i-x_i\,\!</math> (5)		::<math>\frac{\partial F}{\partial a}=2\displaystyle\sum_{i=1}^N(a+by_i-x_i)\,\!</math> (5)

	:and:		:and:

Richard House: /* Example */

2014-01-31T19:58:37Z

Example

← Older revision		Revision as of 19:58, 31 January 2014
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	The plotted line is shown in the next figure.		The plotted line is shown in the next figure.

	[[Image:~~LdaappendixA~~.~~1.gif~~\|center\|500px]]		[[Image:LdaappendixA_1_new.png\|center\|500px]]

	For rank regression on X (RRX) using the same table yields:		For rank regression on X (RRX) using the same table yields:
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	The plotted line is shown in the next figure.		The plotted line is shown in the next figure.

	[[Image:~~LdaappendixA~~.~~2.gif~~\|thumb\|center\|500px]]		[[Image:LdaappendixA_2_new.png\|thumb\|center\|500px]]

	Note that the regression on Y is not necessarily the same as the regression on X. The only time when the two regressions are the same (i.e., will yield the same equation for a line) is when the data lie perfectly on a line.		Note that the regression on Y is not necessarily the same as the regression on X. The only time when the two regressions are the same (i.e., will yield the same equation for a line) is when the data lie perfectly on a line.

Richard House at 18:27, 26 September 2012

2012-09-26T18:27:30Z

Show changes

Richard House: /* Example */

2012-08-24T04:33:30Z

Example

← Older revision		Revision as of 04:33, 24 August 2012
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	::<math>\widehat{a}=\frac{56.5}{8}-1.5484\frac{41.5}{8}</math>		::<math>\widehat{a}=\frac{56.5}{8}-1.5484\frac{41.5}{8}</math>
	::\widehat{a}=-0.97002		::<math>\widehat{a}=-0.97002</math>

	The least squares line is given by:		The least squares line is given by:

Richard House: /* Example */

2012-08-21T22:57:07Z

Example

← Older revision		Revision as of 22:57, 21 August 2012
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	The least squares line is given by:		The least squares line is given by:

	::<math>y=0.77836+0.6243x</math>		::<math>\begin{align}
			y=0.77836+0.6243x
			\end{align}</math>

	The plotted line is shown in the next figure.		The plotted line is shown in the next figure.

Richard House: /* Example */

2012-08-08T01:22:16Z

Example

← Older revision		Revision as of 01:22, 8 August 2012
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	Fit a least squares straight line using regression on X and regression on Y to the following data:		Fit a least squares straight line using regression on X and regression on Y to the following data:

	~~<br>~~

	{\| align="center" border="0" cellspacing="5" width="300" cellpadding="5"		{\| align="center" border="0" cellspacing="5" width="300" cellpadding="5"
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	\| align="left" valign="middle" \| 10		\| align="left" valign="middle" \| 10
	\|}		\|}

	~~<br>~~

	The first step is to generate the following table:		The first step is to generate the following table:

	~~<br>~~		{\| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"

	{\| align="center" ~~border~~="1" cellpadding="5" cellspacing="~~1" width="350~~"
	\|-		\|-
	! scope="col" \| '''<math>i</math>'''		! scope="col" \| '''<math>i</math>'''
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	\| align="center" \|276.25		\| align="center" \|276.25
	\|}		\|}

	~~<br>~~

	From the table then, and for rank regression on Y (RRY):		From the table then, and for rank regression on Y (RRY):

Richard House: /* Example */

2012-08-06T05:46:08Z

Example

← Older revision		Revision as of 05:46, 6 August 2012
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	The plotted line is shown in the next figure.		The plotted line is shown in the next figure.

	[[Image:LdaappendixA.1.gif~~\|thumb~~\|center\|500px]]		[[Image:LdaappendixA.1.gif\|center\|500px]]

	For rank regression on X (RRX) using the same table yields:		For rank regression on X (RRX) using the same table yields:

Richard House: /* Example */

2012-08-06T02:56:17Z

Example

← Older revision		Revision as of 02:56, 6 August 2012
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	[[Image:LdaappendixA.2.gif\|thumb\|center\|500px]]		[[Image:LdaappendixA.2.gif\|thumb\|center\|500px]]

	Note that the regression on Y is not necessarily the same as the regression on X. The only time when the two regressions are the same (i.e. will yield the same equation for a line) is when the data lie perfectly on a line.		Note that the regression on Y is not necessarily the same as the regression on X. The only time when the two regressions are the same (i.e., will yield the same equation for a line) is when the data lie perfectly on a line.

	The correlation coefficient is given by:		The correlation coefficient is given by: