Richard House: Redirected page to Gompertz Models#Linear Regression .28Least Squares.29

2012-08-27T02:11:49Z

Redirected page to Gompertz Models#Linear Regression .28Least Squares.29

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	~~===Linear Regression (Least Squares)===~~		#REDIRECT [[Gompertz_Models#Linear_Regression_.28Least_Squares.29]]
	The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on <math>Y</math> , then the sum of the squares of the vertical deviations from the points to the line is minimized. If the regression is on <math>X</math> , the line is fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized. To illustrate the method, this section presents a regression on <math>Y</math> . Consider the linear model [~~2]:~~

	~~::<math>{{Y}_{i}}={{\widehat{\beta }}_{0}}+{{\widehat{\beta }}_{1}}{{X}_{i1}}+{{\widehat{\beta }}_{2}}{{X}_{i2}}+.~~..~~+{{\widehat{\beta }}_{p}}{{X}_{ip}}</math>~~

	~~<br>~~
	~~or in matrix form where bold letters indicate matrices:~~
	~~<br>~~
	~~::<math>Y=X\beta </math>~~
	~~<br>~~
	~~:where:~~
	~~<br>~~
	~~::<math>Y=\left[ \begin{matrix}~~
	~~{{Y}_{1}} \\~~
	~~{{Y}_{2}} \\~~
	~~\vdots \\~~
	~~{{Y}_{N}} \\~~
	~~\end{matrix} \right~~]~~</math>~~

	~~::<math>X=\left[ \begin{matrix}~~
	~~1 & {{X}_{1,1}} & \cdots & {{X}_{1,p}} \\~~
	~~1 & {{X}_{2,1}} & \cdots & {{X}_{2,p}} \\~~
	~~\vdots & \vdots & \ddots & \vdots \\~~
	~~1 & {{X}_{N,1}} & \cdots & {{X}_{N,p}} \\~~
	~~\end{matrix} \right~~]~~</math>~~
	~~<br>~~
	~~:and:~~
	~~<br>~~
	~~::<math>\beta =\left[ \begin{matrix}~~
	~~{{\beta }_{0}} \\~~
	~~{{\beta }_{1}} \\~~
	~~\vdots \\~~
	~~{{\beta }_{p}} \\~~
	~~\end{matrix} \right]</math>~~

	The vector <math>\beta </math> holds the values of the parameters. Now let <math>\widehat{\beta }</math> be the estimates of these parameters, or the regression coefficients. The vector of estimated regression coefficients is denoted by:

	~~::<math>\widehat{\beta }=\left[ \begin{matrix}~~
	~~{{\widehat{\beta }}_{0}} \\~~
	~~{{\widehat{\beta }}_{1}} \\~~
	~~\vdots \\~~
	~~{{\widehat{\beta }}_{p}} \\~~
	~~\end{matrix} \right]</math>~~

	~~Solving for <math>\beta </math> in Eqn. (linear) requires the analyst to left multiply both sides by the transpose of <math>X</math> , <math>{{X}^{T}}</math> :~~

	~~::<math>({{X}^{T}}X)\widehat{\beta }={{X}^{T}}Y</math>~~

	~~Now the term <math>({{X}^{T}}X)</math> becomes a square and invertible matrix. Then taking it to the other side of the equation gives:~~

	~~::<math>\widehat{\beta }={{(}^{T}}^{-1}{{X}^{T}}Y</math>~~

Nicolette Young: Created page with '===Linear Regression (Least Squares)=== The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on $Y$ , the…'

2012-01-10T20:30:02Z

Created page with '===Linear Regression (Least Squares)=== The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on <math>Y</math> , the…'

New page

===Linear Regression (Least Squares)===
The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on <math>Y</math> , then the sum of the squares of the vertical deviations from the points to the line is minimized. If the regression is on <math>X</math> , the line is fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized. To illustrate the method, this section presents a regression on <math>Y</math> . Consider the linear model [2]:

::<math>{{Y}_{i}}={{\widehat{\beta }}_{0}}+{{\widehat{\beta }}_{1}}{{X}_{i1}}+{{\widehat{\beta }}_{2}}{{X}_{i2}}+...+{{\widehat{\beta }}_{p}}{{X}_{ip}}</math>

 
or in matrix form where bold letters indicate matrices:
 
::<math>Y=X\beta </math>
 
:where:
 
::<math>Y=\left[ \begin{matrix}
{{Y}_{1}} \\
{{Y}_{2}} \\
\vdots \\
{{Y}_{N}} \\
\end{matrix} \right]</math>

::<math>X=\left[ \begin{matrix}
1 & {{X}_{1,1}} & \cdots & {{X}_{1,p}} \\
1 & {{X}_{2,1}} & \cdots & {{X}_{2,p}} \\
\vdots & \vdots & \ddots & \vdots \\
1 & {{X}_{N,1}} & \cdots & {{X}_{N,p}} \\
\end{matrix} \right]</math>
 
:and:
 
::<math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
\vdots \\
{{\beta }_{p}} \\
\end{matrix} \right]</math>

The vector <math>\beta </math> holds the values of the parameters. Now let <math>\widehat{\beta }</math> be the estimates of these parameters, or the regression coefficients. The vector of estimated regression coefficients is denoted by:

::<math>\widehat{\beta }=\left[ \begin{matrix}
{{\widehat{\beta }}_{0}} \\
{{\widehat{\beta }}_{1}} \\
\vdots \\
{{\widehat{\beta }}_{p}} \\
\end{matrix} \right]</math>

Solving for <math>\beta </math> in Eqn. (linear) requires the analyst to left multiply both sides by the transpose of <math>X</math> , <math>{{X}^{T}}</math> :

::<math>({{X}^{T}}X)\widehat{\beta }={{X}^{T}}Y</math>

Now the term <math>({{X}^{T}}X)</math> becomes a square and invertible matrix. Then taking it to the other side of the equation gives:

::<math>\widehat{\beta }={{(}^{T}}^{-1}{{X}^{T}}Y</math>

Template:Linear regression (least squares) gompz - Revision history

Richard House: Redirected page to Gompertz Models#Linear Regression .28Least Squares.29

Nicolette Young: Created page with '===Linear Regression (Least Squares)=== The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on Y , the…'

Nicolette Young: Created page with '===Linear Regression (Least Squares)=== The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on $Y$ , the…'