Template:Weibull parameters Rank Regression on Y: Difference between revisions

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=== Rank Regression on Y ===
#REDIRECT [[The Weibull Distribution]]
Performing rank regression on Y requires that a straight line mathematically be fitted to a set of data points such that the sum of the squares of the vertical deviations from the points to the line is minimized. This is in essence the same methodology as the probability plotting method, except that we use the principle of least squares to determine the line through the points, as opposed to just eyeballing it. The first step is to bring our function into a linear form. For the two-parameter Weibull distribution, the  (cumulative density function) is:
 
::<math> F(t)=1-e^{-\left( \frac{t}{\eta }\right) ^{\beta }}  </math>
 
Taking the natural logarithm of both sides of the equation yields:
 
::<math>\ln[ 1-F(t)] =-( \frac{t}{\eta }) ^{\beta } </math>
 
::<math> \ln{ -\ln[ 1-F(t)]} =\beta \ln ( \frac{t}{ \eta }) </math>
 
or:
 
::<math> \ln \{ -\ln[ 1-F(t)]\} =-\beta \ln (\eta )+\beta \ln (t) </math>
 
Now let:
 
::<math> y = \ln \{ -\ln[ 1-F(t)]\}  </math>
 
::<math> a = − βln(\eta) </math>
 
and:
 
::<math> b= \beta</math>
 
which results in the linear equation of:
 
::<math>y=a+bx</math>
 
The least squares parameter estimation method (also known as regression analysis) was discussed in Chapter 3 and the following equations for regression on Y were derived:
 
::<math> \hat{a}=\frac{\sum\limits_{i=1}^{N}y_{i}}{N}-\hat{b}\frac{ \sum\limits_{i=1}^{N}x_{i}}{N}=\bar{y}-\hat{b}\bar{x} </math>
 
and:
 
::<math> \hat{b}={\frac{\sum\limits_{i=1}^{N}x_{i}y_{i}-\frac{\sum \limits_{i=1}^{N}x_{i}\sum\limits_{i=1}^{N}y_{i}}{N}}{\sum \limits_{i=1}^{N}x_{i}^{2}-\frac{\left( \sum\limits_{i=1}^{N}x_{i}\right) ^{2}}{N}}} </math>
 
In this case the equations for <span class="texhtml">''y''<sub>''i''</sub></span> and <span class="texhtml">''x''<sub>''i''</sub></span> are:
 
::<math> y_{i}=\ln \left\{ -\ln [1-F(t_{i})]\right\} , </math>
 
and:
 
::<span class="texhtml">''x''<sub>''i''</sub> = ln(''t''<sub>''i''</sub>).</span>
 
The <math> F(t_{i})s </math> are estimated from the median ranks.
 
Once <math> \hat{a} </math> and <math> \hat{b} </math> are obtained, then <math> \hat{\beta } </math> and <math> \hat{\eta } </math> can easily be obtained from previous equations.
 
 
'''The Correlation Coefficient'''
 
The correlation coefficient is defined as follows:
 
::<math> \rho ={\frac{\sigma _{xy}}{\sigma _{x}\sigma _{y}}} </math>
 
where, <span class="texhtml">σ<sub>''x'' ''y''</sub> = </span>covariance of  and , <span class="texhtml">σ<sub>''x''</sub> = </span>standard deviation of , and <span class="texhtml">σ<sub>''y''</sub> = </span>standard deviation of . The estimator of <span class="texhtml">ρ</span> is the sample ''correlation coefficient'', <math> \hat{\rho} </math>, given by:
 
::<math> \hat{\rho}=\frac{\sum\limits_{i=1}^{N}(x_{i}-\overline{x})(y_{i}-\overline{y} )}{\sqrt{\sum\limits_{i=1}^{N}(x_{i}-\overline{x})^{2}\cdot \sum\limits_{i=1}^{N}(y_{i}-\overline{y})^{2}}}</math>
<br>
 
'''Example 3: '''
{{Example: 2P Weibull Distribution RRY}}

Latest revision as of 08:32, 8 August 2012

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