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Dataplot Vol 1 Vol 2

FRECHET PLOT

Name:
    FRECHET PLOT
Type:
    Graphics Command
Purpose:
    Generates a Frechet plot.
Description:
    The Frechet plot can be used to determine whether the 2-parameter Frechet distribution is an appropriate distributional model for a set of data.

    The Frechet plot is formed by

    1. Sort the data. Call these points yi.

    2. The x-axis coordinates are \( \ln(y_{i}) \).

    3. The y-axis coordinates are \( -\ln(-\ln(p_{i})) \) where

        \( p_{i} = \frac{i - 0.3}{n + 0.4} \)

    If the data come from a 2-parameter Frechet distribution, then the points on this plot should be approximately linear.

    If you fit a line to the points on this plot, the intercept can be used as an estimate of the scale parameter and the slope can be used as an estimate of the shape parameter. However, the plot is generally used to determine if a 2-parameter Frechet is appropriate. If the plot is approximately linear, then parameter estimates would usually be determined by maximum likelihood using the command

      FRECHET MAXIMUM LIKELIHOOD Y

    Maximum likelihood estimates would generally be preferred over the Frechet plot estimates since its statistical properties are better understood.

    Currently, this plot is supported for the maximum version of the Frechet distribution. It is also only supported for the case with no censoring. It is also not supported for grouped data.

    The characteristics of these components are controlled through the LINE and CHARACTER commands. This is demonstrated in the program examples below.

    If you have groups in the data, you can specify a "highlight" option to draw the points in the different groups with different attributes. For example, this can be used to draw outliers in a different color. This is demonstrated in the second program example below.

Syntax 1:
    FRECHET PLOT <y>             <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
    HIGHLIGHT FRECHET PLOT <y> <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
                <x> is a group-id variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    FRECHET PLOT Y1
    HIGHLIGHT FRECHET PLOT Y1 X
Note:
    The following internal parameters are saved after a FRECHET PLOT. These parameters can be used like any user created parameter by the analyst.

      SHAPE - the slope of the fitted line as an estimate of the shape parameter
      SCAL - the intercept of the fitted line as an estimate of the scale parameter
      BPT1 - the 0.1% point of the best fit distribution
      BPT5 - the 0.5% point of the best fit distribution
      BP1 - the 1% point of the best fit distribution
      BP5 - the 5% point of the best fit distribution
      BP10 - the 10% point of the best fit distribution
      BP20 - the 20% point of the best fit distribution
      BP50 - the 50% point of the best fit distribution
      BP80 - the 80% point of the best fit distribution
      BP90 - the 90% point of the best fit distribution
      BP95 - the 95% point of the best fit distribution
      BP99 - the 99% point of the best fit distribution
      BP995 - the 99.5% point of the best fit distribution
      BP999 - the 99.9% point of the best fit distribution

    The percent point estimates are computed by using the estimates of scale (SCAL) and shape (SHAPE) obtained from the plot and plugging these values into the Frechet percent point function.

Note:
    The Frechet plot is similar in concept to a Frechet probability plot. However, a few distinctions should be noted.

    1. The Frechet plot is based on a 2-parameter Frechet distribution. A Frechet probability plot is based on a 3-parameter Frechet distribution.

    2. The Frechet plot linearization is specific to the Frechet distribution. The probability plot linearization can be used for any distribution. For the probability plot method, the ppcc plot is used to estimate the shape parameter and then the probability plot is used to estimate the location and scale parameters and to assess the goodness of fit.
Default:
    None
Synonyms:
    SUBSET is a synonym for HIGHLIGHT
Related Commands: Reference:
    Karl Bury (1999), "Statistical Distributions in Engineering", Cambridge University Press, chapter 16.
Applications:
    Distributional Modeling
Implementation Date:
    2013/10
Program 1:
     
    . Step 1:  Read the data
    .
    skip 25
    read frechet.dat y
    skip 0
    .
    . Step 2:  Set plot control features
    .
    label case asis
    title case asis
    title offset 2
    character X blank
    line blank solid
    title Frechet Plot for FRECHET.DAT
    y1label -LN(-LN(P(i)))
    x1label LN(y(i))
    x2label P(i) = (i - 0.3)/(n + 0.4)
    .
    . Step 3:  Generate the plot
    .
    frechet plot y
        
    plot generated by sample program
Program 2:
     
    . Demonstrate the HIGHLIGHT option
    .
    . Step 1:  Read the data
    .
    skip 25
    read frechet.dat y
    skip 0
    let n = size y
    let x = 0 for i = 1 1 n
    let x = 1 subset y > 22
    let x = 1 subset y < 14
    .
    . Step 2:  Set plot control features
    .
    label case asis
    title case asis
    title offset 2
    character circle circle blank
    character hw 1 0.75 all
    character fill on on
    character color black red
    line blank blank solid
    title Frechet Plot for FRECHET.DAT
    y1label -LN(-LN(P(i)))
    x1label LN(y(i))
    x2label P(i) = (i - 0.3)/(n + 0.4)
    .
    . Step 3:  Generate the plot
    .
    highlight frechet plot y x
        
    plot generated by sample program

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Date created: 01/31/2015
Last updated: 01/31/2015

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