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:
 LINES = Sets the type for plot lines. CHARACTER = Sets the type for plot characters. PROBABILITY PLOT = Generates a probability plot. PPCC PLOT = = Generates a ppcc plot. NORMAL PLOT = Generates a normal plot. WEIBULL PLOT = Generates a Weibull plot. MAXIMUM LIKELIHOOD = Estimate distribution parameters using maximum likelihood.
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
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

Program 2:

. Demonstrate the HIGHLIGHT option
.
. Step 1:  Read the data
.
skip 25
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


NIST is an agency of the U.S. Commerce Department.

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

Please email comments on this WWW page to alan.heckert.gov.