TRUNCATED INFORMATIVE QUANTILE PLOT

TRUNCATED INFORMATIVE QUANTILE PLOT

Graphics Command

Generates a truncated informative quantile plot.

Given a response variable, the truncated informative quantile plot can be used to identify which location-scale distribution best fits that data. In the context of the MIL-HANDBOOK-17, it is used to distinguish whether the data is most consistent with a normal, a 2-parameter lognormal, or a 2-parameter Weibull distribution.

A univariate location-scale parametric distribution is one where the cumulative distribution function can be described as

\( F(x) = F_{0}(\frac{x-a}{b}) \)

with a and b denoting the location and scale parameters, respectively and F₀ denotes the "standard" distribution (i.e., the location parameter is 0 and the scale parameter is 1). This means the distribution has no shape parameters. The 2-parameter lognormal is handled by taking the log of the data and then using the normal distribution. Similarly, the 2-parameter Weibull is handled by taking the log of the data and using the Gumbel distribution.

The estimated IQ function is defined by

\( \hat{\mbox{IQ}} = \frac{\hat{\mbox{Q}}(u) - \hat{\mbox{Q}}(0.5)} {\hat{\mbox{Q}}(0.75) - \hat{\mbox{Q}}(02.5)} \)

where Q(u) is the estimated quantile function (enter HELP EMPIRICAL QUANTILE PLOT for details). The corresponding exact function, denoted by IQ(u), replaces \( \hat{\mbox{Q}} \) with Q in the above formula. The estimated \( \hat{\mbox{IQ}} \) values are determined from the data while the exact IQ values are determined from a theoretical location-scale distribution (e.g., the normal, uniform, or Gumbel).

The estimated truncated IQ function is defined by

\( \hat{TIQ} = \left( \begin{array}{cl} -1 & \mbox{if} \hspace{0.25in} \hat{\mbox{IQ}} \le -1 \\ \hat{\mbox{IQ}}(u) & \mbox{if} \hspace{0.25in} -1 < \hat{\mbox{IQ}} \le -1 \\ 1 & \mbox{if} \hspace{0.25in} \hat{\mbox{IQ}} > 1 \end{array} \right. \)

This command plots the estimated truncated IQ function versus u and also plots the IQ function versus u for a specified theoretical distribution. The curve based on the data is compared to the curve for the theoretical distribution. The curve based on the data will typically be less smooth. However, the curves can be compared for general shape and tail behavior.

Although the MIL-HANDBOOK-17 is primarily concerned with the normal, 2-parameter lognormal, and 2-parameter Weibull distributions, Dataplot supports this command for 19 different distributions.

These plots are suggested as exploratory data analysis techniques in the MIL-HANDBK-17 (2002 edition). They were originally suggested by Parzen (see References below).

<dist> TRUNCATED INFORMATIVE QUANTILE PLOT <y>
                        <SUBSET/EXCEPT/FOR qualification>
where <dist> specifies the theoretical distribution and is one of:
NORMAL
LOGNORMAL
UNIFORM
GUMBEL
WEIBULL
LOGISTIC
DOUBLE EXPONENTIAL
CAUCHY
SEMICIRCULAR
COSINE
ANGLIT
HYPERBOLIC SECANT
HALF-NORMAL
ARCSINE
EXPONENTIAL
HALF-CAUCHY
SLASH
RAYLEIGH
MAXWELL

            <y> is a response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

Note that the 2-parameter lognormal is handled by taking the log of the data and then using the normal distribution. The 2-parameter Weibull is handled by taking the log of the data and then using the Gumbel distribution. The log of the data will be taken by the Dataplot code, so you should not take the log of the data before entering this command.

NORMAL TRUNCATED INFORMATIVE QUANTILE PLOT Y1
WEIBULL TRUNCATED INFORMATIVE QUANTILE PLOT Y1
LOGNORMAL TIQ PLOT Y1

None

TIQ is a synonym for TRUNCATED INFORATIVE QUANTILE.

LINES	=	Sets the type for plot lines.
CHARACTERS	=	Sets the type for plot characters.
EMPIRICAL QUANTILE PLOT	=	Generate a empirical quantile plot.
EMPIRICAL CDF PLOT	=	Generates a tail area plot.
KAPLAN MEIER PLOT	=	Generates a Kaplan Meier plot.
PROBABILITY PLOT	=	Generates a probability plot.
PLOT	=	Generates a data or function plot.

"MIL-HDBK-17-1F Volume 1: Guidelines for Characterization of Structural Materials", Depeartment of Defense, pp. 8-38, 8-39, 2002.

Parzen (1983), "Informative Quantile Functions and Identification of Probability Distribution Types", Technical Report No. A-26, Texas A&M University.

Reliability

2017/03

 
. Step 1:   Define some default plot control features
.
title offset 2
title case asis
case asis
label case asis
line color blue red
multiplot scale factor 2
multiplot corner coordinates 5 5 95 95
.
. Step 3:   Demonstrate TIQ PLOT
.
y1label IQ(u)
x1label u
title automatic
line solid solid
line color black red
character blank all
.
delete x
let n = 1000
let x1 = normal random numbers for i = 1 1 n
let x2 = exponential random numbers for i = 1 1 n
let x3 = double exponential random numbers for i = 1 1 n
let gamma = 2.7
let x4 = weibull random numbers for i = 1 1 n
multiplot 2 2
.
title ^n Normal Random Numbers
normal tiq plot x1
.
title ^n Exponential Random Numbers
exponential tiq plot x2
.
title ^n Double Exponential Random Numbers
double exponential tiq plot x3
.
title ^n Weibull Random Numbers
weibull tiq plot x4
.
end of multiplot
    

    
    

.
let x1 = uniform random numbers for i = 1 1 n
let x2 = cosine random numbers for i = 1 1 n
let x3 = rayleigh exponential random numbers for i = 1 1 n
let sigma = 2.7
let x4 = lognormal random numbers for i = 1 1 n
multiplot 2 2
.
title ^n Uniform Random Numbers
uniform tiq plot x1
.
title ^n Cosine Random Numbers
cosine tiq plot x2
.
title ^n rayleigh Random Numbers
rayleigh tiq plot x3
.
title ^n Lognormal Random Numbers
lognormal tiq plot x4
.
end of multiplot
    

    
    

.
let x1 = gumbel random numbers for i = 1 1 n
let x2 = logistic random numbers for i = 1 1 n
let x3 = maxwell random numbers for i = 1 1 n
let x4 = slash random numbers for i = 1 1 n
.
multiplot 2 2
.
title ^n Gumbel Random Numbers
gumbel tiq plot x1
.
title ^n Logistic Random Numbers
logistic tiq plot x2
.
title ^n Maxwell Random Numbers
maxwell tiq plot x3
.
title ^n slash Random Numbers
slash tiq plot x4
.
end of multiplot
    

    
    

.
let x1 = cauchy random numbers for i = 1 1 n
let r = 1
let x2 = semi-circular random numbers for i = 1 1 n
let x3 = anglit random numbers for i = 1 1 n
let x4 = arcsine random numbers for i = 1 1 n
multiplot 2 2
.
title ^n Cauchy Random Numbers
cauchy tiq plot x1
.
title ^n Anglit Random Numbers
anglit tiq plot x3
.
title ^n Arcsine Random Numbers
arcsine tiq plot x4
.
end of multiplot
    

    
    

.
let x1 = half-normal random numbers for i = 1 1 n
let x2 = half-cauchy random numbers for i = 1 1 n
let x3 = hyperbolic secant exponential random numbers for i = 1 1 n
multiplot 2 2
.
title ^n Half-Normal Random Numbers
half-normal tiq plot x1
.
title ^n Half-Cauchy Random Numbers
half-cauchy tiq plot x2
.
title ^n Hyperbolic Secant Random Numbers
hyperbolic secant tiq plot x3
.
end of multiplot