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Dataplot Vol 2 Vol 1

INFORMATIVE QUANTILE FUNCTION
TRUNCATED INFORMATIVE QUANTILE FUNCTION

Name:
    INFORMATIVE QUANTILE FUNCTION (LET)
    TRUNCATED INFORMATIVE QUANTILE FUNCTION (LET)
Type:
    Let Subcommand
Purpose:
    Compute the informative quantile function or the truncated informative quantile function.
Description:
    The quantile function is the inverse of the cumulative distribution function, F,

      \( Q(u) = F^{-1}(u) \hspace{0.2in} 0 < u < 1 \)

    Given a set of ordered data, x1x2 ... ≤ xn, an empirical estimate of the quantile function can be obtained from the following piecewise linear function

      \( \hat{Q}(u) = (nu - j + \frac{1}{2}) x_{(j+1)} + (j + \frac{1}{2} - nu) x_{j} \)

      \( \frac{2j - 1}{2n} \le u \le \frac{2j + 1}{2n} \)

    This will be computed for a specified number of equi-spaced points between the lower and upper limits. Dataplot will use the number of points in the sample if this is greater than 1,000. Otherwise 1,000 points will be used.

    The estimated informative quantile (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 described above. 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

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

    The TRUNCATED INFORMATIVE QUANTILE PLOT command is used to plot the truncated informative quantiles of the data along with the truncated informative quantiles of a specific distribution.

Syntax 1:
    LET <y> <u> = INFORMATIVE QUANTILE FUNCTION <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is the response variable;
                <y> is a variable containing the informative quantile function values;
                <u> is a variable containing the values where the informative quantile function is computed;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
    LET <y> <u> = TRUNCATED INFORMATIVE QUANTILE FUNCTION <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is the response variable;
                <y> is a variable containing the truncated informative quantile function values;
                <u> is a variable containing the values where the truncated informative quantile function is computed;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET Y U = INFORMATIVE QUANTILE FUNCTION X
    LET Y U = INFORMATIVE QUANTILE FUNCTION X SUBSET X > 0
    LET Y U = TRUNCATED INFORMATIVE QUANTILE FUNCTION X
Default:
    None
Synonyms:
    None
Related Commands: References:
    "MIL-HDBK-17-1F Volume 1: Guidelines for Characterization of Structural Materials", Depeartment of Defense, pp. 8-36 - 8-39, 2002.

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

Applications:
    Distributional Analysis
Implementation Date:
    2017/03
Program:
     
    . 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 2:   Create 50, 100, 200, and 1000 normal random numbers and
    .           compute the informative quantile funciton
    .
    let nv = data 50 100 200 1000
    let p = sequence 0.01 0.01 .99
    let y2 = norppf(p)
    set write decimals 3
    .
    . Step 3:   Loop through the four case and compute and plot the
    .           informative quantile funciton with overlaid NORPPF
    .
    multiplot 2 2
    loop for k = 1 1 4
        let n = nv(k)
        let x = norm rand numb for i = 1 1 n
        let y u   = informative quantile function x
        let y2 u2 = truncated informative quantile function x
        title N: ^n
        plot y  u and
        plot y2 u2
    end of loop
    end of multiplot
    .
    justification center
    move 50 97
    text Informative Quantile Functions (blue) and Truncated Informative ...
    Quantile Functions (red) for Normal Random Numbers
    move 50 5
    text u
    direction vertical
    move 5 50
    text IQ(u)
        
    plot generated by sample program

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Date created: 07/20/2017
Last updated: 07/20/2017

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