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Dataplot Vol 2 Vol 1

EMPIRICAL QUANTILE FUNCTION

Name:
    EMPIRICAL QUANTILE FUNCTION (LET)
Type:
    Let Subcommand
Purpose:
    Compute the empirical 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.

Syntax:
    LET <y> <u> = EMPIRICAL QUANTILE FUNCTION <x>
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is the response variable;
                <y> is a variable containing the empirical quantile function;
    <u> is a variable containing the values where the empirical quantile function is computed;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET Y U = EMPIRICAL QUANTILE FUNCTION X
    LET Y U = EMPIRICAL QUANTILE FUNCTION X SUBSET X > 0
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, 8-37, 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:
    2017/02
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 empirical quantile funciton
    .
    let nv = data 50 100 200 1000
    let p = sequence 0.01 0.01 .99
    let y2 = norppf(p)
    .
    . Step 3:   Loop through the four cases and compute and plot the
    .           empirical 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 = empirical quantile function x
        title N: ^n
        plot y u and
        plot y2 p
    end of loop
    end of multiplot
    .
    justification center
    move 50 97
    text Empirical Quantile Functions (blue) Overlaid with ...
    NORPPF (red) for Normal Random Numbers
    move 50 5
    text u
    direction vertical
    move 5 50
    text Q(u)
        
    plot generated by sample program

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

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