SED navigation bar go to SED home page go to Dataplot home page go to NIST home page SED Home Page SED Staff SED Projects SED Products and Publications Search SED Pages
Dataplot Vol 1 Vol 2

EMPIRICAL QUANTILE PLOT

Name:
    EMPIRICAL QUANTILE PLOT
Type:
    Graphics Command
Purpose:
    Generates an empirical quantile plot, and optionally, overlays a quantile box plot.
Description:
    The empirical quantile function of a variable is defined as

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

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

    This will be computed for a specified number of equi-spaced points between the lower and upper limits. Dataplot will use the minimum of 1,000 points and the number of points in the sample.

    This plot is essentially the inverse of the empirical CDF plot.

    The empirical quantile plot can be enhanced with an overlaid quantile box plot (see Syntax 4). The QUANTILE BOX PLOT command will first generate the empirical quantile plot and then it will overlay the quantile box plot. The quantile box plot draws the following rectangles:

    1. The rectangle enclosing the 25% and 75% percentiles.
    2. The rectangle enclosing the 12.5% and 87.5% percentiles.
    3. The rectagle enclosing the 6.25% and 93.75% percentiles.
    4. A horizontal line is drawn through the median.

    The quantile box plot can help in assesing the symmetry, tail behavior, and the presence of outliers of the underlying distribution of the data.

    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).

Syntax 1:
    EMPIRICAL QUANTILE PLOT <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
    MULTIPLE EMPIRICAL QUANTILE PLOT <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> ... <yk> is a list of 1 to 30 response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax generates an empirical quantile plot for each listed response variable. These response variables can be matrices.

    The TO syntax is supported for this case.

Syntax 3:
    REPLICATED EMPIRICAL QUANTILE PLOT <y> <x1> <x2>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is a response variable;
                <x1> is the first (required) group-id variables;
                <x2> is the second (optional) group-id variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax peforms a cross-tabulation of <x1> and <x2> and generates the quantile function plot for each unique combination of cross-tabulated values. For example, if X1 has 3 levels and X2 has 2 levels, there will be a total of 6 plots generated.

Syntax 4:
    QUANTILE BOX PLOT <y>             <SUBSET/EXCEPT/FOR qualification> and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The MULTIPLE and REPLICATED options are not supported for this syntax.

Examples:
    EMPIRICAL QUANTILE PLOT Y1
    EMPIRICAL QUANTILE PLOT Y1 SUBSET TAG > 1
    MULTIPLE EMPIRICAL QUANTILE PLOT Y1 Y2 Y3 Y4 Y5
    MULTIPLE EMPIRICAL QUANTILE PLOT Y1 TO Y5
    REPLICATED EMPIRICAL QUANTILE PLOT Y 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, 8-37, 2002.

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

Applications:
    Reliability
Implementation Date:
    2017/02
Program:
    . Step 1:   Read the data
    .
    skip 25
    read weibbury.dat y1
    read exp.dat y2
    read lgn.dat y3
    read gamma.dat y4
    .
    . Step 2:   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:   Generate the empirical quantile plot
    .
    multiplot 2 2
    .
    title WEIBBURY.DAT
    empirical quantile plot y1
    .
    title EXP.DAT
    empirical quantile plot y2
    .
    title LGN.DAT
    empirical quantile plot y3
    .
    title GAMMA.DAT
    empirical quantile plot y4
    .
    end of multiplot
    .
    justification center
    move 50 97
    text Empirical Quantile Plots
    move 50 5
    text u
    direction vertical
    move 5 50
    text Q(u)
    

    plot generated by sample program

    . . Step 4: Demonstrate multiple option . y1label Q(u) x1label u title automatic line color black blue red green . multiple empirical quantile plot y1 to y4

    plot generated by sample program

    . . Step 5: Demonstrate quantile box plot . line color black all line solid dotted dotted dotted dotted quantile box plot y1 . line solid all

    plot generated by sample program

    . . Step 6: Demonstrate replication option . skip 25 read gear.dat y x line color black all line solid all character blank all . replicated empirical quantile plot y x

    plot generated by sample program

Privacy Policy/Security Notice
Disclaimer | FOIA

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

Date created: 06/29/2017
Last updated: 06/29/2017

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