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:
 LINES = Sets the type for plot lines. CHARACTERS = Sets the type for plot characters. TRUNCATED INFORMATIVE QUANTILE PLOT = Generate a truncated informative 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.
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
.
. 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)

.
. 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

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

.
. Step 6:    Demonstrate replication option
.
skip 25