PROC UNIVARIATE: QQPLOT Statement

The UNIVARIATE Procedure

QQPLOT Statement

QQPLOT <variables> < / options> ;

The QQPLOT statement creates quantile-quantile plots (Q-Q plots) and compares ordered variable values with quantiles of a specified theoretical distribution. If the data distribution matches the theoretical distribution, the points on the plot form a linear pattern. Thus, you can use a Q-Q plot to determine how well a theoretical distribution models a set of measurements.

Q-Q plots are similar to probability plots, which you can create with the PROBPLOT statement. Q-Q plots are preferable for graphical estimation of distribution parameters, whereas probability plots are preferable for graphical estimation of percentiles.

You can use any number of QQPLOT statements in the UNIVARIATE procedure. The components of the QQPLOT statement are as follows.

variables

are the variables for which Q-Q plots are created. If you specify a VAR statement, the variables must also be listed in the VAR statement. Otherwise, the variables can be any numeric variables in the input data set. If you do not specify a list of variables, then by default the procedure creates a Q-Q plot for each variable listed in the VAR statement, or for each numeric variable in the DATA= data set if you do not specify a VAR statement. For example, each of the following QQPLOT statements produces two Q-Q plots, one for Length and one for Width:

   proc univariate data=Measures;
      var Length Width;
      qqplot;

   proc univariate data=Measures;
      qqplot Length Width;
   run;

options

specify the theoretical distribution for the plot or add features to the plot. If you specify more than one variable, the options apply equally to each variable. Specify all options after the slash (/) in the QQPLOT statement. You can specify only one option that names the distribution in each QQPLOT statement, but you can specify any number of other options. The distributions available are the beta, exponential, gamma, lognormal, normal, two-parameter Weibull, and three-parameter Weibull. By default, the procedure produces a plot for the normal distribution.

In the following example, the NORMAL option requests a normal Q-Q plot for each variable. The MU= and SIGMA= normal-options request a distribution reference line with intercept 10 and slope 0.3 for each plot, corresponding to a normal distribution with mean $\text{[math]}$ and standard deviation $\text{[math]}$ . The SQUARE option displays the plot in a square frame, and the CTEXT= option specifies the text color.

   proc univariate data=measures;
      qqplot length1 length2 / normal(mu=10 sigma=0.3)
                               square ctext=blue;
   run;

Table 4.67 through Table 4.76 list the QQPLOT options by function. For complete descriptions, see the sections Dictionary of Options and Dictionary of Common Options.

Options can be any of the following:

primary options
secondary options
general options

Distribution Options

Table 4.67 lists primary options for requesting a theoretical distribution. See the section Distributions for Probability and Q-Q Plots for detailed descriptions of these distributions.

Table 4.67 Primary Options for Theoretical Distributions
Option	Description
BETA(beta-options)	specifies beta Q-Q plot for shape parameters $\text{[math]}$ and $\text{[math]}$ specified with mandatory ALPHA= and BETA= beta-options
EXPONENTIAL(exponential-options)	specifies exponential Q-Q plot
GAMMA(gamma-options)	specifies gamma Q-Q plot for shape parameter $\text{[math]}$ specified with mandatory ALPHA= gamma-option
LOGNORMAL(lognormal-options)	specifies lognormal Q-Q plot for shape parameter $\text{[math]}$ specified with mandatory SIGMA= lognormal-option
NORMAL(normal-options)	specifies normal Q-Q plot
WEIBULL(Weibull-options)	specifies three-parameter Weibull Q-Q plot for shape parameter $\text{[math]}$ specified with mandatory C= Weibull-option
WEIBULL2(Weibull2-options)	specifies two-parameter Weibull Q-Q plot

Table 4.68 through Table 4.75 list secondary options that specify distribution parameters and control the display of a distribution reference line. Specify these options in parentheses after the distribution keyword. For example, you can request a normal Q-Q plot with a distribution reference line by specifying the NORMAL option as follows:

   proc univariate;
      qqplot Length / normal(mu=10 sigma=0.3 color=red);
   run;

The MU= and SIGMA= normal-options display a distribution reference line that corresponds to the normal distribution with mean $\text{[math]}$ and standard deviation $\text{[math]}$ , and the COLOR= normal-option specifies the color for the line.

Table 4.68 Secondary Reference Line Options Used with All Distributions
Option	Description
COLOR=	specifies color of distribution reference line
L=	specifies line type of distribution reference line
W=	specifies width of distribution reference line

Table 4.69 Secondary Beta-Options
Option	Description
ALPHA=	specifies mandatory shape parameter $\text{[math]}$
BETA=	specifies mandatory shape parameter $\text{[math]}$
SIGMA=	specifies $\text{[math]}$ for distribution reference line
THETA=	specifies $\text{[math]}$ for distribution reference line

Table 4.70 Secondary Exponential-Options
Option	Description
SIGMA=	specifies $\text{[math]}$ for distribution reference line
THETA=	specifies $\text{[math]}$ for distribution reference line

Table 4.71 Secondary Gamma-Options
Option	Description
ALPHA=	specifies mandatory shape parameter $\text{[math]}$
ALPHADELTA=	specifies change in successive estimates of $\text{[math]}$ at which the Newton-Raphson approximation of $\text{[math]}$ terminates
ALPHAINITIAL=	specifies initial value for $\text{[math]}$ in the Newton-Raphson approximation of $\text{[math]}$
MAXITER=	specifies maximum number of iterations in the Newton-Raphson approximation of $\text{[math]}$
SIGMA=	specifies $\text{[math]}$ for distribution reference line
THETA=	specifies $\text{[math]}$ for distribution reference line

Table 4.72 Secondary Lognormal-Options
Option	Description
SIGMA=	specifies mandatory shape parameter $\text{[math]}$
SLOPE=	specifies slope of distribution reference line
THETA=	specifies $\text{[math]}$ for distribution reference line
ZETA=	specifies $\text{[math]}$ for distribution reference line (slope is $\text{[math]}$ )

Table 4.73 Secondary Normal-Options
Option	Description
MU=	specifies $\text{[math]}$ for distribution reference line
SIGMA=	specifies $\text{[math]}$ for distribution reference line

Table 4.74 Secondary Weibull-Options
Option	Description
C=	specifies mandatory shape parameter c
SIGMA=	specifies $\text{[math]}$ for distribution reference line
THETA=	specifies $\text{[math]}$ for distribution reference line

Table 4.75 Secondary Weibull2-Options
Option	Description
C=	specifies $\text{[math]}$ for distribution reference line (slope is $\text{[math]}$ )
SIGMA=	specifies $\text{[math]}$ for distribution reference line (intercept is $\text{[math]}$ )
SLOPE=	specifies slope of distribution reference line
THETA=	specifies known lower threshold $\text{[math]}$

General Options

Table 4.76 summarizes general options for enhancing Q-Q plots.

Table 4.76 General Graphics Options
Option	Description
ANNOKEY	applies annotation requested in ANNOTATE= data set to key cell only
ANNOTATE=	specifies annotate data set
CAXIS=	specifies color for axis
CFRAME=	specifies color for frame
CFRAMESIDE=	specifies color for filling frame for row labels
CFRAMETOP=	specifies color for filling frame for column labels
CGRID=	specifies color for grid lines
CHREF=	specifies color for HREF= lines
CONTENTS=	specifies table of contents entry for Q-Q plot grouping
CTEXT=	specifies color for text
CVREF=	specifies color for VREF= lines
DESCRIPTION=	specifies description for plot in graphics catalog
FONT=	specifies software font for text
GRID	creates a grid
HEIGHT=	specifies height of text used outside framed areas
HMINOR=	specifies number of horizontal minor tick marks
HREF=	specifies reference lines perpendicular to the horizontal axis
HREFLABELS=	specifies labels for HREF= lines
HREFLABPOS=	specifies vertical position of labels for HREF= lines
INFONT=	specifies software font for text inside framed areas
INHEIGHT=	specifies height of text inside framed areas
INTERTILE=	specifies distance between tiles
LGRID=	specifies a line type for grid lines
LHREF=	specifies line style for HREF= lines
LVREF=	specifies line style for VREF= lines
NADJ=	adjusts sample size when computing percentiles
NAME=	specifies name for plot in graphics catalog
NCOLS=	specifies number of columns in comparative Q-Q plot
NOFRAME	suppresses frame around plotting area
NOHLABEL	suppresses label for horizontal axis
NOVLABEL	suppresses label for vertical axis
NOVTICK	suppresses tick marks and tick mark labels for vertical axis
NROWS=	specifies number of rows in comparative Q-Q plot
PCTLAXIS	displays a nonlinear percentile axis
PCTLMINOR	requests minor tick marks for percentile axis
PCTLSCALE	replaces theoretical quantiles with percentiles
RANKADJ=	adjusts ranks when computing percentiles
ROTATE	switches horizontal and vertical axes
SQUARE	displays plot in square format
VAXIS=	specifies AXIS statement for vertical axis
VAXISLABEL=	specifies label for vertical axis
VMINOR=	specifies number of vertical minor tick marks
VREF=	specifies reference lines perpendicular to the vertical axis
VREFLABELS=	specifies labels for VREF= lines
VREFLABPOS=	specifies horizontal position of labels for VREF= lines
WAXIS=	specifies line thickness for axes and frame
WGRID=	specifies line thickness for grid

Dictionary of Options

The following entries provide detailed descriptions of options in the QQPLOT statement. See the section Dictionary of Common Options for detailed descriptions of options common to all plot statements.

ALPHA=value-list | EST

specifies the mandatory shape parameter $\text{[math]}$ for quantile plots requested with the BETA and GAMMA options. Enclose the ALPHA= option in parentheses after the BETA or GAMMA options. If you specify ALPHA=EST, a maximum likelihood estimate is computed for $\text{[math]}$ .

BETA(ALPHA=value | EST BETA=value | EST <beta-options>)

creates a beta quantile plot for each combination of the required shape parameters $\text{[math]}$ and $\text{[math]}$ specified by the required ALPHA= and BETA= beta-options. If you specify ALPHA=EST and BETA=EST, the procedure creates a plot based on maximum likelihood estimates for $\text{[math]}$ and $\text{[math]}$ . You can specify the SCALE= beta-option as an alias for the SIGMA= beta-option and the THRESHOLD= beta-option as an alias for the THETA= beta-option. To create a plot that is based on maximum likelihood estimates for $\text{[math]}$ and $\text{[math]}$ , specify ALPHA=EST and BETA=EST. See the section Beta Distribution for details.

To obtain graphical estimates of $\text{[math]}$ and $\text{[math]}$ , specify lists of values in the ALPHA= and BETA= beta-options and select the combination of $\text{[math]}$ and $\text{[math]}$ that most nearly linearizes the point pattern. To assess the point pattern, you can add a diagonal distribution reference line corresponding to lower threshold parameter $\text{[math]}$ and scale parameter $\text{[math]}$ with the THETA= and SIGMA= beta-options. Alternatively, you can add a line that corresponds to estimated values of $\text{[math]}$ and $\text{[math]}$ with the beta-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the beta distribution with parameters $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ is a good fit.

BETA=value-list | EST

B=value | EST

specifies the mandatory shape parameter $\text{[math]}$ for quantile plots requested with the BETA option. Enclose the BETA= option in parentheses after the BETA option. If you specify BETA=EST, a maximum likelihood estimate is computed for $\text{[math]}$ .

C=value-list | EST

specifies the shape parameter $\text{[math]}$ for quantile plots requested with the WEIBULL and WEIBULL2 options. Enclose this option in parentheses after the WEIBULL or WEIBULL2 option. C= is a required Weibull-option in the WEIBULL option; in this situation, it accepts a list of values, or if you specify C=EST, a maximum likelihood estimate is computed for $\text{[math]}$ . You can optionally specify C=value or C=EST as a Weibull2-option with the WEIBULL2 option to request a distribution reference line; in this situation, you must also specify Weibull2-option SIGMA=value or SIGMA=EST.

CGRID=color

specifies the color for grid lines when a grid displays on the plot. This option also produces a grid.

EXPONENTIAL<(exponential-options)>

EXP<(exponential-options)>

creates an exponential quantile plot. To assess the point pattern, add a diagonal distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ with the THETA= and SIGMA= exponential-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter $\text{[math]}$ and the scale parameter $\text{[math]}$ with the exponential-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the exponential distribution with parameters $\text{[math]}$ and $\text{[math]}$ is a good fit. You can specify the SCALE= exponential-option as an alias for the SIGMA= exponential-option and the THRESHOLD= exponential-option as an alias for the THETA= exponential-option. See the section Exponential Distribution for details.

GAMMA(ALPHA=value | EST <gamma-options>)

creates a gamma quantile plot for each value of the shape parameter $\text{[math]}$ given by the mandatory ALPHA= gamma-option. If you specify ALPHA=EST, the procedure creates a plot based on a maximum likelihood estimate for $\text{[math]}$ . To obtain a graphical estimate of $\text{[math]}$ , specify a list of values for the ALPHA= gamma-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ with the THETA= and SIGMA= gamma-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter $\text{[math]}$ and the scale parameter $\text{[math]}$ with the gamma-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the gamma distribution with parameters $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ is a good fit. You can specify the SCALE= gamma-option as an alias for the SIGMA= gamma-option and the THRESHOLD= gamma-option as an alias for the THETA= gamma-option. See the section Gamma Distribution for details.

GRID

displays a grid of horizontal lines positioned at major tick marks on the vertical axis.

LGRID=linetype

specifies the line type for the grid requested by the GRID option. By default, LGRID=1, which produces a solid line. The LGRID= option also produces a grid.

LOGNORMAL(SIGMA=value | EST <lognormal-options>)

LNORM(SIGMA=value | EST <lognormal-options>)

creates a lognormal quantile plot for each value of the shape parameter $\text{[math]}$ given by the mandatory SIGMA= lognormal-option. If you specify SIGMA=EST, the procedure creates a plot based on a maximum likelihood estimate for $\text{[math]}$ . To obtain a graphical estimate of $\text{[math]}$ , specify a list of values for the SIGMA= lognormal-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ with the THETA= and ZETA= lognormal-options. Alternatively, you can add a line corresponding to estimated values of the threshold parameter $\text{[math]}$ and the scale parameter $\text{[math]}$ with the lognormal-options THETA=EST and ZETA=EST. Agreement between the reference line and the point pattern indicates that the lognormal distribution with parameters $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ is a good fit. You can specify the THRESHOLD= lognormal-option as an alias for the THETA= lognormal-option and the SCALE= lognormal-option as an alias for the ZETA= lognormal-option. See the section Lognormal Distribution for details, and see Example 4.31 through Example 4.33 for examples that use the LOGNORMAL option.

MU=value | EST

specifies the mean $\text{[math]}$ for a normal quantile plot requested with the NORMAL option. Enclose the MU= normal-option in parentheses after the NORMAL option. The MU= normal-option must be specified with the SIGMA= normal-option, and they request a distribution reference line. You can specify MU=EST to request a distribution reference line with $\text{[math]}$ equal to the sample mean.

NADJ=value

specifies the adjustment value added to the sample size in the calculation of theoretical percentiles. By default, NADJ= $\text{[math]}$ . Refer to Chambers et al. (1983) for additional information.

NORMAL<(normal-options)>

creates a normal quantile plot. This is the default if you omit a distribution option. To assess the point pattern, you can add a diagonal distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ with the MU= and SIGMA= normal-options. Alternatively, you can add a line corresponding to estimated values of $\text{[math]}$ and $\text{[math]}$ with the normal-options MU=EST and SIGMA=EST; the estimates of the mean $\text{[math]}$ and the standard deviation $\text{[math]}$ are the sample mean and sample standard deviation. Agreement between the reference line and the point pattern indicates that the normal distribution with parameters $\text{[math]}$ and $\text{[math]}$ is a good fit. See the section Normal Distribution for details, and see Example 4.28 and Example 4.30 for examples that use the NORMAL option.

PCTLAXIS<(axis-options)>

adds a nonlinear percentile axis along the frame of the Q-Q plot opposite the theoretical quantile axis. The added axis is identical to the axis for probability plots produced with the PROBPLOT statement. When using the PCTLAXIS option, you must specify HREF= values in quantile units, and you cannot use the NOFRAME option. You can specify the following axis-options:

Table 4.77 PCTLAXIS Axis Options
Option	Description
CGRID=	specifies color for grid lines
GRID	draws grid lines at major percentiles
LABEL='string'	specifies label for percentile axis
LGRID=linetype	specifies line type for grid
WGRID=n	specifies line thickness for grid

PCTLMINOR

requests minor tick marks for the percentile axis when you specify PCTLAXIS. The HMINOR option overrides the PCTLMINOR option.

PCTLSCALE

requests scale labels for the theoretical quantile axis in percentile units, resulting in a nonlinear axis scale. Tick marks are drawn uniformly across the axis based on the quantile scale. In all other respects, the plot remains the same, and you must specify HREF= values in quantile units. For a true nonlinear axis, use the PCTLAXIS option or use the PROBPLOT statement.

RANKADJ=value

specifies the adjustment value added to the ranks in the calculation of theoretical percentiles. By default, RANKADJ= $\text{[math]}$ , as recommended by Blom (1958). Refer to Chambers et al. (1983) for additional information.

ROTATE

switches the horizontal and vertical axes so that the theoretical quantiles are plotted vertically while the data are plotted horizontally. Regardless of whether the plot has been rotated, horizontal axis options (such as HAXIS=) still refer to the horizontal axis, and vertical axis options (such as VAXIS=) still refer to the vertical axis. All other options that depend on axis placement adjust to the rotated axes.

SIGMA=value | EST

specifies the parameter $\text{[math]}$ , where $\text{[math]}$ . Alternatively, you can specify SIGMA=EST to request a maximum likelihood estimate for $\text{[math]}$ . The interpretation and use of the SIGMA= option depend on the distribution option with which it is used, as summarized in Table 4.78. Enclose this option in parentheses after the distribution option.

Table 4.78 Uses of the SIGMA= Option
Distribution Option	Use of the SIGMA= Option
BETA EXPONENTIAL GAMMA WEIBULL	THETA= $\text{[math]}$ and SIGMA= $\text{[math]}$ request a distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ .
LOGNORMAL	SIGMA= $\text{[math]}$ requests $\text{[math]}$ quantile plots with shape parameters $\text{[math]}$ . The SIGMA= option must be specified.
NORMAL	MU= $\text{[math]}$ and SIGMA= $\text{[math]}$ request a distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ . SIGMA=EST requests a line with $\text{[math]}$ equal to the sample standard deviation.
WEIBULL2	SIGMA= $\text{[math]}$ and C= $\text{[math]}$ request a distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ .

SLOPE=value | EST

specifies the slope for a distribution reference line requested with the LOGNORMAL and WEIBULL2 options. Enclose the SLOPE= option in parentheses after the distribution option. When you use the SLOPE= lognormal-option with the LOGNORMAL option, you must also specify a threshold parameter value $\text{[math]}$ with the THETA= lognormal-option to request the line. The SLOPE= lognormal-option is an alternative to the ZETA= lognormal-option for specifying $\text{[math]}$ , because the slope is equal to $\text{[math]}$ .

When you use the SLOPE= Weibull2-option with the WEIBULL2 option, you must also specify a scale parameter value $\text{[math]}$ with the SIGMA= Weibull2-option to request the line. The SLOPE= Weibull2-option is an alternative to the C= Weibull2-option for specifying $\text{[math]}$ , because the slope is equal to $\text{[math]}$ .

For example, the first and second QQPLOT statements produce the same quantile plots and the third and fourth QQPLOT statements produce the same quantile plots:

   proc univariate data=Measures;
      qqplot Width / lognormal(sigma=2 theta=0 zeta=0);
      qqplot Width / lognormal(sigma=2 theta=0 slope=1);
      qqplot Width / weibull2(sigma=2 theta=0 c=.25);
      qqplot Width / weibull2(sigma=2 theta=0 slope=4);

SQUARE

displays the quantile plot in a square frame. By default, the frame is rectangular.

THETA=value | EST

THRESHOLD=value | EST

specifies the lower threshold parameter $\text{[math]}$ for plots requested with the BETA, EXPONENTIAL, GAMMA, LOGNORMAL, WEIBULL, and WEIBULL2 options. Enclose the THETA= option in parentheses after a distribution option. When used with the WEIBULL2 option, the THETA= option specifies the known lower threshold $\text{[math]}$ , for which the default is 0. When used with the other distribution options, the THETA= option specifies $\text{[math]}$ for a distribution reference line; alternatively in this situation, you can specify THETA=EST to request a maximum likelihood estimate for $\text{[math]}$ . To request the line, you must also specify a scale parameter.

WEIBULL(C=value | EST <Weibull-options>)

WEIB(C=value | EST <Weibull-options>)

creates a three-parameter Weibull quantile plot for each value of the required shape parameter $\text{[math]}$ specified by the mandatory C= Weibull-option. To create a plot that is based on a maximum likelihood estimate for $\text{[math]}$ , specify C=EST. To obtain a graphical estimate of $\text{[math]}$ , specify a list of values in the C= Weibull-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ with the THETA= and SIGMA= Weibull-options. Alternatively, you can add a line corresponding to estimated values of $\text{[math]}$ and $\text{[math]}$ with the Weibull-options THETA=EST and SIGMA=EST. Agreement between the reference line and the point pattern indicates that the Weibull distribution with parameters $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ is a good fit. You can specify the SCALE= Weibull-option as an alias for the SIGMA= Weibull-option and the THRESHOLD= Weibull-option as an alias for the THETA= Weibull-option. See Example 4.34.

WEIBULL2<(Weibull2-options)>

W2<(Weibull2-options)>

creates a two-parameter Weibull quantile plot. You should use the WEIBULL2 option when your data have a known lower threshold $\text{[math]}$ , which is 0 by default. To specify the threshold value $\text{[math]}$ , use the THETA= Weibull2-option. By default, THETA=0. An advantage of the two-parameter Weibull plot over the three-parameter Weibull plot is that the parameters $\text{[math]}$ and $\text{[math]}$ can be estimated from the slope and intercept of the point pattern. A disadvantage is that the two-parameter Weibull distribution applies only in situations where the threshold parameter is known. To obtain a graphical estimate of $\text{[math]}$ , specify a list of values for the THETA= Weibull2-option and select the value that most nearly linearizes the point pattern. To assess the point pattern, add a diagonal distribution reference line corresponding to $\text{[math]}$ and $\text{[math]}$ with the SIGMA= and C= Weibull2-options. Alternatively, you can add a distribution reference line corresponding to estimated values of $\text{[math]}$ and $\text{[math]}$ with the Weibull2-options SIGMA=EST and C=EST. Agreement between the reference line and the point pattern indicates that the Weibull distribution with parameters $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ is a good fit. You can specify the SCALE= Weibull2-option as an alias for the SIGMA= Weibull2-option and the SHAPE= Weibull2-option as an alias for the C= Weibull2-option. See Example 4.34.

WGRID=n

specifies the line thickness for the grid when producing traditional graphics. The option does not apply to ODS Graphics output.

ZETA=value | EST

specifies a value for the scale parameter $\text{[math]}$ for the lognormal quantile plots requested with the LOGNORMAL option. Enclose the ZETA= lognormal-option in parentheses after the LOGNORMAL option. To request a distribution reference line with intercept $\text{[math]}$ and slope $\text{[math]}$ , specify the THETA= $\text{[math]}$ and ZETA= $\text{[math]}$ .

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