The UNIVARIATE Procedure

Example 4.25 Annotating a Folded Normal Curve

This example shows how to display a fitted curve that is not supported by the HISTOGRAM statement. The offset of an attachment point is measured (in mm) for a number of manufactured assemblies, and the measurements (Offset) are saved in a data set named Assembly. The following statements create the data set Assembly: 

data Assembly;
   label Offset = 'Offset (in mm)';
   input Offset @@;
   datalines;
11.11 13.07 11.42  3.92 11.08  5.40 11.22 14.69  6.27  9.76
 9.18  5.07  3.51 16.65 14.10  9.69 16.61  5.67  2.89  8.13
 9.97  3.28 13.03 13.78  3.13  9.53  4.58  7.94 13.51 11.43
11.98  3.90  7.67  4.32 12.69  6.17 11.48  2.82 20.42  1.01
 3.18  6.02  6.63  1.72  2.42 11.32 16.49  1.22  9.13  3.34
 1.29  1.70  0.65  2.62  2.04 11.08 18.85 11.94  8.34  2.07
 0.31  8.91 13.62 14.94  4.83 16.84  7.09  3.37  0.49 15.19
 5.16  4.14  1.92 12.70  1.97  2.10  9.38  3.18  4.18  7.22
15.84 10.85  2.35  1.93  9.19  1.39 11.40 12.20 16.07  9.23
 0.05  2.15  1.95  4.39  0.48 10.16  4.81  8.28  5.68 22.81
 0.23  0.38 12.71  0.06 10.11 18.38  5.53  9.36  9.32  3.63
12.93 10.39  2.05 15.49  8.12  9.52  7.77 10.70  6.37  1.91
 8.60 22.22  1.74  5.84 12.90 13.06  5.08  2.09  6.41  1.40
15.60  2.36  3.97  6.17  0.62  8.56  9.36 10.19  7.16  2.37
12.91  0.95  0.89  3.82  7.86  5.33 12.92  2.64  7.92 14.06
;

It is decided to fit a folded normal distribution to the offset measurements. A variable X has a folded normal distribution if $X=|Y|$, where Y is distributed as $N(\mu ,\sigma )$. The fitted density is

\[ h(x) = \frac{1}{\sqrt {2\pi } \sigma } \left[ \exp \left( -\frac{(x-\mu )^2}{2\sigma ^2} \right) + \exp \left( -\frac{(x+\mu )^2}{2\sigma ^2} \right) \right] \]

where $x \geq 0$.

You can use SAS/IML to compute preliminary estimates of $\mu $ and $\sigma $ based on a method of moments given by Elandt (1961). These estimates are computed by solving equation (19) Elandt (1961), which is given by

\[ f(\theta ) = \frac{\left( \frac{2}{\sqrt {2\pi }} e^{-\theta ^2/2 } - \theta \left[ 1-2\Phi (\theta ) \right] \right)^2}{1+\theta ^2} = A \]

where $\Phi (\cdot )$ is the standard normal distribution function and

\[ A = \frac{ \bar{x}^2 }{ \frac{1}{n} \sum _{i=1}^ n x_ i^2 } \]

Then the estimates of $\sigma $ and $\mu $ are given by

\[ \begin{array}{lll} \hat{\sigma }_0 & = \sqrt { \frac{ \frac{1}{n} \sum _{i=1}^ n x_ i^2}{1 + \hat{\theta }^2 } } \\ \hat{mu}_0 & = \hat{\theta } \cdot \hat{\sigma }_0 \end{array} \]

Begin by using PROC MEANS to compute the first and second moments and by using the following DATA step to compute the constant A: 

proc means data = Assembly noprint;
   var Offset;
   output out=stat mean=m1 var=var n=n min = min;
run;

* Compute constant A from equation (19) of Elandt (1961);
data stat;
   keep m2 a min;
   set stat;
   a  = (m1*m1);
   m2 = ((n-1)/n)*var + a;
   a  = a/m2;
run;

Next, use the SAS/IML subroutine NLPDD to solve equation (19) by minimizing $(f(\theta )-A)^2$, and compute $\hat{mu}_0$ and $\hat{\sigma }_0$: 

proc iml;
   use stat;
   read all var {m2}  into m2;
   read all var {a}   into a;
   read all var {min} into min;

   * f(t) is the function in equation (19) of Elandt (1961);
   start f(t) global(a);
     y = .39894*exp(-0.5*t*t);
     y = (2*y-(t*(1-2*probnorm(t))))**2/(1+t*t);
     y = (y-a)**2;
     return(y);
   finish;

   * Minimize (f(t)-A)**2 and estimate mu and sigma;
   if ( min < 0 ) then do;
      print "Warning: Observations are not all nonnegative.";
      print "     The folded normal is inappropriate.";
      stop;
      end;
   if ( a < 0.637 ) then do;
      print "Warning: the folded normal may be inappropriate";
      end;
   opt = { 0 0 };
   con = { 1e-6 };
   x0  = { 2.0 };
   tc  = { . . . . . 1e-8 . . . . . . .};
   call nlpdd(rc,etheta0,"f",x0,opt,con,tc);
   esig0 = sqrt(m2/(1+etheta0*etheta0));
   emu0  = etheta0*esig0;

   create prelim var {emu0 esig0 etheta0};
   append;
   close prelim;

   * Define the log likelihood of the folded normal;
   start g(p) global(x);
      y = 0.0;
      do i = 1 to nrow(x);
         z = exp( (-0.5/p[2])*(x[i]-p[1])*(x[i]-p[1]) );
         z = z + exp( (-0.5/p[2])*(x[i]+p[1])*(x[i]+p[1]) );
         y = y + log(z);
      end;
      y = y - nrow(x)*log( sqrt( p[2] ) );
      return(y);
   finish;
   * Maximize the log likelihood with subroutine NLPDD;
   use assembly;
   read all var {offset} into x;
   esig0sq = esig0*esig0;
   x0      = emu0||esig0sq;
   opt     = { 1 0 };
   con     = { . 0.0, .  .  };
   call nlpdd(rc,xr,"g",x0,opt,con);
   emu     = xr[1];
   esig    = sqrt(xr[2]);
   etheta  = emu/esig;
   create parmest var{emu esig etheta};
   append;
   close parmest;
quit;

The preliminary estimates are saved in the data set Prelim, as shown in Output 4.25.1.

Output 4.25.1: Preliminary Estimates of $\mu $, $\sigma $, and $\theta $

The Data Set Prelim

Obs EMU0 ESIG0 ETHETA0
1 6.51735 6.54953 0.99509 




            

            


            
Now, using $\hat{mu}_0$ and $\hat{\sigma }_0$ as initial estimates, call the NLPDD subroutine to maximize the log likelihood, $l(\mu ,\sigma )$, of the folded normal distribution, where, up to a constant, 
            

            

               
\[  l(\mu ,\sigma ) = -n \log \sigma + \sum _{i=1}^ n \log \left[ \exp \left( -\frac{(x_ i-\mu )^2}{2\sigma ^2} \right) + \exp \left( -\frac{(x_ i+\mu )^2}{2\sigma ^2} \right) \right]  \]

            


* Define the log likelihood of the folded normal;
start g(p) global(x);
   y = 0.0;
   do i = 1 to nrow(x);
      z = exp( (-0.5/p[2])*(x[i]-p[1])*(x[i]-p[1]) );
      z = z + exp( (-0.5/p[2])*(x[i]+p[1])*(x[i]+p[1]) );
      y = y + log(z);
   end;
   y = y - nrow(x)*log( sqrt( p[2] ) );
   return(y);
finish;
* Maximize the log likelihood with subroutine NLPDD;
use assembly;
read all var {offset} into x;
esig0sq = esig0*esig0;
x0      = emu0||esig0sq;
opt     = { 1 0 };
con     = { . 0.0, .  .  };
call nlpdd(rc,xr,"g",x0,opt,con);
emu     = xr[1];
esig    = sqrt(xr[2]);
etheta  = emu/esig;
create parmest var{emu esig etheta};
append;
close parmest;
quit;



The data set ParmEst contains the maximum likelihood estimates $\hat{mu}$ and $\hat{\sigma }$ (as well as $\hat{mu}/\hat{\sigma }$), as shown in Output 4.25.2. 
            

            
  

            


Output 4.25.2: Final Estimates of $\mu $, $\sigma $, and $\theta $

               

                  

                     
                     

                        
                        
                        

The Data Set ParmEst
                              
                           



                           
                           

                              
                              









Obs
                                       
                                       EMU
                                       
                                       ESIG
                                       
                                       ETHETA
                                       
                                    



1
                                       
                                       6.66761
                                       
                                       6.39650
                                       
                                       1.04239
                                       
                                    





                           
                        

                        



                     
                     
                  





            

            


            
To annotate the curve on a histogram, begin by computing the width and endpoints of the histogram intervals. The following
               statements save these values in a data set called OutCalc. Note that a plot is not produced at this point. 
            


proc univariate data = Assembly noprint;
   histogram Offset / outhistogram = out normal(noprint) noplot;
run;

data OutCalc (drop = _MIDPT_);
   set out (keep = _MIDPT_) end = eof;
   retain _MIDPT1_ _WIDTH_;
   if _N_ = 1 then _MIDPT1_ = _MIDPT_;
   if eof then do;
      _MIDPTN_ = _MIDPT_;
      _WIDTH_ = (_MIDPTN_ - _MIDPT1_) / (_N_ - 1);
      output;
   end;
run;



Output 4.25.3 provides a listing of the data set OutCalc. The width of the histogram bars is saved as the value of the variable _WIDTH_; the midpoints of the first and last histogram bars are saved as the values of the variables _MIDPT1_ and _MIDPTN_. 
            

            
  

            


Output 4.25.3: The Data Set OutCalc

               

                  

                     
                     

                        
                        
                        

The Data Set OutCalc
                              
                           



                           
                           

                              
                              









Obs
                                       
                                       _MIDPT1_
                                       
                                       _WIDTH_
                                       
                                       _MIDPTN_
                                       
                                    



1
                                       
                                       1.5
                                       
                                       3
                                       
                                       22.5
                                       
                                    





                           
                        

                        



                     
                     
                  





            

            


            
The following statements create an annotate data set named Anno, which contains the coordinates of the fitted curve: 
            


data Anno;
   merge ParmEst OutCalc;
   length function color $ 8;
   function = 'point';
   color    = 'black';
   size     =  2;
   xsys     = '2';
   ysys     = '2';
   when     = 'a';
   constant = 39.894*_width_;;
   left     =  _midpt1_ - .5*_width_;
   right    =  _midptn_ + .5*_width_;
   inc      = (right-left)/100;
   do x = left to right by inc;
      z1 = (x-emu)/esig;
      z2 = (x+emu)/esig;
      y  = (constant/esig)*(exp(-0.5*z1*z1)+exp(-0.5*z2*z2));
      output;
      function = 'draw';
   end;
run;



The following statements read the ANNOTATE= data set and display the histogram and fitted curve: 


title 'Folded Normal Distribution';
ods graphics off;
proc univariate data=assembly noprint;
   histogram Offset / annotate = anno;
run;



Output 4.25.4 displays the histogram and fitted curve. 
            

            
  

            


Output 4.25.4: Histogram with Annotated Folded Normal Curve

               

                  
Histogram with Annotated Folded Normal Curve

               

            

            


            
A sample program for this example, uniex15.sas, is available in the SAS Sample Library for Base SAS software.