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Various Illustrations of Splines in PROC TRANSREG

/****************************************************************/
/*          S A S   S A M P L E   L I B R A R Y                 */
/*                                                              */
/*    NAME: TREGSPL                                             */
/*   TITLE: Various Illustrations of Splines in PROC TRANSREG   */
/* PRODUCT: STAT                                                */
/*  SYSTEM: ALL                                                 */
/*    KEYS: regression analysis, transformations                */
/*   PROCS: TRANSREG                                            */
/*    DATA:                                                     */
/*                                                              */
/* SUPPORT: saswfk                UPDATE: July 25, 2010         */
/*     REF: PROC TRANSREG, DETAILS, SPLINES                     */
/*    MISC:                                                     */
/****************************************************************/

title 'An Illustration of Splines and Knots';

* Create in y a discontinuous function of x.;

data a;
   x = -0.000001;
   do i = 0 to 199;
      if mod(i, 50) = 0 then do;
         c = ((x / 2) - 5)**2;
         if i = 150 then c = c + 5;
         y = c;
      end;
      x = x + 0.1;
      y = y - sin(x - c);
      output;
   end;
run;

ods graphics on;

title2 'A Linear Regression Fit';
proc transreg data=a plots=scatter rsquare;
   model identity(y) = identity(x);
run;

title2 'A Quadratic Polynomial Fit';

proc transreg data=A;
   model identity(y)=spline(x / degree=2);
run;

title2 'A Cubic Spline Fit with Knots at X=5, 10, 15';

proc transreg data=a;
   model identity(y) = spline(x / knots=5 10 15);
run;

data b;             /* A is the data set used by transreg */
   set a(keep=x y);
   x1=x;                       /* x                       */
   x2=x**2;                    /* x squared               */
   x3=x**3;                    /* x cubed                 */
   x4=(x> 5)*((x-5)**3);       /* change in x**3 after  5 */
   x5=(x>10)*((x-10)**3);      /* change in x**3 after 10 */
   x6=(x>15)*((x-15)**3);      /* change in x**3 after 15 */
run;

proc reg;
   model y=x1-x6;
run; quit;

title3 'First - Third Derivatives Discontinuous at X=5, 10, 15';

proc transreg data=a;
   model identity(y) = spline(x / knots=5 5 5 10 10 10 15 15 15);
run;

data b;
   set a(keep=x y);
   x1=x;                        /* x                       */
   x2=x**2;                     /* x squared               */
   x3=x**3;                     /* x cubed                 */
   x4=(x>5)   * (x- 5);         /* change in x    after  5 */
   x5=(x>10)  * (x-10);         /* change in x    after 10 */
   x6=(x>15)  * (x-15);         /* change in x    after 15 */
   x7=(x>5)   * ((x-5)**2);     /* change in x**2 after  5 */
   x8=(x>10)  * ((x-10)**2);    /* change in x**2 after 10 */
   x9=(x>15)  * ((x-15)**2);    /* change in x**2 after 15 */
   x10=(x>5)  * ((x-5)**3);     /* change in x**3 after  5 */
   x11=(x>10) * ((x-10)**3);    /* change in x**3 after 10 */
   x12=(x>15) * ((x-15)**3);    /* change in x**3 after 15 */
run;

proc reg;
   model y=x1-x12;
run; quit;

title3 'Discontinuous Function and Derivatives';

proc transreg data=a;
   model identity(y) = spline(x / knots=5 5 5 5 10 10 10 10
                                        15 15 15 15);
run;

title3 'Four Knots';

proc transreg data=a;
   model identity(y) = spline(x / nknots=4);
run;

title3 'Nine Knots';

proc transreg data=a;
   model identity(y) = spline(x / nknots=9);
run;

title 'An Illustration of Splines and Knots';
title2 'Scoring Spline Variables';

data x;
   do i = 1 to 5000;
      w = normal(7);
      x = normal(7);
      z = normal(7);
      y = w * w + log(5 + x) + sin(z) + normal(7);
      output;
   end;
run;

data z;
   do i = 1 to 5000;
      w = normal(1);
      x = normal(1);
      z = normal(1);
      y = w * w + log(5 + x) + sin(z) + normal(1);
      output;
   end;
run;

proc transreg data=x solve details ss2;
   ods output splinecoef=c;
   model identity(y) = spline(w x z / knots=-1.5 to 1.5 by 0.5
                                      exknots=-5 5);
   output out=d;
run;

proc transreg data=z design;
   model bspl(w x z / knots=-1.5 to 1.5 by 0.5 exknots=-5 5);
   output out=b;
run;

proc score data=b score=c out=o1(rename=(spline=bw w=nw));
   var w:;
run;

proc score data=b score=c out=o2(rename=(spline=bx x=nx));
   var x:;
run;

proc score data=b score=c out=o3(rename=(spline=bz z=nz));
   var z:;
run;

data all;
   merge d(keep=w x z tw tx tz) o1(keep=nw bw)
         o2(keep=nx bx) o3(keep=nz bz);
run;

proc template;
   define statgraph twobytwo;
      begingraph;
         layout lattice / rows=2 columns=2;
            layout overlay;
               seriesplot y=tw x=w  / connectorder=xaxis;
               seriesplot y=bw x=nw / connectorder=xaxis;
            endlayout;
            layout overlay;
               seriesplot y=tx x=x  / connectorder=xaxis;
               seriesplot y=bx x=nx / connectorder=xaxis;
            endlayout;
            layout overlay;
               seriesplot y=tz x=z  / connectorder=xaxis;
               seriesplot y=bz x=nz / connectorder=xaxis;
            endlayout;
         endlayout;
      endgraph;
   end;
run;

proc sgrender data=all template=twobytwo;
run;

data x;
   input x @@;
   datalines;
1 2 3 4 5 6 7 8 9 10
;

ods output details=d;
proc transreg details design;
   model bspline(x / nkn=3);
   output out=y;
run;

%let k = 0;
data d;
   set d;
   length d $ 20;
   retain d ' ';
   if description ne ' ' then d = description;
   if d = 'Degree' then call symput('d', compress(formattedvalue));
   if d = 'Number of Knots'
      then call symput('k', compress(formattedvalue));
   if index(d, 'Knots') and not index(d, 'Number');
   keep d numericvalue;
run;

%let nkn = %eval(&d * 2 + &k); /* total number of knots   */
%let nb  = %eval(&d + 1 + &k); /* number of cols in basis */

proc transpose data=d out=k(drop=_name_) prefix=Knot;
run;

proc print; format k: 20.16;
run;

data b(keep=x:);
   if _n_ = 1 then set k; /* read knots from transreg */
   array k[&nkn] knot1-knot&nkn;        /* knots      */
   array b[&nb] x_0 - x_%eval(&nb - 1); /* basis      */
   array w[%eval(2 * &d)];              /* work       */
   set x;
   do i = 1 to &nb; b[i] = 0; end;

   * find the index of first knot greater than current data value;
   do ki = 1 to &nkn while(k[ki] le x); end;
   kki = ki - &d - 1;

   * make the basis;
   b[1 + kki] = 1;
   do j = 1 to &d;
      w[&d + j] = k[ki + j - 1] - x;
      w[j] = x - k[ki - j];
      s = 0;
      do i = 1 to j;
         t = w[&d + i] + w[j + 1 - i];
         if t ne 0.0 then t = b[i + kki] / t;
         b[i + kki] = s + w[&d + i] * t;
         s = w[j + 1 - i] * t;
      end;
      b[j + 1 + kki] = s;
   end;
run;

proc compare data=y(keep=x:) compare=b
   criterion=1e-12 note nosummary;
   title3 "should be no differences";
run;

title 'Linear and Nonlinear Regression Functions';

* Generate an Artificial Nonlinear Scatter Plot;
data a;
   do i = 1 to 500;
      x = i / 2.5;
      y = -((x/50)-1.5)**2 + sin(x/8) + sqrt(x)/5 + 2*log(x) + cos(x);
      x = x / 21;
      if y > 2 then output;
   end;
run;

ods select fitplot(persist);

title2 'Linear Regression';

proc transreg data=a;
   model identity(y)=identity(x);
run;

title2 'A Monotone Regression Function';

proc transreg data=a;
   model identity(y)=mspline(x / nknots=9);
run;

title2 'A Nonlinear Regression Function';

proc transreg data=a;
   model identity(y)=spline(x / nknots=9);
run;

title2 'A Nonlinear Regression Function, 100 Knots';

proc transreg data=a;
   model identity(y)=spline(x / nknots=100);
run;

ods select all;

title 'Separate Curves, Separate Intercepts';

data a;
   do x = -2 to 3 by 0.025;
      g = 1;
      y = 8*(x*x + 2*cos(x*6)) + 15*normal(7654321);
      output;
      g = 2;
      y = 4*(-x*x + 4*sin(x*4)) - 40 + 15*normal(7654321);
      output;
   end;
run;

ods select fitplot(persist);

title 'Parallel Lines, Separate Intercepts';

proc transreg data=a solve;
   model identity(y)=class(g) identity(x);
run;

title 'Parallel Monotone Curves, Separate Intercepts';

proc transreg data=a;
   model identity(y)=class(g) mspline(x / knots=-1.5 to 2.5 by 0.5);
run;

title 'Parallel Curves, Separate Intercepts';

proc transreg data=a solve;
   model identity(y)=class(g) spline(x / knots=-1.5 to 2.5 by 0.5);
run;

title 'Separate Slopes, Same Intercept';

proc transreg data=a;
   model identity(y)=class(g / zero=none) * identity(x);
run;

title 'Separate Monotone Curves, Same Intercept';

proc transreg data=a;
   model identity(y) = class(g / zero=none) *
                       mspline(x / knots=-1.5 to 2.5 by 0.5);
run;

title 'Separate Curves, Same Intercept';

proc transreg data=a solve;
   model identity(y) = class(g / zero=none) *
                       spline(x / knots=-1.5 to 2.5 by 0.5);
run;

title 'Separate Slopes, Separate Intercepts';

proc transreg data=a;
   model identity(y) = class(g / zero=none) | identity(x);
run;

title 'Separate Monotone Curves, Separate Intercepts';

proc transreg data=a;
   model identity(y) = class(g / zero=none) |
                       mspline(x / knots=-1.5 to 2.5 by 0.5);
run;

title 'Separate Curves, Separate Intercepts';

proc transreg data=a solve;
   model identity(y) = class(g / zero=none) |
                       spline(x / knots=-1.5 to 2.5 by 0.5);
run;
ods select all;