An interpolating spline is not uniquely determined by the set of Y values. To achieve a unique interpolant, , you must specify two constraints on the endpoints of the interval . You can achieve uniqueness by specifying one of the following conditions:

The following statements give three examples of computing an interpolating spline for data. Note that the first and last Y values are the same, so you can ask for a periodic spline.

proc iml;
data = { 0 5, 1 3, 2 5, 3 4, 4 6, 5 7, 6 6, 7 5 };

/* Compute three spline interpolants of the data */
/* (1) a cubic spline with type=zero (the default) */
call spline(fitted,data);

/* (2) A periodic spline */
call spline(periodicFitted,data) type='periodic';

/* (3) A spline with specific slopes at endpoints */
call spline(slopeFitted,data) slope={45 30};

/* write data */
create SplineData from data[colname={"x" "y"}];
append from data;
close SplineData;

/* write fitted interpolants */
fit = fitted || periodicFitted[,2] || slopeFitted[,2];
varNames = {"t" "Interpolant" "Periodic" "EndSlopes"};
create SplineFit from fit[colname=varNames];
append from fit;
close SplineFit;
quit;

/* merge data and plot */
data Spline;
merge SplineData SplineFit;
run;

proc sgplot data=Spline;
title "Spline Interpolation";
scatter x=x y=y;
series x=t y=Interpolant;
series x=t y=Periodic / lineattrs=(color="dark red");
series x=t y=EndSlopes / lineattrs=(color="dark green");
run;

As shown in Figure 23.326, the interpolants pass through each point of the data. They differ from each other by the derivatives at the boundary points, and . The generic interpolant has second derivatives that vanish at the boundary points. The periodic curve has a derivative at that matches the derivative at . The third curve has derivatives that match the given slopes at the boundary points.

Figure 23.326: Three Spline Interpolants with Different Boundary Conditions

You can also use a spline to smooth data. In general, a smoothing spline does not pass through any data pair exactly. A very small value of the smooth smoothing parameter approximates an interpolating polynomial for data in which each unique X value is assigned the mean of the Y values that correspond to that X value. As the smooth parameter gets very large, the spline approximates a linear regression.

The following statements compute two smoothing splines for the same data as in the previous example. The spline coefficients are passed to the SPLINEV function, which evaluates the smoothing spline at the original X values. The smoothing spline does not pass through the original Y values. Also, the smoothing parameter for the periodic spline is smaller, so the periodic spline has more “wiggles” than the corresponding spline with the larger smoothing parameter.

proc iml;
data = { 0 5, 1 3, 2 5, 3 4, 4 6, 5 7, 6 6, 7 5 };

/* Compute spline smoothers of the data. */
call splinec(fitted,coeff,endSlopes,data) smooth=1;

/* Evaluate the smoother at the original X positions */
smoothFit = splinev(coeff, data[,1]);

/* Compute periodic spline smoother of the data. */
call splinec(periodicFitted,coeff,endSlopesP,data)
             smooth=0.1 type="periodic";

/* Evaluate the smoother at the original X positions */
smoothPeriodicFit = splinev(coeff, data[,1]);

/* Compare the two fits */
print smoothFit smoothPeriodicFit;

You can write the smoothers to a SAS data set and merge them with the data, as shown in the previous example. Figure 23.328 shows the resulting graph.

Figure 23.328: Graph of Two Spline Smoothers

A parametric spline can be used to interpolate or smooth data for which there does not seem to be a functional relationship between the X and Y variables. A partition is formed as explained in the documentation for the type parameter. Splines are then used to fit the X and Y values independently.

The following program fits a parametric curve to data that are shaped like an “S.” The variable fitted is returned as a numParam  matrix that contains the ordered pairs that correspond to the parametric spline. These ordered pairs correspond to numParam evenly spaced points in the domain of the parameter .

The purpose of the SPLINEV function is to evaluate (score) an interpolating or smoothing spline at an arbitrary set of points. The following program shows how to construct the parameters that correspond to the original data by using the formula specified in the documentation for the type argument. These parameters are used to construct the evenly spaced parameters that correspond to the data in the fitted matrix.

proc iml;
data = {3 7, 2 7, 1 6, 1 5, 2 4, 3 3, 3 2, 2 1, 1 1};

/* Compute parametric spline interpolant */
numParam = 40;
call splinec(fitted,coeff,endSlopes,data)
             nout=numParam type={"zero" "parametric"};

/* write data */
create SplineData from data[colname={"x" "y"}];
append from data;
close SplineData;

/* write parametric spline values */
create SplineFit from fitted[colname={"xt" "yt"}];
append from fitted;
close SplineFit;


/* Manually reproduce/verify the "fitted" values */
/* (1) Form the parameters mapped onto the data */
t = j(nrow(data),1,0); /* first parameter is zero */
do i = 2 to nrow(t);
   t[i] = t[i-1] + sqrt( (data[i,1]-data[i-1,1])##2 +
                         (data[i,2]-data[i-1,2])##2 );
end;

/* (2) Construct numParam evenly spaced parameters
   between 0 and t[nrow(t)] */
params = do(0, t[nrow(t)], t[nrow(t)]/(numParam-1))`;

/* (3) Evaluate the parametric spline at these points */
fit = splinev(coeff, params);
maxDiff = max(abs(fitted-fit));
print maxDiff; /* should be very small or zero */
quit;

/* merge data and plot */
data Spline;
merge SplineData SplineFit;
run;

proc sgplot data=Spline;
title "Parametric Spline Smoother";
scatter x=x y=y;
series x=xt y=yt / legendlabel="Parametric Spline";
run;

Figure 23.330: Parametric Smoothing Spline

Attempting to evaluate a spline outside its domain of definition results in a missing value. For example, the following statements define a spline on the interval . Attempting to evaluate the spline at points outside this interval ( or ) results in missing values.

proc iml;
data = { 0 5, 1 3, 2 5, 3 4, 4 6, 5 7, 6 6, 7 5 };
call splinec(fitted,coeff,endSlopes,data) slope={45 45};
v =  splinev(coeff,{-1 1 2 3 3.5 4 20});
print v;

One use of splines is to estimate the integral of a function that is known only by its value at a discrete set of points. Many people are familiar with elementary methods of numerical integration such as the left-hand rule, the trapezoid rule, and Simpson’s rule. In the left-hand rule, the integral of discrete data is estimated by the exact integral of a piecewise constant function between the data points. In the trapezoid rule, the integral is estimated by the exact integral of a piecewise linear function that connects the data points. In Simpson’s rule, the integral is estimated as the exact integral of a piecewise quadratic function between the data points.

Because a cubic spline is a sequence of cubic polynomials, it is possible to compute the exact integral of the cubic spline and use this as an estimate for the integral of the discrete data. The next example takes a function defined by discrete data and finds the integral and the first moment of the function.

The implementation of the integrand function (SpleinEval) uses a helpful trick to evaluate a spline at a single point. If you pass in a scalar argument to the SPLINEV function, you get back a vector that represents the evaluation of the spline along evenly spaced points, rather than the spline evaluated at the argument. To avoid this, the following statements evaluate the spline at the vector x // x and then extract the entry in the first row, second column. This number is the value of the spline evaluated at .

proc iml;
x = { 0, 2, 5, 7, 8, 10 };
y = x + 0.1*sin(x);
a = x || y;
call splinec(fit,coeff,endSlopes,a);

start SplineEval(x) global(coeff,power);
   /* The first column of v contains the points of evaluation;
      the second column contains the evaluation. */
   v = x##power # splinev(coeff, x//x);
   return(v[1,2]); /* return spline(x) */
finish;

/* Evaluate the "moment" of a function.
   moment(0) = integral of f(x) dx
   moment(1) = integral of x*f(x) dx 
   moment(2) = integral of x##2 *f(x) dx, etc
   Use QUAD to integrate */
start moment(pow) global(coeff,power);
   power = pow;
   intervals = coeff[,1]; /* left endpts of x intervals */
   call quad(z,"SplineEval", intervals) eps = 1.e-10;
   return( sum(z) );
finish;

mass  = moment(0);   /* to compute the mass */
m     = mass //
       (moment(1)/mass) // /* to compute the mean */
       (moment(2)/mass) ;  /* to compute the meansquare */
print m;


/* Check the previous computation by using Gauss-Legendre
   integration, which is valid for moments up to maxng. */
gauss = {
    -9.3246951420315205e-01      -6.6120938646626448e-01
    -2.3861918608319743e-01       2.3861918608319713e-01
     6.6120938646626459e-01       9.3246951420315183e-01,
     1.713244923791701e-01        3.607615730481388e-01
     4.679139345726905e-01        4.679139345726904e-01
     3.607615730481389e-01        1.713244923791707e-01  };
ngauss = ncol(gauss); /* = 6 */
maxng  =  2*ngauss-4;

start momentGL(pow) global(coeff,gauss,ngauss,maxng);
   if pow >= maxng  then return(.);
   nrow    = nrow(coeff);
   left    = coeff[1:nrow-1,1]; /* left endpt of interval */
   right   = coeff[2:nrow,1];   /* right endpt */
   mid     = 0.5*(left+right);
   interv  = 0.5*(right - left);
   /* scale the weights on each interval */
   wgts    = rowvec( interv*gauss[2,] );
   /* scale the points on each interval */
   pts     = rowvec( interv*gauss[1,] + mid );
   /* evaluate the function */
   eval    = splinev(coeff,pts)[,2]`;
   return( sum( wgts#pts##pow # eval) );
finish;

mass  = momentGL(0);       /* to compute the mass */
m2    = mass // (momentGL(1)/mass) //  (momentGL(2)/mass) ;
print m2; /* should agree with earlier result */