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The TRANSREG Procedure |

The missing portions of variables subjected to SPLINE or MSPLINE transformations are handled the same way as for OPSCORE, MONOTONE, UNTIE, and LINEAR transformations (see the previous section). The nonmissing partition is handled by first creating a B-spline basis of the specified degree with the specified knots for the nonmissing partition of the initial scaling vector and then regressing the target onto the basis. The optimally scaled vector is a linear combination of the B-spline basis vectors using least-squares regression coefficients. An algorithm for generating the B-spline basis is given in de Boor (1978, pp. 134 -135). B-splines are both a computationally accurate and efficient way of constructing a basis for piecewise polynomials; however, they are not the most natural method of describing splines.

Consider an initial scaling vector
*x* = (1 2 3 4 5 6 7 8 9)' and a degree
three spline with interior knots at 3.5 and 6.5.
The B-spline basis for the transformation is the
left matrix in Table 65.5, and the natural piecewise
polynomial spline basis is the right matrix.
The two matrices span the same column space.
The natural basis has an intercept, a linear
term, a quadratic term, a cubic term, and two
more terms since there are two interior knots.
These terms are generated (for knot *k* and **x**
element *x*) by the formula (*x* - *k*)^{3} ×*I*_{(x>k)}.
The indicator variable *I*_{(x>k)} evaluates to 1.0
if *x* is greater than *k* and to 0.0 otherwise.
If knot *k* had been repeated, there would
be a (*x* - *k*)^{2} ×*I*_{(x>k)} term also.
Notice that the fifth column makes no contribution to the curve
before 3.5, makes zero contribution at 3.5 (the transformation
is continuous), and makes an increasing contribution beyond 3.5.
The same pattern of results holds for the last term with knot 6.5.
The coefficient of the fifth column represents the
change in the cubic portion of the curve after 3.5.
The coefficient of the sixth column represents the
change in the cubic portion of the curve after 6.5.

MSPLINE transformations are handled like SPLINE transformations except that constraints are placed on the coefficients to ensure monotonicity. When the coefficients of the B-spline basis are monotonically increasing, the transformation is monotonically increasing. When the polynomial degree is two or less, monotone coefficient splines, integrated splines (Winsberg and Ramsay 1980), and the general class of all monotone splines are equivalent.

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