  A liquid scorecard has liquid characteristics, for which the characteristic
score is a smooth function of the characteristic over a liquid range. The
smooth function is based on B-splines, typically cubic. In contrast, the
characteristic scores for traditional scorecards are step functions of the
characteristics. Previously, there were two ways to control the smoothness of
the liquid characteristic score: (1) coarse classing where the fewer the number
of classes, the smoother the curve; (2) the penalty parameter, which penalizes
the norm of the score coefficient vector. However, in classical cubic spline
fitting theory, a direct measure of curve roughness is used as a penalty term
in the fitting objective function. In this paper, I work out the details of
this concept for our characteristic scores, which are linear functions of a
cubic B-spline basis. The roughness penalty is the integral of the second
derivative squared. As you vary the characteristic smoothness parameter from
zero to infinity, the characteristic score goes from being rough to being very
smooth. As one moves from rough to smooth, the palatable characteristic score
jumps off the page. This is illustrated by a case study. This case study also
shows that smoothness parameters, which maximize validation divergence, do not
always yield the most palatable model.
