Expertise:
Advanced
When do I need monotone curves?
We often know that the process behind the raw data is always increasing
or always decreasing. For example, children and adolescents do not actually
lose height, although actual height measurements may fail to increase due to
measurement error. Time itself always increases, and any meaningful transformation
of time should also increase. Sometimes we need to transform data in a
way that ensures that we can reverse the transformation to recover data values
in the original scale. This implies that the transformation should be strictly
monotone.
Monotonicity is a global smoothing principle in the sense that eliminating
negative first derivatives can stabilize curves, and especially at the upper and
lower boundaries of the independent variable.
Why are the usual smoothing curves hard to keep monotone?
Linear combinations are by their nature unruly. Even if the basis functions
themselves are well behaved, it is hard to prevent certain combinations of coefficients
from making the functional data object that they define nonmonotone,
nonpositive or in some way inadmissible.
We transform the problem. Instead of using a functional data object to
directly fit the data, we use it to define a function that will fit the data and
have the desired properties, in this case monotonicity.
Suppose W(t) is a conventional functional data object, and that
is unconstrained in any way except for W(t_{0}) = 0
where t_{0} is the lower boundary over
which we are smoothing. Then exp[W(t)] is certainly a positive
function. And the indefinite integral of a positive function is always
increasing. Consequently, we define a monotone smoothing function m(t)
to be of the form
m(t) = β_{0} + β_{1}
∫^{t}_{t0} exp[W(u)] du
In this equation, β_{0} and β_{1} are constants
that may or may not need to be estimated from the data. β_{0} is the
value of m(t_{0}) and β_{1} is the slope of
m(t) at t_{0}. In addition to these two constants, we need
to estimate the coefficients that
define the linear combination of basis functions that defines W(t). As
a rule, a Bspline basis will be used, and the constraint W(t_{0}) = 0
can easily be achieved by fixing the first coefficient to be zero.
The estimation necessarily involves numerical optimization of a fitting criterion,
and will therefore involve considerably more computation than the usual
data smoothing process.
