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Landmark registration using function

`landmarkreg`

can no longer be done by using function`smooth.basis`

instead of function`smooth.morph`

. The warping function must be strictly monotonic, and we have found that using`smooth.basis`

too often violates this monotonicity constraint. Function`smooth.morph`

ensures monotonicity and in most in most applications takes negligible computer time to do so.Function

`pcaPACE`

carries out a functional PCA with regularization from the estimate of the covariance surface. Function`scoresPACE`

estimates functional Principal Component scores through Conditional Expectation (PACE).`Smooth.morph`

estimates a warping function when the target of the fit by registration is a functional data object. This function has been extended to work when the target for the fit and the fitted functions have different ranges or domains. The warping also maps each boundary into its target boundary. Similarly`landmarkreg`

uses a small number of discrete values to define the warping, and how has an extra argument,`x0lim`

, that defines the range of the target domain. Since it defaults to the range of the warped domain, it continues to work if not used and the domains have the same range.This function works with multinomial data that evolve over a continuum, such as the value of a latent variable in psychometrics. A multinomial observation consists of a set of probabilities that are in the open interval (0,1) and sum to one. The surprisal value S(P_m) corresponding to a probabity P_m is -log_M(P_m), where M is the number of probabities and is the base of the logarithm. The inverse function is P(S_m) = M^(-S_m).

Surprisal is also known as "self-information" in the field of information theory. It has the characteristics of a true metric: Surprisals can be added, multiplied by positive numbers, and the difference between two surprisal values mean the same thing everywhere along the information. continuum. The unit of the metric is called the "M-bit", the generalization of the familiar "bit" or "2-bit" for binary data. The metric property is not possessed by so-called latent variables because they can be arbitrarily monotonically transformed.

Smoothing surprisal data is much easier and faster than smoothing probabilities since surprisal values are only constrained to be non-negative and are otherwise unbounded.

The function

`smooth.surp`

estimates smooth curves which fit a set of surprisal values and which also satisfy the constraint that their probability versions sum to one.Many functions in the fda package optimize a fitting criterion iteratively. Function

`smooth.monotone`

is an example. The optimisation algorithm used was a rather early design, and many improvements have since been made. In most of our optimisations, we have switched to the algorithm to be found in Press, Teukolsky, Vetterling and Flannery Numerical Recipes volumes. We have noticed a bit improvement in speed, are in the process of upgrading all of our optimisers using this approach.