# eval.surp: Values of a Functional Data Object Defining Surprisal Curves. In fda: Functional Data Analysis

## Description

A surprisal vector of length `M` is minus the log to a positive integer base `M` of a set of `M` multinomial probabilities. Surprisal curves are functions of a one-dimensional index set, such that at any value of the index set the values of the curves are a surprisal vector. See Details below for further explanations.

## Usage

 `1` ```eval.surp(evalarg, Wfdobj, nderiv = 0) ```

## Arguments

 `evalarg` a vector or matrix of argument values at which the functional data object is to be evaluated. `Wfdobj` a functional data object of dimension `M-1` to be evaluated. `nderiv` An integer defining a derivatve of `Wfdobj` in the set `c(0,1,2)`.

## Details

A surprisal `M`-vector is information measured in `M`-bits. Since a multinomial probability vector must sum to one, it follows that the surprisal vector `S` must satisfy the constraint `log_M(sum(M^(-S)) = 0.` That is, surprisal vectors lie within a curved `M-1`-dimensional manifold.

Surprisal curves are defined by a set of unconstrained `M-1` B-spline functional data objects defined over an index set that are transformed into surprisal curves defined over the index set.

Let `C` be a `K` by `M-1` coefficient matrix defining the B-spline curves, where `K` is the number of B-spline basis functions.

Let a `M` by `M-1` matrix `Z` have orthonormal columns. Matrices satisfying these constraints are generated by function `zerobasis()`.

Let `N` by `K` matrix be a matrix of B-spline basis values evaluated at `N` evaluation points using function `eval.basis()`.

Let `N` by `M` matrix `X` = `B * C * t(Z)`.

Then the `N` by `M` matrix `S` of surprisal values is `S` = `-X + outer(log(rowSums(M^X))/log(M),rep(1,M))`.

## Value

A `N` by `M` matrix `S` of surprisal values at points `evalarg`, or their first or second derivatives.

## Author(s)

Juan Li and James Ramsay

## References

Ramsay, J. O., Li J. and Wiberg, M. (2020) Full information optimal scoring. Journal of Educational and Behavioral Statistics, 45, 297-315.

Ramsay, J. O., Li J. and Wiberg, M. (2020) Better rating scale scores with information-based psychometrics. Psych, 2, 347-360.

http://testgardener.azurewebsites.net

`smooth.surp`

## Examples

 `1` ```# see example in man/smooth.surp.Rd ```

fda documentation built on Nov. 17, 2021, 5:14 p.m.