fda (version 6.1.4)

eval.surp: Values of a Functional Data Object Defining Surprisal Curves.

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

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

Value

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

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).

Author

Juan Li and James Ramsay

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)).

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

See Also

smooth.surp

Examples

#  see example in man/smooth.surp.Rd