eval.surp | R Documentation |

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.

```
eval.surp(evalarg, Sfdobj, Zmat, nderiv = 0)
```

`evalarg` |
a vector or matrix of argument values at which the functional data object is to be evaluated. |

`Sfdobj` |
a functional data object of dimension |

`Zmat` |
An |

`nderiv` |
An integer defining a derivatve of |

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))`

.

A `N`

by `M`

matrix `S`

of surprisal values at points
`evalarg`

, or their first or second derivatives.

Juan Li and James Ramsay

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.

`smooth.surp`

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

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