smooth.surp {TestGardener}

Fit data with surprisal smoothing.

Description

Surprisal is -log(probability) where the logarithm is to the base being the dimension M of the multinomial observation vector. The surprisal curves for each question are estimated by fitting the surprisal values of binned data using curves whose values are within the M-1-dimensional surprisal subspace that is within the space of non-negative M-dimensional vectors.

Usage

smooth.surp(argvals, y, Bmat0, Sfd, Zmat, wtvec=NULL, conv=1e-4,
            iterlim=50, dbglev=0)

Arguments

argvals

Argument value array of length N, where N is the number of observed curve values for each curve. It is assumed that that these argument values are common to all observed curves. If this is not the case, you will need to run this function inside one or more loops, smoothing each curve separately.

y

A nbin by M_i matrix of surprisal values to be fit.

Bmat0

A Snbasis by M_i - 1 matrix containing starting values for the iterative optimization of the least squares fit of the surprisal curves to the surprisal data.

Sfd

A functional data object used toinitialize the optimization process.

Zmat

An M by M-1 matrix satisfying Zmat'Zmat <- I} and \code{Zmat'1 <- 0.

wtvec

A vector of weights to be used in the smoothing.

conv

A convergence criterion.

iterlim

the maximum number of iterations allowed in the minimization of error sum of squares.

dbglev

Either 0, 1, or 2. This controls the amount information printed out on each iteration, with 0 implying no output, 1 intermediate output level, and 2 full output. If either level 1 or 2 is specified, it can be helpful to turn off the output buffering feature of S-PLUS.

Value

A named list of class surpFd with these members:

PENSSE

The final value of the penalized fitting criterion.

DPENSSE

The final gradient of the penalized fitting criterion.

D2PENSSE

The final hessian of the fitting criterion.

SSE

The final value of the error sum of squares.

DSSE

The final gradient of the error sum of squares.

D2SSE

The final hessian of the error sum of squares.

DvecSmatDvecB

The final cross derivative DvecSmatDvecX times DvecXmatDvecB of the surprisal curve and the basis coordinates.

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.

See Also

eval.surp, ICC_plot, Sbinsmth

Examples

  oldpar <- par(no.readonly=TRUE)
  #  evaluation points
  x <- seq(-2,2,len=11)
  #  evaluate a standard normal distribution function
  p <- pnorm(x)
  #  combine with 1-p
  mnormp <- cbind(p,1-p)
  M <- 2
  #  convert to surprisal values
  mnorms <- -log2(mnormp)
  #  plot the surprisal values
  matplot(x, mnorms, type="l", lty=c(1,1), col=c(1,1), 
          ylab="Surprisal (2-bits)")
  # add some log-normal error
  mnormdata <- exp(log(mnorms) + rnorm(11)*0.1)
  #  set up a b-spline basis object
  nbasis <- 7
  sbasis <- create.bspline.basis(c(-2,2),nbasis)
  #  define an initial coefficient matrix
  cmat <- matrix(0,7,1)
  #  set up a fd object for suprisal smoothing
  Sfd <- fd(cmat, sbasis)
  Zmat <- matrix(c(1,-1),2,1)
  #  smooth the noisy data
  result <- smooth.surp(x, mnormdata, cmat, Sfd, Zmat)
  #  plot the data and the fits of the two surprisal curves
  xfine <- seq(-2,2,len=51)
  sfine <- eval.surp(xfine, result$Sfd, Zmat)
  matplot(xfine, sfine, type="l", lty=c(1,1), col=c(1,1))
  points(x, mnormdata[,1])
  points(x, mnormdata[,2])
  #  convert the surprisal fit values to probabilities
  pfine <- 2^(-sfine)
  #  check that they sum to one
  apply(pfine,1,sum)
  par(oldpar)