# param: Parametric Model In docopulae: Optimal Designs for Copula Models

## Description

`param` creates an initial parametric model object. Unlike other model statements this function does not perform any computation.

## Usage

 `1` ```param(fisherIf, dDim) ```

## Arguments

 `fisherIf` `function(x, ...)`, where `x` is a vector, usually a point from the design space. It shall evaluate to the Fisher information matrix. `dDim` length of `x`, usually the dimensionality of the design space.

## Value

`param` returns an object of `class` `"param"`. An object of class `"param"` is a list containing at least the following components:

• fisherIf: argument

• x: a row matrix of points where `fisherIf` has already been evaluated.

• fisherI: a list of Fisher information matrices, for each row in `x` respectively.

`fisherI`, `update.param`, `Dsensitivity`, `getM`, `Defficiency`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145``` ```library(copula) dfltNCube = nint_integrateNCube ## prepare for SparseGrid integration ncube = function(dimension) { SparseGrid::createIntegrationGrid('GQU', dimension, 3) } ncube = nint_integrateNCube_SparseGrid(ncube) unlockBinding('nint_integrateNCube', environment(nint_integrate)) assign('nint_integrateNCube', ncube, envir=environment(nint_integrate)) ## general settings numDeriv = FALSE ## build pdf, derivatives etas = function(theta) with(theta, { xx = x^(0:4) c(c(beta1, beta2, beta3) %*% xx[c(1, 2, 3)], # x^c(0, 1, 2) c(beta4, beta5, beta6) %*% xx[c(2, 4, 5)]) # x^c(1, 3, 4) }) copula = claytonCopula() alphas = c('alpha') parNames = c(paste('beta', 1:6, sep=''), alphas) if (numDeriv) { margins = function(y, theta, ...) { e = etas(theta) cbind(dnorm(y, mean=e, sd=1), pnorm(y, mean=e, sd=1)) } f = buildf(margins, copula, parNames=alphas) d2logf = numDeriv2Logf(f) } else { es = list( eta1=quote(theta\$beta1 + theta\$beta2*theta\$x + theta\$beta3*theta\$x^2), eta2=quote(theta\$beta4*theta\$x + theta\$beta5*theta\$x^3 + theta\$beta6*theta\$x^4)) margins = list(list(pdf=substitute(dnorm(y[1], mean=eta1, sd=1), es), cdf=substitute(pnorm(y[1], mean=eta1, sd=1), es)), list(pdf=substitute(dnorm(y[2], mean=eta2, sd=1), es), cdf=substitute(pnorm(y[2], mean=eta2, sd=1), es))) pn = as.list(alphas); names(pn) = alphas # map parameter to variable f = buildf(margins, copula, parNames=pn) cat('building derivatives ...') tt = system.time(d2logf <- Deriv2Logf(f, parNames)) cat('\n') print(tt) } f str(d2logf) ## param model = function(theta) { integrand = function(y, theta, i, j) -d2logf(y, theta, i, j) * f(y, theta) yspace = nint_space(nint_intvDim(-Inf, Inf), nint_intvDim(-Inf, Inf)) fisherIf = function(x) { theta\$x = x ## probability integral transform e = etas(theta) tt = nint_transform(integrand, yspace, list(list( dIdcs=1:2, g=function(y) pnorm(y, mean=e, sd=1), giDg=function(z) { t1 = qnorm(z, mean=e, sd=1) list(t1, dnorm(t1, mean=e, sd=1)) } ))) fisherI(tt\$f, theta, parNames, tt\$space) } return(param(fisherIf, 1)) } theta = list(beta1=1, beta2=1, beta3=1, beta4=1, beta5=1, beta6=1, alpha=iTau(copula, 0.5), x=0) m = model(theta) ## update.param system.time(m <- update(m, matrix(seq(0, 1, length.out=101), ncol=1))) ## find D-optimal design D = Dsensitivity(defaults=list(x=m\$x, desx=m\$x, mod=m)) d <- Wynn(D, 7.0007, maxIter=1e4) d\$tag\$Wynn\$tolBreak dev.new(); plot(d, sensTol=7, main='d') getM(m, d) rd = reduce(d, 0.05) cbind(x=rd\$x, w=rd\$w) dev.new(); plot(rd, main='rd') try(getM(m, rd)) m2 = update(m, rd) getM(m2, rd) ## find Ds-optimal design s = c(alphas, 'beta1', 'beta2', 'beta3') Ds = Dsensitivity(A=s, defaults=list(x=m\$x, desx=m\$x, mod=m)) ds <- Wynn(Ds, 4.0004, maxIter=1e4) ds\$tag\$Wynn\$tolBreak dev.new(); plot(reduce(ds, 0.05), sensTol=4, main='ds') ## create custom design n = 4 d2 = design(x=matrix(seq(0, 1, length.out=n), ncol=1), w=rep(1/n, n)) m = update(m, d2) dev.new(); plot(d2, sensx=d\$x, sens=D(x=d\$x, desx=d2\$x, desw=d2\$w, mod=m), sensTol=7, main='d2') ## compare designs Defficiency(ds, d, m) Defficiency(d, ds, m, A=s) # Ds-efficiency Defficiency(d2, d, m) Defficiency(d2, ds, m) # D-efficiency ## end with nice plot dev.new(); plot(rd, main='rd') assign('nint_integrateNCube', dfltNCube, envir=environment(nint_integrate)) ```