R/ddst.normbounded.test.R

In pbiecek/ddst: Data Driven Smooth Tests

#' Data Driven Smooth Test for Normality;  Bounded Basis Functions
#'
#' Performs data driven smooth test for composite hypothesis of normality
#' Null density is given by
#' \eqn{
#'   f(z;\gamma)=1/(\sqrt{2 \pi}\gamma_2) \exp(-(z-\gamma_1)^2/(2 \gamma_2^2))} for \eqn{z \in R}.
#' We model alternatives similarly as in Kallenberg and Ledwina (1997 a,b) using Legendre's polynomials or cosine basis.
#'
#' @param x a (non-empty) numeric vector of data values
#' @param base a function which returns an orthonormal system, possible choice: \code{ddst.base.legendre} for the Legendre polynomials and \code{ddst.base.cos} for the cosine system
#' @param d.n an integer specifying the maximum dimension considered, only for advanced users
#' @param c a calibrating parameter in the penalty in the model selection rule
#' @param compute.p a logical value indicating whether to compute a p-value or not
#' @param alpha a significance level
#' @param compute.cv a logical value indicating whether to compute a critical value corresponding to the significance level alpha or not
#' @param ... further arguments
#'
#' @aliases ddst.norm.Nk
#' @return
#'   An object of class \code{htest}
#' \item{statistic }{the value of the test statistic.}
#' \item{parameter }{the number of choosen coordinates (k).}
#' \item{method }{a character string indicating the parameters of performed test. }
#' \item{data.name }{a character string giving the name(s) of the data. }
#' \item{p.value }{the p-value for the test, computed only if \code{compute.p = TRUE}.}
#'
#' @export
#'
#' @references
#' Chen, L., Shapiro, S.S. (1995). An alternative test for normality based on normalized spacings. \emph{ J. Statist. Comput. Simulation} 53, 269--288.
#'
#' Inglot, T., Ledwina, T. (2006). Towards data driven selection of a penalty function for data driven Neyman tests. \emph{ Linear Algebra and its Appl.} \bold{ 417}, 579--590.
#'
#' Janic, A. and Ledwina, T. (2008). Data-driven tests for a location-scale family revisited. \emph{ J. Statist. Theory. Pract. Special issue on Modern Goodness of Fit Methods.}.
#'
#' Kallenberg, W.C.M., Ledwina, T. (1997 a). Data driven smooth tests for composite hypotheses: Comparison of powers. \emph{ J. Statist. Comput. Simul.} \bold{ 59}, 101--121.
#'
#' Kallenberg, W.C.M.,  Ledwina, T. (1997 b). Data driven smooth tests when the hypothesis is composite. \emph{ J. Amer. Statist. Assoc.} \bold{ 92}, 1094--1104.
#'
#' @keywords htest
#' @examples
#' set.seed(7)
#' # H0 is true
#' z <- rnorm(100)
#' # let's look on first 10 coordinates
#' d.n <- 10
#' t <- ddst.normbounded.test(z, compute.p = TRUE, d.n = d.n)
#' t
#' plot(t)
#'
#' # H0 is false
#' z <- rexp(100, 1)
#' t <- ddst.normbounded.test(z, compute.p = TRUE, d.n = d.n)
#' t
#' plot(t)
#'
#' # for Tephra data
#' z <- c(-1.748789, -1.75753, -1.740102, -1.740102, -1.731467, -1.765523,
#'       -1.761521, -1.72522, -1.80371, -1.745624, -1.872957, -1.729121,
#'       -1.81529, -1.888637, -1.887761, -1.881645, -1.91518, -1.849769,
#'       -1.755141, -1.665687, -1.764721, -1.736171, -1.736956, -1.737742,
#'       -1.687537, -1.804534, -1.790593, -1.808661, -1.784081, -1.729903,
#'       -1.711263, -1.748789, -1.772755, -1.72756, -1.71358, -1.821116,
#'       -1.839588, -1.839588, -1.830321, -1.807835, -1.747206, -1.788147,
#'       -1.759923, -1.786519, -1.726779, -1.738528, -1.754345, -1.781646,
#'       -1.641949, -1.755936, -1.775175, -1.736956, -1.705103, -1.743255,
#'       -1.82613, -1.826967, -1.780025, -1.684504, -1.751168)
#' t <- ddst.normbounded.test(z, compute.p = TRUE, d.n = d.n)
#' t
#' plot(t)
`ddst.normbounded.test` <-
  function(x,
           base = ddst.base.legendre,
           d.n = 10,
           c = 100,
           compute.p = TRUE,
           alpha = 0.05,
           compute.cv = TRUE,
           ...) {
    # method.name = as.character(substitute(base))
    base = ddst.base.legendre
    method.name = "ddst.base.legendre"

    n = length(x)
    if (n < 5)
      stop("length(x) should be at least 5")
    er1 = mean(x)
    sx  = sort(x)
    H   = qnorm((1:n - 3 / 8) / (n + 1 / 4))
    er2 = mean((sx[-1] - sx[-n]) / (H[-1] - H[-n]))
    pp   = (x - er1) / er2
    tmpp = c(mean(pp), (mean(pp ^ 2) - 1) / 2)

    maxN = max(min(d.n, length(x) - 2, 20), 1)
    u = numeric(maxN)
    for (j in 1:maxN)
      u[j] = ddst.phi(pnorm(x, er1, er2), j, base)

    coord = numeric(d.n)
    for (k in 1:d.n) {
      corrected_u = u[1:k] - t(MMnorm12[[k]]) %*% tmpp
      coord[k] = t(corrected_u) %*% MMnorm[[k]] %*% corrected_u * n
    }

    l = ddst.IIC(coord, n, c)
    attr(l, "names") = "T*"
    t = coord[l]
    attr(t, "names") = "W*T*"
    result = list(statistic = t,
                  parameter = l,
                  coordinates = coord - c(0, coord[-d.n]),
                  method = "Data Driven Smooth Test for Normality - Bounded Basis Functions")
    result$data.name = paste(paste(as.character(substitute(x)), collapse = ""),
                             ", base: ",
                             method.name,
                             ", c: ",
                             c,
                             ", d.n: ",
                             d.n,
                             sep = "")
    class(result) = c("htest", "ddst.test")
    if (compute.p) {
      ns <- as.numeric(rownames(tabNorm))
      ts <- as.numeric(colnames(tabNorm))
      n1 <- min(nrow(tabNorm), nrow(tabNorm) + 1 - sum(ns >= n))
      n2 <- max(1, sum(ns <= n))

      t1 <- min(ncol(tabNorm), ncol(tabNorm) + 1 - sum(ts >= t))
      t2 <- max(1, sum(ts <= t))

      del1 <- (t - ts[t1]) / min(diff(ts[c(t1, t2)]), -10 ^ -6)
      del2 <- (n - ns[n1]) / min(diff(ns[c(n1, n2)]), -10 ^ -6)

      tmpwt <-
        tabNorm[n1, t1] + del2 * diff(tabNorm[c(n1, n2), t1]) + del1 * diff(tabNorm[n1, c(t1, t2)])

      p.val = 10 ^ (-tmpwt * (t ^ 0.71 * log(n)))

      result$p.value = p.val
    }
    result
  }


#
# data
#
`MMnorm` <-
  list(
    structure(22.1875398382093, .Dim = c(1L, 1L)),
    structure(
      c(
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      .Dim = c(3L,
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`MMnorm12` <-
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      ),
      .Dim = c(2L, 9L)
    ),
    structure(
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      .Dim = c(2L, 10L)
    ),
    structure(
      c(
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      .Dim = c(2L, 11L)
    ),
    structure(
      c(
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      .Dim = c(2L, 12L)
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    structure(
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    structure(
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      ),
      .Dim = c(2L, 14L)
    ),
    structure(
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      .Dim = c(2L,
               15L)
    ),
    structure(
      c(
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        0.234330805463596,
        0.0465189445727412,-1.98978137872651,
        -0.806390449226108
      ),
      .Dim = c(2L, 16L)
    ),
    structure(
      c(
        0.977205023723124,
        0,
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      .Dim = c(2L,
               17L)
    ),
    structure(
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      .Dim = c(2L,
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        503.59029517106,
        320.39056589639
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      .Dim = c(2L, 19L)
    ),
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      .Dim = c(2L, 20L)
    )
  )
`ddst.polynomial.fun` <-
  list(function (x)
  {
    w <- 0
    w <- 1 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 3.46410161513775 + x * w
    w <- -1.73205080756888 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 13.4164078649987 + x * w
    w <- -13.4164078649987 + x * w
    w <- 2.23606797749979 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 52.9150262212918 + x * w
    w <- -79.3725393319377 + x * w
    w <- 31.7490157327751 + x * w
    w <- -2.64575131106459 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 210 + x * w
    w <- -420 + x * w
    w <- 270 + x * w
    w <- -60 + x * w
    w <- 3 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 835.78944716956 + x * w
    w <- -2089.4736179239 + x * w
    w <- 1857.30988259902 + x * w
    w <- -696.491205974634 + x * w
    w <- 99.498743710662 + x * w
    w <- -3.3166247903554 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 3331.52937852873 + x * w
    w <- -9994.58813558618 + x * w
    w <- 11357.4865177116 + x * w
    w <- -6057.3261427795 + x * w
    w <- 1514.33153569488 + x * w
    w <- -151.433153569488 + x * w
    w <- 3.60555127546399 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 13292.0788441839 + x * w
    w <- -46522.2759546435 + x * w
    w <- 64415.4590141218 + x * w
    w <- -44732.9576486957 + x * w
    w <- 16266.5300540712 + x * w
    w <- -2927.97540973281 + x * w
    w <- 216.887067387615 + x * w
    w <- -3.87298334620742 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 53064.3694016993 + x * w
    w <- -212257.477606797 + x * w
    w <- 346687.213424435 + x * w
    w <- -297160.468649516 + x * w
    w <- 142865.609927652 + x * w
    w <- -38097.4959807072 + x * w
    w <- 5195.11308827826 + x * w
    w <- -296.863605044472 + x * w
    w <- 4.12310562561767 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 211929.666634948 + x * w
    w <- -953683.499857264 + x * w
    w <- 1795168.94090779 + x * w
    w <- -1832568.29384337 + x * w
    w <- 1099540.97630602 + x * w
    w <- -392693.205823580 + x * w
    w <- 80552.4524766317 + x * w
    w <- -8630.61990821054 + x * w
    w <- 392.300904918661 + x * w
    w <- -4.35889894354068 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 846658.355097261 + x * w
    w <- -4233291.77548631 + x * w
    w <- 9023595.6266945 + x * w
    w <- -10694631.8538601 + x * w
    w <- 7706425.89469334 + x * w
    w <- -3467891.652612 + x * w
    w <- 963303.236836668 + x * w
    w <- -157273.997850885 + x * w
    w <- 13610.2498140189 + x * w
    w <- -504.083326445142 + x * w
    w <- 4.58257569495585 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 3383133.02315354 + x * w
    w <- -18607231.6273445 + x * w
    w <- 44302932.4460583 + x * w
    w <- -59808958.8021787 + x * w
    w <- 50365438.9913083 + x * w
    w <- -27421183.4508234 + x * w
    w <- 9678064.74734945 + x * w
    w <- -2160282.30967622 + x * w
    w <- 288037.641290162 + x * w
    w <- -20574.1172350116 + x * w
    w <- 633.049761077279 + x * w
    w <- -4.79583152331273 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 13520780 + x * w
    w <- -81124680 + x * w
    w <- 213393180 + x * w
    w <- -323323000 + x * w
    w <- 311775750 + x * w
    w <- -199536480 + x * w
    w <- 85765680 + x * w
    w <- -24504480 + x * w
    w <- 4504500 + x * w
    w <- -500500 + x * w
    w <- 30030.0000000000 + x * w
    w <- -780 + x * w
    w <- 5.00000000000002 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 54043102.8876026 + x * w
    w <- -351280168.769417 + x * w
    w <- 1011686886.05592 + x * w
    w <- -1700196016.84398 + x * w
    w <- 1848039148.74345 + x * w
    w <- -1360828827.71109 + x * w
    w <- 691214642.646902 + x * w
    w <- -241925124.926416 + x * w
    w <- 57298055.9036248 + x * w
    w <- -8842292.57771988 + x * w
    w <- 832215.772020695 + x * w
    w <- -42556.4883419674 + x * w
    w <- 945.699740932607 + x * w
    w <- -5.19615242270665 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 216034502.501892 + x * w
    w <- -1512241517.51324 + x * w
    w <- 4732755860.36552 + x * w
    w <- -8737395434.52097 + x * w
    w <- 10572248475.7704 + x * w
    w <- -8810207063.14198 + x * w
    w <- 5171208493.58333 + x * w
    w <- -2149073659.67100 + x * w
    w <- 626813150.737374 + x * w
    w <- -125362630.147475 + x * w
    w <- 16495082.9141414 + x * w
    w <- -1332935.99306193 + x * w
    w <- 58805.9996939088 + x * w
    w <- -1130.88460949825 + x * w
    w <- 5.38516480713452 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 863657799.906573 + x * w
    w <- -6477433499.2993 + x * w
    w <- 21889258032.1149 + x * w
    w <- -44039102469.374 + x * w
    w <- 58718803292.4986 + x * w
    w <- -54653655372.2487 + x * w
    w <- 36435770248.1658 + x * w
    w <- -17567246369.6514 + x * w
    w <- 6110346563.357 + x * w
    w <- -1512156472.74997 + x * w
    w <- 259226823.899994 + x * w
    w <- -29457593.6249993 + x * w
    w <- 2067199.55263153 + x * w
    w <- -79507.6751012128 + x * w
    w <- 1336.26344707920 + x * w
    w <- -5.56776436283004 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 3452943955.96051 + x * w
    w <- -27623551647.6841 + x * w
    w <- 100246760011.757 + x * w
    w <- -218315166247.826 + x * w
    w <- 318062612895.539 + x * w
    w <- -327150116121.126 + x * w
    w <- 244352864510.224 + x * w
    w <- -134259815664.958 + x * w
    w <- 54375225344.3081 + x * w
    w <- -16111177879.795 + x * w
    w <- 3432381374.39111 + x * w
    w <- -510602187.92595 + x * w
    w <- 50654978.9609077 + x * w
    w <- -3117229.4745174 + x * w
    w <- 105470.170190438 + x * w
    w <- -1562.52103985834 + x * w
    w <- 5.74456264653804 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 13805800579.8575 + x * w
    w <- -117349304928.789 + x * w
    w <- 455173061541.969 + x * w
    w <- -1066811862988.99 + x * w
    w <- 1686251009240.66 + x * w
    w <- -1899842803744.48 + x * w
    w <- 1572283699650.60 + x * w
    w <- -970644528865.934 + x * w
    w <- 449372467067.562 + x * w
    w <- -155552007831.079 + x * w
    w <- 39821314004.7563 + x * w
    w <- -7391077220.57976 + x * w
    w <- 964053550.510404 + x * w
    w <- -84270415.2544059 + x * w
    w <- 4586145.04785882 + x * w
    w <- -137584.351435765 + x * w
    w <- 1810.32041362848 + x * w
    w <- -5.91607978309963 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 55201892960.2267 + x * w
    w <- -496817036642.04 + x * w
    w <- 2051144622707.85 + x * w
    w <- -5147970817776.57 + x * w
    w <- 8774950257573.7 + x * w
    w <- -10749314065527.8 + x * w
    w <- 9766849876742.99 + x * w
    w <- -6697268486909.48 + x * w
    w <- 3492971926362.27 + x * w
    w <- -1386099970778.68 + x * w
    w <- 415829991233.604 + x * w
    w <- -93052865171.156 + x * w
    w <- 15198634644.6222 + x * w
    w <- -1753688612.84102 + x * w
    w <- 136155948.201942 + x * w
    w <- -6601500.51888205 + x * w
    w <- 176825.906755769 + x * w
    w <- -2080.30478536199 + x * w
    w <- 6.08276253029823 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 220731101683.863 + x * w
    w <- -2096945465996.7 + x * w
    w <- 9181220688958.53 + x * w
    w <- -24568266473231.6 + x * w
    w <- 44924830122480.7 + x * w
    w <- -59459333985636.2 + x * w
    w <- 58858734652448 + x * w
    w <- -44406813197605.9 + x * w
    w <- 25784601211513.1 + x * w
    w <- -11555321283678.1 + x * w
    w <- 3984593546095.89 + x * w
    w <- -1047896354655.09 + x * w
    w <- 206991872524.462 + x * w
    w <- -30007697496.1498 + x * w
    w <- 3086506028.17541 + x * w
    w <- -214340696.40107 + x * w
    w <- 9319160.71309 + x * w
    w <- -224257.878122487 + x * w
    w <- 2373.09923939139 + x * w
    w <- -6.24499799839841 + x * w
    w
  }, function (x)
  {
    w <- 0
    w <- 882648449733.328 + x * w
    w <- -8826484497333.28 + x * w
    w <- 40850780814580.9 + x * w
    w <- -116102219157230 + x * w
    w <- 226713117138105 + x * w
    w <- -322436433263082 + x * w
    w <- 345467607067588 + x * w
    w <- -284502735232131 + x * w
    w <- 182124857023599 + x * w
    w <- -91062428511799.6 + x * w
    w <- 35543722096541.1 + x * w
    w <- -10770824877739.7 + x * w
    w <- 2507002342232.53 + x * w
    w <- -440791620612.312 + x * w
    w <- 57139654523.8183 + x * w
    w <- -5274429648.35246 + x * w
    w <- 329651853.022028 + x * w
    w <- -12927523.6479227 + x * w
    w <- 281033.122780928 + x * w
    w <- -2689.31217972179 + x * w
    w <- 6.40312423743286 + x * w
    w
  })





`ddst.norm.Nk` <-
  function(x,
           base = ddst.base.legendre,
           Dmax = 10,
           n = length(x)) {
    er1 = mean(x)
    sx  = sort(x)
    H   = qnorm((1:n - 3 / 8) / (n + 1 / 4))
    er2 = mean((sx[-1] - sx[-n]) / (H[-1] - H[-n]))
    pp   = (x - er1) / er2
    tmpp = c(mean(pp), (mean(pp ^ 2) - 1) / 2)
    maxN = max(min(Dmax, length(x) - 2, 20), 1)
    u = numeric(maxN)
    for (j in 1:maxN)
      u[j] = ddst.phi(pnorm(x, er1, er2), j, base)
    coord = numeric(Dmax)
    for (k in 1:Dmax) {
      korekta = u[1:k] - t(MMnorm12[[k]]) %*% tmpp
      coord[k] = t(korekta) %*% MMnorm[[k]] %*% korekta * n
    }
    coord
  }