One population Interval Wise Testing procedure

The function implements the Interval Wise Testing procedure for testing the center of symmetry of a functional population. Functional data are tested locally and unadjusted and adjusted p-value functions are provided. The unadjusted p-value function controls the point-wise error rate. The adjusted p-value function controls the interval-wise error rate.

Usage

ITP1bspline(data, mu = 0, B = 1000, order = 2, nknots = dim(data)[2])

ITP1fourier(data, mu = 0, B = 1000, maxfrequency = floor(dim(data)[2]/2))

IWT1(data, mu = 0, B = 1000, dx = NULL, recycle = TRUE)

Arguments

data

Either pointwise evaluations of the functional data set on a uniform grid, or an fd. If pointwise evaluations are provided, data is a matrix of dimensions c(n, J), with J evaluations on columns and n units on rows.

mu

The center of symmetry under the null hypothesis. Three possibilities are available for mu:

a constant (in this case, a constant function is used);
a J-dimensional vector containing the evaluations on the same grid which data are evaluated;
a fd object containing one function. Defaults to 0.

B

The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. Defaults to 1000L.

order

Order of the B-spline basis expansion. Defaults to 2L.

nknots

Number of knots of the B-spline basis expansion. Defaults to dim(data)[2].

maxfrequency

The maximum frequency to be used in the Fourier basis expansion of data. Defaults to floor(dim(data)[2] / 2), leading to an interpolating expansion.

dx

Used only if an fd object is provided. In this case, dx is the size of the discretization step of the gridused to evaluate functional data. If set to NULL, a grid of size 100L is used. Defaults to NULL.

recycle

Flag used to decide whether the recycled version of the IWT should be used (see Pini and Vantini, 2017 for details). Defaults to TRUE.

Value

An object of class IWT1, which is a list containing at least the following components:

test: String vector indicating the type of test performed. In this case equal to "1pop".
mu: Evaluation on a grid of the center of symmetry under the null hypothesis (as entered by the user).
unadjusted_pval: Evaluation on a grid of the unadjusted p-value function.
pval_matrix: Matrix of dimensions c(p, p) of the p-values of the multivariate tests. The element (i,j) of matrix pval.matrix contains the p-value of the joint NPC test of the components (j,j+1,...,j+(p-i)).
adjusted_pval: Evaluation on a grid of the adjusted p-value function.
data.eval: Evaluation on a grid of the functional data.

References

A. Pini and S. Vantini (2017). The Interval Testing Procedure: Inference for Functional Data Controlling the Family Wise Error Rate on Intervals. Biometrics, 73(3): 835–845.

A. Pini and S. Vantini (2017). Interval-wise testing for functional data. Journal of Nonparametric Statistics, 29(2), 407-424.

Examples

# Performing the IWT for one population
IWT.result <- IWT1(NASAtemp$paris, mu = 4, B = 10L)
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#> 
#> ── creating the p-value matrix: end of row 352 out of 365 ──────────────────────
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#> ── creating the p-value matrix: end of row 353 out of 365 ──────────────────────
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#> ── creating the p-value matrix: end of row 354 out of 365 ──────────────────────
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#> 
#> ── creating the p-value matrix: end of row 356 out of 365 ──────────────────────
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#> ── creating the p-value matrix: end of row 357 out of 365 ──────────────────────
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#> ── creating the p-value matrix: end of row 358 out of 365 ──────────────────────
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#> ── creating the p-value matrix: end of row 359 out of 365 ──────────────────────
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#> ── creating the p-value matrix: end of row 360 out of 365 ──────────────────────
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#> ── creating the p-value matrix: end of row 361 out of 365 ──────────────────────
#> 
#> ── creating the p-value matrix: end of row 362 out of 365 ──────────────────────
#> 
#> ── creating the p-value matrix: end of row 363 out of 365 ──────────────────────
#> 
#> ── creating the p-value matrix: end of row 364 out of 365 ──────────────────────
#> 
#> ── creating the p-value matrix: end of row 365 out of 365 ──────────────────────
#> 
#> ── Interval-Wise Testing completed ─────────────────────────────────────────────

# Plotting the results of the IWT
plot(IWT.result, xrange = c(0, 12), main = 'Paris temperatures')



# Plotting the p-value heatmap
IWTimage(IWT.result, abscissa_range = c(0, 12))


# Selecting the significant components at 5% level
which(IWT.result$adjusted_pval < 0.05)
#>   [1]   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18
#>  [19]  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36
#>  [37]  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54
#>  [55]  55  56  57  58  59  60  61  62  63  64  65  66  67  68  69  70  71  72
#>  [73]  73  74  75  76  77  78  79  80  81  82  83  84  85  86  87  88  89  90
#>  [91]  91  92  93  94  95  96  97  98  99 100 101 102 103 104 105 106 107 108
#> [109] 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
#> [127] 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 257 258 259
#> [145] 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277
#> [163] 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295
#> [181] 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313
#> [199] 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331
#> [217] 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349
#> [235] 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365