# ITP1fourier: One population Interval Testing Procedure with Fourier basis In alessiapini/fdatest: Interval Wise Testing for Functional Data

## Description

The function implements the Interval Testing Procedure for testing the center of symmetry of a functional population evaluated on a uniform grid. Data are represented by means of the Fourier expansion and the significance of each basis coefficient is tested with an interval-wise control of the Family Wise Error Rate.

## Usage

 `1` ```ITP1fourier(data, mu = 0, maxfrequency = floor(dim(data)/2), B = 10000) ```

## Arguments

 `data` Pointwise evaluations of the functional data set on a uniform grid. It 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: either a constant (in this case, a constant function is used) or a `J`-dimensional vector containing the evaluations on the same grid which `data` are evaluated. The default is `mu=0`. `maxfrequency` The maximum frequency to be used in the Fourier basis expansion of data. The default is `floor(dim(data)/2)`, leading to an interpolating expansion. `B` The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is `B=1000`.

## Value

`ITP1fourier` returns an object of `class` "`ITP1`". An object of class "`ITP1`" is a list containing at least the following components:

 `basis` String vector indicating the basis used for the first phase of the algorithm. In this case equal to `"Fourier"`. `test` String vector indicating the type of test performed. In this case equal to `"1pop"`. `mu` Center of symmetry under the null hypothesis (as entered by the user). `coeff` Matrix of dimensions `c(n,p)` of the `p` coefficients of the B-spline basis expansion. Rows are associated to units and columns to the basis index. `pval` Unadjusted p-values for each basis coefficient. `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` Adjusted p-values for each basis coefficient. `labels` Labels indicating the population membership of each data (in this case always equal to `1`). `data.eval` Evaluation on a fine uniform grid of the functional data obtained through the basis expansion. `heatmap.matrix` Heatmap matrix of p-values (used only for plots).

## 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.

See also `plot.ITP1` and `ITPimage` for plotting the results, `ITP1bspline` for ITP based on B-spline basis, `IWT1` for a one-sample test that is not based on an a-priori selected basis expansion.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```# Importing the NASA temperatures data set data(NASAtemp) # Performing the ITP ITP.result <- ITP1fourier(NASAtemp\$milan,maxfrequency=20,B=1000) # Plotting the results of the ITP plot(ITP.result,main='NASA data',xrange=c(1,365),xlab='Day') # Plotting the p-value heatmap ITPimage(ITP.result,abscissa.range=c(1,365)) # Selecting the significant coefficients which(ITP.result\$adjusted.pval < 0.05) ```