Getting started with functional statistical testing

In funStatTest: Statistical Testing for Functional Data

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(funStatTest)

#| include: no

# fix seed for simulations
set.seed(123456)

The funStatTest package implements various statistics for two sample comparison testing regarding functional data.

Authorship and license

This package is developed by:

• Zaineb Smida (ORCID)
• Ghislain Durif (link|ORCID)
• Lionel Cucala (link)

It implements statistics (and related experiments) introduced and used in [@smida2022].

Data simulation

Here are some functions used to simulate clustered trajectories of functional data based on the Karhunen-Loève decomposition.

The functional data simulation process is described in [@smida2022] (section 3.1).

Simulate a single trajectory

simu_vec <- simul_traj(100)
plot(simu_vec, xlab = "point", ylab = "value")

Simulate trajectories from two samples diverging by a delta

simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 10, 
    delta_shape = "constant", distrib = "normal"
)
str(simu_data)

Graphical representation of simulated data

# constant delta
simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 5, 
    delta_shape = "constant", distrib = "normal"
)
plot_simu(simu_data)
# linear delta
simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 5, 
    delta_shape = "linear", distrib = "normal"
)
plot_simu(simu_data)
# quadratic delta
simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 5, 
    delta_shape = "quadratic", distrib = "normal"
)
plot_simu(simu_data)

Statistics

MO median statistic

The $MO$ median statistic [@smida2022] is implemented in the stat_mo() function.

simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 10, 
    delta_shape = "constant", distrib = "normal"
)

MatX <- simu_data$mat_sample1
MatY <- simu_data$mat_sample2

stat_mo(MatX, MatY)

MED median statistic

The $MED$ median statistic [@smida2022] is implemented in the stat_med() function.

simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 10, 
    delta_shape = "constant", distrib = "normal"
)

MatX <- simu_data$mat_sample1
MatY <- simu_data$mat_sample2

stat_med(MatX, MatY)

WMW statistic

The Wilcoxon-Mann-Whitney statistic [@chakraborty2015] (noted $WMW$ in [@smida2022]) is implemented in the stat_wmw() function.

simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 10, 
    delta_shape = "constant", distrib = "normal"
)

MatX <- simu_data$mat_sample1
MatY <- simu_data$mat_sample2

stat_wmw(MatX, MatY)

HKR statistics

The Horváth-Kokoszka-Reeder statistics [@horvath2013] (noted $HKR1$ and $HKR2$ in [@smida2022]) are implemented in the stat_hkr() function.

simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 10, 
    delta_shape = "constant", distrib = "normal"
)

MatX <- simu_data$mat_sample1
MatY <- simu_data$mat_sample2

stat_hkr(MatX, MatY)

CFF statistic

The Cuevas-Febrero-Fraiman statistic [@cuevas2004] (noted $CFF$ in [@smida2022]) is implemented in the stat_cff() function.

simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 10, 
    delta_shape = "constant", distrib = "normal"
)

MatX <- simu_data$mat_sample1
MatY <- simu_data$mat_sample2

stat_cff(MatX, MatY)

Compute multiple statistics

The function comp_stat() allows to compute multiple statistics defined above in a single call on the same data.

simu_data <- simul_data(
    n_point = 100, n_obs1 = 50, n_obs2 = 75, c_val = 10, 
    delta_shape = "constant", distrib = "normal"
)

MatX <- simu_data$mat_sample1
MatY <- simu_data$mat_sample2

res <- comp_stat(MatX, MatY, stat = c("mo", "med", "wmw", "hkr", "cff"))
res

Permutation-based computation of p-values

P-values associated to the different statistics defined above can be computed with the permutation-based method as follow:

# simulate small data for the example
simu_data <- simul_data(
    n_point = 20, n_obs1 = 4, n_obs2 = 5, c_val = 10, 
    delta_shape = "constant", distrib = "normal"
)

MatX <- simu_data$mat_sample1
MatY <- simu_data$mat_sample2
res <- permut_pval(
    MatX, MatY, n_perm = 100, stat = c("mo", "med", "wmw", "hkr", "cff"), 
    verbose = TRUE)
res

:warning: computing p-values based on permutations may take some time (for large data or when using a large number of simulations. :warning:

Simulation-based power analysis

We use our simulation scheme with permutation-based p-values computation to run a power analysis to evaluate the different statistics.

# simulate a few small data for the example
res <- power_exp(
    n_simu = 20, alpha = 0.05, n_perm = 100, 
    stat = c("mo", "med", "wmw", "hkr", "cff"), 
    n_point = 25, n_obs1 = 4, n_obs2 = 5, c_val = 10, delta_shape = "constant", 
    distrib = "normal", max_iter = 10000, verbose = FALSE
)
res$power_res

References

::: {#refs} :::

funStatTest documentation built on May 29, 2024, 10:26 a.m.