# rofanova

The package rofanova implements the robust nonparametric functional ANOVA method (RoFANOVA) proposed by Centofanti et al. (2021). RoFANOVA addresses the functional analysis of variance (FANOVA) problem, which aims to identify the presence of significant differences, in terms of functional mean, among groups of a functional data, by being robust against the presence of possible outliers. It is a permutation test whose test statistics rely on the functional equivariant M-estimator, the functional extension of the classical robust M-estimator, which is based on the functional normalized median absolute deviation (FuNMAD) estimator.

The main function is `rofanova` which implements the RoFANOVA method both for univariate and bi-variate functional data by using several families of loss functions. The functions `fusem` and `funmad` implement the functional equivariant M-estimator and the FuNMAD estimator, respectively.

## Installation

You can install the development version of rofanova from GitHub with:

``````# install.packages("devtools")
devtools::install_github("unina-sfere/rofanova")``````

## Example

This is a basic example which shows you how to apply the main function `rofanova` to perform both one-way and two-way FANOVA when data are univariate functional data. The data are generated as described in the first scenario of the simulation study in Centofanti et al. (2021).

``library(rofanova)``

Then, we generate the data and, just as an example, we fix the number of permutations B to 20.

``````data_out<-simulate_data(scenario="one-way")
label_1=data_out\$label_1
X_fdata<-data_out\$X_fdata
B=20``````

We compute the p-values corresponding to the RoFANOVA test with the median, the Huber, the bisquare, the Hampel, and the optimal loss functions.

``````per_list_median<-rofanova(X_fdata,label_1,B = B,family="median")
pvalue_median<-per_list_median\$pval_vec
per_list_huber<-rofanova(X_fdata,label_1,B = B,family="huber")
pvalue_huber<-per_list_huber\$pval_vec
per_list_bisquare<-rofanova(X_fdata,label_1,B = B,family="bisquare")
pvalue_bisquare<-per_list_bisquare\$pval_vec
per_list_hampel<-rofanova(X_fdata,label_1,B = B,family="hampel")
pvalue_hampel<-per_list_hampel\$pval_vec
per_list_optimal<-rofanova(X_fdata,label_1,B = B,family="optimal")
pvalue_optimal<-per_list_optimal\$pval
pvalues<-c(pvalue_median,pvalue_huber,pvalue_bisquare,pvalue_hampel,pvalue_optimal)
names(pvalues)=c("median", "Huber", "bisquare", "Hampel", "optimal")``````

The p-values for the significance of the main factor are

``````print(pvalues)
#>   median    Huber bisquare   Hampel  optimal
#>     0.70     0.80     0.65     0.65     0.70``````

Similarly, two-way FANOVA can be performed as follows.

``````data_out<-simulate_data(scenario="two-way")
label_1=data_out\$label_1
label_2=data_out\$label_2
X_fdata<-data_out\$X_fdata
B=20
per_list_median<-rofanova(X_fdata,label_1,label_2,B = B,family="median")
pvalue_median<-per_list_median\$pval_vec
per_list_huber<-rofanova(X_fdata,label_1,label_2,B = B,family="huber")
pvalue_huber<-per_list_huber\$pval_vec
per_list_bisquare<-rofanova(X_fdata,label_1,label_2,B = B,family="bisquare")
pvalue_bisquare<-per_list_bisquare\$pval_vec
per_list_hampel<-rofanova(X_fdata,label_1,label_2,B = B,family="hampel")
pvalue_hampel<-per_list_hampel\$pval_vec
per_list_optimal<-rofanova(X_fdata,label_1,label_2,B = B,family="optimal")
pvalue_optimal<-per_list_optimal\$pval
pvalues<-cbind(pvalue_median,pvalue_huber,pvalue_bisquare,pvalue_hampel,pvalue_optimal)
colnames(pvalues)=c("median", "Huber", "bisquare", "Hampel", "optimal") ``````

The p-values for the significance of the whole model, the two main factors and the interaction are

``````print(pvalues)
#>     median Huber bisquare Hampel optimal
#> MOD   0.45  0.40     0.45   0.40    0.15
#> F1    0.65  0.55     0.75   0.55    0.50
#> F2    0.40  0.80     0.70   0.65    0.40
#> INT   0.40  0.30     0.15   0.30    0.15``````

# References

• Centofanti, F., Colosimo, B.M., Grasso, M., Menafoglio, A., Palumbo, B., Vantini, S. (2021). Robust Functional ANOVA with Application to Additive Manufacturing. arXiv preprint arXiv:2112.10643.