localFDA

Overview

Software companion for the paper “Localization processes for functional
data analysis” by Elías, Antonio, Jiménez, Raúl, and Yukich, Joe, (2020)
\. It provides the code for computing localization
processes and localization distances and their application to
classification and outlier detection problems.

Installation

#install the package
devtools::install_github("aefdz/localFDA")

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#load the package
library(localFDA)

Test usage

Load the example data and plot it.

X <- exampleData
n <- ncol(X)
p <- nrow(X)
t <- as.numeric(rownames(X))
#plot the data set
df_functions <- data.frame(ids = rep(colnames(X), each = p),
                           y = c(X),
                           x = rep(t, n)
                           )
functions_plot <- ggplot(df_functions) + 
                  geom_line(aes(x = x, y = y, group = ids, color = ids), 
                            color = "black", alpha = 0.25) + 
                  xlab("t") + theme(legend.position = "none")
functions_plot

Compute kth empirical localization processes

Empirical version of Equation (1) of the paper. For one focal,

focal <- "1"
localizarionProcesses_focal <- localizationProcesses(X, focal)$lc

Plot localization processes of order 1, 50, 100 and 200:

df_lc <- data.frame(k = rep(colnames(localizarionProcesses_focal), each = p),
                           y = c(localizarionProcesses_focal),
                           x = rep(t, n-1)
                           )
lc_plots <- list()
ks <- c(1, 50, 100, 200)
for(i in 1:4){
  lc_plots[[i]] <- functions_plot + 
                   geom_line(data = filter(df_lc, k == paste0("k=", ks[i])), 
                             aes(x = x, y = y, group = k), 
                             color = "blue", size = 1) +
                   geom_line(data = filter(df_functions, ids == focal), 
                             aes(x = x, y = y, group = ids), 
                             color = "red", linetype = "dashed", size = 1)+
                   ggtitle(paste("k = ", ks[i]))
}
wrap_plots(lc_plots)

Compute kth empirical localization distances

Equation (18) of the paper. For one focal,

localizationDistances_focal <- localizationDistances(X, focal)
head(localizationDistances_focal)

##          k=1          k=2          k=3          k=4          k=5          k=6 
## 0.0005082926 0.0011346495 0.0017636690 0.0023955745 0.0030095117 0.0035089220

Plot the localization distances:

df_ld <- data.frame(k = names(localizationDistances_focal),
                           y = localizationDistances_focal,
                           x = 1:c(n-1)
                           )
ldistances_plot <- ggplot(df_ld, aes(x = x, y = y)) + 
                   geom_point() + 
                   ggtitle("Localization distances for one focal") + 
                   xlab("kth") + ylab("L")
ldistances_plot

Sample μ and σ

localizationStatistics_full <- localizationStatistics(X, robustify = TRUE)
#See the mean and sd estimations for k = 1, 100, 200, 400, 600
localizationStatistics_full$trim_mean[c(1, 100, 200, 400, 600)]

##         k=1       k=100       k=200       k=400       k=600 
## 0.001083517 0.098465426 0.184940365 0.350528860 0.526580274

localizationStatistics_full$trim_sd[c(1, 100, 200, 400, 600)]

##          k=1        k=100        k=200        k=400        k=600 
## 0.0005326429 0.0329170846 0.0490732397 0.0686018224 0.0806314699

Classification

X <- classificationData
ids_training <- sample(colnames(X), 90)
ids_testing <- setdiff(colnames(X), ids_training)
trainingSample <- X[,ids_training]
testSample <- X[,ids_testing]; colnames(testSample) <- NULL #blind 
classNames <- c("G1", "G2")
classification_results <- localizationClassifier(trainingSample, testSample, classNames, k_opt = 3)
checking <- data.frame(real_classs = ids_testing, 
                      predicted_class =classification_results$test$predicted_class)
checking

##    real_classs predicted_class
## 1        12_G1              G1
## 2        14_G1              G1
## 3        21_G1              G1
## 4        44_G1              G1
## 5        54_G2              G2
## 6        56_G2              G2
## 7        72_G2              G2
## 8        81_G2              G2
## 9        94_G2              G2
## 10      100_G2              G2

Outlier detection

X <- outlierData
outliers <- outlierLocalizationDistance(X, localrule = 0.95, whiskerrule = 1.5)
outliers$outliers_ld_rule

## [1] "1_magnitude" "1_shape"     "2_magnitude" "2_shape"

Plot results,

df_functions <- data.frame(ids = rep(colnames(X), each = nrow(X)),
                           y = c(X),
                           x = rep(seq(from = 0, to = 1, length.out = nrow(X)), ncol(X)))
functions_plot <- ggplot(df_functions) + 
                  geom_line(aes(x = x, y = y, group = ids), 
                            color = "black") + 
                  xlab("t") + 
  theme(legend.position = "bottom")+
                  geom_line(data = df_functions[df_functions$ids %in% outliers$outliers_ld_rule,], aes(x = x, y = y, group = ids, color = ids), size = 1) +
  guides(color = guide_legend(title="Detected outliers"))
functions_plot

References

Elías, Antonio, Jiménez, Raúl and Yukich, Joe (2020). Localization
processes for functional data analysis
[https://arxiv.org/abs/2007.16059]https://arxiv.org/abs/2007.16059.