Functional Outlier Time Series Detection

A comprehensive R implementation for detecting outliers in functional time series data using various depth-based and bootstrap methods.

Overview

This repository provides multiple approaches for outlier detection in functional time series, extending the original proposal by Raña et al. (2015) with several enhancements:

• Bootstrap-based methods for robust cutoff estimation
• Multivariate extensions using derivative information for shape outlier detection
• Directional outlyingness (DirOut) methods for comprehensive outlier characterization
• Sliding window approaches for local outlier detection

Available Methods

1. Original Proposal

Bootstrap-based outlier detection methods following Raña et al. (2015):

• SmBoD: Standard smoothed bootstrap on data
• MBBo: Moving block bootstrap (for temporal dependence)
• StBo: Stationary bootstrap

Best for: General-purpose outlier detection, especially magnitude outliers.

2. Multivariate Process

Multivariate extension using derivative information:

• Uses original process, first derivative, and second derivative
• Combines information across variables with weighted depth
• Best for: Shape outliers and partially contaminated observations

⚠️ Note: May mask magnitude outliers. Use when you expect shape but not magnitude outliers.

3. DirOut Methods

Directional outlyingness-based detection:

• Two-component outlyingness (average and variance)
• Captures both magnitude and shape outliers
• Best for: Comprehensive outlier detection when both types are possible

4. Sliding Window

Local outlier detection using sliding windows:

• Window-based functional depth and boxplot detection
• Aggregates outlier flags across multiple windows
• Best for: Non-stationary data and local pattern detection

Quick Start

Installation

# Required packages
library(fda)
library(roahd)
library(fda.usc)
library(mrfDepth)
library(fdaoutlier)  # For DirOut methods

Basic Example

# Source required files
source("R/utils.R")
source("R/depths.R")

# Prepare your functional data
data_matrix <- matrix(rnorm(100*50), nrow = 100, ncol = 50)
fdataobj <- fdata(data_matrix)

# Method 1: Original proposal (MBBo)
source("R/original_proposal/bootstrap-procedures.R")
source("R/original_proposal/outlier-detection-procedures.R")

result <- outlier_bootstrap(
    fdataobj = fdataobj,
    boot = MBBo,
    dfunc = MBD
)

print(result$outliers)

Documentation

Each method has detailed documentation with:

• Complete function signatures and parameter descriptions
• Working code examples
• Parameter tuning guidelines
• Troubleshooting tips

See the README files in each directory:

• R/original_proposal/README.md
• R/multivariate_process/README.md
• R/dirout/README.md
• R/sliding_window/README.md

Choosing a Method

Method	Best For	Strengths
Original Proposal	General use, magnitude outliers	Robust, well-tested, multiple bootstrap options
Multivariate Process	Shape outliers, partial contamination	Excellent for shape detection, uses derivative info
DirOut	Comprehensive detection	Two-component outlyingness, captures both types
Sliding Window	Non-stationary data, local patterns	Local context, no bootstrap required

Repository Structure

R/
├── original_proposal/     # Original bootstrap methods
├── multivariate_process/  # Multivariate extension with derivatives
├── dirout/                # Directional outlyingness methods
├── sliding_window/        # Sliding window approach
├── real_data/            # Real data examples
├── depths.R              # Depth function implementations
├── utils.R               # Utility functions
└── simulated-models.R    # Simulation models for testing

References

Raña, P., Aneiros, G. and Vilar, J. (2015) Detection of outliers in functional time series. Environmetrics, 26, 178–191.

License

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