fdars

Functional Data Analysis in Rust - A high-performance R package for functional data analysis with a Rust backend.

What is Functional Data Analysis?

Functional Data Analysis (FDA) is a branch of statistics that deals with data where each observation is a function, curve, or surface rather than a single number or vector. Examples include:

• Temperature curves: Daily temperature recordings over a year for multiple weather stations
• Growth curves: Height measurements of children tracked over time
• Spectroscopy data: Absorbance spectra measured across wavelengths
• Financial trajectories: Stock price movements over trading days
• Medical signals: ECG, EEG, or fMRI time series

Traditional statistical methods treat each time point as a separate variable, losing the inherent smoothness and continuity of the data. FDA treats the entire curve as a single observation, enabling more powerful and interpretable analyses.

Features

fdars is a comprehensive toolkit for functional data analysis with a high-performance Rust backend providing 10-200x speedups over pure R implementations.

Data Representation

• 1D functional data - Curves, time series, spectra
• 2D functional data - Surfaces, images, spatial fields
• Metadata support - Attach IDs and covariates to observations
• Flexible I/O - Create from matrices, arrays, or data frames

Depth & Centrality

Measure how "central" or "typical" each curve is:

• Fraiman-Muniz (FM), Band depth (BD), Modified band depth (MBD)
• Modal depth, Random projection (RP, RT, RPD)
• Functional spatial depth (FSD, KFSD)
• Depth-based median, trimmed mean, trimmed variance

Outlier Detection

Multiple approaches to identify anomalous curves:

• Depth-based trimming and weighting
• Likelihood ratio test (LRT)
• Functional boxplot
• Magnitude-Shape plot (magnitude vs shape outliers)
• Outliergram (MEI vs MBD)

Distance & Similarity

Quantify differences between curves:

• Lp distances (L1, L2, L∞)
• Hausdorff distance
• Dynamic time warping (DTW) and Soft-DTW (differentiable)
• Elastic (Fisher-Rao) distance with amplitude-phase decomposition
• PCA-based and derivative-based semimetrics

Elastic Alignment

Separate amplitude and phase variability in functional data:

• Pairwise elastic alignment via dynamic programming
• Karcher (Frechet) mean in the elastic metric
• Landmark registration (peak, valley, zero-crossing, inflection detection)
• Constrained elastic alignment (landmarks + smooth warping)
• TSRVF tangent space projection for linearized shape analysis
• Alignment quality diagnostics (variance reduction, warp complexity)
• Amplitude-phase decomposition and distance matrices

Tolerance & Confidence Bands

Characterize curve-level variability:

• FPCA bootstrap bands (pointwise and simultaneous)
• Conformal prediction bands (distribution-free)
• Elastic tolerance bands (alignment-based, removes phase variability)
• Phase tolerance bands (warping function variability)
• Joint elastic bands with config (amplitude + phase)
• Exponential family bands (for non-Gaussian data)
• Simultaneous confidence band for the mean (Degras)

Scalar-on-Function Regression

Predict scalar outcomes from functional predictors:

• FPC regression (fregre.lm) — Rust backend, standard errors, GCV
• Principal component regression (fregre.pc)
• Basis expansion regression (fregre.basis)
• Nonparametric kernel regression (fregre.np)
• Multiple functional predictors (fregre.np.multi)
• Mixed scalar + functional (fregre.np.mixed)
• L1 robust regression (fregre.l1) — resistant to outliers
• Huber M-estimation (fregre.huber) — robust with tuning parameter
• Cross-validation for model selection (cv.fdata)

Function-on-Scalar Regression

Predict functional responses from scalar predictors:

• Penalized FOSR (fosr) — ridge penalty, coefficient functions
• FPC-based FOSR (fosr.fpc) — dimension-reduced estimation
• Functional ANOVA (fanova) — permutation F-test for group differences

Elastic Regression

Alignment-aware regression that jointly aligns curves and fits models:

• elastic.regression() — scalar-on-function regression with simultaneous alignment
• elastic.logistic() — elastic logistic regression for binary classification
• elastic.pcr() — principal component regression with elastic FPCA

Elastic FPCA

Separate amplitude and phase variability via PCA:

• vert.fpca() — amplitude (vertical) FPCA on aligned curves
• horiz.fpca() — phase (horizontal) FPCA on warping functions
• joint.fpca() — combined amplitude + phase FPCA with balance parameter

Scalar-on-Shape Regression

Phase-invariant regression using elastic alignment:

• scalar.on.shape() — regression via Fisher-Rao alignment + index functions
• Index methods: identity, polynomial, Nadaraya-Watson
• Separates amplitude from phase variability before regression

Statistical Process Monitoring

Control charts for functional data streams:

• spm.phase1() — Phase I estimation (FPCA, T²/SPE control limits)
• spm.monitor() — Phase II online monitoring with alarm detection
• spm.ewma() — EWMA smoothing for gradual drift detection
• spm.contributions() — T²/SPE contribution diagnostics
• mfpca() — Multivariate FPCA for multiple functional variables
• frcc.phase1(), frcc.monitor() — Functional Regression Control Charts

Elastic Changepoint Detection

Detect structural breaks in functional time series:

• elastic.changepoint() — CUSUM test with permutation p-values
• Three change types: amplitude (shape), phase (timing), fpca (combined)

Functional Classification

Supervised classification of functional data:

• LDA, QDA, k-NN, kernel, DD-plot classifiers (fclassif)
• SVM classification
• Functional logistic regression (functional.logistic)
• Elastic logistic regression (elastic.logistic)
• Cross-validated classification error (fclassif.cv)

Cross-Validation

• cv.fdata() — unified k-fold CV for any fitting function
• Supports regression (RMSE, MAE, R²) and classification (accuracy)
• Repeated k-fold with majority voting / mean aggregation
• Stratified folding for balanced splits

Model Explainability & Diagnostics

Comprehensive toolkit for understanding functional regression models:

• Global: beta decomposition, PDP, ALE, Sobol sensitivity, permutation importance
• Local: LIME, SHAP, counterfactual, anchor rules, saliency maps
• Diagnostics: leverage, Cook's distance, VIF, DFBETAS, LOO
• Domain: significant regions, domain selection, pointwise importance
• Calibration: Brier score, ECE, Hosmer-Lemeshow (logistic models)
• All 22 methods support both fregre.lm and functional.logistic models

Uncertainty Quantification

• Prediction intervals with standard errors
• Conformal prediction intervals (distribution-free)
• Conformal generic wrappers with calibration.indices for pre-trained models
• Bootstrap-based explanation stability

Clustering

Group similar curves together:

• K-means clustering with K-means++ initialization
• Fuzzy C-means with soft membership
• Gaussian Mixture Models (cluster.gmm)
• Automatic selection of optimal k (silhouette, CH, elbow)

Smoothing & Basis Expansion

• Nadaraya-Watson, local linear/polynomial regression
• B-spline and Fourier basis expansions
• P-splines with automatic smoothing parameter selection
• Cross-validation (GCV, AIC, BIC) for basis selection
• Penalized basis smoothing (smooth.basis.fd) with automatic lambda via GCV

Functional Statistics

• Mean, variance, standard deviation, covariance
• Geometric median (L1 median)
• Bootstrap confidence intervals
• Hypothesis testing for functional means
• Functional equivalence test (TOST) via simultaneous confidence bands

Gaussian Process Simulation

Generate synthetic functional data:

• Multiple covariance kernels (Gaussian, Matérn, Exponential, Periodic)
• Kernel composition (addition, multiplication)
• Brownian motion and Ornstein-Uhlenbeck processes

Group Comparison

• Between-group distance matrices (centroid, Hausdorff, depth-based)
• Permutation tests for significant group differences
• Visualization (heatmaps, dendrograms)

Visualization

• Curve plots with categorical/continuous coloring
• Group means and confidence intervals
• Functional boxplots
• FPCA component visualization
• Outlier diagnostic plots

Installation

Prerequisites

• R (>= 4.0)
• Rust toolchain (install from rustup.rs)
• A C compiler (gcc, clang)

From GitHub

# Install remotes if needed
install.packages("remotes")

# Install fdars (with documentation)
remotes::install_github("sipemu/fdars-r", build_vignettes = TRUE)

Note: On Windows, you may need Rtools installed.

From Binary Release (No Rust Required)

Download the pre-built binary from GitHub Releases:

# macOS
install.packages("path/to/fdars_x.y.z.tgz", repos = NULL, type = "mac.binary")

# Windows
install.packages("path/to/fdars_x.y.z.zip", repos = NULL, type = "win.binary")

From Source

# Clone the repository
git clone https://github.com/sipemu/fdars-r.git
cd fdars-r

# Build and install
R CMD build .
R CMD INSTALL fdars_*.tar.gz

Quick Start

library(fdars)

# Create functional data from a matrix (rows = observations, cols = time points)
t <- seq(0, 1, length.out = 100)
X <- matrix(0, 20, 100)
for (i in 1:20) {
  X[i, ] <- sin(2 * pi * t) + rnorm(100, sd = 0.1)
}
fd <- fdata(X, argvals = t)

# Compute depth - measures how "central" each curve is
depths <- depth(fd)  # default: FM method
depths <- depth(fd, method = "mode")  # or specify method

# Find the functional median (most central curve)
median_curve <- median(fd)  # default: FM method

# Detect outliers
outliers <- outliers.depth.trim(fd, trim = 0.1)

# Functional regression: predict scalar y from functional X
y <- rowMeans(X) + rnorm(20, sd = 0.1)
model <- fregre.pc(fd, y, ncomp = 3)
predictions <- predict(model, fd)

# Cluster curves into groups
clusters <- cluster.kmeans(fd, ncl = 2)

# Smooth noisy curves
S <- S.NW(t, h = 0.1)  # Nadaraya-Watson smoother
smoothed <- S %*% X[1, ]

Key Concepts

Functional Data Objects (fdata)

The fdata class stores functional data as a matrix where rows are observations and columns are evaluation points:

fd <- fdata(data_matrix, argvals = time_points, rangeval = c(0, 1))

Identifiers and Metadata

You can attach identifiers and metadata (covariates) to functional data objects:

# Create fdata with IDs and metadata
meta <- data.frame(
  group = factor(c("control", "treatment", ...)),
  age = c(25, 32, ...),
  response = c(0.5, 0.8, ...)
)
fd <- fdata(X, id = paste0("patient_", 1:n), metadata = meta)

# Access fields
fd$id              # Character vector of identifiers
fd$metadata$group  # Access metadata columns

# Subsetting preserves metadata
fd_sub <- fd[1:10, ]  # id and metadata are also subsetted

# View metadata info
print(fd)    # Shows metadata columns
summary(fd)  # Shows metadata types and ranges

Note: If metadata contains an id column or has non-default row names, they must match the fdata identifiers. An error is thrown on mismatch.

Depth Functions

Depth measures how "central" or "typical" a curve is relative to a sample. Higher depth = more central.

Use the unified depth() function with a method parameter:

depth(fd, method = "FM")     # Fraiman-Muniz depth (default)
depth(fd, method = "BD")     # Band depth
depth(fd, method = "MBD")    # Modified band depth
depth(fd, method = "mode")   # Modal depth (kernel density)
depth(fd, method = "RP")     # Random projection depth
depth(fd, method = "RT")     # Random Tukey depth
depth(fd, method = "FSD")    # Functional spatial depth
depth(fd, method = "KFSD")   # Kernel functional spatial depth
depth(fd, method = "RPD")    # Random projection with derivatives

Functional Regression

Predict a scalar response from functional predictors:

• fregre.lm - FPC-based functional linear model (Rust backend)
• fregre.pc - Principal component regression
• fregre.basis - Basis expansion regression
• fregre.np - Nonparametric kernel regression
• elastic.regression - Elastic alignment + regression (joint)
• elastic.pcr - Elastic FPCA + principal component regression

All models support predict() for new data. fregre.lm has full explainability/diagnostics support.

Distance Metrics

Measure similarity between curves using metric() with a method parameter:

metric(fd, method = "lp")        # Lp distance (default, L2 = Euclidean)
metric(fd, method = "hausdorff") # Hausdorff distance
metric(fd, method = "dtw")       # Dynamic time warping
metric(fd, method = "softdtw")   # Soft-DTW (differentiable)
metric(fd, method = "elastic")   # Elastic (Fisher-Rao) distance
metric(fd, method = "amplitude") # Amplitude distance (shape only)
metric(fd, method = "phase")     # Phase distance (timing only)
metric(fd, method = "pca")       # PCA-based semimetric
metric(fd, method = "deriv")     # Derivative-based semimetric

Individual functions are also available: metric.lp, metric.hausdorff, metric.DTW, metric.softDTW, elastic.distance, amplitude.distance, phase.distance, semimetric.pca, semimetric.deriv.

Outlier Detection

Identify unusual curves:

• outliers.depth.trim - Trimmed depth-based detection
• outliers.depth.pond - Weighted depth-based detection
• outliers.lrt - Likelihood ratio test
• outliers.boxplot - Functional boxplot-based detection
• magnitudeshape - Magnitude-Shape outlier detection
• outliergram - Outliergram (MEI vs MBD plot)

Labeling Outliers by ID or Metadata

Both magnitudeshape and outliergram support labeling points by ID or metadata columns:

# Create fdata with IDs and metadata
fd <- fdata(X, id = paste0("patient_", 1:n),
            metadata = data.frame(subject_id = paste0("S", 1:n)))

# Outliergram with custom labels
og <- outliergram(fd)
plot(og, label = "id")           # Label outliers with patient IDs
plot(og, label = "subject_id")   # Label with metadata column
plot(og, label_all = TRUE)       # Label ALL points, not just outliers

# magnitudeshape with custom labels
magnitudeshape(fd, label = "id")        # Label outliers with patient IDs
magnitudeshape(fd, label = NULL)        # No labels

Functional Statistics

• mean(fd) - Functional mean
• var(fd) - Functional variance
• sd(fd) - Functional standard deviation
• cov(fd) - Functional covariance
• gmed(fd) - Geometric median (L1 median via Weiszfeld algorithm)

Covariance Functions and Gaussian Process Generation

Generate synthetic functional data from Gaussian processes with various covariance kernels:

# Smooth samples with Gaussian (squared exponential) kernel
fd_smooth <- make_gaussian_process(n = 20, t = seq(0, 1, length.out = 100),
                                   cov = kernel_gaussian(length_scale = 0.2))

# Rough samples with Matern kernel
fd_rough <- make_gaussian_process(n = 20, t = seq(0, 1, length.out = 100),
                                  cov = kernel_matern(nu = 1.5))

# Periodic samples
fd_periodic <- make_gaussian_process(n = 10, t = seq(0, 2, length.out = 200),
                                     cov = kernel_periodic(period = 0.5))

# Combine kernels: signal + noise
cov_total <- kernel_add(kernel_gaussian(variance = 1), kernel_whitenoise(variance = 0.1))
fd_noisy <- make_gaussian_process(n = 10, t = seq(0, 1, length.out = 100), cov = cov_total)

Available covariance functions:

• kernel_gaussian - Squared exponential (RBF) kernel, infinitely smooth
• kernel_exponential - Exponential kernel (Matern ν=0.5), rough
• kernel_matern - Matern family with smoothness parameter ν
• kernel_brownian - Brownian motion covariance (1D only)
• kernel_linear - Linear kernel
• kernel_polynomial - Polynomial kernel
• kernel_whitenoise - Independent noise at each point
• kernel_periodic - Periodic kernel (1D only)
• kernel_add - Combine kernels by addition
• kernel_mult - Combine kernels by multiplication

Depth-Based Medians and Trimmed Means

Use the unified functions with a method parameter:

# Median (curve with maximum depth)
median(fd)                          # default: FM method
median(fd, method = "mode")         # modal depth-based median

# Trimmed mean (mean of deepest curves)
trimmed(fd, trim = 0.1)             # default: FM method
trimmed(fd, trim = 0.1, method = "RP")  # RP depth-based trimmed mean

# Trimmed variance
trimvar(fd, trim = 0.1)             # default: FM method
trimvar(fd, trim = 0.1, method = "mode")

Visualization

• plot(fd, color = ...) - Plot curves with coloring by numeric or categorical variables
  • show.mean = TRUE - Overlay group mean curves
  • show.ci = TRUE - Show confidence interval ribbons per group
• boxplot.fdata - Functional boxplot with depth-based envelopes
• magnitudeshape - Magnitude-Shape outlier detection and visualization
• outliergram - Outliergram for shape outlier detection (MEI vs MBD plot)
• plot.fdata2pc - FPCA visualization (components, variance, scores)

Group Comparison

• group.distance - Compute distances between groups (centroid, Hausdorff, depth-based)
• group.test - Permutation test for significant group differences
• fequiv.test - Functional equivalence test (TOST) with bootstrap SCB
• plot.group.distance - Visualize group distances (heatmap, dendrogram)

Clustering

• cluster.kmeans - K-means clustering for functional data
• cluster.optim - Optimal k selection using silhouette, CH, or elbow
• cluster.fcm - Fuzzy C-means clustering with soft membership
• cluster.init - K-means++ center initialization

Curve Registration & Alignment

• elastic.align - Elastic alignment via SRSF and dynamic programming
• karcher.mean - Karcher mean in the elastic metric
• landmark.register - Landmark registration (peaks, valleys, etc.)
• elastic.align.constrained - Constrained elastic alignment
• tsrvf.transform - TSRVF tangent space projection
• register.fd - Shift registration using cross-correlation

Feature Extraction

• localavg.fdata - Extract local average features from curves

2D Functional Data (Surfaces)

fdars supports 2D functional data (surfaces/images). The following functions have full 2D support:

Category	Functions
Depth	depth (methods: FM, mode, RP, RT, FSD, KFSD)
Distance	metric.lp, metric.hausdorff, semimetric.pca, semimetric.deriv
Statistics	mean, var, sd, cov, gmed, deriv
Centrality	median, trimmed, trimvar (all methods except BD, MBD, RPD)
Regression	fregre.np (nonparametric)
Visualization	plot (heatmap + contours)

Note: Band depths (BD, MBD), RPD, and DTW do not support 2D data.

# Create 2D functional data (e.g., 10 surfaces on a 20x30 grid)
n <- 10
m1 <- 20
m2 <- 30
s <- seq(0, 1, length.out = m1)
t <- seq(0, 1, length.out = m2)

# Generate surfaces: f(s,t) = sin(2*pi*s) * cos(2*pi*t) + noise
X <- array(0, dim = c(n, m1, m2))
for (i in 1:n) {
  for (si in 1:m1) {
    for (ti in 1:m2) {
      X[i, si, ti] <- sin(2*pi*s[si]) * cos(2*pi*t[ti]) + rnorm(1, sd = 0.1)
    }
  }
}

fd2d <- fdata(X, argvals = list(s, t), fdata2d = TRUE)

# All these work with 2D data:
mean_surface <- mean(fd2d)           # Mean surface
var_surface <- var(fd2d)             # Pointwise variance
depths <- depth(fd2d)                # Depth values
median_surface <- median(fd2d)       # Depth-based median
gmed_surface <- gmed(fd2d)           # Geometric median

# Plot 2D data (heatmap + contours)
plot(fd2d)

Converting DataFrames to 2D fdata

Use df_to_fdata2d() to convert long-format DataFrames to 2D functional data:

# DataFrame structure: id column, s-index column, t-value columns
df <- data.frame(
  id = rep(c("surf1", "surf2"), each = 5),
  s = rep(1:5, 2),
  t1 = rnorm(10), t2 = rnorm(10), t3 = rnorm(10)
)

# Convert to 2D fdata
fd2d <- df_to_fdata2d(df, id_col = 1, s_col = 2)

# With metadata (must have one row per surface)
meta <- data.frame(group = c("A", "B"), value = c(1.5, 2.3))
fd2d <- df_to_fdata2d(df, id_col = 1, s_col = 2, metadata = meta)

Examples

• Wine Quality Analysis with Andrews Curves — A comprehensive walkthrough using the UCI Wine dataset (178 wines, 13 chemicals, 3 cultivars) demonstrating outlier detection, clustering, hypothesis testing, FPCA, and process monitoring. Render with quarto render examples/medium-andrews-wine.qmd.

• Predictive Truck Maintenance with Andrews Curves — Applying the full FDA pipeline to the Scania APS Failure dataset (76,000 trucks, 170 anonymized sensors, binary failure classification) for fleet health monitoring, outlier triage, and sensor-level diagnostics. Render with quarto render examples/scania-aps-failure.qmd.

License

MIT

Author

Simon Mueller

Acknowledgments

• Built with extendr for R-Rust integration
• Uses rayon for parallelization