funMoDisco: Functional Motif Discovery

In funMoDisco: Motif Discovery in Functional Data

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

# Load the package without startup messages
suppressPackageStartupMessages(library(funMoDisco))

Introduction

The discoverMotifs function serves as the core of the package, offering a robust and efficient implementation of two advanced algorithms: ProbKMA(Cremona and Chiaromonte, 2020) and FunBiAlign(Di Iorio, Cremona,and Chiaromonte, 2023). Together, these algorithms provide a comprehensive solution for the detection and clustering of recurring patterns within functional data. In addition to motif discovery, funMoDisco allows users to simulate functional curves with embedded motifs. The package provides several functions for generating synthetic functional data, which can be useful for testing and benchmarking the motif discovery algorithms. These simulated curves are customizable, allowing users to control the number, length, and complexity of the motifs. By the end of this vignette, users will be guide through practical examples and equipped with a solid understanding of how to effectively utilize funMoDisco package for their pattern detection needs.

Overview of discoverMotifs

The discoverMotifs function allows users to:

• Choose to run ProbKMA multiple times with varying numbers of motifs (K) and minimum motif lengths (c), or FunBiAlign specifying the length (portion_len) and the minimum cardinality (min_card) of the motifs.
• Perform clustering based on local alignments of curve segments.
• Control the clustering process through a wide range of hyperparameters.

Common Parameters

The following parameters are shared between both algorithms:

library(knitr)
library(kableExtra)

params <- data.frame(
  Parameter = c("Y0", "method", "stopCriterion", "name", "plot", "worker_number"),
  Description = c(
    "A list of N vectors (for univariate curves) or N matrices (for multivariate curves) representing the curves, where each curve is evaluated on a uniform grid.",
    "A character string specifying which method to use: either 'ProbKMA' or 'funBIalign'.",
    "A character string indicating the convergence criterion based on Bhattacharyya distance between memberships for 'ProbKMA', or ranking criteria for 'funBIalign'.",
    "A character string providing the name of the resulting folder.",
    "A logical value indicating whether to plot the motifs and results.",
    "An integer specifying the number of CPU cores to use for parallel computations. Defaults to cores minus one."
  ),
  Default = c("mandatory", "mandatory", "mandatory", "mandatory", "mandatory", "`detectCores() - 1`")
)

# Create the table with wrapped text and adjustable column widths
kable(params, format = "latex", booktabs = TRUE, longtable = TRUE, escape = TRUE) %>%
  kable_styling(latex_options = c("hold_position", "striped", "scale_down"), 
                position = "center", 
                font_size = 12) %>%
  column_spec(1, italic = TRUE , background = "#e2e3e5") %>%  # Header color for Parameter
  column_spec(2, width = "25em", background = "#e2e3e5") %>%  # Background for Description
  column_spec(3, width = "10em", background = "#e2e3e5") %>%  # Background for Default
  row_spec(0, bold = TRUE, color = "black", background = "#8DADB8") 

Key Arguments for ProbKMA

Below is an overview of the key arguments for the ProbKMA algorithm in the discoverMotifs function:

library(knitr)

# Data for the table
params <- data.frame(
  Parameter = c("K", "c", "diss", "alpha", "Y1", "P0", "S0", "c_max", "w", "m", "iter_max", "quantile", 
                "tol", "iter4elong", "tol4elong", "max_elong", "trials_elong", "deltaJK_elong", "max_gap", 
                "iter4clean", "tol4clean", "quantile4clean", "return_options", "n_subcurves", "sil_threshold", 
                "set_seed", "seed", "exe_print", "transformed", "V_init", "n_init_motif"),
  Description = c(
    "A vector specifying the numbers of motifs to be tested.",
    "A vector specifying the minimum motif lengths to be tested.",
    "Dissimilarity. Possible choices are 'd0_L2', 'd1_L2', 'd0_d1_L2'.",
    "Parameter in [0,1] defining the relative weight of the curve's levels and derivatives. 'alpha'=0 means 'd0_L2', 'alpha'=1 means 'd1_L2'.",
    "A list of N vectors or matrices representing the derivative of the curves. Required if `diss='d0_d1_L2'`.",
    "A matrix specifying the initial membership probabilities. If not specified, it will be randomly generated.",
    "A matrix specifying the initial shift. If not specified, it will be randomly generated.",
    "An integer or a vector of K integers specifying the maximum motif lengths.",
    "Weight vector for the dissimilarity index across dimensions.",
    "Weighting exponent in the least-squares functional method (must be greater than 1).",
    "Maximum number of iterations allowed for the ProbKMA algorithm.",
    "Double specifying quantile probability when 'stopCriterion'=\"quantile\".",
    "Double specifying the tolerance level for the method; iteration stops if the stop criterion is less than 'tol'.",
    "Integer specifying the number of iterations after which motif elongation is performed. If 'iter4elong' > 'iter_max', no elongation is performed.",
    "Tolerance on the Bhattacharyya distance for motif elongation.",
    "Maximum elongation allowed in a single iteration, as a percentage of motif length.",
    "Integer specifying the number of elongation trials (equispaced) on each side of the motif in a single iteration.",
    "Maximum relative increase in the objective function allowed during motif elongation.",
    "Double specifying the maximum gap allowed in each alignment as a percentage of the motif length.",
    "Integer specifying number of iterations after which motif cleaning is performed. If 'iter4clean' > 'iter_max', no cleaning is performed.",
    "Tolerance on the Bhattacharyya distance for motif cleaning.",
    "Dissimilarity quantile used for motif cleaning.",
    "If 'TRUE', the options passed to the method are returned.",
    "Integer specifying the number of splitting subcurves used when the number of curves is equal to one.",
    "Double specifying the threshold value to filter candidate motifs.",
    "If 'TRUE', sets a random seed to ensure reproducibility.",
    "The random seed for initialization (used if set_seed=TRUE).",
    "If 'TRUE' and worker_number is equal to one, prints execution details for each iteration.",
    "A logical value indicating whether to normalize the curve segments to the interval [0,1] before applying the dissimilarity measure.",
    "A list of motif sets provided as specific initializations for clustering rather than using random initializations.",
    "The number of initial motif sets from `V_init` to be used directly as starting points in clustering."
  ),
  Default = c("mandatory", "mandatory", "mandatory", "mandatory", "`NULL`", "`matrix()`", "`matrix()`", "`Inf`", "`1`", "`2`", "`1e3`", "`0.25`", "`1e-8`", "`100`", "`1e-3`", "`0.5`", "`201`", "`0.05`", "`0.2`", "`50`", "`1e-4`", "`0.5`", "`TRUE`", "`10`", "`0.9`", "`FALSE`", "`1`", "`FALSE`", "`NULL`", "`NULL`", "`NULL`")
)

# Create the table with wrapped text and adjustable column widths
kable(params, format = "latex", booktabs = TRUE, longtable = TRUE, escape = TRUE) %>%
  kable_styling(latex_options = c("hold_position", "striped", "scale_down"), 
                position = "center", 
                font_size = 12) %>%
  column_spec(1, italic = TRUE , background = "#e2e3e5") %>%  # Header color for Parameter
  column_spec(2, width = "20em", background = "#e2e3e5") %>%  # Background for Description
  column_spec(3, width = "8em", background = "#e2e3e5") %>%  # Background for Default
  row_spec(0, bold = TRUE, color = "black", background = "#8DADB8") 

Example Usage

Here is an example showing a possible use of discoverMotifs with ProbKMA :

library(funMoDisco)

diss <- 'd0_d1_L2' 
alpha <- 0.5
# run probKMA multiple times (2x3x10=60 times)
K <- c(2,3) 
c <- c(61,51) 
n_init = 10 

data("simulated200") # load simulated data

results = funMoDisco::discoverMotifs(
  Y0 = simulated200$Y0,
  method = "ProbKMA",
  stopCriterion = "max",
  name = './results_ProbKMA_VectorData/',
  plot = TRUE,
  probKMA_options = list(
    Y1 = simulated200$Y1,
    K = K,
    c = c,
    n_init = n_init,
    diss = diss,
    alpha = alpha
  ),
  worker_number = NULL
)

In this scenario, all plots generated during the algorithm's execution will be saved in the folder specified by the 'name' parameter. Additionally, the function handles the entire post-processing phase, including filtering patterns and searching for occurrences within the curves, presenting both intermediate and final results along with their corresponding plots. If the user chooses to only perform the post-processing by adjusting parameters like 'sil_threshold,' it is sufficient to call the same discoverMotifs function with the updated parameters. The algorithm will automatically load the previously computed results (which are computationally expensive) and proceed with the post-processing, returning updated plots and results.

Below is an example of how to call discoverMotifs with a customized V_init and transformed = TRUE:

library(funMoDisco)

# Define parameters
c <- 5                
K <- 2                
n_init <- 2
diss <- 'd0_d1_L2' 
alpha <- 0.5

# Load sample data
data("simulated200")

motif1 <- list(v0 = matrix(runif(c * 4), nrow = c, ncol = 1))
motif2 <- list(v0 = matrix(runif(c * 4), nrow = c, ncol = 1))

# Optional: Include `v1` if using a dissimilarity measure that requires it
motif1$v1 <- matrix(runif(c * 4), nrow = c, ncol = 1)
motif2$v1 <- matrix(runif(c * 4), nrow = c, ncol = 1)

# Define V_init with multiple initial motif sets, matching `K` motifs 
# per initialization
V_init <- list(
  list(motif1, motif2),   # Initialization 1 with 2 motifs
  list(motif1, motif2)    # Initialization 2 with 2 motifs
)

# Run discoverMotifs 
results <- funMoDisco::discoverMotifs(
  Y0 = simulated200$Y0,
  method = "ProbKMA",
  stopCriterion = "max",
  name = './results_ProbKMA_VectorData/',
  plot = TRUE,
  probKMA_options = list(
    Y1 = simulated200$Y1, K = K, c = c, n_init = n_init,
    diss = diss, alpha = alpha, sil_threshold = 0.5,
    V_init = V_init,          
    transformed = TRUE        
  ),
  worker_number = NULL
)

Key Arguments for funBIalign

Below is an overview of the key arguments for the ProbKMA algorithm in the discoverMotifs function:

library(knitr)

# Data for the table
small_params <- data.frame(
  Parameter = c("portion_len", "min_card", "cut_off"),
  Description = c(
    "An integer specifying the length of the curve portions to be aligned.",
    "An integer specifying the minimum cardinality of motifs, i.e., the minimum number of motif occurrences required.",
    "Used when `plot` is set to `TRUE`. A double that specifies the number of top-ranked motifs to keep based on the ranking criteria. In particular, all motifs that rank below the cut_off are retained."
  ),
  Default = c("mandatory", "mandatory", "`NULL`")
)

# Create the table with specified column widths for better fitting
kable(params, format = "latex", booktabs = TRUE, longtable = TRUE, escape = TRUE) %>%
  kable_styling(latex_options = c("hold_position", "striped", "scale_down"), 
                position = "center", 
                font_size = 12) %>%
  column_spec(1, italic = TRUE , background = "#e2e3e5") %>%  # Header color for Parameter
  column_spec(2, width = "18em", background = "#e2e3e5") %>%  # Background for Description
  column_spec(3, width = "8em", background = "#e2e3e5") %>%  # Background for Default
  row_spec(0, bold = TRUE, color = "black", background = "#8DADB8") 

Example Usage

Here is an example showing a possible use of discoverMotifs with funBIalign :

library(funMoDisco)

data("simulated200") # load simulated data

funBialignResult <- funMoDisco::discoverMotifs(
  Y0 = simulated200$Y0,
  method = "FunBIalign",
  stopCriterion = 'fMRS',
  name = './results_FunBialign',
  plot = TRUE,
  funBIalign_options = list(
    portion_len = 60,
    min_card = 3,
    cut_off = 1.0
  )
)

As previously discussed for 'ProbKMA', if the user intends to execute only the post-processing phase related to the re-ranking of discovered motifs, they can simply call the same function, specifying the updated re-ranking criterion and, if necessary, adjusting the new cut_off value.

Motif Simulation

As previously noted, the package offers the capability to generate synthetic curves embedded with patterns. This functionality facilitates the testing of both algorithms and provides a reliable reference benchmark for performance evaluation.

The algorithm begins by generating random curves utilizing B-splines as the foundational tools. Subsequently, it incorporates either random or positional patterns into these curves. Finally, noise is introduced, which can manifest as either pointwise noise or noise applied to the expansion coefficients of the B-splines. This process effectively simulates real-world scenarios in which each measurement is associated with a degree of noise.

Key Arguments for motifSimulationBuilder

'motifSimulationBuilder' represents the first function to be called. In particular, it represents the constructor of the S4 class 'motifSimulation'.

Below is an overview of the key arguments:

library(knitr)

# Data for the table
params <- data.frame(
  Parameter = c("N", "len", "mot_details", "norder", "coeff_min", "coeff_max", "dist_knots", "min_dist_motifs", "distribution"),
  Description = c(
    "The number of background curves to be generated.",
    "The length of the background curves.",
    "A list outlining the definitions of the motifs to be included. Each motif is characterized by its length, a set of coefficients that may be optionally specified, and the number of occurrences. These occurrences can be indicated either by specific positions within the curves or by a total count. In the latter case, the algorithm will randomly position the motifs throughout the curves.",
    "Integer specifying the order of the B-splines.",
    "Additive coefficients to be incorporated into the generation of coefficients for the background curves.",
    "Additive coefficients to be incorporated into the generation of coefficients for the background curves.",
    "Integer specifying the distance between two consecutive knots.",
    "Integer specifying the minimum distance between two consecutive motifs embedded in the same curve.",
    "Distribution from which the coefficients of the background curves are generated. You can choose between a uniform distribution or a beta distribution. Alternatively, you can pass a vector representing the empirical distribution from which you wish to sample."
  ),
  Default = c("mandatory", "mandatory", "mandatory", "3", "`-15`", "`15`", "`10`", "`'norder' * 'dist_knots'`", "`unif`")
)

# Create the table with specified column widths for better fitting
kable(params, format = "latex", booktabs = TRUE, longtable = TRUE, escape = TRUE) %>%
  kable_styling(latex_options = c("hold_position", "striped", "scale_down"), 
                position = "center", 
                font_size = 12) %>%
  column_spec(1, italic = TRUE , background = "#e2e3e5") %>%  # Header color for Parameter
  column_spec(2, width = "20em", background = "#e2e3e5") %>%  # Background for Description
  column_spec(3, width = "8em", background = "#e2e3e5") %>%  # Background for Default
  row_spec(0, bold = TRUE, color = "black", background = "#8DADB8") 

Key Arguments for generateCurves

After calling the constructor of the class, it is then possible to generate the curves with the motifs embedded.

Below is an overview of the key arguments:

library(knitr)

# Data for the table
params <- data.frame(
  Parameter = c("object", "noise_type", "noise_str", "seed_background", "seed_motif", "only_der", "coeff_min_shift", "coeff_max_shift"),
  Description = c(
    "The S4 object first constructed.",
    "A string specifying whether to add pointwise error or coefficients ('pointwise' and 'coeff').",
    "A list corresponding to the number of motifs, specifying the structure of noise to be added for each motif. If 'pointwise' is chosen, the user can specify a list of vectors or matrices indicating the amount of noise for each motif. If 'coeff' is selected, a list of individual values or vectors can be provided.",
    "An integer specifying the seed for background curve generation.",
    "An integer specifying the seed for motif generation.",
    "If 'FALSE', a vertical shift is added to each motif instance.",
    "Minimum vertical shift.",
    "Maximum vertical shift."
  ),
  Default = c("mandatory", "mandatory", "mandatory", "`777`", "`43213`", "`TRUE`", "`-10`", "`10`")
)

# Create the table with specified column widths for better fitting
kable(params, format = "latex", booktabs = TRUE, longtable = TRUE, escape = TRUE) %>%
  kable_styling(latex_options = c("hold_position", "striped", "scale_down"), 
                position = "center", 
                font_size = 12) %>%
  column_spec(1, italic = TRUE , background = "#e2e3e5") %>%  # Header color for Parameter
  column_spec(2, width = "20em", background = "#e2e3e5") %>%  # Background for Description
  column_spec(3, width = "8em", background = "#e2e3e5") %>%  # Background for Default
  row_spec(0, bold = TRUE, color = "black", background = "#8DADB8") 

Key Arguments for plot_motifs

This is the final function to be called. As indicated by its name, it generates summary plots. Each plot displays the background curve, the motif without noise, and the motif with noise highlighted within a shaded region.

Below is an overview of the key arguments:

library(knitr)

# Data for the table
params <- data.frame(
  Parameter = c("object", "curves", "path"),
  Description = c(
    "The S4 object first constructed.",
    "The result of the previous method.",
    "Path specifying the directory where the results will be saved."
  ),
  Default = c("mandatory", "mandatory", "mandatory")
)

# Create the table with specified column widths for better fitting
kable(params, format = "latex", booktabs = TRUE, longtable = TRUE, escape = TRUE) %>%
  kable_styling(latex_options = c("hold_position", "striped", "scale_down"), 
                position = "center", 
                font_size = 12) %>%
  column_spec(1, italic = TRUE , background = "#e2e3e5") %>%  # Header color for Parameter
  column_spec(2, width = "20em", background = "#e2e3e5") %>%  # Background for Description
  column_spec(3, width = "8em", background = "#e2e3e5") %>%  # Background for Default
  row_spec(0, bold = TRUE, color = "black", background = "#8DADB8") 

Examples

The five main types of use are considered below.

0) Special case: No motifs

library(funMoDisco)

mot_len <- 100
mot_details <- NULL # or list()

builder <- funMoDisco::motifSimulationBuilder(N = 20,len = 300,mot_details)

curves <- funMoDisco::generateCurves(builder) 

funMoDisco::plot_motifs(builder,curves,name = "plots_0")

knitr::include_graphics("plots_0.pdf")

1) Set the motif position and add pointwise noise

library(funMoDisco)

mot_len <- 100  

# Struct specifying the motif ID, the number of curves, and the relative knot position
motif_str <- rbind.data.frame(
  c(1, 1, 20),
  c(2, 1, 2),  
  c(1, 3, 1),
  c(1, 2, 1),
  c(1, 2, 15),
  c(1, 4, 1),
  c(2, 5, 1),
  c(2, 7, 1),
  c(2, 17, 1)
)

names(motif_str) <- c("motif_id", "curve", "start_break_pos")

mot1 <- list(
  "len" = mot_len,               # Length
  "coeffs" = NULL,              # Weights for the motif
  "occurrences" = motif_str %>% filter(motif_id == 1)
)

mot2 <- list(
  "len" = mot_len,
  "coeffs" = NULL,
  "occurrences" = motif_str %>% filter(motif_id == 2)
)

mot_details <- list(mot1, mot2)

# MATRIX NOISE 
noise_str <- list(
  rbind(
    rep(2, 100),                      # Constant and identical
    c(rep(0.1, 50), rep(2, 50)),     # SD 0.1 first, SD 1 later
    c(rep(2, 50), rep(0.1, 50)),     # SD 1 first, 0.1 later
    c(seq(2, 0.1, len = 50), rep(0.1, 50))
  ),
  rbind(
    rep(0.0, 100),  
    rep(0.5, 100),  
    rep(1.0, 100),  
    rep(5.0, 100)
  )
)

builder <- funMoDisco::motifSimulationBuilder(
  N = 20,
  len = 300,
  mot_details,
  distribution = 'beta'
)

curves <- funMoDisco::generateCurves(
  builder,
  noise_type = 'pointwise',
  noise_str = noise_str
) 

funMoDisco::plot_motifs(builder, curves, "plots_1")

knitr::include_graphics("plots_1.pdf")

2) Set the motif position and add coeff noise

library(funMoDisco)

mot_len <- 100  

# Struct specifying the motif ID, the number of curves, and the relative knot position
motif_str <- rbind.data.frame(
  c(1, 1, 20),
  c(1, 1, 2),  
  c(1, 3, 1),
  c(1, 2, 1),
  c(1, 2, 15),
  c(1, 4, 1),
  c(1, 5, 1),
  c(1, 7, 1),
  c(2, 17, 1)
)

names(motif_str) <- c("motif_id", "curve", "start_break_pos")

mot1 <- list(
  "len" = mot_len,               # Length
  "coeffs" = NULL,              # Weights for the motif
  "occurrences" = motif_str %>% filter(motif_id == 1)
)

mot2 <- list(
  "len" = mot_len,
  "coeffs" = NULL,
  "occurrences" = motif_str %>% filter(motif_id == 2)
)

mot_details <- list(mot1, mot2)

# VECTOR NOISE 
noise_str <- list(
  c(0.1, 1.0, 5.0),
  c(0.0, 0.0, 0.0)
)

builder <- funMoDisco::motifSimulationBuilder(
  N = 20,
  len = 300,
  mot_details,
  distribution = 'beta'
)

curves <- funMoDisco::generateCurves(
  builder,
  noise_type = 'coeff',
  noise_str,
  only_der = FALSE
)

funMoDisco::plot_motifs(builder, curves, "plots_2")

knitr::include_graphics("plots_2.pdf")

3) Random motif position and add pointwse noise

library(funMoDisco)

mot_len <- 100  

# Define motifs
mot1 <- list(
  "len" = mot_len,              # Length
  "coeffs" = NULL,             # Weights for the motif
  "occurrences" = 5
) 

mot2 <- list(
  "len" = mot_len,
  "coeffs" = NULL,
  "occurrences" = 6
)

mot_details <- list(mot1, mot2)

# Define noise structure
noise_str <- list(
  rbind(rep(2, 100)),  
  rbind(rep(0.5, 100))
)

# Build motif simulation
builder <- funMoDisco::motifSimulationBuilder(
  N = 20,
  len = 300,
  mot_details,
  distribution = 'beta'
)

# Generate curves
curves <- funMoDisco::generateCurves(
  builder,
  noise_type = 'pointwise',
  noise_str,
  only_der = FALSE
)

# Plot motifs
funMoDisco::plot_motifs(builder, curves, "plots_3")

knitr::include_graphics("plots_3.pdf")

4) Random motif position and add coeff noise

library(funMoDisco)

mot_len <- 100  

# Define motifs
mot1 <- list(
  "len" = mot_len,              # Length
  "weights" = NULL,             # Weights for the motif
  "occurrences" = 5
) 

mot2 <- list(
  "len" = mot_len,
  "coeffs" = NULL,
  "occurrences" = 6
)

mot_details <- list(mot1, mot2)

# Define vector noise
noise_str <- list(
  c(0.1, 5.0, 10.0),
  c(0.1, 5.0, 10.0)
)

# Build motif simulation
builder <- funMoDisco::motifSimulationBuilder(
  N = 20,
  len = 300,
  mot_details,
  distribution = 'beta'
)

# Generate curves
curves <- funMoDisco::generateCurves(
  builder,
  noise_type = 'coeff',
  noise_str,
  only_der = FALSE
)

# Plot motifs
funMoDisco::plot_motifs(builder, curves, "plots_4")

knitr::include_graphics("plots_4.pdf")

Additional functions

In addition to the functions previously described, the package includes a helper function that facilitates the direct transformation of the output from 'generateCurves' into a format suitable for 'discoverMotifs'. This function generates a comprehensive list that encompasses all curves, each containing the embedded patterns corresponding to various tested noise levels.

result <- funMoDisco::to_motifDiscovery(curves)

Additionally, a Shiny app is available, serving as a graphical user interface (GUI) that enables users to execute all the previously mentioned functions in a straightforward and intuitive manner. The app consistently provides summary plots, enhancing the user experience.

library(funMoDisco)

# Define motif structure
motif_str <- rbind.data.frame(
  c(1, 1, 20),
  c(1, 1, 2),
  c(1, 3, 1),
  c(1, 2, 1),
  c(1, 2, 15),
  c(1, 4, 1),
  c(1, 5, 1),
  c(1, 7, 1),
  c(2, 17, 1)
)

names(motif_str) <- c("motif_id", "curve", "start_break_pos")

# Define motifs
mot1 <- list(
  "len" = 100,                   # Length
  "weights" = NULL,              # Weights for the motif
  "appearance" = motif_str %>% filter(motif_id == 1)
)

mot2 <- list(
  "len" = 150,
  "weights" = NULL,
  "appearance" = motif_str %>% filter(motif_id == 2)
)

mot_details <- list(mot1, mot2)

# Define noise structure
noise_str <- list(
  rbind(rep(2, 100), c(rep(0.1, 50), rep(2, 50))),
  rbind(rep(0.0, 150), rep(5.0, 150))
)

# Run motif simulation app
funMoDisco::motifSimulationApp(noise_str, mot_details)

Conclusion

The discoverMotifs function is a powerful tool for discovering functional motifs in complex datasets. With its flexibility, users can run multiple initializations, customize clustering parameters, simulate functional curves with motifs, and visualize the results in an intuitive way. Whether using ProbKMA or funBIalign, the funMoDisco package provides a robust solution for analyzing functional data and uncovering hidden patterns.

funMoDisco documentation built on April 16, 2025, 1:10 a.m.