Pairwise inner product for L^2 functions

Description

Calculate all pairwise inner products between elements from L^2 supplied to this function. The integral is approximated by the Trapezoidal rule for uniform grids:

\int_a^b f(x) dx \approx \Delta x \left( \sum_{i=1}^{N-1} f(x_i) + \frac{f(x_N) - f(x_0)}{2} \right)

whereas \{x_i \} is an uniform grid on [a,b] such that a = x_0 < x_1 < \ldots < x_N = b and \Delta x the step size, i.e. \Delta x := x_2 - x_1. Therefore, it is assumed that the functions are evaluated at the same, equidistant grid.

Usage

innerProduct(grid, data)

Arguments

Value

Numeric symmetric matrix containing the approximated pairwise inner products between the functions supplied by data. The entry (i,j) of the result is the inner product between the i-th and j-th column of data.

Examples

# Create orthogonal fourier basis via `fdapace` package
library(fdapace)
basis <- fdapace::CreateBasis(K = 10,
                              type = "fourier")
iP <- innerProduct(grid = seq(0, 1, length.out = 50), # default grid in CreateBasis()
                   data = basis)
round(iP,3)
# Since the basis is orthogonal, the resulting matrix will be the identity matrix.

Minimum regularized covariance trace estimator

Description

Functional outlier detection based on the minimum regularized covariance trace estimator (Oguamalam et al. 2023) as a robust covariance estimator. This estimator uses a generalization of the Mahalanobis distance for the functional setting (Berrendero et al. 2020) and a corresponding theoretical cutoff value.

Usage

mrct(
  data,
  h = 0.75,
  alpha = 0.01,
  initializations = 5,
  subset.iteration = 10,
  seed = 123,
  scaling.iterations = 10,
  scaling.tolerance = 10^(-4),
  criterion = "sum",
  sum.percentage = 0.75
)

Arguments

Value

A list:

References

Berrendero JR, Bueno-Larraz B, Cuevas A (2020). “On Mahalanobis Distance in Functional Settings.” J. Mach. Learn. Res., 21(9), 1–33..

Oguamalam J, Radojičić U, Filzmoser P (2023). “Minimum regularized covariance trace estimator and outlier detection for functional data.” https://doi.org/10.48550/arXiv.2307.13509..

Examples

# Fix seed for reproducibility
set.seed(123)

# Sample outlying indices
cont.ind <- sample(1:50, size=10)

# Generate 50 curves on the interval [0,1] at 50 timepoints with 20% outliers
y <- mrct.rgauss(x.grid=seq(0,1,length.out=50), N=50, model=1,
                 outliers=cont.ind, method="linear")

# Visualize curves (regular curves grey, outliers black)
colormap <- rep("grey",50); colormap[cont.ind] <- "black"
matplot(x=seq(0,1,length.out=50), y=t(y), type="l", lty="solid",
        col=colormap, xlab="t",ylab="")

# Run MRCT
mrct.y <- mrct(data=y, h=0.75, alpha=0.1,
               initializations=10, criterion="sum")

# Visualize alpha-Mahalanobis distance with cutoff (horizontal black line)
# Colors correspond to simulated outliers, shapes to estimated (MRCT) ones
# (circle regular and triangle irregular curves)
shapemap <- rep(1,50); shapemap[mrct.y$theoretical.w] <- 2
plot(x=1:50, y=mrct.y$aMHD.w, col=colormap, pch=shapemap,
     xlab="Index", ylab=expression(alpha*"-MHD"))
abline(h = mrct.y$quant.w)

# If you dont have any information on possible outliers,
# alternatively you could use the S3 method plot.mrct()
mrct.plot(mrct.y)

Integrated square error

Description

Calculates the approximation of the integrated square error between the estimated covariance based on non-outlying curves of a data set determined by the MRCT estimator and the true kernel for one of the three outlier settings in the simulation study of Oguamalam et al. 2023.

Usage

mrct.ise(data, outliers.est, model)

Arguments

Value

Numeric value containing the approximated integrated square error between estimated and theoretical covariance.

References

Oguamalam J, Radojičić U, Filzmoser P (2023). “Minimum regularized covariance trace estimator and outlier detection for functional data.” https://doi.org/10.48550/arXiv.2307.13509..

Examples

# Fix seed for reproducibility
set.seed(124)

# Sample outlying indices
cont.ind <- sample(1:100,size=10)

# Generate 100 curves on the interval [0,1] at 150 timepoints with 20% outliers.
y <- mrct.rgauss(x.grid=seq(0,1,length.out=150), N=100, model=1,
                 outliers=cont.ind, method="linear")
# Run MRCT
mrct.y <- mrct(data=y, h=0.75, alpha=0.1,
               initializations=10, criterion="sum")
# Two additional curves are regarded as outlying according to the algorithm
mrct.y$theoretical.w %in% cont.ind
# Compare the ISE between true kernel and 1) true non-outliers,
# 2) estimated non-outliers and 3) the complete data
ise1 <- mrct.ise(data=y, outliers.est=cont.ind, model=1)
ise2 <- mrct.ise(data=y, outliers.est=mrct.y$theoretical.w, model=1)
ise3 <- mrct.ise(data=y, outliers.est=c(), model=1)
ise1; ise2; ise3

Plot function for result from mrct()

Description

A function for descriptive plots for an object resulting from a call to mrct().

Usage

mrct.plot(mrct.object)

Arguments

Value

Descriptive plots

Examples

# Similar to example in mrct() helpfile
# Fix seed for reproducibility
set.seed(123)

# Sample outlying indices
cont.ind <- sample(1:50, size=10)

# Generate 50 curves on the interval [0,1] at 50 timepoints with 20% outliers
y <- mrct.rgauss(x.grid=seq(0,1,length.out=50), N=50, model=1,
                 outliers=cont.ind, method="linear")

# Visualize curves (regular curves grey, outliers black)
colormap <- rep("grey",50); colormap[cont.ind] <- "black"
matplot(x=seq(0,1,length.out=50), y=t(y), type="l", lty="solid",
        col=colormap, xlab="t",ylab="")

# Run MRCT
mrct.y <- mrct(data=y, h=0.75, alpha=0.1,
               initializations=10, criterion="sum")

# Visualize alpha-Mahalanobis distance
# Colorinfromation according to estimated outliers (grey regular, black irregular)
mrct.plot(mrct.y)

Random sample from Gaussian process

Description

Generate random samples of Gaussian process on a uniform grid for different settings of the simulation study in Oguamalam et al. 2023.

Usage

mrct.rgauss(
  x.grid,
  N,
  seed = 123,
  model,
  outliers,
  sigma = 1,
  l = 1,
  method = "linear"
)

Arguments

Value

Numeric matrix with N rows and length(x.grid) columns containing the randomly generated curves following a Gaussian process. Each observations is a row of the result.

References

Oguamalam J, Radojičić U, Filzmoser P (2023). “Minimum regularized covariance trace estimator and outlier detection for functional data.” https://doi.org/10.48550/arXiv.2307.13509..

Examples

# Fix seed for reproducibility
set.seed(123)

# Sample outlying indices
cont.ind <- sample(1:50,size=10)

# Generate 50 curves on the interval [0,1] at 50 timepoints with 20% outliers
y <- mrct.rgauss(x.grid=seq(0,1,length.out=50), N=50 ,model=1,
                 outliers=cont.ind)

# Visualize curves (regular curves grey, outliers black)
colormap <- rep("grey",50); colormap[cont.ind] <- "black"
matplot(x=seq(0,1,length.out=50), y=t(y), type="l", lty="solid",
        col=colormap, xlab="t",ylab="")

Sparse minimum regularized covariance trace estimator

Description

Robust outlier detection for sparse functional data as a generalization of the minimum regularized covariance trace (MRCT) estimator (Oguamalam et al. 2023). At first the observations are smoothed by a B-spline basis and afterwards the MRCT algorithm is performed with the matrix of basis coefficients.

Usage

mrct.sparse(
  data,
  nbasis = dim(data)[2],
  new.p = dim(data)[2],
  h = 0.75,
  alpha = 0.01,
  initializations = 5,
  seed = 123,
  scaling.iterations = 10,
  scaling.tolerance = 10^(-4),
  criterion = "sum",
  sum.percentage = 0.75
)

Arguments

Value

A list with two entries

References

Oguamalam J, Radojičić U, Filzmoser P (2023). “Minimum regularized covariance trace estimator and outlier detection for functional data.” https://doi.org/10.48550/arXiv.2307.13509..

Examples

# Fix seed for reproducibility
set.seed(123)

# Sample outlying indices
cont.ind <- sample(1:50,size=10)

# Generate 50 sparse curves on the interval [0,1] at 10 timepoints with 20% outliers
y <- mrct.rgauss(x.grid=seq(0,1,length.out=10), N=50, model=1,
                 outliers=cont.ind, method="linear")

# Visualize curves (regular curves grey, outliers black)
colormap <- rep("grey",50); colormap[cont.ind] <- "black"
matplot(x = seq(0,1,length.out=10), y = t(y), type="l", lty="solid",
        col=colormap, xlab="t",ylab="")

# Run sparse MRCT
sparse.mrct.y <- mrct.sparse(data = y, nbasis = 10, h = 0.75, new.p = 50,
                             alpha = 0.1, initializations = 10, criterion = "sum" )

# Visualize smoothed functions
matplot(x=seq(0,1,length.out=50), y=t(sparse.mrct.y$data.smooth),
        type="l", lty="solid", col=colormap, xlab="t", ylab="")

# Visualize alpha-Mahalanobis distance with cutoff (horizontal black line)
# Colors correspond to simulated outliers, shapes to estimated (sparse MRCT) ones
# (circle regular and triangle irregular curves)
shapemap <- rep(1,50); shapemap[sparse.mrct.y$mrct.output$theoretical.w] <- 2
plot(x = 1:50, y = sparse.mrct.y$mrct.output$aMHD.w, col=colormap, pch = shapemap,
     xlab = "Index", ylab = expression(alpha*"-MHD"))
abline(h = sparse.mrct.y$mrct.output$quant.w)

# If you dont have any information on possible outliers,
# alternatively you could use the S3 method plot.mrctsparse()
mrct.sparse.plot(mrct.sparse.object = sparse.mrct.y)

Plot function for result from mrct.sparse()

Description

A function for descriptive plots for an object resulting from a call to mrct.sparse().

Usage

mrct.sparse.plot(
  x = seq(0, 1, length.out = dim(mrct.sparse.object[[2]])[2]),
  mrct.sparse.object
)

Arguments

Value

Descriptive plots.

Examples

# Similar to example in mrct.sparse() helpfile
# Fix seed for reproducibility
set.seed(123)

# Sample outlying indices
cont.ind <- sample(1:50,size=10)

# Generate 50 sparse curves on the interval [0,1] at 10 timepoints with 20% outliers
y <- mrct.rgauss(x.grid=seq(0,1,length.out=10), N=50, model=1,
                 outliers=cont.ind, method="linear")

# Visualize curves (regular curves grey, outliers black)
colormap <- rep("grey",50); colormap[cont.ind] <- "black"
matplot(x = seq(0,1,length.out=10), y = t(y), type="l", lty="solid",
        col=colormap, xlab="t",ylab="")

# Run sparse MRCT
sparse.mrct.y <- mrct.sparse(data = y, nbasis = 10, h = 0.75, new.p = 50,
                             alpha = 0.1, initializations = 10, criterion = "sum" )

# Visualize alpha-Mahalanobis distances and smoothed curves
# Colorinformation according to estimated outliers (grey regular, black irregular)
mrct.sparse.plot(mrct.sparse.object = sparse.mrct.y)