Package 'FunctionalCalibration'

Functional Data Calibration with Splines

Description

This function performs functional calibration based on the following model:

$A_i(x_m) = \displaystyle \sum_{l=1}^{L} y_{il} \alpha_l(x_m) + e_i(x_m), \quad i = 1,...,I, \quad m = 1,...,M = 2^J$

where the functions $\alpha_l(x)$ are estimated using spline basis functions.

In matrix notation, the model is represented as:

$A = \alpha y + e$

Usage

functional_calibration_splines(data, weights, x, n_functions = 10)

Arguments

data	A matrix $M$ x $I$ where each column represents one sample of the aggregated function — the matrix $A$ in the model.
weights	A matrix $L$ x $I$ representing the weight values associated with each sample — the matrix $y$ in the model.
x	A numeric vector of values at which the function is evaluated.
n_functions	Number of spline basis functions to be used for estimating $\alpha_l(x)$.

Value

The function returns a list containing two objects.

alpha

A matrix with the estimated functional coefficients $\alpha$.

Plots

A list of plot objects, each representing the corresponding function $\alpha_l(x)$.

References

Saraiva, M. A., & Dias, R. (2009). Analise não-parametrica de dados funcionais: uma aplicação a quimiometria (Doctoral dissertation, Master’s thesis, Universidade Estadual de Campinas, Campinas).

Examples

functional_calibration_splines(simulated_data_spl$data,
simulated_data_spl$weights, simulated_data_spl$x)
functional_calibration_splines(simulated_data_spl$data,
simulated_data_spl$weights, simulated_data_spl$x, 12)

---

Functional Data Calibration with Wavelets

Description

This function performs functional calibration based on the following model:

$A_i(x_m) = \displaystyle \sum_{l=1}^{L} y_{il} \alpha_l(x_m) + e_i(x_m), \quad i = 1,...,I, \quad m = 1,...,M = 2^J$

where the functions $\alpha_l(x)$ are estimated using wavelet decomposition.

In matrix notation, the model is represented as:

$A = \alpha y + e$

Usage

functional_calibration_wavelets(
  data,
  weights,
  wavelet = "DaubExPhase",
  method = "bayesian",
  tau = 1,
  p = NULL,
  sigma = NULL,
  MC = TRUE,
  type = "soft",
  singular = FALSE,
  corre = FALSE,
  x = NULL
)

Arguments

data	A matrix $M$ x $I$ where each column represents one sample of the aggregated function — the matrix $A$ in the model.
weights	A matrix $L$ x $I$ representing the weight values associated with each sample — the matrix $y$ in the model.
wavelet	A string indicating the wavelet family to be used in the Discrete Wavelet Transform (DWT).
method	A string specifying the shrinkage method applied to the empirical wavelet coefficients. Options are: "bayesian", "universal", "sure", "probability", or "cv".
tau	A numeric value for the $\tau$ parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.
p	A numeric value for the $p$ parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.
sigma	A numeric value for the $\sigma$ parameter in the Bayesian shrinkage. If NULL, it is estimated from the data.
MC	A logical evaluating to TRUE or FALSE indicating if the integrals in the Bayesian shrinkage are approximated using Monte Carlo simulation.
type	A string indicating whether the thresholding should be "soft" or "hard" (applies only when the method is not "bayesian").
singular	A logical evaluating to TRUE or FALSE indicating if it adds a small constant (1e-10) to the diagonal of $yy^T$ to stabilize the matrix inversion.
corre	A logical evaluating to TRUE or FALSE indicating if the errors of the data are correlated. Only available for "bayesian" and "universal" methods.
x	A numeric vector of values at which the function is evaluated. If NULL, the default is the sequence 1:nrow(data).

Value

The function returns a list containing two objects:

alpha

A matrix with the estimated functional coefficients $\alpha$.

Plots

A list of plot objects, each representing the corresponding function $\alpha_l(x)$.

References

dos Santos Sousa, A. R. (2024). A wavelet-based method in aggregated functional data analysis. Monte Carlo Methods and Applications, 30(1), 19-30.

JOHNSTONE, I. M.; SILVERMAN, B. W. Wavelet Threshold Estimators for Data with Correlated Noise. Journal of the Royal Statistical Society: Series B (Statistical Methodology), v. 59, n. 2, p. 319–351, 1997.

Examples

functional_calibration_wavelets(simulated_data_wav$data, simulated_data_wav$weights)
functional_calibration_wavelets(simulated_data_wav$data, simulated_data_wav$weights,
                                tau = 5, p = 0.95, sigma = 0.1, x = simulated_data_wav$x)
functional_calibration_wavelets(simulated_data_wav$data, simulated_data_wav$weights,
                                method = "universal")
functional_calibration_wavelets(simulated_data_cor$data, simulated_data_cor$weights, corre = TRUE)

---

Aggregated Curve Plot

Description

Generates the plot of the aggregated curve based on the functional coefficients and their corresponding weights. The aggregated curve is computed as:

$A(x) = \displaystyle \sum_{l=1}^{L} y_l \alpha_l(x)$

Usage

plot_aggregated_curve(alpha, weights, title = NULL, x = NULL)

Arguments

alpha	A numeric matrix where each column represents the values of a function $\alpha_l(x)$ evaluated at each point in $x$.
weights	A numeric vector with the weight values corresponding to each function $\alpha_l(x)$.
title	A string specifying the title of the plot.
x	A numeric vector of values at which the function is evaluated. If NULL, the default is the sequence 1:nrow(alpha).

Value

The function returns the plot of the aggregated function.

Examples

plot_aggregated_curve(simulated_data_wav$alphas, c(0.7, 0.3))
plot_aggregated_curve(simulated_data_wav$alphas, c(0.7, 0.3),
                      "Aggregated Curve Example", simulated_data_wav$x)

---

Simulated Data

Description

This is a simulated dataset designed to illustrate the functionalities of the package. It contains 100 samples of aggregated data generated from two functions, $\alpha_1(x)$ and $\alpha_2(x)$, with added Gaussian noise $N(0, 0.1)$.

The functions used in the simulation are:

$\alpha_1(x) = \sin(5x) e^{-x^2} \quad \alpha_2(x) = \begin{cases} -2, & x < 0 \\ 0, & 0 \leq x < 1.5 \\ 3, & x \geq 1.5 \end{cases}$

The simulations were performed over an equally spaced grid of 1024 points in the interval [-1, 2]. These functions were linearly combined using random concentrations to generate the samples, with the addition of Gaussian noise.

Usage

simulated_data

Format

An object of class list of length 4.

Value

data

A data frame with 1024 rows and 100 columns.
Each column represents one sample of the aggregated functions with Gaussian noise $N(0, 0.1)$.

weigths

A data frame with 2 rows and 100 columns.
Each column contains the random concentrations used to aggregate the two functions in each sample.

x

A numeric vector of length 1024.
The grid of x-values used in the simulation, equally spaced from -1 to 2.

alphas

A data frame with 1024 rows and 2 columns.
The true values of the functions $\alpha_1(x)$ and $\alpha_2(x)$ evaluated over the x grid.

---

Simulated Data Correlated

Description

This is a simulated dataset designed to illustrate the functionalities of the package. It contains 100 samples of aggregated data generated from two functions, $\alpha_1(x)$ and $\alpha_2(x)$, with added noise AR(1) process with autoregressive parameter 0.5 and Gaussian process noise distributed as $N(0, 0.0075)$.

The functions used in the simulation are:

$\alpha_1(x) = \sum_{j=1}^{11} h_j \left(1 + \left|\frac{x - x_j}{w_j}\right|\right)^{-4} \quad \alpha_2(x) = \sqrt{x(1-x)}\sin\left(\frac{2\pi(1 + 0.05)}{x + 0.05}\right)$

where $x = (0.1,\,0.13,\,0.15,\,0.23,\,0.25,\,0.40,\,0.44,\,0.65,\,0.76,\,0.78,\,0.81)$,

$h = (4,\,5,\,3,\,4,\,5,\,4.2,\,2.1,\,4.3,\,3.1,\,5.1,\,4.2)$,

$w = (0.005,\,0.005,\,0.006,\,0.01,\,0.01,\,0.03,\,0.01,\,0.01,\,0.005,\,0.008,\,0.005)$.

The simulations were performed over an equally spaced grid of 1024 points in the interval [0, 1]. These functions were linearly combined using random concentrations to generate the samples, with the addition of AR(1) noise.

Usage

simulated_data_cor

Format

An object of class list of length 4.

Value

data

A data frame with 1024 rows and 100 columns.
Each column represents one sample of the aggregated functions with AR(1) noise.

weigths

A data frame with 2 rows and 100 columns.
Each column contains the random concentrations used to aggregate the two functions in each sample.

x

A numeric vector of length 1024.
The grid of x-values used in the simulation, equally spaced from 0 to 1.

alphas

A data frame with 1024 rows and 2 columns.
The true values of the functions $\alpha_1(x)$ and $\alpha_2(x)$ evaluated over the x grid.

---

Simulated Data Splines

Description

This is a simulated dataset designed to illustrate the functionalities of the package. It contains 100 samples of aggregated data generated from two functions, $\alpha_1(x)$ and $\alpha_2(x)$, with added Gaussian noise $N(0, 0.01)$.

The functions used in the simulation are:

$\alpha_1(x) = \sin(5x) e^{-x^2} \quad \alpha_2(x) = \frac{1.2 \log(1 + 9x)}{\log(10)} e^{-x}$

The simulations were performed over an equally spaced grid of 1024 points in the interval [0, 1]. These functions were linearly combined using random concentrations to generate the samples, with the addition of Gaussian noise.

Usage

simulated_data_spl

Format

An object of class list of length 4.

Value

data

A data frame with 1024 rows and 100 columns.
Each column represents one sample of the aggregated functions with Gaussian noise.

weigths

A data frame with 2 rows and 100 columns.
Each column contains the random concentrations used to aggregate the two functions in each sample.

x

A numeric vector of length 1024.
The grid of x-values used in the simulation, equally spaced from 0 to 1.

alphas

A data frame with 1024 rows and 2 columns.
The true values of the functions $\alpha_1(x)$ and $\alpha_2(x)$ evaluated over the x grid.

---

Simulated Data Wavelets

Description

This is a simulated dataset designed to illustrate the functionalities of the package. It contains 100 samples of aggregated data generated from two functions, $\alpha_1(x)$ and $\alpha_2(x)$, with added Gaussian noise $N(0, 0.01)$.

The functions used in the simulation are:

$\alpha_1(x) = \sum_{j=1}^{11} h_j \left(1 + \left|\frac{x - x_j}{w_j}\right|\right)^{-4} \quad \alpha_2(x) = \sqrt{x(1-x)}\sin\left(\frac{2\pi(1 + 0.05)}{x + 0.05}\right)$

where $x = (0.1,\,0.13,\,0.15,\,0.23,\,0.25,\,0.40,\,0.44,\,0.65,\,0.76,\,0.78,\,0.81)$,

$h = (4,\,5,\,3,\,4,\,5,\,4.2,\,2.1,\,4.3,\,3.1,\,5.1,\,4.2)$,

$w = (0.005,\,0.005,\,0.006,\,0.01,\,0.01,\,0.03,\,0.01,\,0.01,\,0.005,\,0.008,\,0.005)$.

The simulations were performed over an equally spaced grid of 1024 points in the interval [0, 1]. These functions were linearly combined using random concentrations to generate the samples, with the addition of Gaussian noise.

Usage

simulated_data_wav

Format

An object of class list of length 4.

Value

data

A data frame with 1024 rows and 100 columns.
Each column represents one sample of the aggregated functions with Gaussian noise.

weigths

A data frame with 2 rows and 100 columns.
Each column contains the random concentrations used to aggregate the two functions in each sample.

x

A numeric vector of length 1024.
The grid of x-values used in the simulation, equally spaced from 0 to 1.

alphas

A data frame with 1024 rows and 2 columns.
The true values of the functions $\alpha_1(x)$ and $\alpha_2(x)$ evaluated over the x grid.

---

Weight Estimation

Description

Estimates the weights associated with the functional coefficients $\alpha_l(x)$ using the using Ordinary Least Squares.

The problem can be formulated as:

$A(x) = \displaystyle \sum_{l=1}^{L} y_l \alpha_l(x)$

where $A(x)$ is the aggregated function evaluated at each point $x$, $\alpha_l(x)$ are the functional coefficients, and $y_l$ are the weights to be estimated.

Usage

weight_estimation(data, alpha)

Arguments

data	A numeric vector representing one sample of the aggregated function $A(x)$, evaluated at a grid of points $x$.
alpha	A numeric matrix where each column represents the values of a function $\alpha_l(x)$ evaluated at the same grid of points as data.

Value

The function returns a vector with the estimated weights obtained using Ordinary Least Squares.

Examples

weight_estimation(simulated_data_wav$data[,1], simulated_data_wav$alphas)

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