Package 'NPFD' reference manual

Title:	N-Power Fourier Deconvolution
Description:	Provides tools for non-parametric Fourier deconvolution using the N-Power Fourier Deconvolution (NPFD) method. This package includes methods for density estimation (densprf()) and sample generation (createSample()), enabling users to perform statistical analyses on mixed or replicated data sets.
Authors:	Akin Anarat [aut, cre]
Maintainer:	Akin Anarat <[email protected]>
License:	GPL-3
Version:	1.0.0
Built:	2024-12-05 07:11:19 UTC
Source:	CRAN

Create a Sample from a Centered Distribution

Description

This function creates a sample from a centered distribution based on replicates of mixed data.

Usage

createSample(z1, z2)
createSample(z1, z2)

Arguments

`z1`	A numeric vector where $z_1 = x_1 + y$ .
`z2`	A numeric vector of the same length as $z_1$ where $z_2 = x_2 + y$ .

Value

A numeric vector representing a sample from the centered distribution.

Examples

# Set seed for reproducibility
set.seed(123)

# Generate random data
x1 <- rnorm(1000)
x2 <- rnorm(1000)
y <- rgamma(1000, 10, 2)
z1 <- x1 + y
z2 <- x2 + y

# Use createSample to generate a sample
x <- createSample(z1, z2)

# Perform density estimation
f.x <- stats::density(x, adjust = 1.5)
x.x <- f.x$x
f <- dnorm(x.x)

# Plot the results
plot(NULL, xlim = range(f.x$x), ylim = c(0, max(f, f.x$y)), xlab = "x", ylab = "Density")
lines(x.x, f, col = "blue", lwd = 2)
lines(f.x, col = "orange", lwd = 2)
legend("topright", legend = c(expression(f), expression(f[x])), col = c("blue", "orange"), lwd = 2)
# Set seed for reproducibility
set.seed(123)

# Generate random data
x1 <- rnorm(1000)
x2 <- rnorm(1000)
y <- rgamma(1000, 10, 2)
z1 <- x1 + y
z2 <- x2 + y

# Use createSample to generate a sample
x <- createSample(z1, z2)

# Perform density estimation
f.x <- stats::density(x, adjust = 1.5)
x.x <- f.x$x
f <- dnorm(x.x)

# Plot the results
plot(NULL, xlim = range(f.x$x), ylim = c(0, max(f, f.x$y)), xlab = "x", ylab = "Density")
lines(x.x, f, col = "blue", lwd = 2)
lines(f.x, col = "orange", lwd = 2)
legend("topright", legend = c(expression(f), expression(f[x])), col = c("blue", "orange"), lwd = 2)

N-Power Fourier Deconvolution

Description

Estimates the density $f_y$ , given vectors $x$ and $z$ , where $f_z$ results from the convolution of $f_x$ and $f_y$ .

Usage

deconvolve(
  x = NULL,
  z,
  mode = c("empirical", "denspr"),
  dfx = 5,
  dfz = 5,
  Lx = 10^2,
  Lz = 10^2,
  Ly = 10^2,
  N = 1:100,
  FT.grid = seq(0, 100, 0.1),
  lambda = 1,
  eps = 10^-3,
  delta = 10^-2,
  error = c("unknown", "normal", "laplacian"),
  sigma = NULL,
  calc.error = FALSE,
  plot = FALSE,
  legend = TRUE,
  positive = FALSE
)
deconvolve(
  x = NULL,
  z,
  mode = c("empirical", "denspr"),
  dfx = 5,
  dfz = 5,
  Lx = 10^2,
  Lz = 10^2,
  Ly = 10^2,
  N = 1:100,
  FT.grid = seq(0, 100, 0.1),
  lambda = 1,
  eps = 10^-3,
  delta = 10^-2,
  error = c("unknown", "normal", "laplacian"),
  sigma = NULL,
  calc.error = FALSE,
  plot = FALSE,
  legend = TRUE,
  positive = FALSE
)

Arguments

`x`	Vector of observations for $x$ .
`z`	Vector of observations for $z$ .
`mode`	Deconvolution mode (`empirical` or `denspr`). If `empirical`, the Fourier transforms of $x$ and $z$ are estimated using the empirical form. If `denspr`, they are calculated based on the density estimations using `densprf` (see the package siggenes).
`dfx`	Degrees of freedom for the estimation of $f_x$ if mode is set to `denspr`.
`dfz`	Degrees of freedom for the estimation of $f_z$ if mode is set to `denspr`.
`Lx`	Number of points for $f_x$ -grid if mode is set to `denspr`.
`Lz`	Number of points for $f_z$ -grid if mode is set to `denspr`.
`Ly`	Number of points for $f_y$ -grid.
`N`	Possible power values.
`FT.grid`	Vector of grid for Fourier transformation of $f_x$ and $f_z$ .
`lambda`	Smoothing parameter.
`eps`	Tolerance for convergence.
`delta`	Small margin value.
`error`	Error model (`unknown`, `normal`, `laplacian`). If `unknown`, the Fourier transform of $x$ is calculated based on the mode. If `normal`, the exact form of the Fourier transform of a centered normal distribution with standard deviation sigma is used for $x$ . If `laplacian`, the exact form of the Fourier transform of a centered Laplace distribution with standard deviation sigma is used for $x$ .
`sigma`	Standard deviation for normal or Laplacian error.
`calc.error`	Logical indicating whether to calculate error (10 x ISE between $f_z$ and $f_x * f_y$ ).
`plot`	Logical indicating whether to plot $f_z$ vs. $f_x * f_y$ if `calc.error` is `TRUE`.
`legend`	Logical indicating whether to include a legend in the plot if `calc.error` is `TRUE`.
`positive`	Logical indicating whether to enforce non-negative density estimation.

Value

A list with the following components:

`x`	A vector of $x$ -values of the resulting density estimation.
`y`	A vector of $y$ -values of the resulting density estimation.
`N`	The power used in the deconvolution process.
`error`	The calculated error if `calc.error = TRUE`.

Author(s)

Akin Anarat [email protected]

References

Anarat A., Krutmann, J., and Schwender, H. (2024). A nonparametric statistical method for deconvolving densities in the analysis of proteomic data. Submitted.

Examples

# Deconvolution when mixed data and data from an independent experiment are provided:
set.seed(123)
x <- rnorm(1000)
y <- rgamma(1000, 10, 2)
z <- x + y

f <- function(x) dgamma(x, 10, 2)

independent.x <- rnorm(100)

fy.NPFD <- deconvolve(independent.x, z, calc.error = TRUE, plot = TRUE)
x.x <- fy.NPFD$x
fy <- f(x.x)

# Check power and error values
fy.NPFD$N
fy.NPFD$error

# Plot density functions
plot(NULL, xlim = range(y), ylim = c(0, max(fy, fy.NPFD$y)), xlab = "x", ylab = "Density")
lines(x.x, fy, col = "blue", lwd = 2)
lines(fy.NPFD, col = "orange", lwd = 2)
legend("topright", legend = c(expression(f[y]), expression(f[y]^{NPFD})),
       col = c("blue", "orange"), lwd = c(2, 2))

# For replicated mixed data:
set.seed(123)
x1 <- VGAM::rlaplace(1000, 0, 1/sqrt(2))
x2 <- VGAM::rlaplace(1000, 0, 1/sqrt(2))
y <- rgamma(1000, 10, 2)
z1 <- z <- x1 + y
z2 <- x2 + y

x <- createSample(z1, z2)

fy.NPFD <- deconvolve(x, z, mode = "denspr", calc.error = TRUE, plot = TRUE)
x.x <- fy.NPFD$x
fy <- f(x.x)

# Check power and error values
fy.NPFD$N
fy.NPFD$error

# Plot density functions
plot(NULL, xlim = range(y), ylim = c(0, max(fy, fy.NPFD$y)), xlab = "x", ylab = "Density")
lines(x.x, fy, col = "blue", lwd = 2)
lines(fy.NPFD, col = "orange", lwd = 2)
legend("topright", legend = c(expression(f[y]), expression(f[y]^{NPFD})),
       col = c("blue", "orange"), lwd = c(2, 2))

# When the distribution of x is asymmetric and the sample size is very small:
set.seed(123)
x <- rgamma(5, 4, 2)
y <- rgamma(1000, 10, 2)
z <- x + y

fy.NPFD <- deconvolve(x, z, mode = "empirical", lambda = 2)
x.x <- fy.NPFD$x
fy <- f(x.x)

# Check power value
fy.NPFD$N

# Plot density functions
plot(NULL, xlim = range(y), ylim = c(0, max(fy, fy.NPFD$y)), xlab = "x", ylab = "Density")
lines(x.x, fy, col = "blue", lwd = 2)
lines(fy.NPFD, col = "orange", lwd = 2)
legend("topright", legend = c(expression(f[y]), expression(f[y]^{NPFD})),
       col = c("blue", "orange"), lwd = c(2, 2))
# Deconvolution when mixed data and data from an independent experiment are provided:
set.seed(123)
x <- rnorm(1000)
y <- rgamma(1000, 10, 2)
z <- x + y

f <- function(x) dgamma(x, 10, 2)

independent.x <- rnorm(100)

fy.NPFD <- deconvolve(independent.x, z, calc.error = TRUE, plot = TRUE)
x.x <- fy.NPFD$x
fy <- f(x.x)

# Check power and error values
fy.NPFD$N
fy.NPFD$error

# Plot density functions
plot(NULL, xlim = range(y), ylim = c(0, max(fy, fy.NPFD$y)), xlab = "x", ylab = "Density")
lines(x.x, fy, col = "blue", lwd = 2)
lines(fy.NPFD, col = "orange", lwd = 2)
legend("topright", legend = c(expression(f[y]), expression(f[y]^{NPFD})),
       col = c("blue", "orange"), lwd = c(2, 2))

# For replicated mixed data:
set.seed(123)
x1 <- VGAM::rlaplace(1000, 0, 1/sqrt(2))
x2 <- VGAM::rlaplace(1000, 0, 1/sqrt(2))
y <- rgamma(1000, 10, 2)
z1 <- z <- x1 + y
z2 <- x2 + y

x <- createSample(z1, z2)

fy.NPFD <- deconvolve(x, z, mode = "denspr", calc.error = TRUE, plot = TRUE)
x.x <- fy.NPFD$x
fy <- f(x.x)

# Check power and error values
fy.NPFD$N
fy.NPFD$error

# Plot density functions
plot(NULL, xlim = range(y), ylim = c(0, max(fy, fy.NPFD$y)), xlab = "x", ylab = "Density")
lines(x.x, fy, col = "blue", lwd = 2)
lines(fy.NPFD, col = "orange", lwd = 2)
legend("topright", legend = c(expression(f[y]), expression(f[y]^{NPFD})),
       col = c("blue", "orange"), lwd = c(2, 2))

# When the distribution of x is asymmetric and the sample size is very small:
set.seed(123)
x <- rgamma(5, 4, 2)
y <- rgamma(1000, 10, 2)
z <- x + y

fy.NPFD <- deconvolve(x, z, mode = "empirical", lambda = 2)
x.x <- fy.NPFD$x
fy <- f(x.x)

# Check power value
fy.NPFD$N

# Plot density functions
plot(NULL, xlim = range(y), ylim = c(0, max(fy, fy.NPFD$y)), xlab = "x", ylab = "Density")
lines(x.x, fy, col = "blue", lwd = 2)
lines(fy.NPFD, col = "orange", lwd = 2)
legend("topright", legend = c(expression(f[y]), expression(f[y]^{NPFD})),
       col = c("blue", "orange"), lwd = c(2, 2))

Density Estimation Function

Description

This function estimates the density using a Poisson GLM with natural splines.

Usage

densprf(
  x,
  n.interval = NULL,
  df = 5,
  knots.mode = TRUE,
  type.nclass = c("wand", "scott", "FD"),
  addx = FALSE
)
densprf(
  x,
  n.interval = NULL,
  df = 5,
  knots.mode = TRUE,
  type.nclass = c("wand", "scott", "FD"),
  addx = FALSE
)

Arguments

`x`	Input data vector.
`n.interval`	Number of intervals (optional).
`df`	Degrees of freedom for the splines.
`knots.mode`	Boolean to determine if quantiles should be used for knots.
`type.nclass`	Method for determining number of classes.
`addx`	Add $x$ values (optional).

Details

densprf is a modification of the denspr function from the siggenes package.

For more details, see the documentation in the siggenes package.

Value

The function densprf(x) returns a function that, for a given input z, computes the estimated density evaluated at the position values of z as a result.

Examples

# Set seed for reproducibility
set.seed(123)

# Generate random data
z <- rnorm(1000)

# Apply densprf function
f <- densprf(z)

# Define sequences for evaluation
x1 <- seq(-4, 4, 0.5)
x2 <- seq(-5, 5, 0.1)

# Evaluate the density function at specified points
f1 <- f(x1)
f2 <- f(x2)
# Set seed for reproducibility
set.seed(123)

# Generate random data
z <- rnorm(1000)

# Apply densprf function
f <- densprf(z)

# Define sequences for evaluation
x1 <- seq(-4, 4, 0.5)
x2 <- seq(-5, 5, 0.1)

# Evaluate the density function at specified points
f1 <- f(x1)
f2 <- f(x2)

Package 'NPFD'

Help Index

Create a Sample from a Centered Distribution

Description

Usage

Arguments

Value

Examples

N-Power Fourier Deconvolution

Description

Usage

Arguments

Value

Author(s)

References

Examples

Density Estimation Function

Description

Usage

Arguments

Details

Value

Examples