Package 'dsdp' reference manual

Title:	Density Estimation with Semidefinite Programming
Description:	The models of probability density functions are Gaussian or exponential distributions with polynomial correction terms. Using a maximum likelihood method, 'dsdp' computes parameters of Gaussian or exponential distributions together with degrees of polynomials by a grid search, and coefficient of polynomials by a variant of semidefinite programming. It adopts Akaike Information Criterion for model selection. See a vignette for a tutorial and more on our 'Github' repository <https://github.com/tsuchiya-lab/dsdp/>.
Authors:	Satoshi Kakihara [aut, cre], Takashi Tsuchiya [aut]
Maintainer:	Satoshi Kakihara <[email protected]>
License:	MIT + file LICENSE
Version:	0.1.1
Built:	2024-11-24 06:46:54 UTC
Source:	CRAN

Cumulative distribution function of Expomemtial-based model

Description

A cumulative distribution function(CDF) of Exponential-based model. To access parameters and coefficients in an object emodel of a class expmodel, use emodel$result[k, "lmd1"], emodel$coeffs[[k]] for some index k. This index appears in the leftmost column of estimation table generated by summary(emodel).

Usage

cdf_expmodel(coeff, lmd, x)
cdf_expmodel(coeff, lmd, x)

Arguments

`coeff`	A coefficient vector in increasing order of degrees; the first element is 0th degree, ..., and last element is the largest degree of coefficients.
`lmd`	A rate parameter, which is positive.
`x`	A numeric vector of input.

Value

A numeric vector of CDF of an exponential-based model.

Examples

## Create an object of `expmodel`
emodel <- expmodel(mixexpgamma$n200)
## Estimate with degree 4 and rate parameter 2.0
emodel <- estimate(emodel, 4, 2.0)
## Input vector
x <- seq(0, 12, 0.1)
## Output of PDF in above estimation
yv <- cdf_expmodel(emodel$coeffs[[1]], emodel$result[1, "lmd1"], x)
## Create an object of `expmodel`
emodel <- expmodel(mixexpgamma$n200)
## Estimate with degree 4 and rate parameter 2.0
emodel <- estimate(emodel, 4, 2.0)
## Input vector
x <- seq(0, 12, 0.1)
## Output of PDF in above estimation
yv <- cdf_expmodel(emodel$coeffs[[1]], emodel$result[1, "lmd1"], x)

Cumulative distribution function of Gaussian-based model

Description

A cumulative distribution function(CDF) of Gaussian-based model. To access parameters and coefficients in an object gmodel of a class gaussmodel, use gmodel$result[k, "mu1"], gmodel$result[k, "sig1"], gmodel$coeffs[[k]] for some index k. This index appears in the leftmost column of estimation table generated by summary(gmodel).

Usage

cdf_gaussmodel(coeff, mu, sig, x)
cdf_gaussmodel(coeff, mu, sig, x)

Arguments

`coeff`	A coefficient vector in increasing order of degrees; the first element is 0th degree, ..., and last element is the largest degree of coefficients.
`mu`	A mean of Gaussian distribution.
`sig`	A standard deviation of Gaussian distribution, which is positive.
`x`	A numeric input vector.

Value

A numeric vector of CDF of Gaussian-based model.

Examples

## Create an object of `gaussmodel`
gmodel <- gaussmodel(mix2gauss$n200)
## Estimate with a degree 6, a mean 0, and standard deviations 0.5
gmodel <- estimate(gmodel, 6, 0, 0.5)
## Input vector
x <- seq(-3, 3, 0.1)
## Output of PDF in above estimation
yv <- cdf_gaussmodel(gmodel$coeffs[[1]], gmodel$result[1, "mu1"],
 gmodel$result[1, "sig1"], x)
## Create an object of `gaussmodel`
gmodel <- gaussmodel(mix2gauss$n200)
## Estimate with a degree 6, a mean 0, and standard deviations 0.5
gmodel <- estimate(gmodel, 6, 0, 0.5)
## Input vector
x <- seq(-3, 3, 0.1)
## Output of PDF in above estimation
yv <- cdf_gaussmodel(gmodel$coeffs[[1]], gmodel$result[1, "mu1"],
 gmodel$result[1, "sig1"], x)

Reduce a data set to representatives of bins and their frequencies

Description

Reduce a data set to a named list of 'values' and 'freq', which are representatives of bins and their frequencies, respectively.

Usage

databinning(data, bins = 40)
databinning(data, bins = 40)

Arguments

`data`	A numeric vector of a data set.
`bins`	A positive integer to represent the number of bins.

Value

A named list of values and freq whose length is bins.

Examples

rlst <- databinning(mix2gauss$n200)
rlst <- databinning(mix2gauss$n200)

Compute the mean and the standard deviation of a data set

Description

Compute the mean and the standard deviation of a data set represented by the pair of the numeric vectors data and optionally its frequency vector freq.

Usage

datastats(data, freq = NULL)
datastats(data, freq = NULL)

Arguments

`data`	A numeric vector of a data set.
`freq`	A frequency vector corresponding to the `data` vector. The default value is `NULL`, which means all frequencies are one.

Value

The mean and the standard deviation of a data set.

Examples

## Without a frequency data
datastats(mix2gauss$n200)
## With a frequency data
datastats(mix2gaussHist$n200p, mix2gaussHist$n200f)
## Without a frequency data
datastats(mix2gauss$n200)
## With a frequency data
datastats(mix2gaussHist$n200p, mix2gaussHist$n200f)

dsdp: Density Estimation using Semidefinite Programming

Description

Density estimation with Semidefinite Programming. The models of probability density functions are Gaussian or exponential distributions with polynomial correction terms. Using a maximum likelihood method, it computes parameters of Gaussian or exponential distributions together with degrees of polynomials by a grid search, and coefficients of polynomials by a variant of semidefinite programming. It adopts Akaike Information Criterion for model selection. See vignettes for tutorials and more information.

Generic Method for estimation

Description

This is a generic S3 method for estimation.

Usage

estimate(model, ...)
estimate(model, ...)

Arguments

`model`	An instance of a class model to be estimated.
`...`	additional arguments affecting the `estimate` produced.

Value

An instance of a model with estimated data.

Estimate Exponential-based model `expmodel`

Description

Estimates Exponential-based model expmodel among parameter vectors, deglist, lmdlist. Then it sorts the results by AIC.

Usage

## S3 method for class 'expmodel'
estimate(
  model,
  deglist = deglist,
  lmdlist = lmdlist,
  recompute = FALSE,
  stepsize = NULL,
  verbose = FALSE,
  ...
)
## S3 method for class 'expmodel'
estimate(
  model,
  deglist = deglist,
  lmdlist = lmdlist,
  recompute = FALSE,
  stepsize = NULL,
  verbose = FALSE,
  ...
)

Arguments

`model`	An object of `expmodel` class.
`deglist`	A vector of degrees of polynomials. The element should be positive integers.
`lmdlist`	A vector of rate parameters of Exponential-based models. The element should be larger than 0.
`recompute`	If `TRUE`, recomputes the results for better estimation and accuracy. Parameters whose accuracies had been already attained sufficiently, namely around 1.0e-6, are not included in candidates for recomputing.
`stepsize`	A vector in descending order whose values are between 0 and 1. If a small step size is supplied, it can attain successful estimates, but it might take more iterations.
`verbose`	If `TRUE`, it shows the detailed message of SDP solver.
`...`	Arguments to be passed to or from other methods.

Value

A expmodel object including the estimates. Those estimates are stored in model$result with data.frame format and model$coeffs in list format.

Examples

## Create an expmodel object
emodel <- expmodel(mixexpgamma$n200)
## Estimate a model with parameters
emodel <- estimate(emodel, deglist=c(4,5), lmdlist=c(0.5, 1, 2))
## Create an expmodel object
emodel <- expmodel(mixexpgamma$n200)
## Estimate a model with parameters
emodel <- estimate(emodel, deglist=c(4,5), lmdlist=c(0.5, 1, 2))

Estimate Gaussian-based model `gaussmodel`

Description

Estimates Gaussian-based model gaussmodel among parameter vectors, deglist, mulist, sdlist. Then it sorts the results by AIC.

Usage

## S3 method for class 'gaussmodel'
estimate(
  model,
  deglist = deglist,
  mulist = mulist,
  sdlist = sdlist,
  scaling = FALSE,
  recompute = FALSE,
  stepsize = NULL,
  verbose = FALSE,
  ...
)
## S3 method for class 'gaussmodel'
estimate(
  model,
  deglist = deglist,
  mulist = mulist,
  sdlist = sdlist,
  scaling = FALSE,
  recompute = FALSE,
  stepsize = NULL,
  verbose = FALSE,
  ...
)

Arguments

`model`	An object of a `gaussmodel` class.
`deglist`	A vector of degrees of polynomials. The element should be positive even numbers.
`mulist`	A vector of means for Gaussian-based models.
`sdlist`	A vector of standard deviations for Gaussian-based models. The element should be larger than 0.
`scaling`	A logical scalar, which indicates whether or not it scales means and standard deviations in `mulist` and `sdlist`. The default value is `FALSE`.
`recompute`	If `TRUE`, recomputes the results for better estimation and accuracy. Parameters whose accuracies had been already attained sufficiently, namely around 1.0e-6, are not included in candidates for recomputing.
`stepsize`	A vector in descending order whose values are between 0 and 1. If a small step size is supplied, it can attain successful estimates, but it might take more iterations.
`verbose`	If `TRUE`, it shows the detailed message of SDP solver.
`...`	Arguments to be passed to or from other methods.

Value

A gaussmodel object including the estimates. Those estimates are stored in model$result with data.frame format and model$coeffs in list format.

Examples

## Create an `gaussmodel` object
gmodel <- gaussmodel(mix2gauss$n200)
## Estimate a model with parameters
gmodel <- estimate(gmodel, deglist=c(2, 4), mulist=c(0.0, 0.2),
             sdlist=c(0.75, 1.0))
## Create an `gaussmodel` object
gmodel <- gaussmodel(mix2gauss$n200)
## Estimate a model with parameters
gmodel <- estimate(gmodel, deglist=c(2, 4), mulist=c(0.0, 0.2),
             sdlist=c(0.75, 1.0))

Evaluate a polynomial

Description

Evaluate the polynomial whose coefficients are represented in coeff vector. The order of the coefficient is an increasing order, i.e., coeff[1] is a constant term, and coeff[2] is a coefficient of 1st degree term, etc. Evaluation is done using Horner's method.

Usage

eval_poly(coeff, x)
eval_poly(coeff, x)

Arguments

`coeff`	A coefficient vector in increasing order of degrees; the first element is 0th degree, ..., and the last element is the largest degree of coefficients.
`x`	A numeric input vector.

Value

A vector of values of a polynomial whose coefficient is coeff.

Examples

## Evaluate a polynomial x^2 - 2x + 2 with x = 1, 2, 3.
## 0th, 1st, 2nd degree of coefficients
coeff <- c(2, -2, 1)
x <- c(1, 2, 3)
eval_poly(coeff, x)
## Evaluate a polynomial x^2 - 2x + 2 with x = 1, 2, 3.
## 0th, 1st, 2nd degree of coefficients
coeff <- c(2, -2, 1)
x <- c(1, 2, 3)
eval_poly(coeff, x)

Estimate coefficients of a polynomial in Exponential-based Model

Description

Estimate coefficients of a polynomial in Exponential-based model:

$\mathrm{poly}(x; \alpha) \mathrm{Exp}(x; \lambda)$

, where $\alpha$ is a coefficient vector, $\lambda$ is a rate parameter of an exponential distribution:

$\mathrm{Exp}(x; \lambda) := \lambda e^{-\lambda x}$

Using data and optionally its frequencies freq, and a degree of a polynomial, a rate parameter lmd of an exponential distribution, it computes the coefficients of polynomial, along with Akaike Information Criterion(AIC) and an accuracy information from underlying SDP solver. In general, the smaller the AIC is, the better the model is. An accuracy around 1e-7 is a good indication for a computational result of coefficients estimation.

Usage

exp_est(deg, lmd, data, freq, verbose, stepvec)
exp_est(deg, lmd, data, freq, verbose, stepvec)

Arguments

`deg`	A degree of polynomial, which is positive even integer.
`lmd`	A rate parameter of an exponential distribution, which is positive.
`data`	A numeric vector of a data set to be estimated.
`freq`	A numeric vector of frequencies for a data set `data`. The default value is `NULL`, which indicates that all frequencies are equally one. If `freq` is not `NULL`, then it should be the same length as `data`, and all values should be positive integers.
`verbose`	If `TRUE`, it shows a detail information about SDP solver.
`stepvec`	It designates the stepsize for SDP solver. If the problem is easy, i.e., the number of a data set are small and a degree of a polynomial is small, then, for example, `0.9` might be ok. If it looks difficult, then `c(0.5, 0.3)` might work.

Value

A list of ⁠deg, lmd, aic, accuracy, coefficient vector⁠

Examples

rlst <- exp_est(3, 1.0, mixexpgamma$n200, NULL, FALSE, c(0.7, 0.4))
rlst <- exp_est(3, 1.0, mixexpgamma$n200, NULL, FALSE, c(0.7, 0.4))

Constructor for S3 class `expmodel`

Description

This function is a constructor for S3 class expmodel, which represents Exponential-based model. It usually takes data and optionally freq as arguments and also optionally stepsize Members of interest in practice are result and coeffs, which maintain the information of estimates and coefficients of polynomials, respectively.

Usage

expmodel(data = data, freq = NULL, stepsize = c(0.5, 0.3))
expmodel(data = data, freq = NULL, stepsize = c(0.5, 0.3))

Arguments

`data`	A nonnegative numeric vector of a data set to be estimated.
`freq`	A frequency vector corresponding to the `data` vector. The default value is `NULL`, which means all frequencies are one. If supplied, the length of a vector should be same as `data` and each element should be a nonnegative integer.
`stepsize`	A numeric vector whose element is larger than 0 and smaller than 1, and decreasing order. The default value is `c(0.5, 0.3)`. If you encounter numerical difficulties, decreasing its values, for example, to `c(0.4, 0.2)`, might help to estimate a model.

Value

An object of Exponential-based model expmodel.

Examples

## Create `expmodel` object from a data set `mixexpgamma$n200`.
 emodel <- expmodel(mixexpgamma$n200)
## Create `expmodel` object from a data set `mixExpGammaHist$n800p` and
## its frequencies `mixExpGammaHist$n800f`.
 emodel <- expmodel(mixExpGammaHist$n800p, mixExpGammaHist$n800f)
## Create `expmodel` object from a data set `mixexpgamma$n200`.
 emodel <- expmodel(mixexpgamma$n200)
## Create `expmodel` object from a data set `mixExpGammaHist$n800p` and
## its frequencies `mixExpGammaHist$n800f`.
 emodel <- expmodel(mixExpGammaHist$n800p, mixExpGammaHist$n800f)

Generic Method for evaluate the estimate

Description

This is a generic S3 method for estimate.

Usage

func(model, x, ...)
func(model, x, ...)

Arguments

`model`	An instance of a class model to be evaluated.
`x`	A numeric vector for input.
`...`	additional arguments affecting the `func` produced.

Value

An evaluation of x with model.

Return the evaluation of a vector with Exponential-based model

Description

Evaluate an input vector x with Exponential-based model and return its vector. By default, it evaluate with the best model and its density, but it can designate the model by index and also can evaluate with a cumulative distribution.

Usage

## S3 method for class 'expmodel'
func(model, x, cdf = FALSE, n = 1, ...)
## S3 method for class 'expmodel'
func(model, x, cdf = FALSE, n = 1, ...)

Arguments

`model`	`expmodel` object.
`x`	A numeric vector to be evaluated with a distribution.
`cdf`	A logical scalar whether the evaluation is done with a cumulative distribution or not. A default value is `FALSE`, which means that the evaluation is done with a density.
`n`	The index indicates the estimates. 1, by default, is the best estimate, and 2 is the 2nd best, etc.
`...`	Arguments to be passed to or from other methods.

Value

A numeric vector of the evaluatio of input vector x with a model.

Examples

## Create an `expmodel` object
emodel <- expmodel(mixexpgamma$n200)
## Estimate an model with parameters
emodel <- estimate(emodel, deglist=5, lmdlist=3.75)
## A vector for input
x <- seq(0, 14, by=0.1)
## Density function
y <- func(emodel, x)
## Cumulative distribution
y <- func(emodel, x, cdf=TRUE)
## Create an `expmodel` object
emodel <- expmodel(mixexpgamma$n200)
## Estimate an model with parameters
emodel <- estimate(emodel, deglist=5, lmdlist=3.75)
## A vector for input
x <- seq(0, 14, by=0.1)
## Density function
y <- func(emodel, x)
## Cumulative distribution
y <- func(emodel, x, cdf=TRUE)

Return the evaluation of a vector with Gaussian-based model

Description

Evaluate an input vector x with Gaussian-based model and return its vector. By default, it evaluate with the best model and its density, but it can designate the model by index and also can evaluate with a cumulative distribution.

Usage

## S3 method for class 'gaussmodel'
func(model, x, cdf = FALSE, n = 1, scaling = FALSE, ...)
## S3 method for class 'gaussmodel'
func(model, x, cdf = FALSE, n = 1, scaling = FALSE, ...)

Arguments

`model`	`gaussmodel` object.
`x`	A numeric vector to be evaluated with a distribution.
`cdf`	A logical scalar whether the evaluation is done with a cumulative distribution or not. A default value is `FALSE`, which means that the evaluation is done with a density.
`n`	The index indicates the estimates. 1, by default, is the best estimate, and 2 is the 2nd best, etc.
`scaling`	A logical scalar, which indicates whether or not it scales means and standard deviations in `mulist` and `sdlist`. The default value is `FALSE`.
`...`	Arguments to be passed to or from other methods.

Value

A numeric vector of the evaluatio of input vector x with a model.

Examples

## Create an `gaussmodel` object
gmodel <- gaussmodel(mix2gauss$n200)
## Estimate an model with parameters
gmodel <- estimate(gmodel, deglist=4, mulist=0.15, sdlist=0.73)
## A vector for input
x <- seq(-4, 4, by=0.1)
## Density function
y <- func(gmodel, x)
## Cumulative distribution
y <- func(gmodel, x, cdf=TRUE)
## Create an `gaussmodel` object
gmodel <- gaussmodel(mix2gauss$n200)
## Estimate an model with parameters
gmodel <- estimate(gmodel, deglist=4, mulist=0.15, sdlist=0.73)
## A vector for input
x <- seq(-4, 4, by=0.1)
## Density function
y <- func(gmodel, x)
## Cumulative distribution
y <- func(gmodel, x, cdf=TRUE)

Estimate coefficients of a polynomial in Gaussian-based model

Description

Estimate coefficients of a polynomial in Gaussian-based model:

$\mathrm{poly}(x, \alpha) N(x; \mu, \sigma^2)$

, where $\alpha$ is a coefficient vector, $\mu$ and $\sigma$ are a mean and a standard deviation of Gaussian distribution:

$N(x; \mu, \sigma^2) :=\frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$

Using data and optionally its frequencies freq, and a degree of a polynomial, a mean mu and a standard deviation sig of Gausian distribution, it computes the coefficients of a polynomial, along with Akaike Information Criterion(AIC) and an accuracy information from an underlying SDP solver. In general, the smaller the AIC is, the better the model is. An accuracy around 1e-7 is a good indication for a computational result of coefficients estimation.

Usage

gauss_est(deg, mu, sig, data, freq, verbose, stepsize)
gauss_est(deg, mu, sig, data, freq, verbose, stepsize)

Arguments

`deg`	A degree of polynomial, which is positive even integer.
`mu`	A mean of Gaussian distribution.
`sig`	A standard deviation of Gaussian distribution, which is positive.
`data`	A numeric vector of a data set to be estimated.
`freq`	A numeric vector of frequencies for a data set `data`. The default value is `NULL`, which indicates that all frequencies are equally one. If `freq` is not `NULL`, then it should be the same length as `data`, and all values should be positive integers.
`verbose`	If `TRUE`, it shows a detail information about SDP solver.
`stepsize`	It designates the stepsize for SDP solver. If the problem is easy, i.e., the number of a data set are small and a degree of a polynomial is small, then, for example, `0.9` might be ok. If it looks difficult, then `c(0.5, 0.3)` might work.

Value

A list of deg, mu, sig, aic, accuracy, ⁠coefficient vector⁠.

Examples

rlst <- gauss_est(4, 0, 1, mix2gauss$n200, NULL, FALSE, c(0.7, 0.4))
rlst <- gauss_est(4, 0, 1, mix2gauss$n200, NULL, FALSE, c(0.7, 0.4))

Constructor for S3 class `gaussmodel`

Description

This function is a constructor for S3 class gaussmodel, which represents Gaussian-based model. It usually takes data and optionally freq as arguments and also optionally stepsize. Members of interest in practice are result and coeffs, which maintain the information of estimates and coefficients of polynomials, respectively.

Usage

gaussmodel(data = data, freq = NULL, stepsize = c(0.5, 0.3))
gaussmodel(data = data, freq = NULL, stepsize = c(0.5, 0.3))

Arguments

`data`	A numeric vector of a data set to be estimated.
`freq`	A frequency vector corresponding to the `data` vector. The default value is `NULL`, which means all frequencies are one. If supplied, the length of a vector should be same as `data` and each element should be a nonnegative integer.
`stepsize`	A numeric vector whose element is larger than 0 and smaller than 1, and decreasing order. The default value is `c(0.5, 0.3)`. If you encounter numerical difficulties, decreasing its values, for example, to `c(0.4, 0.2)`, might help to estimate a model.

Value

An object of Gaussian-based model gaussmodel.

Examples

## Create `gaussmodel` object from a data set `mix2gauss$n200`.
 gmodel <- gaussmodel(mix2gauss$n200)
## Create `gaussmodel` object from a data set `mix2gaussHist$n200p` and
## its frequencies `mix2gaussHist$n200f`.
 gmodel <- gaussmodel(mix2gaussHist$n200p, mix2gaussHist$n200f)
## Create `gaussmodel` object from a data set `mix2gauss$n200`.
 gmodel <- gaussmodel(mix2gauss$n200)
## Create `gaussmodel` object from a data set `mix2gaussHist$n200p` and
## its frequencies `mix2gaussHist$n200f`.
 gmodel <- gaussmodel(mix2gaussHist$n200p, mix2gaussHist$n200f)

Compute the mean of a data set

Description

Compute the mean of a data set represented by the pair of the numeric vectors data and optionally its frequency vector freq.

Usage

histmean(data, freq = NULL)
histmean(data, freq = NULL)

Arguments

`data`	A numeric vector of a data set.
`freq`	A frequency vector corresponding to the `data` vector. The default value is `NULL`, which means all frequencies are one.

Value

The mean of a data set.

Examples

## Without a frequency data
histmean(mix2gauss$n200)
## With a frequency data
histmean(mix2gaussHist$n200p, mix2gaussHist$n200f)
## Without a frequency data
histmean(mix2gauss$n200)
## With a frequency data
histmean(mix2gaussHist$n200p, mix2gaussHist$n200f)

Incomplete Gamma Function

Description

Evaluate an incomplete gamma function:

$\gamma(a, x) = \int_{0}^{x} t^{a-1} e^{-t}dt,$

using SLATEC dgami in https://netlib.org/slatec/. When ( $x > 0$ and $a \ge 0$ ) or ( $x \ge 0$ and $a > 0$ ), compute the result, otherwise the value is NaN.

Usage

igamma(a, x)
igamma(a, x)

Arguments

`a`	A positive numeric vector.
`x`	A nonnegative numeric vector with same length as `a`. length.

Value

A vector of values of an incomplete gamma function.

Examples

igamma(1, 1)
igamma(1, 1)

Complementary Incomplete Gamma Function

Description

Evaluate an complementary incomplete gamma function:

$\gamma^{*}(a, x) = \int_{x}^{\infty} t^{a-1} e^{-t}dt,$

using SLATEC dgamic in https://netlib.org/slatec/. When ( $x > 0$ and $a \ge 0$ ) or ( $x \ge 0$ and $a > 0$ ), compute the result, otherwise the value is NaN.

Usage

igammac(a, x)
igammac(a, x)

Arguments

`a`	A numeric vector.
`x`	A nonnegative numeric vector with same length as `a`.

Value

A vector of values of a complementary incomplete gamma function.

Examples

igammac(1, 1)
igammac(1, 1)

Datasets of Mixture of 2 Gaussian Distributions

Description

Dataset generated by mixture of 2 Gaussian Distributions whose density is:

$\frac{0.3}{\sqrt{2 \pi 0.5^2}} \exp\left(\frac{(x+1)^2}{2 \cdot 0.5^2}\right) + \frac{0.7}{\sqrt{2 \pi 0.5^2}} \exp\left(\frac{(x-1)^2}{2 \cdot 0.5^2}\right)$

Usage

mix2gauss
mix2gauss

Format

A list of numeric vectors of Bimodal Gaussian Mixture Model.

n200: A numeric vector with 200 elements.
n400: A numeric vector with 400 elements.
n600: A numeric vector with 600 elements.
n800: A numeric vector with 800 elements.
n1000: A numeric vector with 1000 elements.
n1200: A numeric vector with 1200 elements.

Source

mix2gauss_gen()

A density function of mixed Gaussian distributions

Description

A density function of mixed Gaussian distributions whose density is:

$\frac{0.3}{\sqrt{2 \pi 0.5^2}} \exp\left(\frac{(x+1)^2}{2 \cdot 0.5^2}\right) + \frac{0.7}{\sqrt{2 \pi 0.5^2}} \exp\left(\frac{(x-1)^2}{2 \cdot 0.5^2}\right)$

Usage

mix2gauss_fun(x)
mix2gauss_fun(x)

Arguments

`x`	A numeric vector for arguments of a density function.

Value

A numeric vector of probabilities for a given argument x.

Generate mixed Gaussian random numbers

Description

Generate mix gaussian random numbers whose density is:

$\frac{0.3}{\sqrt{2 \pi 0.5^2}} \exp\left(\frac{(x+1)^2}{2 \cdot 0.5^2}\right) + \frac{0.7}{\sqrt{2 \pi 0.5^2}} \exp\left(\frac{(x-1)^2}{2 \cdot 0.5^2}\right)$

Usage

mix2gauss_gen(n = 100, seed = NULL)
mix2gauss_gen(n = 100, seed = NULL)

Arguments

`n`	The number of random numbers.
`seed`	A seed for random number generator.

Value

A numeric vector of random numbers whose a density is described in Description.

Dataset of Mixture of 2 Gaussian Distributions: Histogram version

Description

Dataset of Mixture of 2 Gaussian Distributions, histogram version, whose density is:

$\frac{0.3}{\sqrt{2 \pi 0.5^2}} \exp\left(\frac{(x+1)^2}{2 \cdot 0.5^2}\right) + \frac{0.7}{\sqrt{2 \pi 0.5^2}} \exp\left(\frac{(x-1)^2}{2 \cdot 0.5^2}\right)$

Usage

mix2gaussHist
mix2gaussHist

Format

A list of numeric vectors of Bimodal Gaussian Mixture Model.

n200: A numeric vector with 200 elements.
n200p: Histogram sample data with 25 bins.
n200f: Histogram frequency data with 25 bins.
n400: A numeric vector with 400 elements.
n400p: Histogram sample data with 50 bins.
n400f: Histogram frequency data with 50 bins.
n800: A numeric vector with 800 elements.
n800p: Histogram sample data with 100 bins.
n800f: Histogram frequency data with 100 bins.

Source

mix2gauss_gen() databinning()

Datasets of Mixture of 3 Gaussian Distributions

Description

Datasets generated by Mixture of 3 Gaussian Distributions whose density is proportional to:

$\exp(\frac{x^2}{2}) + 5\exp(\frac{(x-1)^2}{0.2}) + 3\exp(\frac{(x-1)^2}{0.5}).$

Usage

mix3gauss
mix3gauss

Format

A list of numeric vectors of Unimodal Gaussian Asymmetric Mixture Model.

n200: A numeric vector with 200 elements.
n400: A numeric vector with 400 elements.
n600: A numeric vector with 600 elements.
n800: A numeric vector with 800 elements.
n1000: A numeric vector with 1000 elements.
n1200: A numeric vector with 1200 elements.

Source

mix3gauss_gen()

A density function of mixed gaussian distribution

Description

A density function proportional to:

$\exp(\frac{x^2}{2}) + 5\exp(\frac{(x-1)^2}{0.2}) + 3\exp(\frac{(x-1)^2}{0.5}).$

Usage

mix3gauss_fun(x)
mix3gauss_fun(x)

Arguments

`x`	A numeric vector for arguments of a density function.

Value

A numeric vector of probabilities for a given argument x.

Generate Mixed Gaussian Random Numbers

Description

A random number generator whose density function is proportional to:

$\exp(\frac{x^2}{2}) + 5\exp(\frac{(x-1)^2}{0.2}) + 3\exp(\frac{(x-1)^2}{0.5}).$

Usage

mix3gauss_gen(n = 100, seed = NULL)
mix3gauss_gen(n = 100, seed = NULL)

Arguments

`n`	The number of random numbers.
`seed`	A seed for random number generator.

Value

A numeric vector of probabilities for a given argument x.

Dataset of Mixture of Exponential Distribution and Gamma Distribution

Description

Dataset of mixture of exponential distribution and gamma distribution whose density is:

$0.2(2 e^{-2x}) + 0.8 \frac{x^3}{3!}e^{-x}.$

Usage

mixexpgamma
mixexpgamma

Format

A list of numeric vectors of Mixture of Exponential and Gamma distribution Model.

n200: A numeric vector with 200 elements.
n400: A numeric vector with 400 elements.
n600: A numeric vector with 600 elements.
n800: A numeric vector with 800 elements.
n1000: A numeric vector with 1000 elements.
n1200: A numeric vector with 1200 elements.

Source

mixexpgamma_gen()

A density function of Mixed Exponential and Gamma Distributions

Description

A density function of

$0.2(2 e^{-2x}) + 0.8 \frac{x^3}{3!}e^{-x}.$

Usage

mixexpgamma_fun(x)
mixexpgamma_fun(x)

Arguments

`x`	A numeric vector for arguments of a density function.

Value

A numeric vector of probabilities for a given argument x.

Generate random numbers of Mixed Exponential and Gamma Distributions

Description

Generate random numbers whose density function:

$0.2(2 e^{-2x}) + 0.8 \frac{x^3}{3!}e^{-x}.$

Usage

mixexpgamma_gen(n = 100, seed = NULL)
mixexpgamma_gen(n = 100, seed = NULL)

Arguments

`n`	The number of random numbers.
`seed`	A seed for random number generator.

Value

A numeric vector of probabilities for a given argument x.

Dataset of Mixture of Exponential Distribution and Gamma Distribution: Histogram Version

Description

Dataset of mixture of exponential distribution and gamma distribution, histogram version, whose density is:

$0.2(2 e^{-2x}) + 0.8 \frac{x^3}{3!}e^{-x}.$

Usage

mixExpGammaHist
mixExpGammaHist

Format

A list of numeric vectors of Mixture of Exponential and Gamma distribution Model.

n200: A numeric vector with 200 elements.
n200p: Histogram sample data with 25 bins.
n200f: Histogram frequency data with 25 bins.
n400: A numeric vector with 400 elements.
n400p: Histogram sample data with 50 bins.
n400f: Histogram frequency data with 50 bins.
n800: A numeric vector with 800 elements.
n800p: Histogram sample data with 100 bins.
n800f: Histogram frequency data with 100 bins.

Source

mix2gauss_gen() databinning()

Probability density function of Exponential-based model

Description

A probability density function(PDF) of Exponential-based model. It is an underlying routine for plot.expmodel to compute the values of PDF. To access parameters and coefficients in an object emodel of a class expmodel, use emodel$result[k, "lmd1"], emodel$coeffs[[k]] for some index k. This index appears in the leftmost column of estimation table generated by summary(emodel).

Usage

pdf_expmodel(coeff, lmd, x)
pdf_expmodel(coeff, lmd, x)

Arguments

`coeff`	A coefficient vector in increasing order of degrees; the first element is 0th degree, ..., and last element is the largest degree of coefficients.
`lmd`	A rate parameter of an exponential distribution, which is positive.
`x`	A numeric input vector.

Value

A numeric vector of PDF of an exponential-based model.

Examples

## Create an object of `expmodel`
emodel <- expmodel(mixexpgamma$n200)
## Estimate with degree 4 and rate parameter 2.0
emodel <- estimate(emodel, 4, 2.0)
## Input vector
x <- seq(0, 12, 0.1)
## Output of PDF in above estimation
yv <- pdf_expmodel(emodel$coeffs[[1]], emodel$result[1, "lmd1"], x)
## Create an object of `expmodel`
emodel <- expmodel(mixexpgamma$n200)
## Estimate with degree 4 and rate parameter 2.0
emodel <- estimate(emodel, 4, 2.0)
## Input vector
x <- seq(0, 12, 0.1)
## Output of PDF in above estimation
yv <- pdf_expmodel(emodel$coeffs[[1]], emodel$result[1, "lmd1"], x)

Probability density function of Gaussian-based model

Description

A probability density function(PDF) of a Gaussian model. It is an underlying routine for plot.gaussmodel to compute the values of PDF. To access parameters and coefficients in an object gmodel of a class gaussmodel, use gmodel$result[k, "mu1"], gmodel$result[k, "sig1"], gmodel$coeffs[[k]] for some index k. This index appears in the leftmost column of estimation table generated by summary(gmodel).

Usage

pdf_gaussmodel(coeff, mu, sig, x)
pdf_gaussmodel(coeff, mu, sig, x)

Arguments

`coeff`	A coefficient vector in increasing order of degrees; the first element is 0th degree, ..., and last element is the largest degree of coefficients.
`mu`	A mean of Gaussian distribution.
`sig`	A standard deviation of Gaussian distribution, which is positive.
`x`	A numeric input vector.

Value

A numeric vector of PDF of Gaussian-based distribution.

Examples

## Create an object of `gaussmodel`
gmodel <- gaussmodel(mix2gauss$n200)
## Estimate with a degree 6, a mean 0, and standard deviations 0.5
gmodel <- estimate(gmodel, 6, 0, 0.5)
## Input vector
x <- seq(-3, 3, 0.1)
## Output of PDF in above estimation
yv <- pdf_gaussmodel(gmodel$coeffs[[1]], gmodel$result[1, "mu1"],
 gmodel$result[1, "sig1"], x)
## Create an object of `gaussmodel`
gmodel <- gaussmodel(mix2gauss$n200)
## Estimate with a degree 6, a mean 0, and standard deviations 0.5
gmodel <- estimate(gmodel, 6, 0, 0.5)
## Input vector
x <- seq(-3, 3, 0.1)
## Output of PDF in above estimation
yv <- pdf_gaussmodel(gmodel$coeffs[[1]], gmodel$result[1, "mu1"],
 gmodel$result[1, "sig1"], x)

Plot a histogram and estimated densities/distributions of Exponential-based model object

Description

Plot the histogram and, if available, estimated densities or cumulative distributions of expmodel object.

Usage

## S3 method for class 'expmodel'
plot(
  x,
  cum = FALSE,
  nmax = 4,
  graphs = NULL,
  bins = 40,
  hist = TRUE,
  linesize = 1,
  ...
)
## S3 method for class 'expmodel'
plot(
  x,
  cum = FALSE,
  nmax = 4,
  graphs = NULL,
  bins = 40,
  hist = TRUE,
  linesize = 1,
  ...
)

Arguments

`x`	`expmodel` object.
`cum`	A logical scalar, whether or not it plots cumulative histogram/distributions instead of plain histogram/densities. Default value is `FALSE`.
`nmax`	A maximum number of estimates to be plotted in the graph. The default value is 4.
`graphs`	A vector of indices to be displayed in the graph. These indices appear in the leftmost column of the table in `summary.expmodel`. The default value is `NULL`, and if it is not `NULL`, only the estimated densities designated by `graphs` option appear, and `nmax` is ignored.
`bins`	A number of bins of the histogram.
`hist`	A logical scalar. If `TRUE`, display a histogram, otherwise not. The default value is `TRUE`.
`linesize`	A positive numeric scalar, which indicates the thickness of lines. The default value is `1`.
`...`	Arguments to be passed to or from other methods.

Value

A ggplot2 object.

Examples

## Create `expmodel` object from a data set mixexpgamma$n200
emodel <- expmodel(mixexpgamma$n200)
## Plot it (histogram only)
plot(emodel)
## Create `expmodel` object from a data set mixexpgamma$n200
emodel <- expmodel(mixexpgamma$n200)
## Plot it (histogram only)
plot(emodel)

Plot a histogram and estimated densities/distributions of Gaussian-based model object

Description

Plot the histogram and, if available, estimated densities or cumulative distributions of gaussmodel object.

Usage

## S3 method for class 'gaussmodel'
plot(
  x,
  cum = FALSE,
  nmax = 4,
  graphs = NULL,
  bins = 40,
  hist = TRUE,
  scaling = FALSE,
  linesize = 1,
  ...
)
## S3 method for class 'gaussmodel'
plot(
  x,
  cum = FALSE,
  nmax = 4,
  graphs = NULL,
  bins = 40,
  hist = TRUE,
  scaling = FALSE,
  linesize = 1,
  ...
)

Arguments

`x`	`gaussmodel` object.
`cum`	A logical scalar, whether or not it plots cumulative histogram/distributions instead of plain histogram/densities. Default value is `FALSE`.
`nmax`	A maximum number of estimates to be plotted in the graph. The default value is 4.
`graphs`	A vector of indices to be displayed in the graph. These indices appear in the leftmost column of the table in `estimate.gaussmodel`. The default value is `NULL`, and if it is not `NULL`, only the estimated densities designated by `graphs` option appear, and `nmax` is ignored.
`bins`	A number of bins of the histogram.
`hist`	A logical scalar. If `TRUE`, display a histogram, otherwise not. The default value is `TRUE`.
`scaling`	A logical scalar, which indicates whether or not it scales means and standard deviations in `mulist` and `sdlist`. The default value is `FALSE`.
`linesize`	A positive numeric scalar, which indicates the thickness of lines. The default value is `1`.
`...`	Arguments to be passed to or from other methods.

Value

A ggplot2 object.

Examples

## Create `gaussmodel` object from a data set mix2gauss$n200
gmodel <- gaussmodel(mix2gauss$n200)
## Plot it (histogram only)
plot(gmodel)
## Create `gaussmodel` object from a data set mix2gauss$n200
gmodel <- gaussmodel(mix2gauss$n200)
## Plot it (histogram only)
plot(gmodel)

Substitute a coefficient of polynomial

Description

Substitute a coefficient of a polynomial with a*x + b. For a polynomial with a coefficient vector $poly(x; coeff)$ , compute the coefficient vector of

$poly(a*x + b; coeff).$

Usage

polyaxb(coeff, c, a, b)
polyaxb(coeff, c, a, b)

Arguments

`coeff`	A coefficient vector in increasing order of degrees; the first element is 0th degree, ..., and the last element is the largest degree of coefficients.
`c`	A multiple factor of constant to be applied to all coefficients.
`a`	A coefficient of 1st degree of `a*x + b`.
`b`	A coefficient of 0th degree of `a*x + b`.

Value

A substituted coefficient.

Examples

coeff <- c(2, -2, 1)
a <- 1.1
b <- 1.2
coeff1 <- c(b, a)
coeff2 <- polyaxb(coeff, 1, a, b)
xv <- c(1, 2, 3)
## a*x + b
yv1 <- eval_poly(coeff1, xv)
## polynomial(a*x + b, coeff)
yv2 <- eval_poly(coeff, yv1)
## polynomial(x, coeff2)
yv <- eval_poly(coeff2, xv)
## This value is nearly 0 in the presence of rounding errors
yv - yv2
coeff <- c(2, -2, 1)
a <- 1.1
b <- 1.2
coeff1 <- c(b, a)
coeff2 <- polyaxb(coeff, 1, a, b)
xv <- c(1, 2, 3)
## a*x + b
yv1 <- eval_poly(coeff1, xv)
## polynomial(a*x + b, coeff)
yv2 <- eval_poly(coeff, yv1)
## polynomial(x, coeff2)
yv <- eval_poly(coeff2, xv)
## This value is nearly 0 in the presence of rounding errors
yv - yv2

printf

Description

printf

Usage

printf(...)
printf(...)

Arguments

...

Any number of arguments to be printed.

Value

None.

Summary of Exponential-based `expmodel` object.

Description

Summary of expmodel object, including a mean and quantiles. If some estimation has done, also print out estimates, up to nmax number of them.

Usage

## S3 method for class 'expmodel'
summary(object, nmax = 10, estonly = FALSE, ...)
## S3 method for class 'expmodel'
summary(object, nmax = 10, estonly = FALSE, ...)

Arguments

`object`	`expmodel` object.
`nmax`	A number of estimates to show in the summary. The default is 10.
`estonly`	Show only the results of estimates. The default value is `FALSE`.
`...`	Arguments to be passed to or from other methods.

Value

None.

Examples

## Create expmodel object from a data set mixexpgamma$n200
emodel <- expmodel(mixexpgamma$n200)
## Print a summary of an object
summary(emodel)
## Create expmodel object from a data set mixexpgamma$n200
emodel <- expmodel(mixexpgamma$n200)
## Print a summary of an object
summary(emodel)

Summary of Gaussian-based model `gaussmodel` object

Description

Summary of gaussmodel object, including a mean and a standard deviation and quantiles. If some estimation has done, also print out estimates, up to nmax number of them.

Usage

## S3 method for class 'gaussmodel'
summary(object, nmax = 10, estonly = FALSE, ...)
## S3 method for class 'gaussmodel'
summary(object, nmax = 10, estonly = FALSE, ...)

Arguments

`object`	`gaussmodel` object.
`nmax`	A number of estimates to show in the summary. The default is 10.
`estonly`	Show only the results of estimates. The default value is `FALSE`.
`...`	Arguments to be passed to or from other methods.

Value

None.

Examples

## Create gaussmodel object from a data set mix2gauss$n200
gmodel <- gaussmodel(mix2gauss$n200)
## Print a summary of an object
summary(gmodel)
## Create gaussmodel object from a data set mix2gauss$n200
gmodel <- gaussmodel(mix2gauss$n200)
## Print a summary of an object
summary(gmodel)

Package 'dsdp'

Help Index

Cumulative distribution function of Expomemtial-based model

Description

Usage

Arguments

Value

See Also

Examples

Cumulative distribution function of Gaussian-based model

Description

Usage

Arguments

Value

See Also

Examples

Reduce a data set to representatives of bins and their frequencies

Description

Usage

Arguments

Value

See Also

Examples

Compute the mean and the standard deviation of a data set

Description

Usage

Arguments

Value

See Also

Examples

dsdp: Density Estimation using Semidefinite Programming

Description

Generic Method for estimation

Description

Usage

Arguments

Value

Estimate Exponential-based model expmodel

Description

Usage

Arguments

Value

See Also

Examples

Estimate Gaussian-based model gaussmodel

Description

Usage

Arguments

Value

See Also

Examples

Evaluate a polynomial

Description

Usage

Arguments

Value

See Also

Examples

Estimate coefficients of a polynomial in Exponential-based Model

Description

Usage

Arguments

Value

See Also

Examples

Constructor for S3 class expmodel

Description

Usage

Arguments

Value

See Also

Examples

Generic Method for evaluate the estimate

Description

Usage

Arguments

Value

Return the evaluation of a vector with Exponential-based model

Description

Usage

Estimate Exponential-based model `expmodel`

Estimate Gaussian-based model `gaussmodel`

Constructor for S3 class `expmodel`

Constructor for S3 class `gaussmodel`