hpaDist: Probabilities and Moments Hermite Polynomial Approximation
In hpa: Distributions Hermite Polynomial Approximation
hpaDist R Documentation
Probabilities and Moments Hermite Polynomial Approximation
Description
Approximation of truncated, marginal and conditional densities,
moments and cumulative probabilities of multivariate distributions via
Hermite polynomial based approach proposed by Gallant and Nychka in 1987.
Density approximating function is scale adjusted product of two terms.
The first one is squared multivariate polynomial of pol_degrees
degrees with pol_coefficients coefficients vector.
The second is product of independent normal random variables' densities with
expected values and standard deviations given by mean and sd
vectors correspondingly. Approximating function satisfies properties of
density function thus generating a broad family of distributions.
Characteristics of these distributions
(moments, quantiles, probabilities and so on)
may provide accurate approximations to characteristic of other
distributions. Moreover it is usually possible to provide arbitrary close
approximation by the means of polynomial degrees increase.
Usage
dhpa(
x,
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
phpa(
x,
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
ihpa(
x_lower = numeric(0),
x_upper = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
ehpa(
x = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
expectation_powers = numeric(0),
is_parallel = FALSE,
is_validation = TRUE
)
etrhpa(
tr_left = numeric(0),
tr_right = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
mean = numeric(0),
sd = numeric(0),
expectation_powers = numeric(0),
is_parallel = FALSE,
is_validation = TRUE
)
dtrhpa(
x,
tr_left = numeric(0),
tr_right = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
itrhpa(
x_lower = numeric(0),
x_upper = numeric(0),
tr_left = numeric(0),
tr_right = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
dhpaDiff(
x,
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
type = "pol_coefficients",
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
ehpaDiff(
x = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
expectation_powers = numeric(0),
type = "pol_coefficients",
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
ihpaDiff(
x_lower = numeric(0),
x_upper = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
type = "pol_coefficients",
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
qhpa(
p,
x = matrix(1, 1),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0)
)
rhpa(
n,
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
mean = numeric(0),
sd = numeric(0)
)
Arguments
x
numeric matrix of function arguments and
conditional values. Note that x rows are points (observations)
while random vectors components (variables) are columns.
pol_coefficients
numeric vector of polynomial coefficients.
pol_degrees
non-negative integer vector of polynomial
degrees (orders).
given_ind
logical or numeric vector indicating whether corresponding
random vector component is conditioned. By default it is a logical
vector of FALSE values. If give_ind[i] equals TRUE or
i then i-th column of x matrix will contain
conditional values.
omit_ind
logical or numeric vector indicating whether corresponding
random component is omitted. By default it is a logical vector
of FALSE values. If omit_ind[i] equals TRUE or i
then values in i-th column of x matrix will be ignored.
mean
numeric vector of expected values.
sd
positive numeric vector of standard deviations.
is_parallel
if TRUE then multiple cores will be
used for some calculations. It usually provides speed advantage for
large enough samples (about more than 1000 observations).
log
logical; if TRUE then probabilities p are given as log(p)
or derivatives will be given respect to log(p).
is_validation
logical value indicating whether function input
arguments should be validated. Set it to FALSE for slight
performance boost (default value is TRUE).
x_lower
numeric matrix of lower integration limits.
Note that x_lower rows are observations while variables are columns.
x_upper
numeric matrix of upper integration limits.
Note that x_upper rows are observations while variables
are columns.
expectation_powers
integer vector of random vector components powers.
tr_left
numeric matrix of left (lower) truncation limits.
Note that tr_left rows are observations while variables are columns.
If tr_left and tr_right are single row matrices then the same
truncation limits will be applied to all observations that are determined
by the first rows of these matrices.
tr_right
numeric matrix of right (upper) truncation limits.
Note that tr_right rows are observations while variables are columns.
If tr_left and tr_right are single row matrices then the same
truncation limits will be applied to all observations that are determined
by the first rows of these matrices.
type
determines the partial derivatives to be included into the
gradient. If type="pol_coefficients" then gradient will contain
partial derivatives respect to polynomial coefficients listed in the
same order as pol_coefficients. Other available types are
type = "mean" and type = "sd".
For function dhpaDiff it is possible to take
gradient respect to the x points setting type="x".
For function ihpaDiff it is possible to take
gradient respect to the x lower and upper points setting
type = "x_lower" or type = "upper" correspondingly.
In order to get full gradient please set type="all".
p
numeric vector of probabilities
n
positive integer representing the number of observations
Details
It is possible to approximate densities
dhpa, cumulative probabilities
phpa, ihpa, moments
ehpa as well as their truncated
dtrhpa, itrhpa,
etrhpa forms
and gradients dhpaDiff, ihpaDiff.
Note that phpa is special of ihpa
where x
corresponds to x_upper while x_lower is matrix of
negative infinity values. So phpa intended to approximate
cumulative
distribution functions while ihpa approximates
probabilities that
random vector components will be between values determined by rows of
x_lower and x_upper matrices. Further details are given below.
Since density approximating function is non-negative and integrates
to 1 it is density function for some m-variate
random vector \xi. Approximating function f_{\xi }(x)
has the following form:
f_{\xi }(x) = f_{\xi }(x;\mu, \sigma, \alpha) =
\frac{1}{\psi }\prod\limits_{t=1}^{m}\phi
({x}_{t};{\mu }_{t},{\sigma }_{t}){{\left( \sum\limits_{{i}_{1}=0}^{{K}_{1}}
{...}\sum\limits_{{i}_{m}=0}^{{K}_{m}}{{{\alpha }_{({{i}_{1}},...,{{i}_{m}})
}}\prod\limits_{r=1}^{m}x_{r}^{{{i}_{r}}}} \right)}^{2}}
\psi =\sum\limits_{{i}_{1}=0}^{{K}_{1}}{...}\sum
\limits_{{i}_{m}=0}^{{K}_{m}}{\sum\limits_{{j}_{1}=0}^{{K}_{1}}
{...}\sum\limits_{{j}_{m}=0}^{{K}_{m}}{{{\alpha }_{({i}_{1},
\cdots,{i}_{m})}}{{\alpha }_{({j}_{1},\cdots,{j}_{m})}}\prod
\limits_{r=1}^{m}\mathcal{M}({i}_{r}+{j}_{r};{{\mu }_{r}},{\sigma }_{r})}},
where:
x = (x_{1},...x_{m}) - is vector of arguments i.e. rows
of x matrix in dhpa.
{\alpha }_{({i}_{1},\cdots,{i}_{m})} - is polynomial coefficient
corresponding to pol_coefficients[k] element. In order to investigate
correspondence between k and ({i}_{1},\cdots,{i}_{m}) values
please see 'Examples' section below or polynomialIndex
function 'Details', 'Value' and 'Examples' sections. Note that if m=1
then pol_coefficients[k] simply corresponds to \alpha_{k-1}.
(K_{1},...,K_{m}) - are polynomial degrees (orders) provided via
pol_degrees argument so pol_degrees[i] determines K_{i}.
\phi
(.;{\mu }_{t},{\sigma }_{t}) - is normal random variable density function
where \mu_{t} and \sigma_{t} are mean and standard deviation
determined by mean[t] and sd[t] arguments values.
\mathcal{M}(q;{{\mu }_{r}},{\sigma }_{r}) - is q-th order
moment of normal random variable with mean {\mu }_{r} and standard
deviation {\sigma }_{r}. Note that function
normalMoment allows to calculate and differentiate normal
random variable's moments.
\psi - constant term insuring that f_{\xi }(x) is
density function.
Therefore dhpa allows to calculate f_{\xi}(x)
values at points
determined by rows of x matrix given polynomial
degrees pol_degrees (K) as well as mean (\mu),
sd (\sigma) and pol_coefficients (\alpha)
parameters values. Note that mean, sd and pol_degrees are
m-variate vectors while pol_coefficients has
prod(pol_degrees + 1) elements.
Cumulative probabilities could be approximated as follows:
P\left(\underline{x}_{1}\leq\xi_{1}\leq\overline{x}_{1},...,
\underline{x}_{m}\leq\xi_{m}\leq\overline{x}_{m}\right) =
= \bar{F}_{\xi}(\underline{x},\bar{x}) =
\bar{F}_{\xi}(\underline{x},\bar{x};\mu, \sigma, \alpha) =
\frac{1}{\psi }
\prod\limits_{t=1}^{m}(\Phi ({{{\bar{x}}}_{t}};{{\mu }_{t}},
{{\sigma }_{t}})-\Phi ({{{\underline{x}}}_{t}};{{\mu }_{t}},
{{\sigma }_{t}})) *
* \sum\limits_{{{i}_{1}}=0}^{{{K}_{1}}}{...}
\sum\limits_{{{i}_{m}}=0}^{{{K}_{m}}}{\sum\limits_{{{j}_{1}}=0}^{{{K}_{1}}}
{...}\sum\limits_{{{j}_{m}}=0}^{{{K}_{m}}}
{{{\alpha }_{({{i}_{1}},...,{{i}_{m}})}}{{\alpha }_{({{j}_{1}},...,{{j}_{m}})
}}}}\prod\limits_{r=1}^{m}\mathcal{M}_{TR}\left({i}_{r}+{j}_{r};
\underline{x}_{r},\overline{x}_{r},\mu_{r},\sigma_{r}\right)
where:
\Phi
(.;{\mu }_{t},{\sigma }_{t}) - is normal random variable's cumulative
distribution function where \mu_{t} and \sigma_{t} are mean and
standard deviation determined by mean[t] and sd[t] arguments
values.
\mathcal{M}_{TR}(q;
\underline{x}_{r},\overline{x}_{r},\mu_{r},\sigma_{r}) - is
q-th order
moment of truncated (from above by \overline{x}_{r} and from below by
\underline{x}_{r})
normal random variable with mean {\mu }_{r} and standard
deviation {\sigma }_{r}. Note that function
truncatedNormalMoment allows to calculate and
differentiate truncated normal random variable's moments.
\overline{x} = (\overline{x}_{1},...,\overline{x}_{m}) -
vector of upper integration limits
i.e. rows of x_upper matrix in ihpa.
\underline{x} = (\underline{x}_{1},...,\underline{x}_{m}) -
vector of lower integration limits
i.e. rows of x_lower matrix in ihpa.
Therefore ihpa allows to calculate interval distribution
function \bar{F}_{\xi}(\underline{x},\bar{x})
values at points determined by rows of x_lower (\underline{x})
and x_upper (\overline{x}) matrices.
The rest of the arguments are similar to dhpa.
Expected value powered product approximation is as follows:
E\left( \prod\limits_{t=1}^{m}\xi_{t}^{{{k}_{t}}} \right)=
\frac{1}{\psi }\sum\limits_{{{i}_{1}}=0}^{{{K}_{1}}}{...}
\sum\limits_{{{i}_{m}}=0}^{{{K}_{m}}}
{\sum\limits_{{{j}_{1}}=0}^{{{K}_{1}}}{...}
\sum\limits_{{{j}_{m}}=0}^{{{K}_{m}}}
{{{\alpha }_{({{i}_{1}},...,{{i}_{m}})}}
{{\alpha }_{({{j}_{1}},...,{{j}_{m}})}}}}
\prod\limits_{r=1}^{m}\mathcal{M}({{i}_{r}}+{{j}_{r}}+{{k}_{t}};
{{\mu }_{r}},{{\sigma }_{r}})
where (k_{1},...,k_{m}) are integer powers determined by
expectation_powers argument of ehpa so
expectation_powers[t] assigns k_{t}. Note that argument x
in ehpa allows to determined conditional values.
Expanding polynomial degrees (K_{1},...,K_{m}) it is possible to
provide arbitrary close approximation to density of some m-variate
random vector \xi^{\star}. So actually f_{\xi}(x)
approximates f_{\xi^{\star}}(x). Accurate approximation requires
appropriate mean, sd and pol_coefficients values
selection. In order to get sample estimates of these parameters please apply
hpaML function.
In order to perform calculation for marginal distribution of some
\xi components please provide omitted
components via omit_ind argument.
For examples if ones assume m=5-variate distribution
and wants to deal with 1-st, 3-rd, and 5-th components
only i.e. (\xi_{1},\xi_{3},\xi_{5}) then set
omit_ind = c(FALSE, TRUE, FALSE, TRUE, FALSE)
indicating that \xi_{2} and \xi_{4} should be 'omitted' from
\xi since 2-nd and 4-th values of omit_ind are
TRUE.
Then x still should be 5 column matrix but
values in 2-nd and 4-th columns will not affect
calculation results. Meanwhile note that marginal distribution of t
components of \xi usually do not coincide with any marginal
distribution generated by t-variate density approximating function.
In order to perform calculation for conditional distribution i.e. given
fixed values for some \xi components please provide these
components via given_ind argument.
For example if ones assume m=5-variate distribution
and wants to deal with 1-st, 3-rd, and 5-th components
given fixed values (suppose 8 and 10) for the other two components i.e.
(\xi|\xi_{2} = 8, \xi_{4} = 10) then set
given_ind = c(FALSE, TRUE, FALSE, TRUE, FALSE) and
x[2] = 8, x[4] = 10 where for simplicity it is assumed that
x is single row 5 column matrix; it is possible to provide
different conditional values for the same components simply setting different
values to different x rows.
Note that it is possible to combine given_ind and omit_ind
arguments. However it is wrong to set both given_ind[i] and
omit_ind[i] to TRUE. Also at least one value should be
FALSE both for given_ind and omit_ind.
In order to consider truncated distribution of \xi i.e.
\left(\xi|\overline{a}_{1}\leq\xi_{1}\leq\overline{b}_{1},
\cdots,\overline{a}_{m}\leq\xi_{m}\leq\overline{b}_{m}\right)
please set lower (left) truncation points \overline{a} and
upper (right) truncation points \overline{b} via tr_left
and tr_right arguments correspondingly. Note that if lower truncation
points are negative infinite and upper truncation points are positive
infinite then dtrhpa, itrhpa and
etrhpa are similar to dhpa,
ihpa and ehpa correspondingly.
In order to calculate Jacobian of f_{\xi }(x;\mu, \sigma, \alpha)
and \bar{F}_{\xi}(\underline{x},\bar{x};\mu, \sigma, \alpha) w.r.t
all ore some particular parameters please apply dhpaDiff
and ihpaDiff functions correspondingly specifying
parameters of interest via type argument. If x or
x_lower and x_upper are single row matrices then gradients
will be calculated.
For further information please see 'Examples' section. Note that examples
are given separately for each function.
If given_ind and (or) omit_ind are numeric vectors
then they are insensitive to the order of elements.
For example given_ind = c(5, 2, 3) is similar
to given_ind = c(2, 3, 5).
Densities Hermite polynomial approximation approach has been
proposed by A. Gallant and D. W. Nychka in 1987. The main idea is to
approximate unknown distribution density with scaled Hermite polynomial.
For more information please refer to the literature listed below.
Value
Functions dhpa, phpa and
dtrhpa return vector of probabilities of length
nrow(x).
Functions ihpa and
itrhpa return vector of probabilities of length
nrow(x_upper).
If x argument has not been provided or is a single row
matrix then function
ehpa returns moment value. Otherwise it returns vector of
length nrow(x) containing moments values.
If tr_left and tr_right arguments are single row matrices then
function etrhpa returns moment value.
Otherwise it returns vector of length
max(nrow(tr_left), nrow(tr_right)) containing moments values.
Functions dhpaDiff and ihpaDiff
return Jacobin matrix. The number
of columns depends on type argument. The number of rows is
nrow(x) for dhpaDiff and
nrow(x_upper) for
ihpaDiff
If mean or sd are not specified they assume the default
values of m-dimensional vectors of 0 and 1, respectively.
If x_lower is not specified then it is the matrix of the
same size as x_upper containing negative infinity values only. If
expectation_powers is not specified then it is m-dimensional
vector of 0 values.
Please see 'Details' section for additional information.
References
A. Gallant and D. W. Nychka (1987) <doi:10.2307/1913241>
Examples
## Example demonstrating dhpa function application.
## Let's approximate some three random variables (i.e. X1, X2 and X3)
## joint density function at points x = (0,1, 0.2, 0.3) and
## y = (0.5, 0.8, 0.6) with Hermite polynomial of (1, 2, 3) degrees which
## polynomial coefficients equal 1 except coefficient related to x1*(x^3)
## polynomial element which equals 2. Also suppose that normal density
## related mean vector equals (1.1, 1.2, 1.3) while standard deviations
## vector is (2.1, 2.2, 2.3).
# Prepare initial values
x <- matrix(c(0.1, 0.2, 0.3), nrow = 1) # x point as a single row matrix
y <- matrix(c(0.5, 0.8, 0.6), nrow = 1) # y point as a single row matrix
x_y <- rbind(x, y) # matrix which rows are x and y
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power", "x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate density approximation
# at point x (note that x should be a matrix)
dhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# at points x and y
dhpa(x = x_y,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.5 i.e. X2 = 0.5.
# Substitute x and y second components with conditional value 0.5
x <- matrix(c(0.1, 0.5, 0.3), nrow = 1) # or simply x[2] <- 0.5
y <- matrix(c(0.4, 0.5, 0.6), nrow = 1) # or simply y[2] <- 0.5
x_y <- rbind(x, y)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) density approximation
# at point x
dhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# at points x and y
dhpa(x = x_y,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution conditioned on the
# second component 0.5 value i.e. (X3 | X2 = 0.5).
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on x2 = 0.5) marginal (for x3) density approximation
# at point x
dhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind)
# at points x and y
dhpa(x = x_y,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind)
## Example demonstrating phpa function application.
## Let's approximate some three random variables (X1, X2, X3)
## joint cumulative distribution function (cdf) at point (0,1, 0.2, 0.3)
## with Hermite polynomial of (1, 2, 3) degrees which polynomial
## coefficients equal 1 except coefficient related to x1*(x^3) polynomial
## element which equals 2. Also suppose that normal density related
## mean vector equals (1.1, 1.2, 1.3) while standard deviations
## vector is (2.1, 2.2, 2.3).
## Prepare initial values
x <- matrix(c(0.1, 0.2, 0.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate cdf approximation at point x
phpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.5
# Substitute x second component with conditional value 0.5
x <- matrix(c(0.1, 0.5, 0.3), nrow = 1) # or simply x[2] <- 0.5
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) cdf approximation at point x
phpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.5 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) cdf
# approximation at point x
phpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind)
## Example demonstrating ihpa function application.
## Let's approximate some three random variables (X1, X2, X3) joint interval
## distribution function (intdf) at lower and upper points (0,1, 0.2, 0.3)
## and (0,4, 0.5, 0.6) correspondingly with Hermite polynomial of (1, 2, 3)
## degrees which polynomial coefficients equal 1 except coefficient related
## to x1*(x^3) polynomial element which equals 2. Also suppose that normal
## density related mean vector equals (1.1, 1.2, 1.3) while standard
## deviations vector is (2.1, 2.2, 2.3).
## Prepare initial values
x_lower <- matrix(c(0.1, 0.2, 0.3), nrow=1)
x_upper <- matrix(c(0.4, 0.5, 0.6), nrow=1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate intdf approximation at points x_lower and x_upper
ihpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.7
# Substitute x second component with conditional value 0.7
x_upper <- matrix(c(0.4, 0.7, 0.6), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) intdf approximation
# at points x_lower and x_upper
ihpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.7 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3)
# intdf approximation at points x_lower and x_upper
ihpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind)
## Example demonstrating ehpa function application.
## Let's approximate some three random variables (X1, X2, X3) powered product
## expectation for powers (3, 2, 1) with Hermite polynomial of (1, 2, 3)
## degrees which polynomial coefficients equal 1 except coefficient
## related to x1*(x^3) polynomial element which equals 2.
## Also suppose that normal density related mean vector equals
## (1.1, 1.2, 1.3) while standard deviations vector is (2.1, 2.2, 2.3).
# Prepare initial values
expectation_powers = c(3,2,1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
#Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate expected powered product approximation
ehpa(pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers)
# Condition second component to be 0.5
# Substitute x second component with conditional value 0.5
x <- matrix(c(NA, 0.5, NA), nrow = 1)
#Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) expected powered product approximation
ehpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.5 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) expected powered
# product approximation at points x_lower and x_upper
ehpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers,
given_ind = given_ind,
omit_ind = omit_ind)
## Example demonstrating etrhpa function application.
## Let's approximate some three truncated random variables (X1, X2, X3)
## powered product expectation for powers (3, 2, 1) with Hermite polynomial
## of (1,2,3) degrees which polynomial coefficients equal 1 except
## coefficient related to x1*(x^3) polynomial element which equals 2. Also
## suppose that normal density related mean vector equals (1.1, 1.2, 1.3)
## while standard deviations vector is (2.1, 2.2, 2.3). Suppose that lower
## and upper truncation points are (-1.1,-1.2,-1.3) and (1.1,1.2,1.3)
## correspondingly.
# Prepare initial values
expectation_powers = c(3,2,1)
tr_left = matrix(c(-1.1,-1.2,-1.3), nrow = 1)
tr_right = matrix(c(1.1,1.2,1.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate expected powered product approximation for truncated distribution
etrhpa(pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers,
tr_left = tr_left, tr_right = tr_right)
## Example demonstrating dtrhpa function application.
## Let's approximate some three random variables (X1, X2, X3) joint density
## function at point (0,1, 0.2, 0.3) with Hermite polynomial of (1,2,3)
## degrees which polynomial coefficients equal 1 except coefficient related
## to x1*(x^3) polynomial element which equals 2. Also suppose that normal
## density related mean vector equals (1.1, 1.2, 1.3) while standard
## deviations vector is (2.1, 2.2, 2.3). Suppose that lower and upper
## truncation points are (-1.1,-1.2,-1.3) and (1.1,1.2,1.3) correspondingly.
# Prepare initial values
x <- matrix(c(0.1, 0.2, 0.3), nrow=1)
tr_left = matrix(c(-1.1,-1.2,-1.3), nrow = 1)
tr_right = matrix(c(1.1,1.2,1.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate density approximation at point x
dtrhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
tr_left = tr_left,
tr_right = tr_right)
# Condition second component to be 0.5
# Substitute x second component with conditional value 0.5
x <- matrix(c(0.1, 0.5, 0.3), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on x2 = 0.5) density approximation at point x
dtrhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
tr_left = tr_left, tr_right = tr_right)
# Consider third component marginal distribution
# conditioned on the second component 0.5 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3)
# density approximation at point x
dtrhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
tr_left = tr_left, tr_right = tr_right)
## Example demonstrating itrhpa function application.
## Let's approximate some three truncated random variables (X1, X2, X3) joint
## interval distribution function at lower and upper points (0,1, 0.2, 0.3)
## and (0,4, 0.5, 0.6) correspondingly with Hermite polynomial of (1 ,2, 3)
## degrees which polynomial coefficients equal 1 except coefficient
## related to x1*(x^3) polynomial element which equals 2. Also suppose
## that normal density related mean vector equals (1.1, 1.2, 1.3) while
## standard deviations vector is (2.1, 2.2, 2.3). Suppose that lower and
## upper truncation are (-1.1,-1.2,-1.3) and (1.1,1.2,1.3) correspondingly.
# Prepare initial values
x_lower <- matrix(c(0.1, 0.2, 0.3), nrow=1)
x_upper <- matrix(c(0.4, 0.5, 0.6), nrow=1)
tr_left = matrix(c(-1.1,-1.2,-1.3), nrow = 1)
tr_right = matrix(c(1.1,1.2,1.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate intdf approximation at points x_lower and x_upper
itrhpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
tr_left = tr_left, tr_right = tr_right)
# Condition second component to be 0.7
# Substitute x second component with conditional value 0.7
x_upper <- matrix(c(0.4, 0.7, 0.6), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) intdf
# approximation at points x_lower and x_upper
itrhpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
tr_left = tr_left, tr_right = tr_right)
# Consider third component marginal distribution
# conditioned on the second component 0.7 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) intdf
# approximation at points x_lower and x_upper
itrhpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
tr_left = tr_left, tr_right = tr_right)
## Example demonstrating dhpaDiff function application.
## Let's approximate some three random variables (X1, X2, X3) joint density
## function at point (0,1, 0.2, 0.3) with Hermite polynomial of (1,2,3)
## degrees which polynomial coefficients equal 1 except coefficient related
## to x1*(x^3) polynomial element which equals 2. Also suppose that normal
## density related mean vector equals (1.1, 1.2, 1.3) while standard
## deviations vector is (2.1, 2.2, 2.3). In this example let's calculate
## density approximating function's gradient respect to various parameters
# Prepare initial values
x <- matrix(c(0.1, 0.2, 0.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate density approximation gradient
# respect to polynomial coefficients at point x
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.5
# Substitute x second component with conditional value 0.5
x <- matrix(c(0.1, 0.5, 0.3), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on x2 = 0.5) density approximation's
# gradient respect to polynomial coefficients at point x
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.5 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) density
# approximation's gradient respect to:
# polynomial coefficients
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind)
# mean
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind,
type = "mean")
# sd
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind,
type = "sd")
# x
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind,
type = "x")
## Example demonstrating ehpaDiff function application.
## Let's approximate some three random variables (X1, X2, X3) expectation
## of the form E((X1 ^ 3) * (x2 ^ 1) * (X3 ^ 2)) and calculate the gradient
# Distribution parameters
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
pol_coefficients_n <- prod(pol_degrees + 1)
pol_coefficients <- rep(1, pol_coefficients_n)
# Set powers for expectation
expectation_powers <- c(3, 1, 2)
# Calculate expectation approximation gradient
# respect to all parameters
ehpaDiff(pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers,
type = "all")
# Let's calculate gradient of E(X1 ^ 3 | (X2 = 1, X3 = 2))
x <- c(0, 1, 2) # x[1] may be arbitrary (not NA) values
expectation_powers <- c(3, 0, 0) # expectation_powers[2:3] may be
# arbitrary (not NA) values
given_ind <- c(2, 3)
ehpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
expectation_powers = expectation_powers,
type = "all")
## Example demonstrating ihpaDiff function application.
## Let's approximate some three random variables (X1, X2, X3 ) joint interval
## distribution function (intdf) at lower and upper points (0,1, 0.2, 0.3)
## and (0,4, 0.5, 0.6) correspondingly with Hermite polynomial of (1, 2, 3)
## degrees which polynomial coefficients equal 1 except coefficient
## related to x1*(x^3) polynomial element which equals 2.
## Also suppose that normal density related mean vector equals
## (1.1, 1.2, 1.3) while standard deviations vector is (2.1, 2.2, 2.3).
## In this example let's calculate interval distribution approximating
## function gradient respect to polynomial coefficients.
# Prepare initial values
x_lower <- matrix(c(0.1, 0.2, 0.3), nrow=1)
x_upper <- matrix(c(0.4, 0.5, 0.6), nrow=1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate intdf approximation gradient respect to
# polynomial coefficients at points x_lower and x_upper
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.7
# Substitute x second component with conditional value 0.7
x_upper <- matrix(c(0.4, 0.7, 0.6), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) intdf approximation
# respect to polynomial coefficients at points x_lower and x_upper
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.7 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) intdf approximation
# respect to:
# polynomial coefficients
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind)
# mean
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
type = "mean")
# sd
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
type = "sd")
# x_lower
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
type = "x_lower")
# x_upper
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
type = "x_upper")
## Examples demonstrating qhpa function application.
## Sub-example 1 - univariate distribution
## Consider random variable X
# Distribution parameters
mean <- 1
sd <- 2
pol_degrees <- 2
pol_coefficients <- c(1, 0.1, -0.01)
# The level of quantile
p <- 0.7
# Calculate quantile of X
qhpa(p = p,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
## Sub-example 2 - marginal distribution
## Consider random vector (X1, X2) and quantile of X1
# Distribution parameters
mean <- c(1, 1.2)
sd <- c(2, 3)
pol_degrees <- c(2, 2)
pol_coefficients <- c(1, 0.1, -0.01, 0.2, 0.012,
0.0013, 0.0042, 0.00025, 0)
# The level of quantile
p <- 0.7
# Calculate quantile of X1
qhpa(p = p,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
omit_ind = 2) # set omitted variable index
## Sub-example 3 - marginal and conditional distribution
## Consider random vector (X1, X2, X3) and
## quantiles of X1|X3 and X1|(X2,X3)
mean <- c(1, 1.2, 0.9)
sd <- c(2, 3, 2.5)
pol_degrees <- c(1, 1, 1)
pol_coefficients <- c(1, 0.1, -0.01, 0.2, 0.012,
0.0013, 0.0042, 0.00025)
# The level of quantile
p <- 0.7
# Calculate quantile of X1|X3 = 0.2
qhpa(p = p,
x = matrix(c(NA, NA, 0.2), nrow = 1), # set any values to
# unconditioned and
# omitted components
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
omit_ind = 2, # set omitted variable index
given_ind = 3) # set conditioned variable index
# Calculate quantile of X1|(X2 = 0.5, X3 = 0.2)
qhpa(p = p,
x = matrix(c(NA, 0.5, 0.2), nrow = 1), # set any values to
# unconditioned and
# omitted components
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = c(2, 3)) # set conditioned
# variables indexes
## Examples demonstrating rhpa function application.
# Set seed for reproducibility
set.seed(123)
# Distribution parameters
mean <- 1
sd <- 2
pol_degrees <- 2
pol_coefficients <- c(1, 0.1, -0.01)
# Simulate two observations from this distribution
rhpa(n = 2,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
hpa documentation built on May 31, 2023, 8:25 p.m.
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| hpaDist | R Documentation |
Probabilities and Moments Hermite Polynomial Approximation
Description
Approximation of truncated, marginal and conditional densities, moments and cumulative probabilities of multivariate distributions via Hermite polynomial based approach proposed by Gallant and Nychka in 1987.
Density approximating function is scale adjusted product of two terms.
The first one is squared multivariate polynomial of pol_degrees
degrees with pol_coefficients coefficients vector.
The second is product of independent normal random variables' densities with
expected values and standard deviations given by mean and sd
vectors correspondingly. Approximating function satisfies properties of
density function thus generating a broad family of distributions.
Characteristics of these distributions
(moments, quantiles, probabilities and so on)
may provide accurate approximations to characteristic of other
distributions. Moreover it is usually possible to provide arbitrary close
approximation by the means of polynomial degrees increase.
Usage
dhpa(
x,
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
phpa(
x,
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
ihpa(
x_lower = numeric(0),
x_upper = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
ehpa(
x = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
expectation_powers = numeric(0),
is_parallel = FALSE,
is_validation = TRUE
)
etrhpa(
tr_left = numeric(0),
tr_right = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
mean = numeric(0),
sd = numeric(0),
expectation_powers = numeric(0),
is_parallel = FALSE,
is_validation = TRUE
)
dtrhpa(
x,
tr_left = numeric(0),
tr_right = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
itrhpa(
x_lower = numeric(0),
x_upper = numeric(0),
tr_left = numeric(0),
tr_right = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
dhpaDiff(
x,
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
type = "pol_coefficients",
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
ehpaDiff(
x = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
expectation_powers = numeric(0),
type = "pol_coefficients",
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
ihpaDiff(
x_lower = numeric(0),
x_upper = numeric(0),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0),
type = "pol_coefficients",
is_parallel = FALSE,
log = FALSE,
is_validation = TRUE
)
qhpa(
p,
x = matrix(1, 1),
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
given_ind = numeric(0),
omit_ind = numeric(0),
mean = numeric(0),
sd = numeric(0)
)
rhpa(
n,
pol_coefficients = numeric(0),
pol_degrees = numeric(0),
mean = numeric(0),
sd = numeric(0)
)
Arguments
x |
numeric matrix of function arguments and
conditional values. Note that |
pol_coefficients |
numeric vector of polynomial coefficients. |
pol_degrees |
non-negative integer vector of polynomial degrees (orders). |
given_ind |
logical or numeric vector indicating whether corresponding
random vector component is conditioned. By default it is a logical
vector of |
omit_ind |
logical or numeric vector indicating whether corresponding
random component is omitted. By default it is a logical vector
of |
mean |
numeric vector of expected values. |
sd |
positive numeric vector of standard deviations. |
is_parallel |
if |
log |
logical; if |
is_validation |
logical value indicating whether function input
arguments should be validated. Set it to |
x_lower |
numeric matrix of lower integration limits.
Note that |
x_upper |
numeric matrix of upper integration limits.
Note that |
expectation_powers |
integer vector of random vector components powers. |
tr_left |
numeric matrix of left (lower) truncation limits.
Note that |
tr_right |
numeric matrix of right (upper) truncation limits.
Note that |
type |
determines the partial derivatives to be included into the
gradient. If |
p |
numeric vector of probabilities |
n |
positive integer representing the number of observations |
Details
It is possible to approximate densities
dhpa, cumulative probabilities
phpa, ihpa, moments
ehpa as well as their truncated
dtrhpa, itrhpa,
etrhpa forms
and gradients dhpaDiff, ihpaDiff.
Note that phpa is special of ihpa
where x
corresponds to x_upper while x_lower is matrix of
negative infinity values. So phpa intended to approximate
cumulative
distribution functions while ihpa approximates
probabilities that
random vector components will be between values determined by rows of
x_lower and x_upper matrices. Further details are given below.
Since density approximating function is non-negative and integrates
to 1 it is density function for some m-variate
random vector \xi. Approximating function f_{\xi }(x)
has the following form:
f_{\xi }(x) = f_{\xi }(x;\mu, \sigma, \alpha) =
\frac{1}{\psi }\prod\limits_{t=1}^{m}\phi
({x}_{t};{\mu }_{t},{\sigma }_{t}){{\left( \sum\limits_{{i}_{1}=0}^{{K}_{1}}
{...}\sum\limits_{{i}_{m}=0}^{{K}_{m}}{{{\alpha }_{({{i}_{1}},...,{{i}_{m}})
}}\prod\limits_{r=1}^{m}x_{r}^{{{i}_{r}}}} \right)}^{2}}
\psi =\sum\limits_{{i}_{1}=0}^{{K}_{1}}{...}\sum
\limits_{{i}_{m}=0}^{{K}_{m}}{\sum\limits_{{j}_{1}=0}^{{K}_{1}}
{...}\sum\limits_{{j}_{m}=0}^{{K}_{m}}{{{\alpha }_{({i}_{1},
\cdots,{i}_{m})}}{{\alpha }_{({j}_{1},\cdots,{j}_{m})}}\prod
\limits_{r=1}^{m}\mathcal{M}({i}_{r}+{j}_{r};{{\mu }_{r}},{\sigma }_{r})}},
where:
x = (x_{1},...x_{m}) - is vector of arguments i.e. rows
of x matrix in dhpa.
{\alpha }_{({i}_{1},\cdots,{i}_{m})} - is polynomial coefficient
corresponding to pol_coefficients[k] element. In order to investigate
correspondence between k and ({i}_{1},\cdots,{i}_{m}) values
please see 'Examples' section below or polynomialIndex
function 'Details', 'Value' and 'Examples' sections. Note that if m=1
then pol_coefficients[k] simply corresponds to \alpha_{k-1}.
(K_{1},...,K_{m}) - are polynomial degrees (orders) provided via
pol_degrees argument so pol_degrees[i] determines K_{i}.
\phi
(.;{\mu }_{t},{\sigma }_{t}) - is normal random variable density function
where \mu_{t} and \sigma_{t} are mean and standard deviation
determined by mean[t] and sd[t] arguments values.
\mathcal{M}(q;{{\mu }_{r}},{\sigma }_{r}) - is q-th order
moment of normal random variable with mean {\mu }_{r} and standard
deviation {\sigma }_{r}. Note that function
normalMoment allows to calculate and differentiate normal
random variable's moments.
\psi - constant term insuring that f_{\xi }(x) is
density function.
Therefore dhpa allows to calculate f_{\xi}(x)
values at points
determined by rows of x matrix given polynomial
degrees pol_degrees (K) as well as mean (\mu),
sd (\sigma) and pol_coefficients (\alpha)
parameters values. Note that mean, sd and pol_degrees are
m-variate vectors while pol_coefficients has
prod(pol_degrees + 1) elements.
Cumulative probabilities could be approximated as follows:
P\left(\underline{x}_{1}\leq\xi_{1}\leq\overline{x}_{1},...,
\underline{x}_{m}\leq\xi_{m}\leq\overline{x}_{m}\right) =
= \bar{F}_{\xi}(\underline{x},\bar{x}) =
\bar{F}_{\xi}(\underline{x},\bar{x};\mu, \sigma, \alpha) =
\frac{1}{\psi }
\prod\limits_{t=1}^{m}(\Phi ({{{\bar{x}}}_{t}};{{\mu }_{t}},
{{\sigma }_{t}})-\Phi ({{{\underline{x}}}_{t}};{{\mu }_{t}},
{{\sigma }_{t}})) *
* \sum\limits_{{{i}_{1}}=0}^{{{K}_{1}}}{...}
\sum\limits_{{{i}_{m}}=0}^{{{K}_{m}}}{\sum\limits_{{{j}_{1}}=0}^{{{K}_{1}}}
{...}\sum\limits_{{{j}_{m}}=0}^{{{K}_{m}}}
{{{\alpha }_{({{i}_{1}},...,{{i}_{m}})}}{{\alpha }_{({{j}_{1}},...,{{j}_{m}})
}}}}\prod\limits_{r=1}^{m}\mathcal{M}_{TR}\left({i}_{r}+{j}_{r};
\underline{x}_{r},\overline{x}_{r},\mu_{r},\sigma_{r}\right)
where:
\Phi
(.;{\mu }_{t},{\sigma }_{t}) - is normal random variable's cumulative
distribution function where \mu_{t} and \sigma_{t} are mean and
standard deviation determined by mean[t] and sd[t] arguments
values.
\mathcal{M}_{TR}(q;
\underline{x}_{r},\overline{x}_{r},\mu_{r},\sigma_{r}) - is
q-th order
moment of truncated (from above by \overline{x}_{r} and from below by
\underline{x}_{r})
normal random variable with mean {\mu }_{r} and standard
deviation {\sigma }_{r}. Note that function
truncatedNormalMoment allows to calculate and
differentiate truncated normal random variable's moments.
\overline{x} = (\overline{x}_{1},...,\overline{x}_{m}) -
vector of upper integration limits
i.e. rows of x_upper matrix in ihpa.
\underline{x} = (\underline{x}_{1},...,\underline{x}_{m}) -
vector of lower integration limits
i.e. rows of x_lower matrix in ihpa.
Therefore ihpa allows to calculate interval distribution
function \bar{F}_{\xi}(\underline{x},\bar{x})
values at points determined by rows of x_lower (\underline{x})
and x_upper (\overline{x}) matrices.
The rest of the arguments are similar to dhpa.
Expected value powered product approximation is as follows:
E\left( \prod\limits_{t=1}^{m}\xi_{t}^{{{k}_{t}}} \right)=
\frac{1}{\psi }\sum\limits_{{{i}_{1}}=0}^{{{K}_{1}}}{...}
\sum\limits_{{{i}_{m}}=0}^{{{K}_{m}}}
{\sum\limits_{{{j}_{1}}=0}^{{{K}_{1}}}{...}
\sum\limits_{{{j}_{m}}=0}^{{{K}_{m}}}
{{{\alpha }_{({{i}_{1}},...,{{i}_{m}})}}
{{\alpha }_{({{j}_{1}},...,{{j}_{m}})}}}}
\prod\limits_{r=1}^{m}\mathcal{M}({{i}_{r}}+{{j}_{r}}+{{k}_{t}};
{{\mu }_{r}},{{\sigma }_{r}})
where (k_{1},...,k_{m}) are integer powers determined by
expectation_powers argument of ehpa so
expectation_powers[t] assigns k_{t}. Note that argument x
in ehpa allows to determined conditional values.
Expanding polynomial degrees (K_{1},...,K_{m}) it is possible to
provide arbitrary close approximation to density of some m-variate
random vector \xi^{\star}. So actually f_{\xi}(x)
approximates f_{\xi^{\star}}(x). Accurate approximation requires
appropriate mean, sd and pol_coefficients values
selection. In order to get sample estimates of these parameters please apply
hpaML function.
In order to perform calculation for marginal distribution of some
\xi components please provide omitted
components via omit_ind argument.
For examples if ones assume m=5-variate distribution
and wants to deal with 1-st, 3-rd, and 5-th components
only i.e. (\xi_{1},\xi_{3},\xi_{5}) then set
omit_ind = c(FALSE, TRUE, FALSE, TRUE, FALSE)
indicating that \xi_{2} and \xi_{4} should be 'omitted' from
\xi since 2-nd and 4-th values of omit_ind are
TRUE.
Then x still should be 5 column matrix but
values in 2-nd and 4-th columns will not affect
calculation results. Meanwhile note that marginal distribution of t
components of \xi usually do not coincide with any marginal
distribution generated by t-variate density approximating function.
In order to perform calculation for conditional distribution i.e. given
fixed values for some \xi components please provide these
components via given_ind argument.
For example if ones assume m=5-variate distribution
and wants to deal with 1-st, 3-rd, and 5-th components
given fixed values (suppose 8 and 10) for the other two components i.e.
(\xi|\xi_{2} = 8, \xi_{4} = 10) then set
given_ind = c(FALSE, TRUE, FALSE, TRUE, FALSE) and
x[2] = 8, x[4] = 10 where for simplicity it is assumed that
x is single row 5 column matrix; it is possible to provide
different conditional values for the same components simply setting different
values to different x rows.
Note that it is possible to combine given_ind and omit_ind
arguments. However it is wrong to set both given_ind[i] and
omit_ind[i] to TRUE. Also at least one value should be
FALSE both for given_ind and omit_ind.
In order to consider truncated distribution of \xi i.e.
\left(\xi|\overline{a}_{1}\leq\xi_{1}\leq\overline{b}_{1},
\cdots,\overline{a}_{m}\leq\xi_{m}\leq\overline{b}_{m}\right)
please set lower (left) truncation points \overline{a} and
upper (right) truncation points \overline{b} via tr_left
and tr_right arguments correspondingly. Note that if lower truncation
points are negative infinite and upper truncation points are positive
infinite then dtrhpa, itrhpa and
etrhpa are similar to dhpa,
ihpa and ehpa correspondingly.
In order to calculate Jacobian of f_{\xi }(x;\mu, \sigma, \alpha)
and \bar{F}_{\xi}(\underline{x},\bar{x};\mu, \sigma, \alpha) w.r.t
all ore some particular parameters please apply dhpaDiff
and ihpaDiff functions correspondingly specifying
parameters of interest via type argument. If x or
x_lower and x_upper are single row matrices then gradients
will be calculated.
For further information please see 'Examples' section. Note that examples are given separately for each function.
If given_ind and (or) omit_ind are numeric vectors
then they are insensitive to the order of elements.
For example given_ind = c(5, 2, 3) is similar
to given_ind = c(2, 3, 5).
Densities Hermite polynomial approximation approach has been proposed by A. Gallant and D. W. Nychka in 1987. The main idea is to approximate unknown distribution density with scaled Hermite polynomial. For more information please refer to the literature listed below.
Value
Functions dhpa, phpa and
dtrhpa return vector of probabilities of length
nrow(x).
Functions ihpa and
itrhpa return vector of probabilities of length
nrow(x_upper).
If x argument has not been provided or is a single row
matrix then function
ehpa returns moment value. Otherwise it returns vector of
length nrow(x) containing moments values.
If tr_left and tr_right arguments are single row matrices then
function etrhpa returns moment value.
Otherwise it returns vector of length
max(nrow(tr_left), nrow(tr_right)) containing moments values.
Functions dhpaDiff and ihpaDiff
return Jacobin matrix. The number
of columns depends on type argument. The number of rows is
nrow(x) for dhpaDiff and
nrow(x_upper) for
ihpaDiff
If mean or sd are not specified they assume the default
values of m-dimensional vectors of 0 and 1, respectively.
If x_lower is not specified then it is the matrix of the
same size as x_upper containing negative infinity values only. If
expectation_powers is not specified then it is m-dimensional
vector of 0 values.
Please see 'Details' section for additional information.
References
A. Gallant and D. W. Nychka (1987) <doi:10.2307/1913241>
Examples
## Example demonstrating dhpa function application.
## Let's approximate some three random variables (i.e. X1, X2 and X3)
## joint density function at points x = (0,1, 0.2, 0.3) and
## y = (0.5, 0.8, 0.6) with Hermite polynomial of (1, 2, 3) degrees which
## polynomial coefficients equal 1 except coefficient related to x1*(x^3)
## polynomial element which equals 2. Also suppose that normal density
## related mean vector equals (1.1, 1.2, 1.3) while standard deviations
## vector is (2.1, 2.2, 2.3).
# Prepare initial values
x <- matrix(c(0.1, 0.2, 0.3), nrow = 1) # x point as a single row matrix
y <- matrix(c(0.5, 0.8, 0.6), nrow = 1) # y point as a single row matrix
x_y <- rbind(x, y) # matrix which rows are x and y
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power", "x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate density approximation
# at point x (note that x should be a matrix)
dhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# at points x and y
dhpa(x = x_y,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.5 i.e. X2 = 0.5.
# Substitute x and y second components with conditional value 0.5
x <- matrix(c(0.1, 0.5, 0.3), nrow = 1) # or simply x[2] <- 0.5
y <- matrix(c(0.4, 0.5, 0.6), nrow = 1) # or simply y[2] <- 0.5
x_y <- rbind(x, y)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) density approximation
# at point x
dhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# at points x and y
dhpa(x = x_y,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution conditioned on the
# second component 0.5 value i.e. (X3 | X2 = 0.5).
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on x2 = 0.5) marginal (for x3) density approximation
# at point x
dhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind)
# at points x and y
dhpa(x = x_y,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind)
## Example demonstrating phpa function application.
## Let's approximate some three random variables (X1, X2, X3)
## joint cumulative distribution function (cdf) at point (0,1, 0.2, 0.3)
## with Hermite polynomial of (1, 2, 3) degrees which polynomial
## coefficients equal 1 except coefficient related to x1*(x^3) polynomial
## element which equals 2. Also suppose that normal density related
## mean vector equals (1.1, 1.2, 1.3) while standard deviations
## vector is (2.1, 2.2, 2.3).
## Prepare initial values
x <- matrix(c(0.1, 0.2, 0.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate cdf approximation at point x
phpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.5
# Substitute x second component with conditional value 0.5
x <- matrix(c(0.1, 0.5, 0.3), nrow = 1) # or simply x[2] <- 0.5
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) cdf approximation at point x
phpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.5 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) cdf
# approximation at point x
phpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind)
## Example demonstrating ihpa function application.
## Let's approximate some three random variables (X1, X2, X3) joint interval
## distribution function (intdf) at lower and upper points (0,1, 0.2, 0.3)
## and (0,4, 0.5, 0.6) correspondingly with Hermite polynomial of (1, 2, 3)
## degrees which polynomial coefficients equal 1 except coefficient related
## to x1*(x^3) polynomial element which equals 2. Also suppose that normal
## density related mean vector equals (1.1, 1.2, 1.3) while standard
## deviations vector is (2.1, 2.2, 2.3).
## Prepare initial values
x_lower <- matrix(c(0.1, 0.2, 0.3), nrow=1)
x_upper <- matrix(c(0.4, 0.5, 0.6), nrow=1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate intdf approximation at points x_lower and x_upper
ihpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.7
# Substitute x second component with conditional value 0.7
x_upper <- matrix(c(0.4, 0.7, 0.6), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) intdf approximation
# at points x_lower and x_upper
ihpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.7 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3)
# intdf approximation at points x_lower and x_upper
ihpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind)
## Example demonstrating ehpa function application.
## Let's approximate some three random variables (X1, X2, X3) powered product
## expectation for powers (3, 2, 1) with Hermite polynomial of (1, 2, 3)
## degrees which polynomial coefficients equal 1 except coefficient
## related to x1*(x^3) polynomial element which equals 2.
## Also suppose that normal density related mean vector equals
## (1.1, 1.2, 1.3) while standard deviations vector is (2.1, 2.2, 2.3).
# Prepare initial values
expectation_powers = c(3,2,1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
#Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate expected powered product approximation
ehpa(pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers)
# Condition second component to be 0.5
# Substitute x second component with conditional value 0.5
x <- matrix(c(NA, 0.5, NA), nrow = 1)
#Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) expected powered product approximation
ehpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.5 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) expected powered
# product approximation at points x_lower and x_upper
ehpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers,
given_ind = given_ind,
omit_ind = omit_ind)
## Example demonstrating etrhpa function application.
## Let's approximate some three truncated random variables (X1, X2, X3)
## powered product expectation for powers (3, 2, 1) with Hermite polynomial
## of (1,2,3) degrees which polynomial coefficients equal 1 except
## coefficient related to x1*(x^3) polynomial element which equals 2. Also
## suppose that normal density related mean vector equals (1.1, 1.2, 1.3)
## while standard deviations vector is (2.1, 2.2, 2.3). Suppose that lower
## and upper truncation points are (-1.1,-1.2,-1.3) and (1.1,1.2,1.3)
## correspondingly.
# Prepare initial values
expectation_powers = c(3,2,1)
tr_left = matrix(c(-1.1,-1.2,-1.3), nrow = 1)
tr_right = matrix(c(1.1,1.2,1.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate expected powered product approximation for truncated distribution
etrhpa(pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers,
tr_left = tr_left, tr_right = tr_right)
## Example demonstrating dtrhpa function application.
## Let's approximate some three random variables (X1, X2, X3) joint density
## function at point (0,1, 0.2, 0.3) with Hermite polynomial of (1,2,3)
## degrees which polynomial coefficients equal 1 except coefficient related
## to x1*(x^3) polynomial element which equals 2. Also suppose that normal
## density related mean vector equals (1.1, 1.2, 1.3) while standard
## deviations vector is (2.1, 2.2, 2.3). Suppose that lower and upper
## truncation points are (-1.1,-1.2,-1.3) and (1.1,1.2,1.3) correspondingly.
# Prepare initial values
x <- matrix(c(0.1, 0.2, 0.3), nrow=1)
tr_left = matrix(c(-1.1,-1.2,-1.3), nrow = 1)
tr_right = matrix(c(1.1,1.2,1.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate density approximation at point x
dtrhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
tr_left = tr_left,
tr_right = tr_right)
# Condition second component to be 0.5
# Substitute x second component with conditional value 0.5
x <- matrix(c(0.1, 0.5, 0.3), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on x2 = 0.5) density approximation at point x
dtrhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
tr_left = tr_left, tr_right = tr_right)
# Consider third component marginal distribution
# conditioned on the second component 0.5 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3)
# density approximation at point x
dtrhpa(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
tr_left = tr_left, tr_right = tr_right)
## Example demonstrating itrhpa function application.
## Let's approximate some three truncated random variables (X1, X2, X3) joint
## interval distribution function at lower and upper points (0,1, 0.2, 0.3)
## and (0,4, 0.5, 0.6) correspondingly with Hermite polynomial of (1 ,2, 3)
## degrees which polynomial coefficients equal 1 except coefficient
## related to x1*(x^3) polynomial element which equals 2. Also suppose
## that normal density related mean vector equals (1.1, 1.2, 1.3) while
## standard deviations vector is (2.1, 2.2, 2.3). Suppose that lower and
## upper truncation are (-1.1,-1.2,-1.3) and (1.1,1.2,1.3) correspondingly.
# Prepare initial values
x_lower <- matrix(c(0.1, 0.2, 0.3), nrow=1)
x_upper <- matrix(c(0.4, 0.5, 0.6), nrow=1)
tr_left = matrix(c(-1.1,-1.2,-1.3), nrow = 1)
tr_right = matrix(c(1.1,1.2,1.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate intdf approximation at points x_lower and x_upper
itrhpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
tr_left = tr_left, tr_right = tr_right)
# Condition second component to be 0.7
# Substitute x second component with conditional value 0.7
x_upper <- matrix(c(0.4, 0.7, 0.6), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) intdf
# approximation at points x_lower and x_upper
itrhpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
tr_left = tr_left, tr_right = tr_right)
# Consider third component marginal distribution
# conditioned on the second component 0.7 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) intdf
# approximation at points x_lower and x_upper
itrhpa(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
tr_left = tr_left, tr_right = tr_right)
## Example demonstrating dhpaDiff function application.
## Let's approximate some three random variables (X1, X2, X3) joint density
## function at point (0,1, 0.2, 0.3) with Hermite polynomial of (1,2,3)
## degrees which polynomial coefficients equal 1 except coefficient related
## to x1*(x^3) polynomial element which equals 2. Also suppose that normal
## density related mean vector equals (1.1, 1.2, 1.3) while standard
## deviations vector is (2.1, 2.2, 2.3). In this example let's calculate
## density approximating function's gradient respect to various parameters
# Prepare initial values
x <- matrix(c(0.1, 0.2, 0.3), nrow = 1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate density approximation gradient
# respect to polynomial coefficients at point x
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.5
# Substitute x second component with conditional value 0.5
x <- matrix(c(0.1, 0.5, 0.3), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on x2 = 0.5) density approximation's
# gradient respect to polynomial coefficients at point x
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.5 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) density
# approximation's gradient respect to:
# polynomial coefficients
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind)
# mean
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind,
type = "mean")
# sd
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind,
type = "sd")
# x
dhpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
omit_ind = omit_ind,
type = "x")
## Example demonstrating ehpaDiff function application.
## Let's approximate some three random variables (X1, X2, X3) expectation
## of the form E((X1 ^ 3) * (x2 ^ 1) * (X3 ^ 2)) and calculate the gradient
# Distribution parameters
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
pol_coefficients_n <- prod(pol_degrees + 1)
pol_coefficients <- rep(1, pol_coefficients_n)
# Set powers for expectation
expectation_powers <- c(3, 1, 2)
# Calculate expectation approximation gradient
# respect to all parameters
ehpaDiff(pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
expectation_powers = expectation_powers,
type = "all")
# Let's calculate gradient of E(X1 ^ 3 | (X2 = 1, X3 = 2))
x <- c(0, 1, 2) # x[1] may be arbitrary (not NA) values
expectation_powers <- c(3, 0, 0) # expectation_powers[2:3] may be
# arbitrary (not NA) values
given_ind <- c(2, 3)
ehpaDiff(x = x,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind,
expectation_powers = expectation_powers,
type = "all")
## Example demonstrating ihpaDiff function application.
## Let's approximate some three random variables (X1, X2, X3 ) joint interval
## distribution function (intdf) at lower and upper points (0,1, 0.2, 0.3)
## and (0,4, 0.5, 0.6) correspondingly with Hermite polynomial of (1, 2, 3)
## degrees which polynomial coefficients equal 1 except coefficient
## related to x1*(x^3) polynomial element which equals 2.
## Also suppose that normal density related mean vector equals
## (1.1, 1.2, 1.3) while standard deviations vector is (2.1, 2.2, 2.3).
## In this example let's calculate interval distribution approximating
## function gradient respect to polynomial coefficients.
# Prepare initial values
x_lower <- matrix(c(0.1, 0.2, 0.3), nrow=1)
x_upper <- matrix(c(0.4, 0.5, 0.6), nrow=1)
mean <- c(1.1, 1.2, 1.3)
sd <- c(2.1, 2.2, 2.3)
pol_degrees <- c(1, 2, 3)
# Create polynomial powers and indexes correspondence matrix
pol_ind <- polynomialIndex(pol_degrees)
# Set all polynomial coefficients to 1
pol_coefficients <- rep(1, ncol(pol_ind))
pol_degrees_n <- length(pol_degrees)
# Assign coefficient 2 to the polynomial element (x1 ^ 1)*(x2 ^ 0)*(x3 ^ 2)
pol_coefficients[apply(pol_ind, 2, function(x) all(x == c(1, 0, 2)))] <- 2
# Visualize correspondence between polynomial
# elements and their coefficients
as.data.frame(rbind(pol_ind, pol_coefficients),
row.names = c("x1 power", "x2 power",
"x3 power", "coefficients"),
optional = TRUE)
printPolynomial(pol_degrees, pol_coefficients)
# Calculate intdf approximation gradient respect to
# polynomial coefficients at points x_lower and x_upper
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
# Condition second component to be 0.7
# Substitute x second component with conditional value 0.7
x_upper <- matrix(c(0.4, 0.7, 0.6), nrow = 1)
# Set TRUE to the second component indicating that it is conditioned
given_ind <- c(FALSE, TRUE, FALSE)
# Calculate conditional (on X2 = 0.5) intdf approximation
# respect to polynomial coefficients at points x_lower and x_upper
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind)
# Consider third component marginal distribution
# conditioned on the second component 0.7 value
# Set TRUE to the first component indicating that it is omitted
omit_ind <- c(TRUE, FALSE, FALSE)
# Calculate conditional (on X2 = 0.5) marginal (for X3) intdf approximation
# respect to:
# polynomial coefficients
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind)
# mean
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
type = "mean")
# sd
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
type = "sd")
# x_lower
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
type = "x_lower")
# x_upper
ihpaDiff(x_lower = x_lower, x_upper = x_upper,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = given_ind, omit_ind = omit_ind,
type = "x_upper")
## Examples demonstrating qhpa function application.
## Sub-example 1 - univariate distribution
## Consider random variable X
# Distribution parameters
mean <- 1
sd <- 2
pol_degrees <- 2
pol_coefficients <- c(1, 0.1, -0.01)
# The level of quantile
p <- 0.7
# Calculate quantile of X
qhpa(p = p,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
## Sub-example 2 - marginal distribution
## Consider random vector (X1, X2) and quantile of X1
# Distribution parameters
mean <- c(1, 1.2)
sd <- c(2, 3)
pol_degrees <- c(2, 2)
pol_coefficients <- c(1, 0.1, -0.01, 0.2, 0.012,
0.0013, 0.0042, 0.00025, 0)
# The level of quantile
p <- 0.7
# Calculate quantile of X1
qhpa(p = p,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
omit_ind = 2) # set omitted variable index
## Sub-example 3 - marginal and conditional distribution
## Consider random vector (X1, X2, X3) and
## quantiles of X1|X3 and X1|(X2,X3)
mean <- c(1, 1.2, 0.9)
sd <- c(2, 3, 2.5)
pol_degrees <- c(1, 1, 1)
pol_coefficients <- c(1, 0.1, -0.01, 0.2, 0.012,
0.0013, 0.0042, 0.00025)
# The level of quantile
p <- 0.7
# Calculate quantile of X1|X3 = 0.2
qhpa(p = p,
x = matrix(c(NA, NA, 0.2), nrow = 1), # set any values to
# unconditioned and
# omitted components
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
omit_ind = 2, # set omitted variable index
given_ind = 3) # set conditioned variable index
# Calculate quantile of X1|(X2 = 0.5, X3 = 0.2)
qhpa(p = p,
x = matrix(c(NA, 0.5, 0.2), nrow = 1), # set any values to
# unconditioned and
# omitted components
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd,
given_ind = c(2, 3)) # set conditioned
# variables indexes
## Examples demonstrating rhpa function application.
# Set seed for reproducibility
set.seed(123)
# Distribution parameters
mean <- 1
sd <- 2
pol_degrees <- 2
pol_coefficients <- c(1, 0.1, -0.01)
# Simulate two observations from this distribution
rhpa(n = 2,
pol_coefficients = pol_coefficients,
pol_degrees = pol_degrees,
mean = mean, sd = sd)
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