pprobability {exams.forge}

Polynomial Probability

Description

Creates for each value of a discrete random variable, a polynomial and estimates the least squares and the maximum likelihood solution. The following conditions stand:

• If sample is not given then the sample contains each x value once.

• If sample is an integer, then it is interpreted as the sample size and a sample is generated by rmultinom(1, sample, ddiscrete(runif(length(x)))).

• If sample is a vector, it is interpreted in such a way that the corresponding x[i] value occurs i times in the sample. Thus, sum(sample) is the sample size.

• If coeff is a polylist of length(x), then these polynomials are taken.

• If coeff is a matrix with length(x), columns and power+1 rows, then the columns are interpreted as the coefficients of a polynomial.

• Otherwise coeff is interpreted as a vector from which the coefficient is sampled. The intercepts are sampled via ddiscrete(runif(length(x)), zero=zero). If coeff is not given then it is ensured that the least squares and the maximum likelihood solution exists and the estimated probabilities are between zero and one. Otherwise, the results may contain NA or the estimated probabilities are outside the interval [0;1].

Usage

pprobability(
  x,
  power = 1,
  zero = FALSE,
  coef = round(seq(-1, 1, by = 0.1), 1),
  sample = rep(1, length(x)),
  pl = NULL,
  tol = 1e-09
)

polynomial_probability(
  x,
  power = 1,
  zero = FALSE,
  coef = round(seq(-1, 1, by = 0.1), 1),
  sample = rep(1, length(x)),
  pl = NULL,
  tol = 1e-09
)

Arguments

x

numeric: values of a discrete random variable

power

integer: the degree for the polynomials (default: 1), must be larger 0

zero

logical: are zero coefficients and zero samples allowed? (default: FALSE)

coef

matrix: for each degree coefficients to sample from (default: seq(-1, 1, by=0.1))

sample

integer: number of x values in the sample or sample size (default: rep(1, length(x)))

pl

polylist: a list of polynomials which describes the probability for x (default: NULL)

tol

numeric: tolerance to detect zero values (default: 1e-9)

Value

A list with the components:

• p: the polynomials for the probabilities

• ep: the expected value as polynomial

• x: the values for the discrete random variable, the same as the input x

• sample: the sample given or generated

• LS$pi: the summands for the least squares problem

• LS$pl: the summands for the least squares problem in LaTeX

• LS$pf: the sum of LS$pi

• LS$df: the derivative of LS$pf

• LS$pest: the estimated parameter, minimum of LS$pf

• LS$p: the estimated probabilities

• ML$pi: the factors for the maximum likelihood problem

• ML$pl: the summands for the maximum likelihood problem in LaTeX

• ML$pf: the product of ML$pi

• ML$df: the derivative of ML$pf

• ML$pest: the estimated parameter, maximum of ML$pf

• ML$p: the estimated probabilities

Examples

# linear polynomials
pprobability(0:2)
pprobability(0:2, power=1)
# constant polynomials, some NAs are generated
pprobability(0:3, power=0)
# polynomials generated from a different set
pprobability(0:2, coef=seq(-2, 2, by=0.1))
pprobability(0:2, 0, coef=seq(-2, 2, by=0.1))
# polynomials (x, x, 1-2*x) are used
pprobability(0:2, 0, coef=matrix(c(0.4, 0.4, 0.3), ncol=3))
pprobability(0:2, 1, coef=polylist(c(0,1), c(0,1), c(1, -2)))