R/Beta.R
In Amalan-ConStat/R-fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD
#' Beta Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Beta Distribution bounded between [0,1]
#'
#' @usage
#' dBETA(p,a,b)
#'
#' @param p vector of probabilities.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded Beta distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{B_p(a,b)}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{a}{a+b} }
#' \deqn{var[P]= \frac{ab}{(a+b)^2(a+b+1)} }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \prod_{i=0}^{r-1} (\frac{a+i}{a+b+i}) }
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B_p(a,b)=\int^p_0 t^{a-1} (1-t)^{b-1}\,dt} is
#' incomplete beta integrals and \eqn{B(a,b)} is the beta function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{dBETA} gives a list format consisting
#'
#' \code{pdf} probability density values in vector form.
#'
#' \code{mean} mean of the Beta distribution.
#'
#' \code{var} variance of the Beta distribution.
#'
#' @references
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis,
#' 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' @seealso
#' \code{\link[stats]{Beta}}
#'
#' or
#'
#' \url{https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,4))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dBETA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dBETA(seq(0,1,by=0.01),2,3)$pdf #extracting the pdf values
#' dBETA(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dBETA(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pBETA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pBETA(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#' mazBETA(1.4,3,2) #acquiring the moment about zero values
#' mazBETA(2,3,2)-mazBETA(1,3,2)^2 #acquiring the variance for a=3,b=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazBETA(1.9,5.5,6)
#'
#' @export
dBETA<-function(p,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b))) | any(is.infinite(c(p,a,b))) | any(is.nan(c(p,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, if not providing error message and
#stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
ans<-NULL
#for each input values in the vector necessary calculations and conditions are applied
for (i in 1:length(p))
{
if(p[i]<0 |p[i]>1)
{
stop("Invalid values in the input")
}
else
{
ans[i]<-(p[i]^(a-1)*(1-p[i])^(b-1))/beta(a,b)
}
}
}
mean<-a/(a+b) #according to theory the mean value
variance<-(a*b)/(((a+b)^2)*(a+b+1)) #according to theory the variance value
# generating an output in list format consisting pdf,mean and variance
output<-list("pdf"=ans,"mean"=mean,"var"=variance)
return(output)
}
}
#' Beta Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Beta Distribution bounded between [0,1].
#'
#' @usage
#' pBETA(p,a,b)
#'
#' @param p vector of probabilities.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded beta distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{B_p(a,b)}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{a}{a+b} }
#' \deqn{var[P]= \frac{ab}{(a+b)^2(a+b+1)} }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \prod_{i=0}^{r-1} (\frac{a+i}{a+b+i}) }
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B_p(a,b)=\int^p_0 t^{a-1} (1-t)^{b-1}\,dt} is
#' incomplete beta integrals and \eqn{B(a,b)} is the beta function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{pBETA} gives the cumulative density values in vector form.
#'
#' @references
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis,
#' 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' @seealso
#' \code{\link[stats]{Beta}}
#'
#' or
#'
#' \url{https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,4))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dBETA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dBETA(seq(0,1,by=0.01),2,3)$pdf #extracting the pdf values
#' dBETA(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dBETA(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pBETA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pBETA(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#' mazBETA(1.4,3,2) #acquiring the moment about zero values
#' mazBETA(2,3,2)-mazBETA(1,3,2)^2 #acquiring the variance for a=3,b=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazBETA(1.9,5.5,6)
#'
#' @export
pBETA<-function(p,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b))) | any(is.infinite(c(p,a,b))) | any(is.nan(c(p,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero and if not providing an error message
#and stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
ans<-NULL
val<-NULL
#the equation contains partial beta integration, below is the integral function
Bp<-function(q)
{
(q^(a-1))*((1-q)^(b-1))
}
#for each input values in the vector necessary calculations and conditions are applied
for(i in 1:length(p))
{
if(p[i]<0 | p[i]>1)
{
stop("Invalid values in the input")
}
else
{
#integrating the above mentioned function under limits of zero and vector p
val<-stats::integrate(Bp,lower = 0,upper = p[i])
ans[i]<-val$value/beta(a,b)
}
}
#generating an ouput vector of cumulative probability values
return(ans)
}
}
}
#' Beta Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Beta Distribution bounded between [0,1].
#'
#' @usage
#' mazBETA(r,a,b)
#'
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param r vector of moments.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded beta distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{B_p(a,b)}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{a}{a+b} }
#' \deqn{var[P]= \frac{ab}{(a+b)^2(a+b+1)} }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \prod_{i=0}^{r-1} (\frac{a+i}{a+b+i}) }
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B_p(a,b)=\int^p_0 t^{a-1} (1-t)^{b-1}\,dt} is
#' incomplete beta integrals and \eqn{B(a,b)} is the beta function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{mazBETA} gives the moments about zero in vector form.
#'
#' @references
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis,
#' 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' @seealso
#' \code{\link[stats]{Beta}}
#'
#' or
#'
#' \url{https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,4))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dBETA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dBETA(seq(0,1,by=0.01),2,3)$pdf #extracting the pdf values
#' dBETA(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dBETA(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pBETA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pBETA(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#' mazBETA(1.4,3,2) #acquiring the moment about zero values
#' mazBETA(2,3,2)-mazBETA(1,3,2)^2 #acquiring the variance for a=3,b=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazBETA(1.9,5.5,6)
#'
#' @export
mazBETA<-function(r,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(r,a,b))) | any(is.infinite(c(r,a,b))) | any(is.nan(c(r,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, and if not providing an error
#message and stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#the moments cannot be a decimal value therefore converting it into an integer
r<-as.integer(r)
ans<-NULL
#for each input values in the vector necessary calculations and conditions are applied
for(i in 1:length(r))
{
#checking if moment values are less than or equal to zero and creating
# an error message as well as stopping the function progress
if(r[i]<=0)
{
stop("Moments cannot be less than or equal to zero")
}
else
{
j<-0:(r[i]-1)
ans[i]<-prod((a+j)/(a+b+j))
}
}
#generating an ouput vector of moment about zero values
return(ans)
}
}
}
#' Beta-Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Beta-Binomial Distribution.
#'
#' @usage
#' dBetaBin(x,n,a,b)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' Mixing Beta distribution with Binomial distribution will create the Beta-Binomial
#' distribution. The probability function and cumulative probability function can be
#' constructed and are denoted below.
#'
#' The cumulative probability function is the summation of probability function values.
#'
#' \deqn{P_{BetaBin}(x)= {n \choose x} \frac{B(a+x,n+b-x)}{B(a,b)} }
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{BetaBin}[x]= \frac{na}{a+b} }
#' \deqn{Var_{BetaBin}[x]= \frac{(nab)}{(a+b)^2} \frac{(a+b+n)}{(a+b+1)} }
#' \deqn{over dispersion= \frac{1}{a+b+1} }
#'
#' Defined as \code{B(a,b)} is the beta function.
#'
#' @return
#' The output of \code{dBetaBin} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of the Beta-Binomial Distribution.
#'
#' \code{var} variance of the Beta-Binomial Distribution.
#'
#' \code{over.dis.para} over dispersion value of the Beta-Binomial Distribution.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,0.2)
#' plot(0,0,main="Beta-binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
#' for (i in 1:5)
#' {
#' lines(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dBetaBin(0:10,10,4,.2)$pdf #extracting the pdf values
#' dBetaBin(0:10,10,4,.2)$mean #extracting the mean
#' dBetaBin(0:10,10,4,.2)$var #extracting the variance
#' dBetaBin(0:10,10,4,.2)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pBetaBin(0:10,10,4,.2) #acquiring the cumulative probability values
#'
#' @export
dBetaBin<-function(x,n,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,n,a,b))) | any(is.infinite(c(x,n,a,b))) |any(is.nan(c(x,n,a,b))))
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are less than or equal zero ,
#if so providing an error message and stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#checking if at any chance the binomial random variable is greater than binomial trial value
#if so providing an error message and stopping the function from progress
if(max(x)>n)
{
stop("Binomial random variable cannot be greater than binomial trial value")
}
#checking if any random variable or trial value is negative if so providig an error message
#and stopping the function progress
else if(any(x<0) | n<0)
{
stop("Binomial random variable or binomial trial value cannot be negative")
}
ans<-NULL
#for each random variable in the input vector below calculations occur
for (i in 1:length(x))
{
ans[i]<-choose(n,x[i])*(beta(a+x[i],n+b-x[i])/beta(a,b))
}
}
}
mean<-n*(a/(a+b)) #according to theory the mean
variance<-n*((a*b)/(a+b)^2)*((a+b+n)/(a+b+1)) #according to theory variance
ove.dis.par<-1/(a+b+1) #according to theory overdispersion value
# generating an output in list format consisting pdf,mean,variance and overdispersion value
output<-list('pdf'=ans,'mean'=mean,'var'=variance,
'over.dis.para'=ove.dis.par)
return(output)
}
#' Beta-Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Beta-Binomial Distribution.
#'
#' @usage
#' pBetaBin(x,n,a,b)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' Mixing Beta distribution with Binomial distribution will create the Beta-Binomial
#' distribution. The probability function and cumulative probability function can be
#' constructed and are denoted below.
#'
#' The cumulative probability function is the summation of probability function values.
#'
#' \deqn{P_{BetaBin}(x)= {n \choose x} \frac{B(a+x,n+b-x)}{B(a,b)} }
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{BetaBin}[x]= \frac{na}{a+b} }
#' \deqn{Var_{BetaBin}[x]= \frac{(nab)}{(a+b)^2} \frac{(a+b+n)}{(a+b+1)} }
#' \deqn{over dispersion= \frac{1}{a+b+1} }
#'
#' Defined as \code{B(a,b)} is the beta function.
#'
#' @return
#' The output of \code{pBetaBin} gives cumulative probability values in vector form.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,0.2)
#' plot(0,0,main="Beta-binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
#' for (i in 1:5)
#' {
#' lines(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dBetaBin(0:10,10,4,.2)$pdf #extracting the pdf values
#' dBetaBin(0:10,10,4,.2)$mean #extracting the mean
#' dBetaBin(0:10,10,4,.2)$var #extracting the variance
#' dBetaBin(0:10,10,4,.2)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pBetaBin(0:10,10,4,.2) #acquiring the cumulative probability values
#'
#' @export
pBetaBin<-function(x,n,a,b)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative proability function
#values are calculated
for(i in 1:length(x))
{
j<-0:x[i]
ans[i]<-sum(dBetaBin(j,n,a,b)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of Beta-Binomial Distribution
#'
#' This function will calculate the Negative Log Likelihood value when the vector of binomial random
#' variables and vector of corresponding frequencies are given with the shape parameters a and b.
#'
#' @usage
#' NegLLBetaBin(x,freq,a,b)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' \deqn{0 < a,b }
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,...}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{NegLLBetaBin} will produce a single numeric value.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' NegLLBetaBin(No.D.D,Obs.fre.1,.3,.4) #acquiring the negative log likelihood value
#'
#' @export
NegLLBetaBin<-function(x,freq,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,freq,a,b))) | any(is.infinite(c(x,freq,a,b)))
|any(is.nan(c(x,freq,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if any of the random variables of frequencies are less than zero if so
#creating an error message as well as stopping the function progress
if( any(c(x,freq)< 0) )
{
stop("Binomial random variable or frequency values cannot be negative")
}
#checking if shape parameters are less than or equal to zero
#if so creating an error message as well as stopping the function progress
else if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
i<-1:sum(freq)
term1<-sum(log(choose(n,data[i])))
term2<-sum(log(beta(a+data[i],n+b-data[i])))
BetaBinLL<-term1+term2-sum(freq)*log(beta(a,b))
#calculating the negative log likelihood value and representing as a single output value
return(-BetaBinLL)
}
}
}
#' Estimating the shape parameters a and b for Beta-Binomial Distribution
#'
#' The functions will estimate the shape parameters using the maximum log likelihood method and
#' moment generating function method for the Beta-Binomial distribution when the binomial
#' random variables and corresponding frequencies are given.
#'
#' @usage
#' EstMLEBetaBin(x,freq,a,b,...)
#'
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#'
#' @details
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' \code{EstMLEBetaBin} here is used as a wrapper for the \code{mle2} function of \pkg{bbmle} package
#' therefore output is of class of mle2.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' estimate <- EstMLEBetaBin(No.D.D,Obs.fre.1,a=0.1,b=0.1)
#'
#' bbmle::coef(estimate) #extracting the parameters
#'
#' #estimating the parameters using moment generating function methods
#' EstMGFBetaBin(No.D.D,Obs.fre.1)
#'
#'@export
EstMLEBetaBin<-function(x,freq,a,b,...)
{
suppressWarnings2 <-function(expr, regex=character())
{
withCallingHandlers(expr, warning=function(w)
{
if (length(regex) == 1 && length(grep(regex, conditionMessage(w))))
{
invokeRestart("muffleWarning")
}
} )
}
suppressWarnings2(bbmle::mle2(.EstMLEBetaBin,data=list(x=x,freq=freq),
start = list(a=a,b=b),...),"NaN")
}
.EstMLEBetaBin<-function(x,freq,a,b)
{
#with respective to using bbmle package function mle2 there is no need impose any restrictions
#therefor the output is directly a single numeric value for the negative log likelihood value of
#beta binomial distribution
n<-max(x)
data<-rep(x,freq)
i<-1:sum(freq)
term1<-sum(log(choose(n,data[i])))
term2<-sum(log(beta(a+data[i],n+b-data[i])))
BetaBinLL<-term1+term2-sum(freq)*log(beta(a,b))
return(-BetaBinLL)
}
#' Estimating the shape parameters a and b for Beta-Binomial Distribution
#'
#' The functions will estimate the shape parameters using the maximum log likelihood method and
#' moment generating function method for the Beta-Binomial distribution when the binomial
#' random variables and corresponding frequencies are given.
#'
#' @usage
#' EstMGFBetaBin(x,freq)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#'
#' @details
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{EstMGFBetaBin} will produce the class \code{mgf} format consisting
#'
#' \code{a} shape parameter of beta distribution representing for alpha
#'
#' \code{b} shape parameter of beta distribution representing for beta
#'
#' \code{min} Negative loglikelihood value
#'
#' \code{AIC} AIC value
#'
#' \code{call} the inputs for the function
#'
#' Methods \code{print}, \code{summary}, \code{coef} and \code{AIC} can be used to extract
#' specific outputs.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' estimate <- EstMLEBetaBin(No.D.D,Obs.fre.1,a=0.1,b=0.1)
#'
#' bbmle::coef(estimate) #extracting the parameters
#'
#' #estimating the parameters using moment generating function methods
#' results <- EstMGFBetaBin(No.D.D,Obs.fre.1)
#'
#' # extract the estimated parameters and summary
#' coef(results)
#' summary(results)
#'
#' AIC(results) #show the AIC value
#'
#' @export
EstMGFBetaBin<-function(x,freq)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,freq))) | any(is.infinite(c(x,freq))) | any(is.nan(c(x,freq))))
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
#creating necessary equations
m1<-sum(data)/length(data)
m2<-sum(data^2)/length(data)
#constructing equations according to theory for a(alpha) and b(beta)
a<-((n*m1-m2)*m1)/(n*(m2-m1-m1^2)+m1^2)
b<-((n*m1-m2)*(n-m1))/(n*(m2-m1-m1^2)+m1^2)
#generating an output of list for shape parameters a(alpha) and b(beta)
NegLL<-NegLLBetaBin(x,freq,a,b)
AICvalue<-2*2+(2*NegLL)
argument<-match.call()
ans<-list("a"=a,"b"=b,"min"=NegLL,"AIC"=AICvalue,"call"=argument)
class(ans)<-"mgf"
return(ans)
}
}
#' @method EstMGFBetaBin default
#' @export
EstMGFBetaBin.default<-function(x,freq)
{
est<-EstMGFBetaBin(x,freq)
return(est)
}
#' @method print mgf
#' @export
print.mgf<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nCoefficients: \n")
coeff<-c(x$a,x$b)
names(coeff)<-c("a","b")
print(coeff)
}
#' @method summary mgf
#' @export
summary.mgf<-function(object,...)
{
cat("Coefficients: \n a \t b \n", object$a,object$b)
cat("\n\nNegative Log-likelihood : ",object$min)
cat("\n\nAIC : ",object$AIC)
}
#' @method coef mgf
#' @export
coef.mgf<-function(object,...)
{
cat(" \t a \t b \n", object$a,object$b)
}
#' @method AIC mgf
#' @export
AIC.mgf<-function(object,...)
{
return(object$AIC)
}
#' Fitting the Beta-Binomial Distribution when binomial random variable, frequency and shape
#' parameters a and b are given
#'
#' The function will fit the Beta-Binomial distribution when random variables, corresponding
#' frequencies and shape parameters are given. It will provide the expected frequencies, chi-squared
#' test statistics value, p value, degree of freedom and over dispersion value so that it can be
#' seen if this distribution fits the data.
#'
#' @usage fitBetaBin(x,obs.freq,a,b)
#'
#' @param x vector of binomial random variables.
#' @param obs.freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' \deqn{0 < a,b}
#' \deqn{x = 0,1,2,...,n}
#' \deqn{obs.freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{fitBetaBin} gives the class format \code{fitBB} and \code{fit} consisting a list
#'
#' \code{bin.ran.var} binomial random variables.
#'
#' \code{obs.freq} corresponding observed frequencies.
#'
#' \code{exp.freq} corresponding expected frequencies.
#'
#' \code{statistic} chi-squared test statistics.
#'
#' \code{df} degree of freedom.
#'
#' \code{p.value} probability value by chi-squared test statistic.
#'
#' \code{fitBB} fitted values of \code{dBetaBin}.
#'
#' \code{NegLL} Negative Log Likelihood value.
#'
#' \code{a} estimated value for alpha parameter as a.
#'
#' \code{b} estimated value for alpha parameter as b.
#'
#' \code{AIC} AIC value.
#'
#' \code{over.dis.para} over dispersion value.
#'
#' \code{call} the inputs of the function.
#'
#' Methods \code{summary}, \code{print}, \code{AIC}, \code{residuals} and \code{fitted} can be
#' used to extract specific outputs.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLEBetaBin(No.D.D,Obs.fre.1,0.1,0.1)
#'
#' bbmle::coef(parameters) #extracting the parameters a and b
#' aBetaBin <- bbmle::coef(parameters)[1] #assigning the parameter a
#' bBetaBin <- bbmle::coef(parameters)[2] #assigning the parameter b
#'
#' #fitting when the random variable,frequencies,shape parameter values are given.
#' fitBetaBin(No.D.D,Obs.fre.1,aBetaBin,bBetaBin)
#'
#' #estimating the parameters using moment generating function methods
#' results <- EstMGFBetaBin(No.D.D,Obs.fre.1)
#' results
#'
#' aBetaBin1 <- results$a #assigning the estimated a
#' bBetaBin1 <- results$b #assigning the estimated b
#'
#' #fitting when the random variable,frequencies,shape parameter values are given.
#' BB <- fitBetaBin(No.D.D,Obs.fre.1,aBetaBin1,bBetaBin1)
#'
#' #extracting the expected frequencies
#' fitted(BB)
#'
#' #extracting the residuals
#' residuals(BB)
#'
#' @export
fitBetaBin<-function(x,obs.freq,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,obs.freq,a,b))) | any(is.infinite(c(x,obs.freq,a,b))) |
any(is.nan(c(x,obs.freq,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dBetaBin(x,max(x),a,b)
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$pdf
#using the estimated probability values the expected frequencies are calculated
exp.freq<-round((sum(obs.freq)*est.prob),2)
#chi-squared test statistics is calculated with observed frequency and expected frequency
statistic<-sum(((obs.freq-exp.freq)^2)/exp.freq)
#degree of freedom is calculated
df<-length(x)-3
#p value of chi-squared test statistic is calculated
p.value<-1-stats::pchisq(statistic,df)
#all the above information is mentioned as a message below
#and if the user wishes they can print or not to
#checking if df is less than or equal to zero
if(df<0 | df==0)
{
stop("Degrees of freedom cannot be less than or equal to zero")
}
#checking if any of the expected frequencies are less than five and greater than zero, if so
#a warning message is provided in interpreting the results
if(min(exp.freq)<5 && min(exp.freq) > 0)
{
message("Chi-squared approximation may be doubtful because expected frequency is less than 5")
}
#checking if expected frequency is zero, if so providing a warning message in interpreting
#the results
if(min(exp.freq)==0)
{
message("Chi-squared approximation is not suitable because expected frequency approximates to zero")
}
#calculating Negative Log Likelihood value and AIC
NegLL<-NegLLBetaBin(x,obs.freq,a,b)
AICvalue<-2*2+NegLL
#the final output is in a list format containing the calculated values
final<-list("bin.ran.var"=x,"obs.freq"=obs.freq,"exp.freq"=exp.freq,
"statistic"=round(statistic,4),"df"=df,"p.value"=round(p.value,4),
"fitBB"=est,"NegLL"=NegLL,"a"=a,"b"=b, "AIC"=AICvalue,
"over.dis.para"=est$over.dis.para,"call"=match.call())
class(final)<-c("fitBB","fit")
return(final)
}
}
#' @method fitBetaBin default
#' @export
fitBetaBin.default<-function(x,obs.freq,a,b)
{
est<-fitBetaBin(x,obs.freq,a,b)
return(est)
}
#' @method print fitBB
#' @export
print.fitBB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Beta-Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated a parameter :",x$a, " ,estimated b parameter :",x$b,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n\t
over dispersion :",x$over.dis.para,"\n")
}
#' @method summary fitBB
#' @export
summary.fitBB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Beta-Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated a parameter :",object$a," ,estimated b parameter :",object$b,"\n\t
X-squared :",object$statistic," ,df :",object$df," ,p-value :",object$p.value,"\n\t
over dispersion :",object$over.dis.para,"\n\t
Negative Loglikehood value :",object$NegLL,"\n\t
AIC value :",object$AIC,"\n")
}
Amalan-ConStat/R-fitODBOD documentation built on July 4, 2019, 4:17 p.m.
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#' Beta Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Beta Distribution bounded between [0,1]
#'
#' @usage
#' dBETA(p,a,b)
#'
#' @param p vector of probabilities.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded Beta distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{B_p(a,b)}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{a}{a+b} }
#' \deqn{var[P]= \frac{ab}{(a+b)^2(a+b+1)} }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \prod_{i=0}^{r-1} (\frac{a+i}{a+b+i}) }
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B_p(a,b)=\int^p_0 t^{a-1} (1-t)^{b-1}\,dt} is
#' incomplete beta integrals and \eqn{B(a,b)} is the beta function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{dBETA} gives a list format consisting
#'
#' \code{pdf} probability density values in vector form.
#'
#' \code{mean} mean of the Beta distribution.
#'
#' \code{var} variance of the Beta distribution.
#'
#' @references
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis,
#' 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' @seealso
#' \code{\link[stats]{Beta}}
#'
#' or
#'
#' \url{https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,4))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dBETA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dBETA(seq(0,1,by=0.01),2,3)$pdf #extracting the pdf values
#' dBETA(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dBETA(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pBETA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pBETA(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#' mazBETA(1.4,3,2) #acquiring the moment about zero values
#' mazBETA(2,3,2)-mazBETA(1,3,2)^2 #acquiring the variance for a=3,b=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazBETA(1.9,5.5,6)
#'
#' @export
dBETA<-function(p,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b))) | any(is.infinite(c(p,a,b))) | any(is.nan(c(p,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, if not providing error message and
#stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
ans<-NULL
#for each input values in the vector necessary calculations and conditions are applied
for (i in 1:length(p))
{
if(p[i]<0 |p[i]>1)
{
stop("Invalid values in the input")
}
else
{
ans[i]<-(p[i]^(a-1)*(1-p[i])^(b-1))/beta(a,b)
}
}
}
mean<-a/(a+b) #according to theory the mean value
variance<-(a*b)/(((a+b)^2)*(a+b+1)) #according to theory the variance value
# generating an output in list format consisting pdf,mean and variance
output<-list("pdf"=ans,"mean"=mean,"var"=variance)
return(output)
}
}
#' Beta Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Beta Distribution bounded between [0,1].
#'
#' @usage
#' pBETA(p,a,b)
#'
#' @param p vector of probabilities.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded beta distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{B_p(a,b)}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{a}{a+b} }
#' \deqn{var[P]= \frac{ab}{(a+b)^2(a+b+1)} }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \prod_{i=0}^{r-1} (\frac{a+i}{a+b+i}) }
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B_p(a,b)=\int^p_0 t^{a-1} (1-t)^{b-1}\,dt} is
#' incomplete beta integrals and \eqn{B(a,b)} is the beta function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{pBETA} gives the cumulative density values in vector form.
#'
#' @references
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis,
#' 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' @seealso
#' \code{\link[stats]{Beta}}
#'
#' or
#'
#' \url{https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,4))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dBETA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dBETA(seq(0,1,by=0.01),2,3)$pdf #extracting the pdf values
#' dBETA(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dBETA(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pBETA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pBETA(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#' mazBETA(1.4,3,2) #acquiring the moment about zero values
#' mazBETA(2,3,2)-mazBETA(1,3,2)^2 #acquiring the variance for a=3,b=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazBETA(1.9,5.5,6)
#'
#' @export
pBETA<-function(p,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b))) | any(is.infinite(c(p,a,b))) | any(is.nan(c(p,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero and if not providing an error message
#and stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
ans<-NULL
val<-NULL
#the equation contains partial beta integration, below is the integral function
Bp<-function(q)
{
(q^(a-1))*((1-q)^(b-1))
}
#for each input values in the vector necessary calculations and conditions are applied
for(i in 1:length(p))
{
if(p[i]<0 | p[i]>1)
{
stop("Invalid values in the input")
}
else
{
#integrating the above mentioned function under limits of zero and vector p
val<-stats::integrate(Bp,lower = 0,upper = p[i])
ans[i]<-val$value/beta(a,b)
}
}
#generating an ouput vector of cumulative probability values
return(ans)
}
}
}
#' Beta Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Beta Distribution bounded between [0,1].
#'
#' @usage
#' mazBETA(r,a,b)
#'
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param r vector of moments.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded beta distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{B_p(a,b)}{B(a,b)} } ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{a}{a+b} }
#' \deqn{var[P]= \frac{ab}{(a+b)^2(a+b+1)} }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \prod_{i=0}^{r-1} (\frac{a+i}{a+b+i}) }
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B_p(a,b)=\int^p_0 t^{a-1} (1-t)^{b-1}\,dt} is
#' incomplete beta integrals and \eqn{B(a,b)} is the beta function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{mazBETA} gives the moments about zero in vector form.
#'
#' @references
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis,
#' 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' @seealso
#' \code{\link[stats]{Beta}}
#'
#' or
#'
#' \url{https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,4))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dBETA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dBETA(seq(0,1,by=0.01),2,3)$pdf #extracting the pdf values
#' dBETA(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dBETA(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pBETA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pBETA(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#' mazBETA(1.4,3,2) #acquiring the moment about zero values
#' mazBETA(2,3,2)-mazBETA(1,3,2)^2 #acquiring the variance for a=3,b=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazBETA(1.9,5.5,6)
#'
#' @export
mazBETA<-function(r,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(r,a,b))) | any(is.infinite(c(r,a,b))) | any(is.nan(c(r,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, and if not providing an error
#message and stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#the moments cannot be a decimal value therefore converting it into an integer
r<-as.integer(r)
ans<-NULL
#for each input values in the vector necessary calculations and conditions are applied
for(i in 1:length(r))
{
#checking if moment values are less than or equal to zero and creating
# an error message as well as stopping the function progress
if(r[i]<=0)
{
stop("Moments cannot be less than or equal to zero")
}
else
{
j<-0:(r[i]-1)
ans[i]<-prod((a+j)/(a+b+j))
}
}
#generating an ouput vector of moment about zero values
return(ans)
}
}
}
#' Beta-Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Beta-Binomial Distribution.
#'
#' @usage
#' dBetaBin(x,n,a,b)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' Mixing Beta distribution with Binomial distribution will create the Beta-Binomial
#' distribution. The probability function and cumulative probability function can be
#' constructed and are denoted below.
#'
#' The cumulative probability function is the summation of probability function values.
#'
#' \deqn{P_{BetaBin}(x)= {n \choose x} \frac{B(a+x,n+b-x)}{B(a,b)} }
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{BetaBin}[x]= \frac{na}{a+b} }
#' \deqn{Var_{BetaBin}[x]= \frac{(nab)}{(a+b)^2} \frac{(a+b+n)}{(a+b+1)} }
#' \deqn{over dispersion= \frac{1}{a+b+1} }
#'
#' Defined as \code{B(a,b)} is the beta function.
#'
#' @return
#' The output of \code{dBetaBin} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of the Beta-Binomial Distribution.
#'
#' \code{var} variance of the Beta-Binomial Distribution.
#'
#' \code{over.dis.para} over dispersion value of the Beta-Binomial Distribution.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,0.2)
#' plot(0,0,main="Beta-binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
#' for (i in 1:5)
#' {
#' lines(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dBetaBin(0:10,10,4,.2)$pdf #extracting the pdf values
#' dBetaBin(0:10,10,4,.2)$mean #extracting the mean
#' dBetaBin(0:10,10,4,.2)$var #extracting the variance
#' dBetaBin(0:10,10,4,.2)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pBetaBin(0:10,10,4,.2) #acquiring the cumulative probability values
#'
#' @export
dBetaBin<-function(x,n,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,n,a,b))) | any(is.infinite(c(x,n,a,b))) |any(is.nan(c(x,n,a,b))))
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are less than or equal zero ,
#if so providing an error message and stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#checking if at any chance the binomial random variable is greater than binomial trial value
#if so providing an error message and stopping the function from progress
if(max(x)>n)
{
stop("Binomial random variable cannot be greater than binomial trial value")
}
#checking if any random variable or trial value is negative if so providig an error message
#and stopping the function progress
else if(any(x<0) | n<0)
{
stop("Binomial random variable or binomial trial value cannot be negative")
}
ans<-NULL
#for each random variable in the input vector below calculations occur
for (i in 1:length(x))
{
ans[i]<-choose(n,x[i])*(beta(a+x[i],n+b-x[i])/beta(a,b))
}
}
}
mean<-n*(a/(a+b)) #according to theory the mean
variance<-n*((a*b)/(a+b)^2)*((a+b+n)/(a+b+1)) #according to theory variance
ove.dis.par<-1/(a+b+1) #according to theory overdispersion value
# generating an output in list format consisting pdf,mean,variance and overdispersion value
output<-list('pdf'=ans,'mean'=mean,'var'=variance,
'over.dis.para'=ove.dis.par)
return(output)
}
#' Beta-Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Beta-Binomial Distribution.
#'
#' @usage
#' pBetaBin(x,n,a,b)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' Mixing Beta distribution with Binomial distribution will create the Beta-Binomial
#' distribution. The probability function and cumulative probability function can be
#' constructed and are denoted below.
#'
#' The cumulative probability function is the summation of probability function values.
#'
#' \deqn{P_{BetaBin}(x)= {n \choose x} \frac{B(a+x,n+b-x)}{B(a,b)} }
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{BetaBin}[x]= \frac{na}{a+b} }
#' \deqn{Var_{BetaBin}[x]= \frac{(nab)}{(a+b)^2} \frac{(a+b+n)}{(a+b+1)} }
#' \deqn{over dispersion= \frac{1}{a+b+1} }
#'
#' Defined as \code{B(a,b)} is the beta function.
#'
#' @return
#' The output of \code{pBetaBin} gives cumulative probability values in vector form.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,0.2)
#' plot(0,0,main="Beta-binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
#' for (i in 1:5)
#' {
#' lines(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dBetaBin(0:10,10,4,.2)$pdf #extracting the pdf values
#' dBetaBin(0:10,10,4,.2)$mean #extracting the mean
#' dBetaBin(0:10,10,4,.2)$var #extracting the variance
#' dBetaBin(0:10,10,4,.2)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pBetaBin(0:10,10,4,.2) #acquiring the cumulative probability values
#'
#' @export
pBetaBin<-function(x,n,a,b)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative proability function
#values are calculated
for(i in 1:length(x))
{
j<-0:x[i]
ans[i]<-sum(dBetaBin(j,n,a,b)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of Beta-Binomial Distribution
#'
#' This function will calculate the Negative Log Likelihood value when the vector of binomial random
#' variables and vector of corresponding frequencies are given with the shape parameters a and b.
#'
#' @usage
#' NegLLBetaBin(x,freq,a,b)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' \deqn{0 < a,b }
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,...}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{NegLLBetaBin} will produce a single numeric value.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' NegLLBetaBin(No.D.D,Obs.fre.1,.3,.4) #acquiring the negative log likelihood value
#'
#' @export
NegLLBetaBin<-function(x,freq,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,freq,a,b))) | any(is.infinite(c(x,freq,a,b)))
|any(is.nan(c(x,freq,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if any of the random variables of frequencies are less than zero if so
#creating an error message as well as stopping the function progress
if( any(c(x,freq)< 0) )
{
stop("Binomial random variable or frequency values cannot be negative")
}
#checking if shape parameters are less than or equal to zero
#if so creating an error message as well as stopping the function progress
else if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
i<-1:sum(freq)
term1<-sum(log(choose(n,data[i])))
term2<-sum(log(beta(a+data[i],n+b-data[i])))
BetaBinLL<-term1+term2-sum(freq)*log(beta(a,b))
#calculating the negative log likelihood value and representing as a single output value
return(-BetaBinLL)
}
}
}
#' Estimating the shape parameters a and b for Beta-Binomial Distribution
#'
#' The functions will estimate the shape parameters using the maximum log likelihood method and
#' moment generating function method for the Beta-Binomial distribution when the binomial
#' random variables and corresponding frequencies are given.
#'
#' @usage
#' EstMLEBetaBin(x,freq,a,b,...)
#'
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#'
#' @details
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' \code{EstMLEBetaBin} here is used as a wrapper for the \code{mle2} function of \pkg{bbmle} package
#' therefore output is of class of mle2.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' estimate <- EstMLEBetaBin(No.D.D,Obs.fre.1,a=0.1,b=0.1)
#'
#' bbmle::coef(estimate) #extracting the parameters
#'
#' #estimating the parameters using moment generating function methods
#' EstMGFBetaBin(No.D.D,Obs.fre.1)
#'
#'@export
EstMLEBetaBin<-function(x,freq,a,b,...)
{
suppressWarnings2 <-function(expr, regex=character())
{
withCallingHandlers(expr, warning=function(w)
{
if (length(regex) == 1 && length(grep(regex, conditionMessage(w))))
{
invokeRestart("muffleWarning")
}
} )
}
suppressWarnings2(bbmle::mle2(.EstMLEBetaBin,data=list(x=x,freq=freq),
start = list(a=a,b=b),...),"NaN")
}
.EstMLEBetaBin<-function(x,freq,a,b)
{
#with respective to using bbmle package function mle2 there is no need impose any restrictions
#therefor the output is directly a single numeric value for the negative log likelihood value of
#beta binomial distribution
n<-max(x)
data<-rep(x,freq)
i<-1:sum(freq)
term1<-sum(log(choose(n,data[i])))
term2<-sum(log(beta(a+data[i],n+b-data[i])))
BetaBinLL<-term1+term2-sum(freq)*log(beta(a,b))
return(-BetaBinLL)
}
#' Estimating the shape parameters a and b for Beta-Binomial Distribution
#'
#' The functions will estimate the shape parameters using the maximum log likelihood method and
#' moment generating function method for the Beta-Binomial distribution when the binomial
#' random variables and corresponding frequencies are given.
#'
#' @usage
#' EstMGFBetaBin(x,freq)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#'
#' @details
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary error
#' messages will be provided to go further.
#'
#' @return
#' The output of \code{EstMGFBetaBin} will produce the class \code{mgf} format consisting
#'
#' \code{a} shape parameter of beta distribution representing for alpha
#'
#' \code{b} shape parameter of beta distribution representing for beta
#'
#' \code{min} Negative loglikelihood value
#'
#' \code{AIC} AIC value
#'
#' \code{call} the inputs for the function
#'
#' Methods \code{print}, \code{summary}, \code{coef} and \code{AIC} can be used to extract
#' specific outputs.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' estimate <- EstMLEBetaBin(No.D.D,Obs.fre.1,a=0.1,b=0.1)
#'
#' bbmle::coef(estimate) #extracting the parameters
#'
#' #estimating the parameters using moment generating function methods
#' results <- EstMGFBetaBin(No.D.D,Obs.fre.1)
#'
#' # extract the estimated parameters and summary
#' coef(results)
#' summary(results)
#'
#' AIC(results) #show the AIC value
#'
#' @export
EstMGFBetaBin<-function(x,freq)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,freq))) | any(is.infinite(c(x,freq))) | any(is.nan(c(x,freq))))
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
#creating necessary equations
m1<-sum(data)/length(data)
m2<-sum(data^2)/length(data)
#constructing equations according to theory for a(alpha) and b(beta)
a<-((n*m1-m2)*m1)/(n*(m2-m1-m1^2)+m1^2)
b<-((n*m1-m2)*(n-m1))/(n*(m2-m1-m1^2)+m1^2)
#generating an output of list for shape parameters a(alpha) and b(beta)
NegLL<-NegLLBetaBin(x,freq,a,b)
AICvalue<-2*2+(2*NegLL)
argument<-match.call()
ans<-list("a"=a,"b"=b,"min"=NegLL,"AIC"=AICvalue,"call"=argument)
class(ans)<-"mgf"
return(ans)
}
}
#' @method EstMGFBetaBin default
#' @export
EstMGFBetaBin.default<-function(x,freq)
{
est<-EstMGFBetaBin(x,freq)
return(est)
}
#' @method print mgf
#' @export
print.mgf<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nCoefficients: \n")
coeff<-c(x$a,x$b)
names(coeff)<-c("a","b")
print(coeff)
}
#' @method summary mgf
#' @export
summary.mgf<-function(object,...)
{
cat("Coefficients: \n a \t b \n", object$a,object$b)
cat("\n\nNegative Log-likelihood : ",object$min)
cat("\n\nAIC : ",object$AIC)
}
#' @method coef mgf
#' @export
coef.mgf<-function(object,...)
{
cat(" \t a \t b \n", object$a,object$b)
}
#' @method AIC mgf
#' @export
AIC.mgf<-function(object,...)
{
return(object$AIC)
}
#' Fitting the Beta-Binomial Distribution when binomial random variable, frequency and shape
#' parameters a and b are given
#'
#' The function will fit the Beta-Binomial distribution when random variables, corresponding
#' frequencies and shape parameters are given. It will provide the expected frequencies, chi-squared
#' test statistics value, p value, degree of freedom and over dispersion value so that it can be
#' seen if this distribution fits the data.
#'
#' @usage fitBetaBin(x,obs.freq,a,b)
#'
#' @param x vector of binomial random variables.
#' @param obs.freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#'
#' @details
#' \deqn{0 < a,b}
#' \deqn{x = 0,1,2,...,n}
#' \deqn{obs.freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{fitBetaBin} gives the class format \code{fitBB} and \code{fit} consisting a list
#'
#' \code{bin.ran.var} binomial random variables.
#'
#' \code{obs.freq} corresponding observed frequencies.
#'
#' \code{exp.freq} corresponding expected frequencies.
#'
#' \code{statistic} chi-squared test statistics.
#'
#' \code{df} degree of freedom.
#'
#' \code{p.value} probability value by chi-squared test statistic.
#'
#' \code{fitBB} fitted values of \code{dBetaBin}.
#'
#' \code{NegLL} Negative Log Likelihood value.
#'
#' \code{a} estimated value for alpha parameter as a.
#'
#' \code{b} estimated value for alpha parameter as b.
#'
#' \code{AIC} AIC value.
#'
#' \code{over.dis.para} over dispersion value.
#'
#' \code{call} the inputs of the function.
#'
#' Methods \code{summary}, \code{print}, \code{AIC}, \code{residuals} and \code{fitted} can be
#' used to extract specific outputs.
#'
#' @references
#' Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical
#' research methodology, 8(1), p.58.
#'
#' Available at: \url{http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \url{http://linkinghub.elsevier.com/retrieve/pii/0167947396900158}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \url{http://www.apsnet.org/publications/phytopathology/backissues/Documents/1993Abstracts/Phyto_83_759.htm}
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLEBetaBin(No.D.D,Obs.fre.1,0.1,0.1)
#'
#' bbmle::coef(parameters) #extracting the parameters a and b
#' aBetaBin <- bbmle::coef(parameters)[1] #assigning the parameter a
#' bBetaBin <- bbmle::coef(parameters)[2] #assigning the parameter b
#'
#' #fitting when the random variable,frequencies,shape parameter values are given.
#' fitBetaBin(No.D.D,Obs.fre.1,aBetaBin,bBetaBin)
#'
#' #estimating the parameters using moment generating function methods
#' results <- EstMGFBetaBin(No.D.D,Obs.fre.1)
#' results
#'
#' aBetaBin1 <- results$a #assigning the estimated a
#' bBetaBin1 <- results$b #assigning the estimated b
#'
#' #fitting when the random variable,frequencies,shape parameter values are given.
#' BB <- fitBetaBin(No.D.D,Obs.fre.1,aBetaBin1,bBetaBin1)
#'
#' #extracting the expected frequencies
#' fitted(BB)
#'
#' #extracting the residuals
#' residuals(BB)
#'
#' @export
fitBetaBin<-function(x,obs.freq,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,obs.freq,a,b))) | any(is.infinite(c(x,obs.freq,a,b))) |
any(is.nan(c(x,obs.freq,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dBetaBin(x,max(x),a,b)
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$pdf
#using the estimated probability values the expected frequencies are calculated
exp.freq<-round((sum(obs.freq)*est.prob),2)
#chi-squared test statistics is calculated with observed frequency and expected frequency
statistic<-sum(((obs.freq-exp.freq)^2)/exp.freq)
#degree of freedom is calculated
df<-length(x)-3
#p value of chi-squared test statistic is calculated
p.value<-1-stats::pchisq(statistic,df)
#all the above information is mentioned as a message below
#and if the user wishes they can print or not to
#checking if df is less than or equal to zero
if(df<0 | df==0)
{
stop("Degrees of freedom cannot be less than or equal to zero")
}
#checking if any of the expected frequencies are less than five and greater than zero, if so
#a warning message is provided in interpreting the results
if(min(exp.freq)<5 && min(exp.freq) > 0)
{
message("Chi-squared approximation may be doubtful because expected frequency is less than 5")
}
#checking if expected frequency is zero, if so providing a warning message in interpreting
#the results
if(min(exp.freq)==0)
{
message("Chi-squared approximation is not suitable because expected frequency approximates to zero")
}
#calculating Negative Log Likelihood value and AIC
NegLL<-NegLLBetaBin(x,obs.freq,a,b)
AICvalue<-2*2+NegLL
#the final output is in a list format containing the calculated values
final<-list("bin.ran.var"=x,"obs.freq"=obs.freq,"exp.freq"=exp.freq,
"statistic"=round(statistic,4),"df"=df,"p.value"=round(p.value,4),
"fitBB"=est,"NegLL"=NegLL,"a"=a,"b"=b, "AIC"=AICvalue,
"over.dis.para"=est$over.dis.para,"call"=match.call())
class(final)<-c("fitBB","fit")
return(final)
}
}
#' @method fitBetaBin default
#' @export
fitBetaBin.default<-function(x,obs.freq,a,b)
{
est<-fitBetaBin(x,obs.freq,a,b)
return(est)
}
#' @method print fitBB
#' @export
print.fitBB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Beta-Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated a parameter :",x$a, " ,estimated b parameter :",x$b,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n\t
over dispersion :",x$over.dis.para,"\n")
}
#' @method summary fitBB
#' @export
summary.fitBB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Beta-Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated a parameter :",object$a," ,estimated b parameter :",object$b,"\n\t
X-squared :",object$statistic," ,df :",object$df," ,p-value :",object$p.value,"\n\t
over dispersion :",object$over.dis.para,"\n\t
Negative Loglikehood value :",object$NegLL,"\n\t
AIC value :",object$AIC,"\n")
}
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