Nothing
R/Beta.R
In fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD
Defines functions
summary.fitBB
print.fitBB
fitBetaBin.default
fitBetaBin
AIC.mgf
coef.mgf
summary.mgf
print.mgf
EstMGFBetaBin.default
EstMGFBetaBin
.EstMLEBetaBin
EstMLEBetaBin
NegLLBetaBin
pBetaBin
dBetaBin
mazBETA
pBETA
dBETA
Documented in
dBETA
dBetaBin
EstMGFBetaBin
EstMLEBetaBin
fitBetaBin
mazBETA
NegLLBetaBin
pBETA
pBetaBin
#' 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: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' @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)
}
}
}
# generating an output in list format consisting pdf,mean and variance
return(list("pdf"=ans,"mean"=a/(a+b),"var"=(a*b)/(((a+b)^2)*(a+b+1)) ))
}
}
#' 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: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' @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
#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(function(q){ (q^(a-1))*((1-q)^(b-1)) },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: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' @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
{
ans[i]<-prod((a+(0:(r[i]-1)))/(a+b+(0:(r[i]-1))))
}
}
#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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' 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))
}
}
}
# generating an output in list format consisting pdf,mean,variance and overdispersion value
return(list('pdf'=ans,'mean'=n*(a/(a+b)) ,
'var'=n*((a*b)/(a+b)^2)*((a+b+n)/(a+b+1)) ,
'over.dis.para'=1/(a+b+1)))
}
#' 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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pBetaBin(0:10,10,4,.2) #acquiring the cumulative probability values
#'
#' @export
pBetaBin<-function(x,n,a,b)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative proability function
#values are calculated
for(i in 1:length(x))
{
ans[i]<-sum(dBetaBin(0:x[i],n,a,b)$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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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
data<-rep(x,freq)
#calculating the negative log likelihood value and representing as a single output value
return(-(sum(log(choose(max(x),data[1:sum(freq)]))) +
sum(log(beta(a+data[1:sum(freq)],max(x)+b-data[1:sum(freq)]))) -
sum(freq)*log(beta(a,b))))
}
}
}
#' 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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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
data<-rep(x,freq)
return(-(sum(log(choose(max(x),data[1:sum(freq)]))) +
sum(log(beta(a+data[1:sum(freq)],max(x)+b-data[1:sum(freq)]))) -
sum(freq)*log(beta(a,b))))
}
#' 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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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
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<-((max(x)*m1-m2)*m1)/(max(x)*(m2-m1-m1^2)+m1^2)
b<-((max(x)*m1-m2)*(max(x)-m1))/(max(x)*(m2-m1-m1^2)+m1^2)
#generating an output of list for shape parameters a(alpha) and b(beta)
ans<-list("a"=a,"b"=b,"min"=NegLLBetaBin(x,freq,a,b),
"AIC"=2*2+(2*NegLLBetaBin(x,freq,a,b)),"call"=match.call())
class(ans)<-"mgf"
return(ans)
}
}
#' @method EstMGFBetaBin default
#' @export
EstMGFBetaBin.default<-function(x,freq)
{
return(EstMGFBetaBin(x,freq))
}
#' @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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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)
odp<-est$over.dis.para; names(odp)<-NULL
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$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")
}
NegLL<-NegLLBetaBin(x,obs.freq,a,b)
names(NegLL)<-NULL
#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"=2*2+2*NegLL,
"over.dis.para"=odp,"call"=match.call())
class(final)<-c("fitBB","fit")
return(final)
}
}
#' @method fitBetaBin default
#' @export
fitBetaBin.default<-function(x,obs.freq,a,b)
{
return(fitBetaBin(x,obs.freq,a,b))
}
#' @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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Defines functions summary.fitBB print.fitBB fitBetaBin.default fitBetaBin AIC.mgf coef.mgf summary.mgf print.mgf EstMGFBetaBin.default EstMGFBetaBin .EstMLEBetaBin EstMLEBetaBin NegLLBetaBin pBetaBin dBetaBin mazBETA pBETA dBETA
Documented in dBETA dBetaBin EstMGFBetaBin EstMLEBetaBin fitBetaBin mazBETA NegLLBetaBin pBETA pBetaBin
#' 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: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' @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)
}
}
}
# generating an output in list format consisting pdf,mean and variance
return(list("pdf"=ans,"mean"=a/(a+b),"var"=(a*b)/(((a+b)^2)*(a+b+1)) ))
}
}
#' 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: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' @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
#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(function(q){ (q^(a-1))*((1-q)^(b-1)) },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: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' @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
{
ans[i]<-prod((a+(0:(r[i]-1)))/(a+b+(0:(r[i]-1))))
}
}
#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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' 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))
}
}
}
# generating an output in list format consisting pdf,mean,variance and overdispersion value
return(list('pdf'=ans,'mean'=n*(a/(a+b)) ,
'var'=n*((a*b)/(a+b)^2)*((a+b+n)/(a+b+1)) ,
'over.dis.para'=1/(a+b+1)))
}
#' 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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pBetaBin(0:10,10,4,.2) #acquiring the cumulative probability values
#'
#' @export
pBetaBin<-function(x,n,a,b)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative proability function
#values are calculated
for(i in 1:length(x))
{
ans[i]<-sum(dBetaBin(0:x[i],n,a,b)$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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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
data<-rep(x,freq)
#calculating the negative log likelihood value and representing as a single output value
return(-(sum(log(choose(max(x),data[1:sum(freq)]))) +
sum(log(beta(a+data[1:sum(freq)],max(x)+b-data[1:sum(freq)]))) -
sum(freq)*log(beta(a,b))))
}
}
}
#' 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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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
data<-rep(x,freq)
return(-(sum(log(choose(max(x),data[1:sum(freq)]))) +
sum(log(beta(a+data[1:sum(freq)],max(x)+b-data[1:sum(freq)]))) -
sum(freq)*log(beta(a,b))))
}
#' 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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Available at: \doi{10.1016/0167-9473(96)90015-8}.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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
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<-((max(x)*m1-m2)*m1)/(max(x)*(m2-m1-m1^2)+m1^2)
b<-((max(x)*m1-m2)*(max(x)-m1))/(max(x)*(m2-m1-m1^2)+m1^2)
#generating an output of list for shape parameters a(alpha) and b(beta)
ans<-list("a"=a,"b"=b,"min"=NegLLBetaBin(x,freq,a,b),
"AIC"=2*2+(2*NegLLBetaBin(x,freq,a,b)),"call"=match.call())
class(ans)<-"mgf"
return(ans)
}
}
#' @method EstMGFBetaBin default
#' @export
EstMGFBetaBin.default<-function(x,freq)
{
return(EstMGFBetaBin(x,freq))
}
#' @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: \doi{10.1186/1471-2288-8-58}.
#'
#' Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.
#'
#' Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease
#' Incidence. Phytopathology, 83(9), p.759.
#'
#' Available at: \doi{10.1094/PHYTO-83-759}
#'
#' @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)
odp<-est$over.dis.para; names(odp)<-NULL
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$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")
}
NegLL<-NegLLBetaBin(x,obs.freq,a,b)
names(NegLL)<-NULL
#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"=2*2+2*NegLL,
"over.dis.para"=odp,"call"=match.call())
class(final)<-c("fitBB","fit")
return(final)
}
}
#' @method fitBetaBin default
#' @export
fitBetaBin.default<-function(x,obs.freq,a,b)
{
return(fitBetaBin(x,obs.freq,a,b))
}
#' @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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