# R/Triangle.R In Amalan-ConStat/R-fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD

#### Documented in dTRIdTriBinEstMLETriBinfitTriBinmazTRINegLLTriBinpTRIpTriBin

#' Triangular Distribution bounded between [0,1]
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moments about zero values for the
#' Triangular Distribution bounded between [0,1]
#'
#' @usage
#' dTRI(p,mode)
#'
#' @param p                vector of probabilities
#' @param mode             single value for mode
#'
#' @details
#' Setting \eqn{min=0} and \eqn{max=1} \eqn{mode=c} in the triangular distribution
#' a unit bounded triangular distribution can be obtained. The probability density function
#' and cumulative density function of a unit bounded triangular distribution with random
#' variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{2p}{c} } ;            \eqn{0 \le p < c}
#' \deqn{g_{P}(p)= \frac{2(1-p)}{(1-c)} } ;    \eqn{c \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^2}{c} } ;           \eqn{0 \le p < c}
#' \deqn{G_{P}(p)= 1-\frac{(1-p)^2}{(1-c)} } ; \eqn{c \le p \le 1}
#' \deqn{0 \le mode=c \le 1}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{(a+b+c)}{3}= \frac{(1+c)}{3} }
#' \deqn{var[P]= \frac{a^2+b^2+c^2-ab-ac-bc}{18}= \frac{(1+c^2-c)}{18} }
#'
#' Moments about zero is denoted as
#' \deqn{E[P^r]= \frac{2c^{r+2}}{c(r+2)}+\frac{2(1-c^{r+1})}{(1-c)(r+1)}+\frac{2(c^{r+2}-1)}{(1-c)(r+2)} }
#' \eqn{r = 1,2,3,...}
#'
#' \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{dTRI} gives a list format consisting
#'
#' \code{pdf}             probability density values in vector form
#'
#' \code{mean}            mean of the unit bounded triangular distribution
#'
#' \code{variance}        variance of the unit bounded triangular distribution
#'
#' @references
#' Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society,
#' Series A, 120:148-191.
#'
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley
#'
#' Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical
#' Modeling. Boston: Birkhuser Boston, pp. 21-33.
#'
#' Available at: \url{http://dx.doi.org/10.1007/978-0-8176-4626-4_2} .
#'
#' Okagbue, H. et al., 2014. Using the Average of the Extreme Values of a Triangular Distribution for a
#' Transformation, and Its Approximant via the Continuous Uniform Distribution. British Journal of Mathematics
#' & Computer Science, 4(24), pp.3497-3507.
#'
#' Available at: \url{http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427} .
#'
#' @seealso
#'
#'
#' ---------------
#'
#'
#'
#' @examples
#' #plotting the random variables and probability values
#' col<-rainbow(4)
#' x<-seq(0.2,0.8,by=0.2)
#' plot(0,0,main="Probability density graph",xlab="Random variable",
#' ylab="Probability density values",xlim = c(0,1),ylim = c(0,3))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dTRI(seq(0,1,by=0.01),x[i])$pdf,col = col[i]) #' } #' #' dTRI(seq(0,1,by=0.05),0.3)$pdf     #extracting the pdf values
#' dTRI(seq(0,1,by=0.01),0.3)$mean #extracting the mean #' dTRI(seq(0,1,by=0.01),0.3)$var     #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col<-rainbow(4)
#' x<-seq(0.2,0.8,by=0.2)
#' 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),pTRI(seq(0,1,by=0.01),x[i]),col = col[i])
#' }
#'
#' pTRI(seq(0,1,by=0.05),0.3)      #acquiring the cumulative probability values
#' mazTRI(1.4,.3)                  #acquiring the moment about zero values
#' mazTRI(2,.3)-mazTRI(1,.3)^2     #variance for when is mode 0.3
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazTRI(1.9,0.5)
#'
#' @export
dTRI<-function(p,mode)
{
#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,mode))) | any(is.infinite(c(p,mode))) | any(is.nan(c(p,mode))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if mode parameter is between zero and one , if not providing an error message and
#stopping the function progress
if(mode < 0 | mode > 1)
{
stop("Mode cannot be less than zero or greater than one")
}
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 if(0<=p[i] && p[i]<mode)
{
ans[i]<-(2*p[i]) /mode
}
else if(mode<=p[i] && p[i]<=1)
{
ans[i]<-(2*(1-p[i]))/(1-mode)
}
}
}
}
mean<-(1+mode)/3          #according to theory the mean value
variance<-(1+mode^2-mode)/18        #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)
}

#' Triangular Distribution bounded between [0,1]
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moments about zero values for the
#' Triangular Distribution bounded between [0,1]
#'
#' @usage
#' pTRI(p,mode)
#'
#' @param p                vector of probabilities
#' @param mode             single value for mode
#'
#' @details
#' Setting \eqn{min=0} and \eqn{max=1} \eqn{mode=c} in the triangular distribution
#' a unit bounded triangular distribution can be obtained. The probability density function
#' and cumulative density function of a unit bounded triangular distribution with random
#' variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{2p}{c} } ;            \eqn{0 \le p < c}
#' \deqn{g_{P}(p)= \frac{2(1-p)}{(1-c)} } ;    \eqn{c \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^2}{c} } ;           \eqn{0 \le p < c}
#' \deqn{G_{P}(p)= 1-\frac{(1-p)^2}{(1-c)} } ; \eqn{c \le p \le 1}
#' \deqn{0 \le mode=c \le 1}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{(a+b+c)}{3}= \frac{(1+c)}{3} }
#' \deqn{var[P]= \frac{a^2+b^2+c^2-ab-ac-bc}{18}= \frac{(1+c^2-c)}{18} }
#'
#' Moments about zero is denoted as
#' \deqn{E[P^r]= \frac{2c^{r+2}}{c(r+2)}+\frac{2(1-c^{r+1})}{(1-c)(r+1)}+\frac{2(c^{r+2}-1)}{(1-c)(r+2)} }
#' \eqn{r = 1,2,3,...}
#'
#' \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{pTRI} gives the cumulative density values in vector form
#'
#' @references
#' Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society,
#' Series A, 120:148-191.
#'
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley
#'
#' Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical
#' Modeling. Boston: Birkhuser Boston, pp. 21-33.
#'
#' Available at: \url{http://dx.doi.org/10.1007/978-0-8176-4626-4_2} .
#'
#' Okagbue, H. et al., 2014. Using the Average of the Extreme Values of a Triangular Distribution for a
#' Transformation, and Its Approximant via the Continuous Uniform Distribution. British Journal of Mathematics
#' & Computer Science, 4(24), pp.3497-3507.
#'
#' Available at: \url{http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427} .
#'
#' @seealso
#'
#'
#' ---------------
#'
#'
#'
#' @examples
#' #plotting the random variables and probability values
#' col<-rainbow(4)
#' x<-seq(0.2,0.8,by=0.2)
#' plot(0,0,main="Probability density graph",xlab="Random variable",
#' ylab="Probability density values",xlim = c(0,1),ylim = c(0,3))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dTRI(seq(0,1,by=0.01),x[i])$pdf,col = col[i]) #' } #' #' dTRI(seq(0,1,by=0.05),0.3)$pdf     #extracting the pdf values
#' dTRI(seq(0,1,by=0.01),0.3)$mean #extracting the mean #' dTRI(seq(0,1,by=0.01),0.3)$var     #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col<-rainbow(4)
#' x<-seq(0.2,0.8,by=0.2)
#' 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),pTRI(seq(0,1,by=0.01),x[i]),col = col[i])
#' }
#'
#' pTRI(seq(0,1,by=0.05),0.3)      #acquiring the cumulative probability values
#' mazTRI(1.4,.3)                  #acquiring the moment about zero values
#' mazTRI(2,.3)-mazTRI(1,.3)^2     #variance for when is mode 0.3
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazTRI(1.9,0.5)
#'
#' @export
pTRI<-function(p,mode)
{
#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,mode))) | any(is.infinite(c(p,mode))) | any(is.nan(c(p,mode))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if mode parameter is between zero and one , if not providing an error message and
#stopping the function progress
if(mode < 0 | mode > 1)
{
stop("Mode cannot be less than zero or greater than one")
}
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 if(0<=p[i] && p[i]<mode)
{
ans[i]<-(p[i])^2/mode
}
else if(mode<= p[i] && p[i]<=1)
{
ans[i]<-1-((1-p[i])^2/(1-mode))
}
}
}
#generating an ouput vector of cumulative probability values
return(ans)
}
}

#' Triangular Distribution bounded between [0,1]
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moments about zero values for the
#' Triangular Distribution bounded between [0,1]
#'
#' @usage
#' mazTRI(r,mode)
#'
#' @param mode             single value for mode
#' @param r                vector of moments
#'
#' @details
#' Setting \eqn{min=0} and \eqn{max=1} \eqn{mode=c} in the triangular distribution
#' a unit bounded triangular distribution can be obtained. The probability density function
#' and cumulative density function of a unit bounded triangular distribution with random
#' variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{2p}{c} } ;            \eqn{0 \le p < c}
#' \deqn{g_{P}(p)= \frac{2(1-p)}{(1-c)} } ;    \eqn{c \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^2}{c} } ;           \eqn{0 \le p < c}
#' \deqn{G_{P}(p)= 1-\frac{(1-p)^2}{(1-c)} } ; \eqn{c \le p \le 1}
#' \deqn{0 \le mode=c \le 1}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{(a+b+c)}{3}= \frac{(1+c)}{3} }
#' \deqn{var[P]= \frac{a^2+b^2+c^2-ab-ac-bc}{18}= \frac{(1+c^2-c)}{18} }
#'
#' Moments about zero is denoted as
#' \deqn{E[P^r]= \frac{2c^{r+2}}{c(r+2)}+\frac{2(1-c^{r+1})}{(1-c)(r+1)}+\frac{2(c^{r+2}-1)}{(1-c)(r+2)} }
#' \eqn{r = 1,2,3,...}
#'
#' \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{mazTRI} give the moments about zero in vector form.
#'
#' @references
#' Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society,
#' Series A, 120:148-191.
#'
#' Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol. 2,
#' Wiley Series in Probability and Mathematical Statistics, Wiley
#'
#' Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical
#' Modeling. Boston: Birkhuser Boston, pp. 21-33.
#'
#' Available at: \url{http://dx.doi.org/10.1007/978-0-8176-4626-4_2} .
#'
#' Okagbue, H. et al., 2014. Using the Average of the Extreme Values of a Triangular Distribution for a
#' Transformation, and Its Approximant via the Continuous Uniform Distribution. British Journal of Mathematics
#' & Computer Science, 4(24), pp.3497-3507.
#'
#' Available at: \url{http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427} .
#'
#' @seealso
#'
#'
#' ---------------
#'
#'

#'
#' @examples
#' #plotting the random variables and probability values
#' col<-rainbow(4)
#' x<-seq(0.2,0.8,by=0.2)
#' plot(0,0,main="Probability density graph",xlab="Random variable",
#' ylab="Probability density values",xlim = c(0,1),ylim = c(0,3))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dTRI(seq(0,1,by=0.01),x[i])$pdf,col = col[i]) #' } #' #' dTRI(seq(0,1,by=0.05),0.3)$pdf     #extracting the pdf values
#' dTRI(seq(0,1,by=0.01),0.3)$mean #extracting the mean #' dTRI(seq(0,1,by=0.01),0.3)$var     #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col<-rainbow(4)
#' x<-seq(0.2,0.8,by=0.2)
#' 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),pTRI(seq(0,1,by=0.01),x[i]),col = col[i])
#' }
#'
#' pTRI(seq(0,1,by=0.05),0.3)      #acquiring the cumulative probability values
#' mazTRI(1.4,.3)                  #acquiring the moment about zero values
#' mazTRI(2,.3)-mazTRI(1,.3)^2     #variance for when is mode 0.3
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazTRI(1.9,0.5)
#'
#' @export
mazTRI<-function(r,mode)
{
#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,mode))) | any(is.infinite(c(r,mode))) | any(is.nan(c(r,mode))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if mode parameter is between zero and one , if not providing an error message and
#stopping the function progress
if(mode < 0 | mode > 1)
{
stop("Mode cannot be less than zero or greater than one")
}
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 if so
#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]<-((2*(mode^(r[i]+2)))/(mode*(r[i]+2)))+((2*(1-mode^(r[i]+1)))/((r[i]+1)*(1-mode)))+
((2*(mode^(r[i]+2)-1))/((r[i]+2)*(1-mode)))
}
}
#generating an ouput vector of moment about zero values
return(ans)
}
}
}

#' Triangular Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Triangular Binomial distribution.
#'
#' @usage
#' dTriBin(x,n,mode)
#'
#' @param x        vector of binomial random variables
#' @param n        single value for no of binomial trials
#' @param mode     single value for mode
#'
#' @details
#' Mixing unit bounded triangular distribution with binomial distribution will create
#' Triangular 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_{TriBin}(x)= 2 {n \choose x}(c^{-1}B_c(x+2,n-x+1)+(1-c)^{-1}B(x+1,n-x+2)-(1-c)^{-1}B_c(x+1,n-x+2))}
#' \deqn{0 < mode=c < 1}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3...}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{TriiBin}[x]= \frac{n(1+c)}{3} }
#' \deqn{Var_{TriBin}[x]= \frac{n(n+3)}{18}-\frac{n(n-3)c(1-c)}{18} }
#' \deqn{over dispersion= \frac{(1-c+c^2)}{2(2+c-c^2)} }
#'
#' Defined as \eqn{B_c(a,b)=\int^c_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{dTriBin} gives a list format consisting
#'
#' \code{pdf}             probability function values in vector form
#'
#' \code{mean}            mean of the Triangular Binomial Distribution
#'
#' \code{var}             variance of the Triangular Binomial Distribution
#'
#' \code{over.dis.para}   over dispersion value of the Triangular Binomial Distribution
#'
#' @references
#' Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society,
#' Series A, 120:148-191.
#'
#' Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical
#' Modeling. Boston: Birkhuser Boston, pp. 21-33.
#'
#' Available at: \url{http://dx.doi.org/10.1007/978-0-8176-4626-4_2} .
#'
#' Okagbue, H. et al., 2014. Using the Average of the Extreme Values of a Triangular Distribution for a
#' Transformation, and Its Approximant via the Continuous Uniform Distribution. British Journal of Mathematics
#' & Computer Science, 4(24), pp.3497-3507.
#'
#' Available at: \url{http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427} .
#'
#' @examples
#' #plotting the random variables and probability values
#' col<-rainbow(7)
#' x<-seq(0.1,0.7,by=0.1)
#' plot(0,0,main="Triangular binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,.3))
#' for (i in 1:7)
#' {
#' lines(0:10,dTriBin(0:10,10,x[i])$pdf,col = col[i],lwd=2.85) #' points(0:10,dTriBin(0:10,10,x[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dTriBin(0:10,10,.4)$pdf #extracting the pdf values #' dTriBin(0:10,10,.4)$mean       #extracting the mean
#' dTriBin(0:10,10,.4)$var #extracting the variance #' dTriBin(0:10,10,.4)$over.dis.para  #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col<-rainbow(7)
#' x<-seq(0.1,0.7,by=0.1)
#' plot(0,0,main="Triangular binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:7)
#' {
#' lines(0:10,pTriBin(0:10,10,x[i]),col = col[i],lwd=2.85)
#' points(0:10,pTriBin(0:10,10,x[i]),col = col[i],pch=16)
#' }
#' pTriBin(0:10,10,.4)    #acquiring the cumulative probability values
#'
#' @export
dTriBin<-function(x,n,mode)
{
#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,mode))) | any(is.nan(c(x,n,mode))) |any(is.infinite(c(x,n,mode))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if mode parameter is less than or equal to zero and more than or equal to one,
#if so providing an error message and stopping the function progress
if(mode <= 0 | mode >= 1)
{
stop("Mode cannot be less than or equal to zero or greater than or equal to one")
}
else
{
#the equation contains a term of beta partial integration, below provides the function for
#integration
Bp1<-function(q)
{
(q^(a-1))*((1-q)^(b-1))
}
#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 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")
}
else
{
a<-NULL
b<-NULL

integrate1<-NULL
integrate2<-NULL
ans<-NULL
#for each random variable in the input vector below calculations occur
for(i in 1:length(x))
{
#setting necessary values for alpha and beta and integrating the first term with the help
#of R function
a<-x[i]+2
b<-n-x[i]+1
integrate1<-stats::integrate(Bp1,lower=0,upper=mode)
#setting necessary values for alpha and beta and integrating the second term with the help
#of R function
a<-x[i]+1
b<-n-x[i]+2
integrate2<-stats::integrate(Bp1,lower=0,upper=mode)

ans[i]<-2*choose(n,x[i])*(((1/mode)*(integrate1$value))+((1/(1-mode))*beta(x[i]+1,n-x[i]+2))- ((1/(1-mode))*(integrate2$value)))
}
}
}
mean<-n*(1+mode)/3        #according to theory the mean
variance<-n*(n+3)/18 - n*(n-3)*mode*(1-mode)/18         #according to theory variance
ove.dis.par<-0.5*(1-mode+mode^2)/(2+mode-mode^2)        #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)
}
}

#' Triangular Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Triangular Binomial distribution.
#'
#' @usage
#' pTriBin(x,n,mode)
#'
#' @param x        vector of binomial random variables
#' @param n        single value for no of binomial trials
#' @param mode     single value for mode
#'
#' @details
#' Mixing unit bounded triangular distribution with binomial distribution will create
#' Triangular 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_{TriBin}(x)= 2 {n \choose x}(c^{-1}B_c(x+2,n-x+1)+(1-c)^{-1}B(x+1,n-x+2)-(1-c)^{-1}B_c(x+1,n-x+2))}
#' \deqn{0 < mode=c < 1}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3...}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{TriiBin}[x]= \frac{n(1+c)}{3} }
#' \deqn{Var_{TriBin}[x]= \frac{n(n+3)}{18}-\frac{n(n-3)c(1-c)}{18} }
#' \deqn{over dispersion= \frac{(1-c+c^2)}{2(2+c-c^2)} }
#'
#' Defined as \eqn{B_c(a,b)=\int^c_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{pTriBin} gives cumulative probability function values in vector form.
#'
#' @references
#' Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society,
#' Series A, 120:148-191.
#'
#' Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical
#' Modeling. Boston: Birkhuser Boston, pp. 21-33.
#'
#' Available at: \url{http://dx.doi.org/10.1007/978-0-8176-4626-4_2} .
#'
#' Okagbue, H. et al., 2014. Using the Average of the Extreme Values of a Triangular Distribution for a
#' Transformation, and Its Approximant via the Continuous Uniform Distribution. British Journal of Mathematics
#' & Computer Science, 4(24), pp.3497-3507.
#'
#' Available at: \url{http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427} .
#'
#' @examples
#' #plotting the random variables and probability values
#' col<-rainbow(7)
#' x<-seq(0.1,0.7,by=0.1)
#' plot(0,0,main="Triangular binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,.3))
#' for (i in 1:7)
#' {
#' lines(0:10,dTriBin(0:10,10,x[i])$pdf,col = col[i],lwd=2.85) #' points(0:10,dTriBin(0:10,10,x[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dTriBin(0:10,10,.4)$pdf #extracting the pdf values #' dTriBin(0:10,10,.4)$mean       #extracting the mean
#' dTriBin(0:10,10,.4)$var #extracting the variance #' dTriBin(0:10,10,.4)$over.dis.para  #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col<-rainbow(7)
#' x<-seq(0.1,0.7,by=0.1)
#' plot(0,0,main="Triangular binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:7)
#' {
#' lines(0:10,pTriBin(0:10,10,x[i]),col = col[i],lwd=2.85)
#' points(0:10,pTriBin(0:10,10,x[i]),col = col[i],pch=16)
#' }
#' pTriBin(0:10,10,.4)    #acquiring the cumulative probability values
#'
#' @export
pTriBin<-function(x,n,mode)
{
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(dTriBin(j,n,mode)$pdf) } #generating an ouput vector of cumulative probability function values return(ans) } #' Negative Log Likelihood value of Triangular 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 mode value. #' #' @usage #' NegLLTriBin(x,freq,mode) #' #' @param x vector of binomial random variables #' @param freq vector of frequencies #' @param mode single value for mode #' #' @details #' \deqn{0 < mode=c < 1} #' \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{NegLLTriBin} will produce a single numeric value #' #' @references #' Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society, #' Series A, 120:148-191. #' #' Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical #' Modeling. Boston: Birkhuser Boston, pp. 21-33. #' #' Available at: \url{http://dx.doi.org/10.1007/978-0-8176-4626-4_2} . #' #' Okagbue, H. et al., 2014. Using the Average of the Extreme Values of a Triangular Distribution for a #' Transformation, and Its Approximant via the Continuous Uniform Distribution. British Journal of Mathematics #' & Computer Science, 4(24), pp.3497-3507. #' #' Available at: \url{http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427} . #' #' @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 #' NegLLTriBin(No.D.D,Obs.fre.1,.023) #acquiring the Negative log likelihood value #' #' @export NegLLTriBin<-function(x,freq,mode) { #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,mode))) | any(is.infinite(c(x,freq,mode))) |any(is.nan(c(x,freq,mode))) ) { 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 a 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 the mode parameter is less than or equal to zero or more than or equal to one #if so creating a error message as well as stopping the function progress else if(mode <= 0 | mode >= 1) { stop("Mode cannot be less than zero or greater than one") } else { #as the equation contains partial beta integration here an integrative function is written inte<-function(a,b) { Bp<-function(q) { ((q^(a-1))*((1-q)^(b-1))) } return(Bp) } #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]))) temp1<-NULL temp2<-NULL a1<-NULL b1<-NULL a2<-NULL b2<-NULL term2<-NULL #creating an instance for the output of the integration temp<-function(a,b) { stats::integrate(inte(a,b),lower=0,upper=mode)$value
}
#doing the calculations for all binomial random variables under assigned data vector
for(i in 1:sum(freq))
{
a1[i]<-data[i]+2
b1[i]<-n-data[i]+1

a2[i]<-data[i]+1
b2[i]<-n-data[i]+2

temp1[i]<-temp(a1[i],b1[i])
temp2[i]<-temp(a2[i],b2[i])

term2[i]<-log((mode^(-1)*temp1[i])+((1-mode)^(-1)*beta(a2[i],b2[i]))-((1-mode)^(-1)*temp2[i]))
}
TriBinLL<-term1+sum(term2)+sum(freq)*log(2)

#calculating the negative log likelihood value and representing as a single output value
return(-TriBinLL)
}
}
}

#' Estimating the mode value for Triangular Binomial Distribution
#'
#' The function will estimate the mode value using the maximum log likelihood method for the
#' triangular binomial distribution when the binomial random variables and corresponding frequencies
#' are given
#'
#' @usage
#' EstMLETriBin(x,freq)
#'
#' @param x                  vector of binomial random variables
#' @param freq               vector of frequencies
#'
#' @details
#' \deqn{0 < mode=c < 1}
#' \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{EstMLETriBin} will produce a list format consisting
#'
#' \code{NegLLTriBin}  Negative log likelihood value for Triangular Binomial Distribution
#'
#' \code{mode}         Estimated mode value
#'
#' @references
#' Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society,
#' Series A, 120:148-191.
#'
#' Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical
#' Modeling. Boston: Birkhuser Boston, pp. 21-33.
#'
#' Available at: \url{http://dx.doi.org/10.1007/978-0-8176-4626-4_2} .
#'
#' Okagbue, H. et al., 2014. Using the Average of the Extreme Values of a Triangular Distribution for a
#' Transformation, and Its Approximant via the Continuous Uniform Distribution. British Journal of Mathematics
#' & Computer Science, 4(24), pp.3497-3507.
#'
#' Available at: \url{http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427} .
#'
#' @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
#'
#' EstMLETriBin(No.D.D,Obs.fre.1)$mode #estimating the mode value and extracting the mode value #' #' @export EstMLETriBin<-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 { #below looping function is to find the best estimated mode parameter which minimizes the #negative log likelihood value by increasing the decimal point to precision of six looping<-function(x,freq,startmode,endmode,repmode,itmode) { #for a given starting value and end value with the sequence function of R, a seq of mode values #are created mode<-seq(startmode,endmode,by=repmode) #create a matrix with one column and itmode as rows value1<-matrix(ncol=1,nrow=itmode) #name the row names as the above mode values rownames(value1)<-mode #now for each row using the mode value calculate the negative log likelihood values #and save them in the vector for (i in 1:itmode) { value1[i,1]<-NegLLTriBin(x,freq,mode[i]) } #find the minimum value of the matrix minimum<-min(value1,na.rm=TRUE) #which is the minimum negative loglikelihood value TriBinNegLL<-minimum #finding which row(mode value) gives the minimum negative loglikelihood value #and save it as inds inds<-which(value1==min(value1,na.rm=TRUE),arr.ind = TRUE) #acquire the name of the row which will give the mode value, assign it to rnames rnames<-as.numeric(rownames(value1)[inds[,1]]) #generate the output as a list format where TriBinNegLL is the minimum negative loglikelihood #value and mode is the corresponding estimated mode parameter value. output<-list("TriBinNegLL"=TriBinNegLL,"mode"=rnames) return(output) } #consider the mode values from 0.1 to 0.9 estimate the best mode value in between 0.1 and 0.9 for first decimal point answer1<-looping(x,freq,0.1,0.9,0.1,9) #assign the found best estimated mode value to mode1 mode1<-answer1$mode
#consider the second decimal point of mode1, now estimate the best mode value
#assign the found best estimated mode1 value to mode2
mode2<-answer2$mode #consider the third decimal point of mode2, now estimate the best mode value answer3<-looping(x,freq,mode2-0.005,mode2+0.004,0.001,10) #assign the found best estimated mode 2 value to mode3 mode3<-answer3$mode
#consider the fourth decimal point of mode3, now estimate the best mode value
#assign the found best estimated mode 3 value to mode 4
mode4<-answer4$mode #consider the fifth decimal point of mode4, now estimate the best mode value answer5<-looping(x,freq,mode4-0.00005,mode4+0.00004,0.00001,10) #assign the found best estimated mode 4 value to mode 5 mode5<-answer5$mode
#consider the sixth decimal point of mode5, now estimate the best mode value

#finally the found best estimated mode5 value to modefin and find the corresponding negative
#log likelihood value as well
modefin<-answerfin$mode ; TriBinNegLLfin<-answerfin$TriBinNegLL
#generate the output as a list format where TriBinNegLL is the minimum negative loglikelihood
#value and mode is the corresponding estimated mode parameter value.
output<-list("NegLLTriBin"=TriBinNegLLfin,"mode"=modefin)
return(output)
}
}

#' Fitting the Triangular Binomial Distribution when binomial random variable, frequency and mode
#' value are given
#'
#' The function will fit the triangular binomial distribution when random variables, corresponding
#' frequencies and mode parameter 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 fitTriBin(x,obs.freq,mode,print)
#'
#' @param x                  vector of binomial random variables
#' @param obs.freq           vector of frequencies
#' @param mode               single value for mode
#' @param print              logical value for print or not
#'
#' @details
#' \deqn{0 < mode=c < 1}
#' \deqn{x = 0,1,2,...}
#' \deqn{0 < mode < 1}
#' \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{fitTriBin} gives a list format consisting
#'
#' \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 value
#'
#' \code{df} degree of freedom
#'
#' \code{p.value} probability value by chi-squared test statistic
#'
#' \code{over.dis.para} over dispersion value.
#'
#' @references
#' Horsnell, G. (1957). Economic acceptance sampling schemes. Journal of the Royal Statistical Society,
#' Series A, 120:148-191.
#'
#' Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical
#' Modeling. Boston: Birkhuser Boston, pp. 21-33.
#'
#' Available at: \url{http://dx.doi.org/10.1007/978-0-8176-4626-4_2} .
#'
#' Okagbue, H. et al., 2014. Using the Average of the Extreme Values of a Triangular Distribution for a
#' Transformation, and Its Approximant via the Continuous Uniform Distribution. British Journal of Mathematics
#' & Computer Science, 4(24), pp.3497-3507.
#'
#' Available at: \url{http://www.sciencedomain.org/abstract.php?iid=699&id=6&aid=6427} .
#'
#' @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
#'
#' modeTriBin=EstMLETriBin(No.D.D,Obs.fre.1)$mode #assigning the extracted the mode value #' #fitting when the random variable,frequencies,mode value are given. #' fitTriBin(No.D.D,Obs.fre.1,modeTriBin) #' #' fitTriBin(No.D.D,Obs.fre.1,modeTriBin,FALSE)$exp.freq  #extracting the expected frequencies
#' @export
fitTriBin<-function(x,obs.freq,mode,print=T)
{
#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,mode))) | any(is.infinite(c(x,obs.freq,mode))) |
any(is.nan(c(x,obs.freq,mode))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#for given random variables and mode parameter calculating the estimated probability values
est.prob<-dTriBin(x,max(x),mode)$pdf #using the estimated probability values the expected frequencies are calculated exp.freq<-round((sum(obs.freq)*est.prob),2) #chi-squared test statistics is calculated with observed frequency and expected frequency statistic<-sum(((obs.freq-exp.freq)^2)/exp.freq) #degree of freedom is calculated df<-length(x)-2 #p value of chi-squared test statistic is calculated p.value<-1-stats::pchisq(statistic,df) #all the above information is mentioned as a message below #and if the user wishes they can print or not to if(print==TRUE) { cat("\nChi-squared test for Triangular Binomial Distribution \n\n Observed Frequency : ",obs.freq,"\n expected Frequency : ",exp.freq,"\n X-squared =",round(statistic,4),"df =",df," p-value =",round(p.value,4),"\n over dispersion =",dTriBin(x,max(x),mode)$over.dis.para,"\n")
}
#checking if df is less than or equal to zero
if(df<0 | df==0)
{
warning("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)
{
warning("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)
{
warning("Chi-squared approximation is not suitable because expected frequency approximates to zero")
}
#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),
"over.dis.para"=dTriBin(x,max(x),mode)\$over.dis.para)

}
}

Amalan-ConStat/R-fitODBOD documentation built on Oct. 1, 2018, 7:13 p.m.