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Diff for /OpenXM/src/R/r-packages/hgm/man/hgm.cwishart.Rd between version 1.2 and 1.10

version 1.2, 2013/03/01 05:27:08 version 1.10, 2015/03/27 02:36:30
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 % $OpenXM: OpenXM/src/R/r-packages/hgm/man/hgm.cwishart.Rd,v 1.1 2013/02/23 07:00:21 takayama Exp $  % $OpenXM: OpenXM/src/R/r-packages/hgm/man/hgm.cwishart.Rd,v 1.9 2015/03/26 11:54:13 takayama Exp $
 \name{hgm.cwishart}  \name{hgm.pwishart}
 \alias{hgm.cwishart}  \alias{hgm.pwishart}
 %- Also NEED an '\alias' for EACH other topic documented here.  %- Also NEED an '\alias' for EACH other topic documented here.
 \title{  \title{
     The function hgm.cwishart evaluates the cumulative distribution function      The function hgm.pwishart evaluates the cumulative distribution function
   of random wishart matrix.    of random wishart matrices.
 }  }
 \description{  \description{
     The function hgm.cwishart evaluates the cumulative distribution function      The function hgm.pwishart evaluates the cumulative distribution function
   of random wishart matrix of size m times m.    of random wishart matrices of size m times m.
 }  }
 \usage{  \usage{
 hgm.cwishart(m,n,beta,x0,approxdeg,h,dp,x)  hgm.pwishart(m,n,beta,q0,approxdeg,h,dp,q,mode,method,
               err,automatic,assigned_series_error,verbose)
 }  }
 %- maybe also 'usage' for other objects documented here.  %- maybe also 'usage' for other objects documented here.
 \arguments{  \arguments{
   \item{m}{The dimension of the Wishart matrix.}    \item{m}{The dimension of the Wishart matrix.}
   \item{n}{The degree of freedome (a parameter of the Wishart distribution)}    \item{n}{The degree of freedome (a parameter of the Wishart distribution)}
   \item{beta}{The eigenvalues of the inverse of the covariant matrix    \item{beta}{The eigenvalues of the inverse of the covariant matrix /2
 (a parameter of the Wishart distribution)  (a parameter of the Wishart distribution).
       The beta is equal to inverse(sigma)/2.
   }    }
   \item{x0}{The point to evaluate the matrix hypergeometric series. x0>0}    \item{q0}{The point to evaluate the matrix hypergeometric series. q0>0}
   \item{approxdeg}{    \item{approxdeg}{
     Zonal polynomials upto the approxdeg are calculated to evaluate      Zonal polynomials upto the approxdeg are calculated to evaluate
    values near the origin. A zonal polynomial is determined by a given     values near the origin. A zonal polynomial is determined by a given
Line 32  hgm.cwishart(m,n,beta,x0,approxdeg,h,dp,x)
Line 34  hgm.cwishart(m,n,beta,x0,approxdeg,h,dp,x)
   \item{dp}{    \item{dp}{
     Sampling interval of solutions by the Runge-Kutta method.      Sampling interval of solutions by the Runge-Kutta method.
   }    }
   \item{x}{    \item{q}{
     The first value of this function is the Prob(L1 < x)      The second value y[0] of this function is the Prob(L1 < q)
     where L1 is the first eigenvalue of the Wishart matrix.      where L1 is the first eigenvalue of the Wishart matrix.
   }    }
     \item{mode}{
       When mode=c(1,0,0), it returns the evaluation
       of the matrix hypergeometric series and its derivatives at x0.
       When mode=c(1,1,(m^2+1)*p), intermediate values of P(L1 < x) with respect to
       p-steps of x are also returned.  Sampling interval is controled by dp.
     }
     \item{method}{
       a-rk4 is the default value.
       When method="a-rk4", the adaptive Runge-Kutta method is used.
       Steps are automatically adjusted by err.
     }
     \item{err}{
       When err=c(e1,e2), e1 is the absolute error and e2 is the relative error.
       As long as NaN is not returned, it is recommended to set to
       err=c(0.0, 1e-10), because initial values are usually very small.
     }
     \item{automatic}{
       automatic=1 is the default value.
       If it is 1, the degree of the series approximation will be increased until
       |(F(i)-F(i-1))/F(i-1)| < assigned_series_error where
       F(i) is the degree i approximation of the hypergeometric series
       with matrix argument.
       Step sizes for the Runge-Kutta method are also set automatically from
       the assigned_series_error if it is 1.
     }
     \item{assigned_series_error}{
       assigned_series_error=0.00001 is the default value.
     }
     \item{verbose}{
       verbose=0 is the default value.
       If it is 1, then steps of automatic degree updates and several parameters
       are output to stdout and stderr.
     }
 }  }
 \details{  \details{
   It is evaluated by the Koev-Edelman algorithm when x is near the origin and    It is evaluated by the Koev-Edelman algorithm when x is near the origin and
   by the HGM when x is far from the origin.    by the HGM when x is far from the origin.
   We can obtain more accurate result when the variables h, x0 are smaller    We can obtain more accurate result when the variables h is smaller,
     x0 is relevant value (not very big, not very small),
   and the approxdeg is more larger.    and the approxdeg is more larger.
     A heuristic method to set parameters x0, h, approxdeg properly
     is to make x larger and to check if the y[0] approaches to 1.
 %  \code{\link[RCurl]{postForm}}.  %  \code{\link[RCurl]{postForm}}.
 }  }
 \value{  \value{
 The output is x, y[0], ..., y[2^m],  The output is x, y[0], ..., y[2^m] in the default mode,
 y[0] is the value of the cumulative distribution  y[0] is the value of the cumulative distribution
 function P(L1 < x) at x.  y[1],...,y[2^m] are some derivatives.  function P(L1 < x) at x.  y[1],...,y[2^m] are some derivatives.
 See the reference below.  See the reference below.
 }  }
 \references{  \references{
 H.Hashiguchi, Y.Numata, N.Takayama, A.Takemura,  H.Hashiguchi, Y.Numata, N.Takayama, A.Takemura,
 Holonomic gradient method for the distribution function of the largest root of a Wishart matrix  Holonomic gradient method for the distribution function of the largest root of a Wishart matrix,
 \url{http://arxiv.org/abs/1201.0472},  Journal of Multivariate Analysis, 117, (2013) 296-312,
   \url{http://dx.doi.org/10.1016/j.jmva.2013.03.011},
 }  }
 \author{  \author{
 Nobuki Takayama  Nobuki Takayama
 }  }
 \note{  \note{
 %%  ~~further notes~~  This function does not work well under the following cases:
   1. The beta (the set of eigenvalues)
   is degenerated or is almost degenerated.
   2. The beta is very skew, in other words, there is a big eigenvalue
   and there is also a small eigenvalue.
   The error control is done by a heuristic method.
   The obtained value is not validated automatically.
 }  }
   
 %% ~Make other sections like Warning with \section{Warning }{....} ~  %% ~Make other sections like Warning with \section{Warning }{....} ~
   
 \seealso{  %\seealso{
 %%\code{\link{oxm.matrix_r2tfb}}  %%%\code{\link{oxm.matrix_r2tfb}}
 }  %}
 \examples{  \examples{
 ## =====================================================  ## =====================================================
 ## Example 1. Computing normalization constant of the Fisher distribution on SO(3)  ## Example 1.
 ## =====================================================  ## =====================================================
 hgm.cwishart(m=3,n=5,beta=c(1,2,3),x=10)  hgm.pwishart(m=3,n=5,beta=c(1,2,3),q=10)
   ## =====================================================
   ## Example 2.
   ## =====================================================
   b<-hgm.pwishart(m=4,n=10,beta=c(1,2,3,4),q0=1,q=10,approxdeg=20,mode=c(1,1,(16+1)*100));
   c<-matrix(b,ncol=16+1,byrow=1);
   #plot(c)
 }  }
 % Add one or more standard keywords, see file 'KEYWORDS' in the  % Add one or more standard keywords, see file 'KEYWORDS' in the
 % R documentation directory.  % R documentation directory.
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