Annotation of OpenXM/src/R/r-packages/hgm/man/hgm.cwishart.Rd, Revision 1.16
1.16 ! takayama 1: % $OpenXM: OpenXM/src/R/r-packages/hgm/man/hgm.cwishart.Rd,v 1.15 2016/03/01 07:29:18 takayama Exp $
1.4 takayama 2: \name{hgm.pwishart}
3: \alias{hgm.pwishart}
1.1 takayama 4: %- Also NEED an '\alias' for EACH other topic documented here.
5: \title{
1.4 takayama 6: The function hgm.pwishart evaluates the cumulative distribution function
1.7 takayama 7: of random wishart matrices.
1.1 takayama 8: }
9: \description{
1.4 takayama 10: The function hgm.pwishart evaluates the cumulative distribution function
1.7 takayama 11: of random wishart matrices of size m times m.
1.1 takayama 12: }
13: \usage{
1.9 takayama 14: hgm.pwishart(m,n,beta,q0,approxdeg,h,dp,q,mode,method,
1.12 takayama 15: err,automatic,assigned_series_error,verbose,autoplot)
1.1 takayama 16: }
17: %- maybe also 'usage' for other objects documented here.
18: \arguments{
1.2 takayama 19: \item{m}{The dimension of the Wishart matrix.}
20: \item{n}{The degree of freedome (a parameter of the Wishart distribution)}
1.3 takayama 21: \item{beta}{The eigenvalues of the inverse of the covariant matrix /2
22: (a parameter of the Wishart distribution).
23: The beta is equal to inverse(sigma)/2.
1.1 takayama 24: }
1.4 takayama 25: \item{q0}{The point to evaluate the matrix hypergeometric series. q0>0}
1.1 takayama 26: \item{approxdeg}{
1.2 takayama 27: Zonal polynomials upto the approxdeg are calculated to evaluate
28: values near the origin. A zonal polynomial is determined by a given
29: partition (k1,...,km). We call the sum k1+...+km the degree.
1.1 takayama 30: }
31: \item{h}{
1.2 takayama 32: A (small) step size for the Runge-Kutta method. h>0.
1.1 takayama 33: }
34: \item{dp}{
1.2 takayama 35: Sampling interval of solutions by the Runge-Kutta method.
1.15 takayama 36: When autoplot=1 or dp is negative, it is automatically set.
37: if it is 0, no sample is stored.
1.2 takayama 38: }
1.4 takayama 39: \item{q}{
40: The second value y[0] of this function is the Prob(L1 < q)
1.2 takayama 41: where L1 is the first eigenvalue of the Wishart matrix.
1.1 takayama 42: }
1.3 takayama 43: \item{mode}{
44: When mode=c(1,0,0), it returns the evaluation
1.11 takayama 45: of the matrix hypergeometric series and its derivatives at q0.
1.13 takayama 46: When mode=c(1,1,(2^m+1)*p), intermediate values of P(L1 < x) with respect to
1.3 takayama 47: p-steps of x are also returned. Sampling interval is controled by dp.
1.12 takayama 48: When autoplot=1, it is automatically set.
1.3 takayama 49: }
50: \item{method}{
1.5 takayama 51: a-rk4 is the default value.
1.3 takayama 52: When method="a-rk4", the adaptive Runge-Kutta method is used.
53: Steps are automatically adjusted by err.
54: }
55: \item{err}{
56: When err=c(e1,e2), e1 is the absolute error and e2 is the relative error.
1.13 takayama 57: This parameter controls the adative Runge-Kutta method.
58: If the output is absurd, you may get a correct answer by setting, e.g.,
1.14 takayama 59: err=c(1e-(xy+5), 1e-10) or by increasing q0 when initial value at q0 is very small as 1e-xy.
1.3 takayama 60: }
1.5 takayama 61: \item{automatic}{
62: automatic=1 is the default value.
63: If it is 1, the degree of the series approximation will be increased until
64: |(F(i)-F(i-1))/F(i-1)| < assigned_series_error where
65: F(i) is the degree i approximation of the hypergeometric series
66: with matrix argument.
1.6 takayama 67: Step sizes for the Runge-Kutta method are also set automatically from
68: the assigned_series_error if it is 1.
1.5 takayama 69: }
70: \item{assigned_series_error}{
71: assigned_series_error=0.00001 is the default value.
72: }
73: \item{verbose}{
74: verbose=0 is the default value.
75: If it is 1, then steps of automatic degree updates and several parameters
76: are output to stdout and stderr.
77: }
1.12 takayama 78: \item{autoplot}{
79: autoplot=0 is the default value.
1.14 takayama 80: If it is 1, then this function outputs an input for plot
81: (which is equivalent to setting the 3rd argument of the mode parameter properly).
1.12 takayama 82: When ans is the output, ans[1,] is c(q,prob at q,...), ans[2,] is c(q0,prob at q0,...), and ans[3,] is c(q0+q/100,prob at q/100,...), ...
1.14 takayama 83: When the adaptive Runge-Kutta method is used, the step size h may change
84: automatically,
85: which makes the sampling period change, in other words, the sampling points
86: q0+q/100, q0+2*q/100, q0+3*q/100, ... may change.
87: In this case, the output matrix may contain zero rows in the tail or overfull.
88: In case of the overful, use the mode option to get the all result.
1.12 takayama 89: }
1.1 takayama 90: }
91: \details{
1.2 takayama 92: It is evaluated by the Koev-Edelman algorithm when x is near the origin and
93: by the HGM when x is far from the origin.
1.3 takayama 94: We can obtain more accurate result when the variables h is smaller,
1.11 takayama 95: q0 is relevant value (not very big, not very small),
1.2 takayama 96: and the approxdeg is more larger.
1.11 takayama 97: A heuristic method to set parameters q0, h, approxdeg properly
1.3 takayama 98: is to make x larger and to check if the y[0] approaches to 1.
1.1 takayama 99: % \code{\link[RCurl]{postForm}}.
100: }
101: \value{
1.3 takayama 102: The output is x, y[0], ..., y[2^m] in the default mode,
1.2 takayama 103: y[0] is the value of the cumulative distribution
104: function P(L1 < x) at x. y[1],...,y[2^m] are some derivatives.
105: See the reference below.
1.1 takayama 106: }
107: \references{
1.2 takayama 108: H.Hashiguchi, Y.Numata, N.Takayama, A.Takemura,
1.6 takayama 109: Holonomic gradient method for the distribution function of the largest root of a Wishart matrix,
110: Journal of Multivariate Analysis, 117, (2013) 296-312,
1.16 ! takayama 111: \doi{10.1016/j.jmva.2013.03.011},
1.1 takayama 112: }
113: \author{
114: Nobuki Takayama
115: }
116: \note{
1.7 takayama 117: This function does not work well under the following cases:
118: 1. The beta (the set of eigenvalues)
119: is degenerated or is almost degenerated.
120: 2. The beta is very skew, in other words, there is a big eigenvalue
121: and there is also a small eigenvalue.
122: The error control is done by a heuristic method.
123: The obtained value is not validated automatically.
1.1 takayama 124: }
125:
126: %% ~Make other sections like Warning with \section{Warning }{....} ~
127:
1.10 takayama 128: %\seealso{
129: %%%\code{\link{oxm.matrix_r2tfb}}
130: %}
1.1 takayama 131: \examples{
132: ## =====================================================
1.3 takayama 133: ## Example 1.
1.1 takayama 134: ## =====================================================
1.4 takayama 135: hgm.pwishart(m=3,n=5,beta=c(1,2,3),q=10)
1.3 takayama 136: ## =====================================================
137: ## Example 2.
138: ## =====================================================
1.4 takayama 139: 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));
1.3 takayama 140: c<-matrix(b,ncol=16+1,byrow=1);
141: #plot(c)
1.12 takayama 142: ## =====================================================
143: ## Example 3.
144: ## =====================================================
145: c<-hgm.pwishart(m=4,n=10,beta=c(1,2,3,4),q0=1,q=10,approxdeg=20,autoplot=1);
146: #plot(c)
1.1 takayama 147: }
148: % Add one or more standard keywords, see file 'KEYWORDS' in the
149: % R documentation directory.
150: \keyword{ Cumulative distribution function of random wishart matrix }
151: \keyword{ Holonomic gradient method }
152: \keyword{ HGM }
153:
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