Annotation of OpenXM/src/R/r-packages/hgm/man/hgm.c2wishart.Rd, Revision 1.7
1.7 ! takayama 1: % $OpenXM: OpenXM/src/R/r-packages/hgm/man/hgm.c2wishart.Rd,v 1.6 2016/02/16 02:17:00 takayama Exp $
1.1 takayama 2: \name{hgm.p2wishart}
3: \alias{hgm.p2wishart}
4: %- Also NEED an '\alias' for EACH other topic documented here.
5: \title{
6: The function hgm.p2wishart evaluates the cumulative distribution function
1.3 takayama 7: of the largest eigenvalues of W1*inverse(W2).
1.1 takayama 8: }
9: \description{
10: The function hgm.p2wishart evaluates the cumulative distribution function
11: of the largest eigenvalues of W1*inverse(W2) where W1 and W2 are Wishart
12: matrices of size m*m of the freedom n1 and n2 respectively.
13: }
14: \usage{
15: hgm.p2wishart(m,n1,n2,beta,q0,approxdeg,h,dp,q,mode,method,
1.2 takayama 16: err,automatic,assigned_series_error,verbose,autoplot)
1.1 takayama 17: }
18: %- maybe also 'usage' for other objects documented here.
19: \arguments{
20: \item{m}{The dimension of the Wishart matrix.}
21: \item{n1}{The degree of freedome of the Wishart distribution S1}
22: \item{n2}{The degree of freedome of the Wishart distribution S2}
23: \item{beta}{The eigenvalues of inverse(S2)*S1 where S1 and S2 are
24: covariant matrices of W1 and W2 respectively.
25: }
26: \item{q0}{The point to evaluate the matrix hypergeometric series. q0>0}
27: \item{approxdeg}{
28: Zonal polynomials upto the approxdeg are calculated to evaluate
29: values near the origin. A zonal polynomial is determined by a given
30: partition (k1,...,km). We call the sum k1+...+km the degree.
31: }
32: \item{h}{
33: A (small) step size for the Runge-Kutta method. h>0.
34: }
35: \item{dp}{
36: Sampling interval of solutions by the Runge-Kutta method.
1.7 ! takayama 37: When autoplot=1 or dp is negative, it is automatically set.
! 38: if it is 0, no sample is stored.
1.1 takayama 39: }
40: \item{q}{
41: The second value y[0] of this function is the Prob(L1 < q)
42: where L1 is the first eigenvalue of the Wishart matrix.
43: }
44: \item{mode}{
45: When mode=c(1,0,0), it returns the evaluation
1.3 takayama 46: of the matrix hypergeometric series and its derivatives at q0.
1.5 takayama 47: When mode=c(1,1,(2^m+1)*p), intermediate values of P(L1 < x) with respect to
1.1 takayama 48: p-steps of x are also returned. Sampling interval is controled by dp.
1.3 takayama 49: When autoplot=1, mode is automatically set.
1.1 takayama 50: }
51: \item{method}{
52: a-rk4 is the default value.
53: When method="a-rk4", the adaptive Runge-Kutta method is used.
54: Steps are automatically adjusted by err.
55: }
56: \item{err}{
57: When err=c(e1,e2), e1 is the absolute error and e2 is the relative error.
1.5 takayama 58: This parameter controls the adative Runge-Kutta method.
59: If the output is absurd, you may get a correct answer by setting, e.g.,
1.6 takayama 60: err=c(1e-(xy+5), 1e-10) or by increasing q0 when initial value at q0 is very small as 1e-xy.
1.1 takayama 61: }
62: \item{automatic}{
63: automatic=1 is the default value.
64: If it is 1, the degree of the series approximation will be increased until
65: |(F(i)-F(i-1))/F(i-1)| < assigned_series_error where
66: F(i) is the degree i approximation of the hypergeometric series
67: with matrix argument.
68: Step sizes for the Runge-Kutta method are also set automatically from
69: the assigned_series_error if it is 1.
70: }
71: \item{assigned_series_error}{
72: assigned_series_error=0.00001 is the default value.
73: }
74: \item{verbose}{
75: verbose=0 is the default value.
76: If it is 1, then steps of automatic degree updates and several parameters
77: are output to stdout and stderr.
78: }
1.2 takayama 79: \item{autoplot}{
80: autoplot=0 is the default value.
1.6 takayama 81: If it is 1, then this function outputs an input for plot
82: (which is equivalent to setting the 3rd argument of the mode parameter properly).
1.4 takayama 83: 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,...), ...
1.6 takayama 84: When the adaptive Runge-Kutta method is used, the step size h may change
85: automatically,
86: which makes the sampling period change, in other words, the sampling points
87: q0+q/100, q0+2*q/100, q0+3*q/100, ... may change.
88: In this case, the output matrix may contain zero rows in the tail or overfull.
89: In case of the overful, use the mode option to get the all result.
1.2 takayama 90: }
1.1 takayama 91: }
92: \details{
93: It is evaluated by the Koev-Edelman algorithm when x is near the origin and
94: by the HGM when x is far from the origin.
95: We can obtain more accurate result when the variables h is smaller,
1.3 takayama 96: q0 is relevant value (not very big, not very small),
1.1 takayama 97: and the approxdeg is more larger.
1.3 takayama 98: A heuristic method to set parameters q0, h, approxdeg properly
1.1 takayama 99: is to make x larger and to check if the y[0] approaches to 1.
100: % \code{\link[RCurl]{postForm}}.
101: }
102: \value{
103: The output is x, y[0], ..., y[2^m] in the default mode,
104: y[0] is the value of the cumulative distribution
105: function P(L1 < x) at x. y[1],...,y[2^m] are some derivatives.
106: See the reference below.
107: }
108: \references{
109: H.Hashiguchi, N.Takayama, A.Takemura,
110: in preparation.
111: }
112: \author{
113: Nobuki Takayama
114: }
115: \note{
116: This function does not work well under the following cases:
117: 1. The beta (the set of eigenvalues)
118: is degenerated or is almost degenerated.
119: 2. The beta is very skew, in other words, there is a big eigenvalue
120: and there is also a small eigenvalue.
121: The error control is done by a heuristic method.
122: The obtained value is not validated automatically.
123: }
124:
125: %% ~Make other sections like Warning with \section{Warning }{....} ~
126:
127: %\seealso{
128: %%%\code{\link{oxm.matrix_r2tfb}}
129: %}
130: \examples{
131: ## =====================================================
132: ## Example 1.
133: ## =====================================================
134: hgm.p2wishart(m=3,n1=5,n2=10,beta=c(1,2,4),q=4)
135: ## =====================================================
136: ## Example 2.
137: ## =====================================================
1.2 takayama 138: b<-hgm.p2wishart(m=3,n1=5,n2=10,beta=c(1,2,4),q0=0.3,q=20,approxdeg=20,mode=c(1,1,(8+1)*1000));
139: c<-matrix(b,ncol=8+1,byrow=1);
140: #plot(c)
141: ## =====================================================
142: ## Example 3.
143: ## =====================================================
144: c<-hgm.p2wishart(m=3,n1=5,n2=10,beta=c(1,2,4),q0=0.3,q=20,approxdeg=20,autoplot=1);
1.1 takayama 145: #plot(c)
146: }
147: % Add one or more standard keywords, see file 'KEYWORDS' in the
148: % R documentation directory.
149: \keyword{ Cumulative distribution function of random two wishart matrices }
150: \keyword{ Holonomic gradient method }
151: \keyword{ HGM }
152:
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