The function hgm.pwishart evaluates the cumulative distribution function of random wishart matrices.

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Description

The function hgm.pwishart evaluates the cumulative distribution function of random wishart matrices of size m times m.

Usage

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hgm.pwishart(m,n,beta,q0,approxdeg,h,dp,q,mode,method,
            err,automatic,assigned_series_error,verbose,autoplot)

Arguments

m

The dimension of the Wishart matrix.

n

The degree of freedome (a parameter of the Wishart distribution)

beta

The eigenvalues of the inverse of the covariant matrix /2 (a parameter of the Wishart distribution). The beta is equal to inverse(sigma)/2.

q0

The point to evaluate the matrix hypergeometric series. q0>0

approxdeg

Zonal polynomials upto the approxdeg are calculated to evaluate values near the origin. A zonal polynomial is determined by a given partition (k1,...,km). We call the sum k1+...+km the degree.

h

A (small) step size for the Runge-Kutta method. h>0.

dp

Sampling interval of solutions by the Runge-Kutta method. When autoplot=1 or dp is negative, it is automatically set. if it is 0, no sample is stored.

q

The second value y[0] of this function is the Prob(L1 < q) where L1 is the first eigenvalue of the Wishart matrix.

mode

When mode=c(1,0,0), it returns the evaluation of the matrix hypergeometric series and its derivatives at q0. When mode=c(1,1,(2^m+1)*p), intermediate values of P(L1 < x) with respect to p-steps of x are also returned. Sampling interval is controled by dp. When autoplot=1, it is automatically set.

method

a-rk4 is the default value. When method="a-rk4", the adaptive Runge-Kutta method is used. Steps are automatically adjusted by err.

err

When err=c(e1,e2), e1 is the absolute error and e2 is the relative error. This parameter controls the adative Runge-Kutta method. If the output is absurd, you may get a correct answer by setting, e.g., err=c(1e-(xy+5), 1e-10) or by increasing q0 when initial value at q0 is very small as 1e-xy.

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.

assigned_series_error

assigned_series_error=0.00001 is the default value.

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.

autoplot

autoplot=0 is the default value. If it is 1, then this function outputs an input for plot (which is equivalent to setting the 3rd argument of the mode parameter properly). 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,...), ... When the adaptive Runge-Kutta method is used, the step size h may change automatically, which makes the sampling period change, in other words, the sampling points q0+q/100, q0+2*q/100, q0+3*q/100, ... may change. In this case, the output matrix may contain zero rows in the tail or overfull. In case of the overful, use the mode option to get the all result.

Details

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. We can obtain more accurate result when the variables h is smaller, q0 is relevant value (not very big, not very small), and the approxdeg is more larger. A heuristic method to set parameters q0, h, approxdeg properly is to make x larger and to check if the y[0] approaches to 1.

Value

The output is x, y[0], ..., y[2^m] in the default mode, y[0] is the value of the cumulative distribution function P(L1 < x) at x. y[1],...,y[2^m] are some derivatives. See the reference below.

Note

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.

Author(s)

Nobuki Takayama

References

H.Hashiguchi, Y.Numata, N.Takayama, A.Takemura, Holonomic gradient method for the distribution function of the largest root of a Wishart matrix, Journal of Multivariate Analysis, 117, (2013) 296-312, http://dx.doi.org/10.1016/j.jmva.2013.03.011,

Examples

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## =====================================================
## Example 1. 
## =====================================================
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)
## =====================================================
## Example 3. 
## =====================================================
c<-hgm.pwishart(m=4,n=10,beta=c(1,2,3,4),q0=1,q=10,approxdeg=20,autoplot=1);
#plot(c)