Expected Euler Characteristic
Papers
- N.Takayama, L.Jiu, S.Kuriki, Y.Zhang,
Computations of the Expected Euler Characteristic for the Largest Eigenvalue of a Real Wishart Matrix,
arxiv:1903.10099
-
References for the HGM and the HGD
Programs
- R program to make a simulation:
matnorm.r .
In order to run the program, install the package mnormt by
install.packages("mnormt");
- The computer algebra
Risa/Asir
programs for Example 5 and other examples:
float-exyi5b.rr ,
exyi5c.rr [construct series sol by ODE ann3.txt and E_vals for x=3.7,3.71,...],
ode-yi3.rr ,
tk_series ,
ann3.txt ,
- Mathematica program to evalute the approximate formula of Theorem 3
in the Appendix:
ec-selb.m .
Example 5
$m=n=2$ case with
$M = \pmatrix{1 & 0 \cr
2 & 3 \cr}$
and
$S={\rm diag}(10^3,10^2)$ where $\Sigma = S^{-1}$.
Simulation by R.
source("matnorm.r");
ans<-seq(0,10);
k<-1;
for (i in seq(3.8,3.81,by=0.001)) {
ans[k]<-yiex5(x=i,try=100000);
print(i); print(ans[k]);
k<-k+1;
}
print(ans)
Approximation by our Theorem 1 and ODE solution (HGM)
on the computer algebra system
Risa/Asir
load("float-exyi5b.rr");
Example: Appendix, Theorem 3
Evalute $P(\lambda_1 > 20)$ for $10 \times 10$ matrix ($m=n=10$)
with $M=0$ and $\Sigma=I_{10}/s$, $s=0.1$.
Simulation by R with 100000 random matrices.
source("matnorm.r");
selb_sim(m=10,s=0.1,x=20,try=100000)
Evaluation by Mathematica and our approximate formula by the Euler
characteristic heuristic.
<<ec-selb.m
ecSelb[10,0.1,20]
Last update: April 6, 2020