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# chitwo2lc_sorm

## PURPOSE

SORM-approximation of the crossing intensity for the noncentral Chi^2 process

## SYNOPSIS

mu=chitwo2lc_sorm(u,g,b,S22,S12,ass,n0,epsi)

## DESCRIPTION

```  CHITWO2LC_SORM  SORM-approximation of the crossing intensity for the noncentral Chi^2 process

The noncentral Chi^2 process is defined in IR-(27).
X(t)=sum beta_j Z_j(t) + gamma_j Z_j(t)^2

CALL:  mu = chitwo2lc_sorm(x0,gamma,beta,S12,S22,ass);

mu   = two column vector containing: levels in the first column,
the SORM-approximation of crossing intensitity for quadratic sea
in the second column.
u    = column with levels, note that u can not take value zero.
g,b    = the vectors containing constants defining the process
approximation of the cdf of the quadratic sea.
S12    = covariance between vector Z(t) and Z'(t), where Z(t)=(Z_1(t),...,Z_n(t)).
S22    = covariance between vector Z'(t) and Z'(t)., where Z(t)=(Z_1(t),...,Z_n(t)).
ass    = 0, SORM, 0< FORM, >0 higher order very slow,  optional ass=0;
n0    = parameter used to find a starting point in the optimization, default value n0=10
epsi    = tolerance level in the optimization, default value epsi=1e-9

Example: S=jonswap; [gamma beta S12 S22]=dirsp2chitwo(S.S,S.w);
L0=sqrt(sum(beta.^2)); u=(0.01:0.1:4*L0)'; u=[(-4*L0:0.1:-0.1)';u];
mu = chitwo2lc_sorm(u,gamma,beta,S22,S12);
semilogy(mu(:,1),mu(:,2))```

## CROSS-REFERENCE INFORMATION

This function calls:
 mindist Finds minimal distance to the origin on the surface b'*x+x'*diag(g)*x=u rqlf_asympt Gives first two terms in an asymptotic expansion of the error Display message and abort function. kron Kronecker tensor product.
This function is called by:

## SOURCE CODE

```001 function  mu=chitwo2lc_sorm(u,g,b,S22,S12,ass,n0,epsi)
002 % CHITWO2LC_SORM  SORM-approximation of the crossing intensity for the noncentral Chi^2 process
003 %
004 %  The noncentral Chi^2 process is defined in IR-(27).
005 %           X(t)=sum beta_j Z_j(t) + gamma_j Z_j(t)^2
006 %
007 %  CALL:  mu = chitwo2lc_sorm(x0,gamma,beta,S12,S22,ass);
008 %
009 %       mu   = two column vector containing: levels in the first column,
010 %              the SORM-approximation of crossing intensitity for quadratic sea
011 %              in the second column.
012 %       u    = column with levels, note that u can not take value zero.
013 %     g,b    = the vectors containing constants defining the process
014 %              approximation of the cdf of the quadratic sea.
015 %     S12    = covariance between vector Z(t) and Z'(t), where Z(t)=(Z_1(t),...,Z_n(t)).
016 %     S22    = covariance between vector Z'(t) and Z'(t)., where Z(t)=(Z_1(t),...,Z_n(t)).
017 %     ass    = 0, SORM, 0< FORM, >0 higher order very slow,  optional ass=0;
018 %      n0    = parameter used to find a starting point in the optimization, default value n0=10
019 %    epsi    = tolerance level in the optimization, default value epsi=1e-9
020 %
021 %   Example: S=jonswap; [gamma beta S12 S22]=dirsp2chitwo(S.S,S.w);
022 %            L0=sqrt(sum(beta.^2)); u=(0.01:0.1:4*L0)'; u=[(-4*L0:0.1:-0.1)';u];
023 %            mu = chitwo2lc_sorm(u,gamma,beta,S22,S12);
024 %            semilogy(mu(:,1),mu(:,2))
025 %
026
027
028 % References:
029 %             K. Breitung, (1988), "Asymptotic crossing rates for stationary {G}aussian vector
030 %                                   processes.", Stochastic Processes and
031 %                                   their Applications, 29,pp. 195-207.
032 %
033 %             U. Machado, I. Rychlik (2002) "Wave statistics in nonlinear sea", Extremes, 6, pp. 125--146.
034 %             Hagberg, O. (2005) PhD - thesis, Dept. of Math. Statistics, Univ. of Lund.
035 % Baxevani, A.,  Hagberg, O. and Rychlik, I.  Note on the distribution of extreme waves crests (OMAE 2005).
036
037
038 %   Calls: mindist
039 % Revised by I.R 14.04.05
040 % By OH.         24.10.04
041 %
042 %------------------------------------------------------------------------------------
043
044 if nargin<7
045    n0=10;
046 end
047 if nargin<8
048    epsi=1e-9;
049 end
050 %
051 % if ass>0 one is using second order assymptotics, see PhD thesis of Hagberg
052 %
053 if nargin<6
054    ass=0;
055 end
056
057 n=length(u);
058 x0=[];
059 mu=[];
060 for i=1:n;
061     xx1=mindist(g,b,u(i),n0,epsi);
062     x0=[x0 xx1];
063     if ass>0
064         [Cb0 Cb1]=rqlf_asympt(u(i),g,b,S22,S12);
065         %mu=[mu; u(i) muu];
066         mu=[mu; u(i) 0.5*(Cb0+Cb1)];
067     end
068 end
069 if ass<=0
070
071 g=g(:);
072 b=b(:);
073 d=length(g);
074 [d1,n]=size(x0);
075 if d1~=d
076    error('the observations of x0 should be stored as columns')
077 end
078 b0=sqrt(sum(x0.^2,1));
079
080 G=g*(-2*b0./sqrt(sum((kron(b,ones(1,n))+2*kron(g,ones(1,n)).* ...
081               x0).^2)));
082 % cofactor matrix:
083 D=zeros(size(x0));
084 for k=1:d
085    D(k,:)=prod(1+G([1:k-1 k+1:d],:),1);
086 end
087
088 cc=[sqrt((sum(x0.*(S22*x0),1)+sum(G.*(S12'*x0).^2,1))) sqrt(sum(D.*x0.^2,1))];
089
090 mu=exp(-b0.^2/2).*sqrt((sum(x0.*(S22*x0),1)+sum(G.*(S12'*x0).^2,1))./(sum(D.*x0.^2,1)))/pi/2;
091
092 mu1=exp(-b0.^2/2);%.*sqrt((sum(x0.*((S22)*x0),1))./(sum(x0.^2,1)))/2/pi;
093 mu2=exp(-b0.^2/2).*sqrt((sum(x0.*((S22)*x0),1))./(sum(x0.^2,1)))/2/pi;
094
095 cc=sqrt((sum(x0.*(S22*x0),1)+sum(G.*(S12'*x0).^2,1))./(sum(D.*x0.^2,1)))/pi/2;
096 mu=[u mu'];
097 if ass<0
098  mu=[u mu2'];
099 end
100 end```

Mathematical Statistics
Centre for Mathematical Sciences
Lund University with Lund Institute of Technology

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