Re: Gradient and Hessian matrix of cumulative normal ditribution
- To: mathgroup at smc.vnet.net
- Subject: [mg67942] Re: Gradient and Hessian matrix of cumulative normal ditribution
- From: "Valeri Astanoff" <astanoff at yahoo.fr>
- Date: Mon, 17 Jul 2006 06:52:03 -0400 (EDT)
- References: <e95aub$gu9$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hello Pratim,
For a numerical computation of the hessian at a given point,
you could proceed this way :
In[1]:=<<Statistics`
In[2]:=gradHess[fun_, {x1_,x2_}, {vx1_,vx2_}, eps_]:=
Module[{data,f},
data=Table[{u1,u2,fun /. x1 -> u1 /. x2 -> u2},
{u1, vx1-eps, vx1+eps,eps},
{u2, vx2-eps, vx2+eps,eps}]//Flatten[#,1]&;
f=Interpolation[data, InterpolationOrder -> {2,2}];
{grad[f[x1,x2],{x1,x2}], hess[f[x1,x2],{x1,x2}]}/.
x1 -> vx1 /. x2 -> vx2
];
grad[fun_,{x1_,x2_}] := {D[fun,x1],D[fun,x2]};
hess[fun_,{x1_,x2_}] := grad[grad[fun,{x1,x2}],{x1,x2}];
Example with your data :
In[5]:=r = {{1, 0.2}, {0.2, 1}};
ndist = MultinormalDistribution[{0, 0}, r];
cdf=CDF[ndist,{x1,x2}] ;
In[8]:=gradHess[cdf,{x1,x2},{1,0.8},0.001]
Out[8]={{0.176603,0.233018},{{-0.192939,0.0816784},{0.0816784,-0.20275}}}
With a higher precision, hessian is slightly modified :
In[9]:=gradHess[cdf,{x1,x2},{1,0.8},0.000001]
Out[9]={{0.176603,0.233018},{{-0.192604,0.0816569},{0.0816569,-0.20259}}}
hth
Valeri Astanoff