I am implementing Baum-Welch algorithm in Matlab from this wikipedia link : Baum-Welch algorithm. It is a part of my volatility forcasting in financial time series.
I have two questions :
1: in the last of update step in wikipedia page, It has been told that "These steps are now repeated iteratively until a desired level of convergence.".
So how to declare a condition to stop the loop? in addition what variables should be evaluated to see if they are acceptable?
2:If you pay attention in the Wikipedia's formulas for kesi:
kesi = (alhpa(i,t) * a(i,j) * beta(j,t+1) * b(j,t+1) ) / sum over states for alpha(states,T)
There are two factors that are scaled in the numerator (alpha and beta) and one in denominator ( just alpha ) . So they will dont cancel each others effect. I have implemented the equations (in a loop that repeats 20 times,for example) and I've done the scaling procedure. but the "transition probability matrix" and "initial distribution" and "emission matrix" gets NaN values!!!
I've read this question's answer Baum-Welch many possible observations. I've did scaling based on tutorial mentioned there.
Here is my code :
n = 3; % number of sataes
T = 20; % number of observations
%do some initializing things with random values and computing gamma,kesi...
index=20;
while index>=0
pi_star = gamma(:,1)';
P_star = zeros(n,n);
for i_2=1:n
makhraj = sum(gamma(i_2,:));
for j=1:n
sorat = sum(kesi(i_2,j,:));
P_star(i_2,j) =(sorat) / (makhraj) ;
end
end
Q_star=zeros(n,T);
for t=1:T
for i_2=1:n
makhraj = sum(gamma(i_2,:));
sorat=0;
for h=1:T
if Obs(h) == Obs(t)
sorat = sorat + (gamma(i_2,t));
end
end
Q_star(i_2,t) = (sorat)/(makhraj);
end
end
%computing the forward probabilities
for i_2=1:n
alpha(1,i_2) = pi_star(1,i_2)*Q_star(i_2,1);
end
for t=2:T
for j=1:n
alpha(t,j) = (alpha(t-1,:)*(P_star(:,j))) * Q_star(j,t) ;
end
end
%%% scaling forward probabilities
for t=1:T
c = 1 / sum(alpha(t,:));
for i2=1:n
alpha(t,i2) = alpha(t,i2) * c;
end
end
%computes backward probabilitis
for t=(T-1):(-1) : 1
rightVector=Q_star(:,t+1).* beta( t+1 ,:)' ;
beta ( t , : ) = P_star* rightVector ;
end
%%% scaling backward probabilities
for t=1:T
d = 1 / sum(beta(t,:));
for i2=1:n
beta(t,i2) = beta(t,i2) * d;
end
end
%computing gamma variable
sigma_ab = zeros(1,T);
for t=1:T
for j=1:n
sigma_ab(1,t) = sigma_ab(1,t) + (alpha(t,j)*beta(t,j));
end
end
for t=1:T
for j=1:n
gamma(j,t) = ((alpha(t,j)*beta(t,j))/sigma_ab(1,t));
end
end
%computing kesi
makhraj_k = zeros(1,T);
for t=1:T
for i_2=1:n
makhraj_k(1,t) = makhraj_k(1,t) + alpha(t,i_2);
end
end
for t=1:T-1
for i_2=1:n
for j=1:n
kesi(i_2,j,t) = (alpha(t,i_2)*P_star(i_2,j)*beta(t+1,j)*Q_star(j,t+1))/makhraj_k(1,t);
end
end
end
index = index -1;
end %end while
So what should I do now for this scaling problem? is this NaN values because scaling issue or something else?
Thanks for your time.
