Transition Matrix -ve states and sum not 1, or 1 with negative probablities

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Masood Abbasi 2018-4-30

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https://ww2.mathworks.cn/matlabcentral/answers/398227-transition-matrix-ve-states-and-sum-not-1-or-1-with-negative-probablities

编辑： Masood Abbasi 2018-4-30

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I have implemented following equation in Matalb

This equation is for calculating transition Matrix Probabilities (Pi,i+j) of this Markov Chain

The code i have written is as follows, where Nt=2:

                    for i=0:Nt  % sum over all i's
   for d=0:Nt %sum over all d's
      for d1=0:min(i,d) %sum for all possible d1's
            for a=0:d  %sum for all a's 
             %index=sprintf("Index  i=%d j=%d i=%d i+j=%d \n",i,d1-a, i,i+d1-a)
                for k=a:d
                    if (i==0 || d1==i)
                        p=pf;
                    else
                        p=pa;
                    end
                    A= nchoosek(d,k) * (PSI^k) *( (1-PSI)^(d-k) )* nchoosek(k,a) *(p^a)*(1-p)^(k-a);
                                     if( Nt-i > d-d1)
                                            %A= nchoosek(d,k) * PSI *( (1-PSI)^(d-k) )* nchoosek(k,a) *(p^a)*(1-p)^(k-a);  
                                            M=nchoosek(i,d1)* nchoosek(Nt-i,d-d1) *(Pi1^d)* (1-Pi1)^(Nt-d) * A ;
end
%make transition matrix array
                                     %newIndex=sprintf("Index  i=%d j=%d i=%d i+j=%d \n",i,d1-a, i,i+d1-a)
                                     %values=sprintf("Index  i=%d d=%d d1=%d a=%d ,k=%d j=%d i=%d i+j=%d \n",i,d,d1,a,k,d1-a, i,i+d1-a)
                                    if ( (i+d1-a) > 0 && (i+d1-a <= Nt)) %( (i<=Nt) && (i+d1-a <=Nt) && (i+d1-a >=0)  )
                                       % newIndex=sprintf("Index  i=%d j=%d i=%d i+j=%d \n",i,d1-a, i,i+d1-a);
                                       TM(i+1,i+d1-a+1)=TM(i+1,i+d1-a+1)+M;   % for indexing +1
                                    end                                     
                end                                 
            end        
        end        
    end    
end

Following are observations from this equation 1) I get negative index sometimes eg. for i+j i.e. Pi,i+j =P0,-1 we dont have '-1' as state. 2) The transition Matrix sums to 1 for about 75% rows. When i ran this code with Nt=40. i was getting sum of rows=1 for 35 rows only. With Nt=10, first row becomes all 0's and 4 rows have sum equal to '1'. 3) In some cases sum of Rows element =1 when having negative probabilities like 1.234 -0.234 0.0 = 1

Can any expert tell me if i have mistakes in code or else.

Thank you