如何找到马尔可夫链概率？

Question

I am trying to find the probability that the chain jumps from state k-1 to state 1 before it hits state k . Can anyone spot my mistake?

I tried to simulate the markov chain, but i want to make a code that allows me to find probability of k ={1, 2, 3, ........17} . But I can really not get the code.

This is the error message I always get

Error in while (X[i] > 1 && X[i] < k) { : 
  missing value where TRUE/FALSE needed

Here is my code:

k <- 17
{   p <- 0.5
q <- 0.1
P <- matrix (0, nrow = k, ncol = k, byrow = TRUE)
for (i in 1:k)
{   for (j in 1:k)
    {   if (i == 1 && i == j)
        {   P[i,j] <- 1
        }
        else if (i == k && i == j)
        {   P[i,j] <- 1
        }
        else if (i == j)
        {   P[i,j] <- p*(1-q)
        }
        else if (j == k && i != 1)
        {   P[i,j] <- q
        }   
        else if (i == j+1 && i != k)
        {   P[i,j] <- (1-p)*(1-q)
        }
    }
}
P
X <- (k-1) 
trials <- 1000
hits <- 0 #counter for no. of hits 
for (i in 1:trials)
{   i <- 1 #no. of steps
    while(X[i] > 1 && X[i] < k)
    {   Y <- runif(1) #uniform samples
        p1 <- P[X[i],] #calculating the p-value
        p1 <- cumsum(p1)
        # changes in the chain
        if(Y <= p1[1])
        {   X[i+1] = 1}
        else if(Y <= p1[2])
        {   X[i+1] = 2}
        else if(Y <= p1[3])
        {   X[i+1] = 3}
        else if(Y <= p1[4])
        {   X[i+1] = 4}
        else if(Y <= p1[5])
        {   X[i+1] = 5}
        else if(Y <= p1[6])
        {   X[i+1] = 6}
        else if(Y <= p1[7])
        {   X[i+1] = 7}
        else if(Y <= p1[8])
        {   X[i+1] = 8}
        else if(Y <= p1[9])
        {   X[i+1] = 9}
        else if(Y <= p1[10])
        {   X[i+1] = 10}
        else if(Y <= p1[11])
        {   X[i+1] = 11}
        else if(Y <= p1[12])
        {   X[i+1] = 12}
        else if(Y <= p1[13])
        {   X[i+1] = 13}
        else if(Y <= p1[14])
        {   X[i+1] = 14}
        else if(Y <= p1[15])
        {   X[i+1] = 15}
        else if(Y <= p1[16])
        {   X[i+1] = 16}
        else if(Y <= p1[17])
        {   X[i+1] <= 17}
        i <- i+1
    }
    if(X[i]==1)
    {   hits <- hits+1}
    else
    {   hits <- hits+0}
}

Probability <- hits/trials
Probability
}

Answer 1

I think the line

i <- 1 #no. of steps

should not be there. Try this:

k <- 17
{   p <- 0.5
q <- 0.1
P <- matrix (0, nrow = k, ncol = k, byrow = TRUE)
for (i in 1:k)
{   for (j in 1:k)
    {   if (i == 1 && i == j)
        {   P[i,j] <- 1
        }
        else if (i == k && i == j)
        {   P[i,j] <- 1
        }
        else if (i == j)
        {   P[i,j] <- p*(1-q)
        }
        else if (j == k && i != 1)
        {   P[i,j] <- q
        }   
        else if (i == j+1 && i != k)
        {   P[i,j] <- (1-p)*(1-q)
        }
    }
}
P
X <- (k-1) 
trials <- 1000
hits <- 0 #counter for no. of hits 
for (i in 1:trials)
{
    while(X[i] > 1 && X[i] < k)
    {   Y <- runif(1) #uniform samples
        p1 <- P[X[i],] #calculating the p-value
        p1 <- cumsum(p1)
        # changes in the chain
        if(Y <= p1[1])
        {   X[i+1] = 1}
        else if(Y <= p1[2])
        {   X[i+1] = 2}
        else if(Y <= p1[3])
        {   X[i+1] = 3}
        else if(Y <= p1[4])
        {   X[i+1] = 4}
        else if(Y <= p1[5])
        {   X[i+1] = 5}
        else if(Y <= p1[6])
        {   X[i+1] = 6}
        else if(Y <= p1[7])
        {   X[i+1] = 7}
        else if(Y <= p1[8])
        {   X[i+1] = 8}
        else if(Y <= p1[9])
        {   X[i+1] = 9}
        else if(Y <= p1[10])
        {   X[i+1] = 10}
        else if(Y <= p1[11])
        {   X[i+1] = 11}
        else if(Y <= p1[12])
        {   X[i+1] = 12}
        else if(Y <= p1[13])
        {   X[i+1] = 13}
        else if(Y <= p1[14])
        {   X[i+1] = 14}
        else if(Y <= p1[15])
        {   X[i+1] = 15}
        else if(Y <= p1[16])
        {   X[i+1] = 16}
        else if(Y <= p1[17])
        {   X[i+1] <= 17}
        i <- i+1
    }
    if(X[i]==1)
    {   hits <- hits+1}
    else
    {   hits <- hits+0}
}

Probability <- hits/trials
Probability
}

Answer 2

You're setting X to k-1. In R, that's treated as a vector of length 1. As soon as i reaches 2, X[i] return an index error, because X does not have a second element.

Further notes: using the same index in two different nesting levels is very bad form. Also, when you start having a massive list of if-then-else statements, it's time to rethink your code. In this case, you could just subset 1:17 on p1[i] >=Y, take the minimum value, and then set X to that.