计算概率

Question

我有这个数据：

simulated_states = c("A", "E", "B", "B", "A", "C", "D", "A", "B", "D", "A", "D", 
"D", "E", "D", "D", "D", "E", "A", "A", "A", "B", "A", "C", "C", 
"D", "A", "A", "D", "A", "D", "A", "A", "A", "C", "C", "D", "A", 
"C", "C", "D", "E", "C", "C", "C", "E", "B", "A", "E", "E", "C", 
"C", "D", "E", "C", "E", "E", "A", "E", "B", "A", "A", "E", "E", 
"C", "E", "C", "C", "C", "D", "E", "D", "C", "D", "A", "B", "B", 
"E", "B", "A", "E", "C", "C", "D", "B", "B", "A", "C", "B", "A", 
"D", "A", "D", "E", "C", "D", "D", "A", "A", "C")

我知道如何计算转移概率：

calculate_transition_probs <- function(states) {
  
  transitions <- data.frame(
    from = states,
    to = c(states[-1], NA) 
  )
  
  transition_counts <- table(transitions, useNA = "always")
  transition_df <- as.data.frame(transition_counts)
  colnames(transition_df) <- c("from", "to", "count")
  transition_df <- transition_df[!is.na(transition_df$to), ]
  
  transition_df <- transition_df %>%
    group_by(from) %>%
    mutate(percent = count / sum(count) * 100) %>%
    ungroup()
  
  transition_df <- transition_df[, c("from", "to", "count", "percent")]
  transition_df <- transition_df[order(transition_df$from, transition_df$to), ]
  
  return(transition_df)
}

transition_probs <- calculate_transition_probs(simulated_states)

结果如下所示：

 from to count   percent
    A  A     7 26.923077
    A  B     3 11.538462
    A  C     6 23.076923
    A  D     5 19.230769
    A  E     5 19.230769
    B  A     7 58.333333
    B  B     3 25.000000
    B  C     0  0.000000
    B  D     1  8.333333
    B  E     1  8.333333
    C  A     0  0.000000
    C  B     1  4.545455
    C  C     9 40.909091
    C  D     9 40.909091
    C  E     3 13.636364
    D  A     9 42.857143
    D  B     1  4.761905
    D  C     1  4.761905
    D  D     4 19.047619
    D  E     6 28.571429
    E  A     2 11.111111
    E  B     4 22.222222
    E  C     7 38.888889
    E  D     2 11.111111
    E  E     3 16.666667

现在，我想扩展它来计算 n 步概率的转移概率。

例如

• 2 步骤：从 = (A,A) 给出 to = A，从 = (A,B) 给出 to = A，从 = (A,C) 给出 to = A ..... to = B 从 = 给出(A,B), to = B 由 = (B,B) 等给出
• 3 个步骤：从 = (A,A,A) 给出 to = A，从 = (A,B,A) 给出 to = A，等等
• N 个步骤：从 = (A,A,A...A) 等给出 to = A

如何编写一个函数来执行 n 步？

Answer 1

您的

transition_counts

可以通过将条目除以行总和来转换为转换矩阵（为了简单起见，我将其称为

A

）：

A <- transition_counts / rowSums(transition_counts)

那么，两步转移概率就是简单的

A %*% A
    to
from         A          B         C         D         E
   A 0.2435780 0.12229330 0.2404796 0.2137937 0.1798554
   B 0.3478582 0.15229446 0.1709910 0.1581451 0.1707112
   C 0.2169913 0.07974223 0.2398662 0.2642168 0.1991834
   D 0.2565411 0.13608217 0.2385630 0.1738936 0.1949202
   E 0.2256817 0.12838088 0.2548378 0.2386595 0.1524402

请注意，行总和仍然为 1。然后是 3 步

A %*% A %*% A

或者为了简化，我们可以使用

expm

包，它具有方便的

%^%

功能：

library(expm)
A %^% 3

这个函数可以让你计算第n步。

A %^% 10
    to
from         A         B         C         D         E
   A 0.2494011 0.1199961 0.2341925 0.2148659 0.1815444
   B 0.2494024 0.1199966 0.2341917 0.2148651 0.1815442
   C 0.2494006 0.1199957 0.2341927 0.2148664 0.1815446
   D 0.2494013 0.1199962 0.2341924 0.2148656 0.1815445
   E 0.2494010 0.1199962 0.2341926 0.2148660 0.1815442

上式接近稳态，由解给出：

qr.solve(rbind(t(A) - diag(5), rep(1, 5)), c(rep(0,5), 1))

#            A         B         C         D         E 
#    0.2494012 0.1199961 0.2341924 0.2148659 0.1815444