在R中对转移矩阵随机变量取整：确保行和为1的方案

核心需求 #

• 把转移矩阵的十进制概率转换为分母固定为36的分数形式
• 强制所有原本为0/36的项转为1/36
• 调整取整逻辑，优先对每行中概率最高的项进行修正，确保每行分数之和严格为1（即分子总和为36）

实现代码 #

# 定义转移矩阵
transmat <- structure(c(0.730926989335521, 0.474022677271958, 0.0091743119266055, 
                        0.326860841423948, 0.97411003236246, 0.926605504587156, 0.926605504587156, 
                        0.961722488038278, 0.926605504587156, 0.238720262510254, 0.40632932814351, 
                        0.926605504587156, 0.326860841423948, 0.00323624595469256, 0.0091743119266055, 
                        0.0091743119266055, 0.00478468899521532, 0.0091743119266055, 
                        7.45767767917071e-05, 0.0170925706549332, 0.0091743119266055, 
                        0.00323624595469256, 0.00323624595469256, 0.0091743119266055, 
                        0.0091743119266055, 0.00478468899521532, 0.0091743119266055, 
                        0.00753225445596242, 0.0340159079370452, 0.0091743119266055, 
                        0.00323624595469256, 0.00323624595469256, 0.0091743119266055, 
                        0.0091743119266055, 0.00478468899521532, 0.0091743119266055, 
                        0.00753225445596242, 0.0340159079370452, 0.0091743119266055, 
                        0.00323624595469256, 0.00323624595469256, 0.0091743119266055, 
                        0.0091743119266055, 0.00478468899521532, 0.0091743119266055, 
                        7.45767767917071e-05, 0.0170925706549332, 0.0091743119266055, 
                        0.00323624595469256, 0.00323624595469256, 0.0091743119266055, 
                        0.0091743119266055, 0.00478468899521532, 0.0091743119266055, 
                        7.45767767917071e-05, 0.0170925706549332, 0.0091743119266055, 
                        0.00323624595469256, 0.00323624595469256, 0.0091743119266055, 
                        0.0091743119266055, 0.00478468899521532, 0.0091743119266055, 
                        0.00753225445596242, 0.00016923337282112, 0.0091743119266055, 
                        0.326860841423948, 0.00323624595469256, 0.0091743119266055, 0.0091743119266055, 
                        0.00478468899521532, 0.0091743119266055, 0.00753225445596242, 
                        0.00016923337282112, 0.0091743119266055, 0.00323624595469256, 
                        0.00323624595469256, 0.0091743119266055, 0.0091743119266055, 
                        0.00478468899521532, 0.0091743119266055), dim = c(9L, 9L), dimnames = list(
                          c("00", "02", "13", "18", "25", "50", "60", "80", "95"), 
                          c("00", "02", "13", "18", "25", "50", "60", "80", "95")))

# 定义行处理函数，满足需求
process_row <- function(row) {
  # 1. 将概率值转换为36倍的原始分子
  raw_numerators <- row * 36
  # 2. 强制所有分子至少为1（对应1/36）
  numerators <- pmax(raw_numerators, 1)
  # 3. 计算当前分子总和与目标36的差值
  sum_diff <- sum(numerators) - 36
  
  if (sum_diff > 0) {
    # 若总和超过36，从概率最高的项开始减1，直到总和为36
    sorted_indices <- order(-raw_numerators)
    numerators[sorted_indices[1:sum_diff]] <- numerators[sorted_indices[1:sum_diff]] - 1
  } else if (sum_diff < 0) {
    # 若总和不足36，从概率最高的项开始加1，直到总和为36
    sorted_indices <- order(-raw_numerators)
    numerators[sorted_indices[1:abs(sum_diff)]] <- numerators[sorted_indices[1:abs(sum_diff)]] + 1
  }
  
  # 转换为分数字符串格式
  paste0(numerators, "/36")
}

# 应用函数到转移矩阵的每一行
fractions_matrix <- t(apply(transmat, 1, process_row))

# 验证每行分子和是否为36
row_sums <- apply(fractions_matrix, 1, function(x) sum(as.integer(sub("/36", "", x))))
print(row_sums) # 应输出全36

代码说明 #

1. 强制最小分子: 使用pmax(raw_numerators,1)确保所有项的分子至少为1，直接满足0/36转1/36的要求。
2. 总和修正逻辑:
   • 若强制最小分子后总和超过36，从概率最高的项依次减1，尽可能降低对整体概率分布的影响。
   • 若总和不足36（极端情况），同样优先给概率最高的项加1，符合需求中的优先调整规则。
3. 格式转换: 将最终分子拼接为x/36的字符串格式，保持统一的分数形式。

内容的提问来源于stack exchange，提问作者Geoarch