From 92bdba6d69909fc212a48fbc7abf1f98a046b27b Mon Sep 17 00:00:00 2001 From: xiaowuhu Date: Fri, 1 Apr 2022 12:52:44 +0800 Subject: [PATCH] Xiaowu/20220330 (#718) * update * up * duao * update --- .../A7-强化学习/draft/三门问题/ThreeDoors.py | 15 +- .../draft/三门问题/images/ThreeDoors2.png | Bin 65661 -> 0 bytes .../三门问题/{images => img}/ThreeDoors1.png | Bin .../draft/三门问题/img/ThreeDoors2.png | Bin 0 -> 68038 bytes .../A7-强化学习/draft/三门问题/三门问题.md | 10 +- .../A7-强化学习/draft/概率图问题/CarData.txt | 1 + .../draft/概率图问题/CarData_backup.txt | 1 + .../概率图问题/RentCar-0-GenerateData.py | 51 ++ .../draft/概率图问题/RentCar-1-A-B.py | 35 ++ .../draft/概率图问题/RentCar-2-Matrix.py | 37 ++ .../draft/概率图问题/RentCar-3-OneStep.py | 19 + .../draft/概率图问题/RentCar-4-KStep.py | 23 + .../draft/概率图问题/RentCar-5-Convergence.py | 22 + .../A7-强化学习/draft/概率图问题/img/Car1.png | Bin 0 -> 44156 bytes .../A7-强化学习/draft/概率图问题/img/Car2.png | Bin 0 -> 81110 bytes .../A7-强化学习/draft/概率图问题/租车问题.md | 439 ++++++++++++++++++ 16 files changed, 644 insertions(+), 9 deletions(-) delete mode 100644 基础教程/A7-强化学习/draft/三门问题/images/ThreeDoors2.png rename 基础教程/A7-强化学习/draft/三门问题/{images => img}/ThreeDoors1.png (100%) create mode 100644 基础教程/A7-强化学习/draft/三门问题/img/ThreeDoors2.png create mode 100644 基础教程/A7-强化学习/draft/概率图问题/CarData.txt create mode 100644 基础教程/A7-强化学习/draft/概率图问题/CarData_backup.txt create mode 100644 基础教程/A7-强化学习/draft/概率图问题/RentCar-0-GenerateData.py create mode 100644 基础教程/A7-强化学习/draft/概率图问题/RentCar-1-A-B.py create mode 100644 基础教程/A7-强化学习/draft/概率图问题/RentCar-2-Matrix.py create mode 100644 基础教程/A7-强化学习/draft/概率图问题/RentCar-3-OneStep.py create mode 100644 基础教程/A7-强化学习/draft/概率图问题/RentCar-4-KStep.py create mode 100644 基础教程/A7-强化学习/draft/概率图问题/RentCar-5-Convergence.py create mode 100644 基础教程/A7-强化学习/draft/概率图问题/img/Car1.png create mode 100644 基础教程/A7-强化学习/draft/概率图问题/img/Car2.png create mode 100644 基础教程/A7-强化学习/draft/概率图问题/租车问题.md diff --git a/基础教程/A7-强化学习/draft/三门问题/ThreeDoors.py b/基础教程/A7-强化学习/draft/三门问题/ThreeDoors.py index 4ce00fbb..9da0b08b 100644 --- a/基础教程/A7-强化学习/draft/三门问题/ThreeDoors.py +++ b/基础教程/A7-强化学习/draft/三门问题/ThreeDoors.py @@ -32,16 +32,23 @@ def try_once(n_doors: int): return no_change_but_win, win_after_change -if __name__ == "__main__": + +def try_n_doors(n_doors): total = 100000 - n_doors = 8 n_win_0 = 0 n_win_1 = 0 for i in range(total): no_change_but_win, win_after_change = try_once(n_doors) n_win_0 += win_after_change n_win_1 += no_change_but_win - + print(str.format("{0}扇门:", n_doors)) print(n_win_0, n_win_1) - print(str.format("更换选择而中奖的概率={0} \n不更换而中奖的概率={1}", + print(str.format("更换选择而中奖的概率={0} \n不换选择而中奖的概率={1}", n_win_0/total, n_win_1/total)) + +if __name__ == "__main__": + n_doors = 3 + try_n_doors(n_doors) + + n_doors = 8 + try_n_doors(n_doors) diff --git a/基础教程/A7-强化学习/draft/三门问题/images/ThreeDoors2.png b/基础教程/A7-强化学习/draft/三门问题/images/ThreeDoors2.png deleted file mode 100644 index 0ab40912b302c3034988999471d69473da3b08ce..0000000000000000000000000000000000000000 GIT binary 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"B", "A", "D", "C", "B", "A", "B", "A", "B", "A", "B", "A", "D", "B", "A", "B", "A", "D", "C", "B", "A", "A", "D", "B", "A", "B", "A", "D", "D", "B", "A", "B", "C", "B", "A", "D", "D", "B", "A", "B", "A", "D", "D", "C", "B", "C", "B", "C", "B", "A", "D", "B", "A", "B", "A", "D", "C", "B", "C", "B", "C", "B", "A", "A", "D", "B", "C", "B", "C", "B", "A", "D", "C", "B", "A", "D", "B", "A", "D", "D", "C", "B", "A", "D", "D", "D"] \ No newline at end of file diff --git a/基础教程/A7-强化学习/draft/概率图问题/RentCar-0-GenerateData.py b/基础教程/A7-强化学习/draft/概率图问题/RentCar-0-GenerateData.py new file mode 100644 index 00000000..4df1fc30 --- /dev/null +++ b/基础教程/A7-强化学习/draft/概率图问题/RentCar-0-GenerateData.py @@ -0,0 +1,51 @@ +import numpy as np +import os +import json + +# 作为目标的转移概率矩阵 +P = np.array([ + [0.1, 0.3, 0.0, 0.6], + [0.8, 0.0, 0.2, 0.0], + [0.0, 0.9, 0.1, 0.0], + [0.0, 0.3, 0.3, 0.4] +]) + +# 采样 +def sample(n_samples, n_states, start_state): + states = [i for i in range(n_states)] + # 状态转移序列 + X = [] + # 开始采样 + X.append(start_state) + current = start_state + for i in range(n_samples): + next = np.random.choice(states, p=P[current]) + X.append(next) + current = next + #endfor + return X + +def save_file(X, file_name): + # 把0123变成ABCD + Y = [chr(x+65) for x in X] + #print(Y) + # 保存Y到文件 + json_list = json.dumps(Y) + file = open(file_name, "w") + file.write(json_list) + file.close() + + +if __name__ == "__main__": + # 采样数量 + n_samples = 10000 + # 状态空间 + n_states = 4 + # 起始状态(从0开始) + start_state = 1 + X = sample(n_samples, n_states, start_state) + #print(X) + # 保存文件 + root = os.path.split(os.path.realpath(__file__))[0] + file_name = os.path.join(root, "CarData.txt") + save_file(X, file_name) diff --git a/基础教程/A7-强化学习/draft/概率图问题/RentCar-1-A-B.py b/基础教程/A7-强化学习/draft/概率图问题/RentCar-1-A-B.py new file mode 100644 index 00000000..c9e4b1bb --- /dev/null +++ b/基础教程/A7-强化学习/draft/概率图问题/RentCar-1-A-B.py @@ -0,0 +1,35 @@ +import numpy as np +import json +import os + +# 计算从a_i转移到a_j的概率 +def calculate_P(X, a_i, a_j): + n_i = 0 + n_j = 0 + for x in range(len(X)-1): + if a_i == X[x]: + n_i += 1 + if a_j == X[x+1]: + n_j += 1 + print(n_i, n_j, n_j/n_i) + +def open_file(file_name): + file = open(file_name, "r") + lines = file.read() + file.close() + data_list = json.loads(lines) + return data_list + +if __name__ == "__main__": + # 状态空间 + n_states = 4 + # 读取文件 + root = os.path.split(os.path.realpath(__file__))[0] + file_name = os.path.join(root, "CarData.txt") + data_list = open_file(file_name) + # 把 ABCD 变成 0123 + X = [ord(x)-65 for x in data_list] + # 计算转移矩阵 + calculate_P(X, 0, 1) # 1代表B店，0代表A店 + + diff --git a/基础教程/A7-强化学习/draft/概率图问题/RentCar-2-Matrix.py b/基础教程/A7-强化学习/draft/概率图问题/RentCar-2-Matrix.py new file mode 100644 index 00000000..d9bca7d9 --- /dev/null +++ b/基础教程/A7-强化学习/draft/概率图问题/RentCar-2-Matrix.py @@ -0,0 +1,37 @@ +import numpy as np +import json +import os + +def calculate_Matrix(n_states, X): + P_counter = np.zeros((n_states, n_states)) + for i in range(len(X)-1): + a_i = X[i] + a_j = X[i+1] + P_counter[a_i, a_j] += 1 + #endfor + # 计算各列之和 + sum = np.sum(P_counter, axis=1, keepdims=True) + print("各个状态出现的次数:\n",sum) + P = P_counter / sum + return P + +def open_file(file_name): + file = open(file_name, "r") + lines = file.read() + file.close() + data_list = json.loads(lines) + return data_list + +if __name__ == "__main__": + # 状态空间 + n_states = 4 + # 读取文件 + root = os.path.split(os.path.realpath(__file__))[0] + file_name = os.path.join(root, "CarData.txt") + data_list = open_file(file_name) + # 把 ABCD 变成 0123 + X = [ord(x)-65 for x in data_list] + # 计算转移矩阵 + P = calculate_Matrix(n_states, X) + print("概率转移矩阵:") + print(np.around(P, 1)) diff --git a/基础教程/A7-强化学习/draft/概率图问题/RentCar-3-OneStep.py b/基础教程/A7-强化学习/draft/概率图问题/RentCar-3-OneStep.py new file mode 100644 index 00000000..363ea77c --- /dev/null +++ b/基础教程/A7-强化学习/draft/概率图问题/RentCar-3-OneStep.py @@ -0,0 +1,19 @@ +import numpy as np + +P = np.array([ + [0.1, 0.3, 0.0, 0.6], + [0.8, 0.0, 0.2, 0.0], + [0.0, 0.9, 0.1, 0.0], + [0.0, 0.3, 0.3, 0.4] +]) + +def calculate_day(X, P, day): + X_curr = X.copy() + for i in range(day): + print(str.format("day {0}: {1} ", i, X_curr)) + X_next = np.dot(X_curr, P) + X_curr = X_next.copy() + +if __name__=="__main__": + X = np.array([0,1,0,0]) + calculate_day(X, P, 6) diff --git a/基础教程/A7-强化学习/draft/概率图问题/RentCar-4-KStep.py b/基础教程/A7-强化学习/draft/概率图问题/RentCar-4-KStep.py new file mode 100644 index 00000000..3b7c76e8 --- /dev/null +++ b/基础教程/A7-强化学习/draft/概率图问题/RentCar-4-KStep.py @@ -0,0 +1,23 @@ +import numpy as np + +P = np.array([ + [0.1, 0.3, 0.0, 0.6], + [0.8, 0.0, 0.2, 0.0], + [0.0, 0.9, 0.1, 0.0], + [0.0, 0.3, 0.3, 0.4] +]) + +# 计算K步转移概率矩阵 +def K_step_matrix(P, K): + Pk=P.copy() + for i in range(K-1): + Pk=np.dot(P,Pk) + #print(Pk) + return Pk + +if __name__=="__main__": + X = np.array([0,1,0,0]) + P5 = K_step_matrix(P, 5) + print("5步转移矩阵:\n", P5) + X5 = np.dot(X, P5) + print("第 5 天的情况：", X5) diff --git a/基础教程/A7-强化学习/draft/概率图问题/RentCar-5-Convergence.py b/基础教程/A7-强化学习/draft/概率图问题/RentCar-5-Convergence.py new file mode 100644 index 00000000..494ef211 --- /dev/null +++ b/基础教程/A7-强化学习/draft/概率图问题/RentCar-5-Convergence.py @@ -0,0 +1,22 @@ +import numpy as np + +P = np.array([ + [0.1, 0.3, 0.0, 0.6], + [0.8, 0.0, 0.2, 0.0], + [0.0, 0.9, 0.1, 0.0], + [0.0, 0.3, 0.3, 0.4] +]) + +def Check_Convergence(P): + P_curr = P.copy() + for i in range(100000): + P_next=np.dot(P,P_curr) + print("迭代次数 =",i+1) + print(P_next) + if np.allclose(P_curr, P_next): + break + P_curr = P_next + return P_next + +if __name__=="__main__": + Pn = Check_Convergence(P) diff --git a/基础教程/A7-强化学习/draft/概率图问题/img/Car1.png b/基础教程/A7-强化学习/draft/概率图问题/img/Car1.png new file mode 100644 index 0000000000000000000000000000000000000000..326c5b3938808244933a12388b6d18073928ad1b GIT binary patch literal 44156 zcmce;c{r4R^gm3BFsT$ms1zalRv2V0YxaF#!;CQYtrD^m5tC&o+t|0k7)se2`#y%U 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zXug>t6=CyuSC0OGoWwCkMAH3nE#LK$R73qQZTL<4bz6;N+^qY-2YDgA!#Mju|BRn1q$*Grjsu2$5tlZ-gRlD)B%$`0Fa&4S~(uMFQx-XDeOyDyAw^b=& z7zdjP^*OKR>=*7KHDD^=pN5_r;Hu~EZO@~RY)XSQ2?udgFH?%a-HME)>`-yxk45mc zJd*&Jw4s8OP&G+Xe@dV^tpY8OBtlD}%PH0n!6vnjW_x21 zR){)uXt`%BgZK118DO1cFS302RV^2jP$r2T!MTtaqsE=|E4k-4%j3$=1B7(Fot^1!nMV~cUyU2xDm@_MmE;Ae`9SjY zN`T`Mq=Z0ec+B v?hLgEN-Bc?fX( + +图 1 + +### 3 解决经理的问题 + +重复一下需求：公司经理想知道如果有一天（命名为第 0 天）这辆车从 B 号店出租了，2 天后的早晨会最有可能在哪个店出现？5 天后又会如何？ + +一般情况下，读者会根据图 1 从 B 店出发，根据概率顺藤摸瓜地计算出第 1 天的情况，再计算出第 2 天的情况。 + +- 第 1 天早晨出现在各门店的概率是：$[0.8,\ 0.0,\ 0.2,\ 0.0]$。 +- 第 2 天早晨出现在各门店的概率是......有点儿复杂，我们绘制出图 2 来帮助整理思路。 + + + +图 2 + +从图2一眼就可以看出来，第 2 天早晨该车出现在各门店的概率就是两个连续的概率之乘积，比如： +- 第 1 天 + - 出现在A店（橙色）的概率是 0.8； + - 出现在C店（红色）的概率是 0.2； + - 但是不可能出现在B,D店； + +- 第 2 天 + - 由于A店有0.3的概率还到B店，所以出现在B店的概率是$0.8 \times 0.3=0.24$； + - 由于C店有0.9的概率还到B店，所以出现在B店的概率是$0.2 \times 0.9=0.18$； + +所以，该车第3天早晨出现在B店的概率是 $0.24+0.18=0.42$。出现在其它店的数字也可以同理得到。 + +其它店的所有情况列在表 1 中，便于统计，数字的颜色和图2 是一一对应的，表示是哪个门店，方便读者对照理解。 + +表1 + +|从$\rightarrow$到|A店|B店|C店|D店|第1天| +|:-:|-|-|-|-|-| +|**A店**|$\color{orange}{0.8\times0.1=0.08}$|$\color{orange}{0.8\times0.3=0.24}$|$\color{orange}{0.8\times0.0=0.00}$|$\color{orange}{0.8\times0.6=0.48}$|$\color{orange}{0.8}$| +|**B店**|$\color{green}{0.0\times0.8=0.0}$|$\color{green}{0.0\times0.0=0.0}$|$\color{green}{0.0\times0.2=0.0}$|$\color{green}{0.0\times0.0=0.0}$|$\color{green}{0.0}$| +|**C店**|$\color{red}{0.2\times0.0=0.00}$|$\color{red}{0.2\times0.9=0.18}$|$\color{red}{0.2\times0.1=0.02}$|$\color{red}{0.2\times0.0=0.00}$|$\color{red}{0.2}$| +|**D店**|$\color{blue}{0.0\times0.0=0.0}$|$\color{blue}{0.0\times0.3=0.0}$|$\color{blue}{0.0\times0.3=0.0}$|$\color{blue}{0.0\times0.4=0.0}$|$\color{blue}{0.0}$| +|**第2天**|$\color{orange}{0.08}$|$\color{green}{0.24+0.18=0.42}$|$\color{red}{0.02}$|$\color{blue}{0.48}$|$1.0$| + +数据解读： + +- 表 1 除去表头，中间部分的 4x4 区域，和图2的数据是一致的。 +- 最后一列是中间4列的和，所以正好是第1天的出现概率。 +- 最后一行是中间4行的和，所以是第2天的出现概率。 + +OK! 2 天后的问题解决了，那么 5 天后呢？这么计算太麻烦了，我们引入转移概率矩阵的概念来帮助解决问题。 + + +### 4 转移概率矩阵 + +把图1变成表2，方便使用计算机来解决问题。 + +表2 + +|P: 从$\rightarrow$到|A|B|C|D||输出总和| +|:-:|-|-|-|-|-|-| +|**A**|0.1|0.3|0.0|0.6||1.0| +|**B**|0.8|0.0|0.2|0.0||1.0| +|**C**|0.0|0.9|0.1|0.0||1.0| +|**D**|0.0|0.3|0.3|0.4||1.0| +|||||||| +|输入总和|0.9|1.5|0.6|1.0|4.0| + +- 是一个 $n \times n$ 的方阵（中间的 4x4 区域）。 +- 矩阵各元素都是非负的，即：$0 \le P_{i,j} \le 1$。 +- 各行元素（输出概率）之和为 1，即：$\sum^n_{j=1} P_{i,j}=1$。 +- 对于输入概率总和没有限制，但是最终的总和（右下角）在行列上肯定应该一致，都是4.0（因为有个 4 个门店），即：$\sum_{i=1}^n \sum_{j=1}^n P_{i,j} = n$。 + +如果从原始数据计算的话，可以用更简单的代码一次搞定所有门店的数据，不需要像前面的代码那样一个个地计算： + +```Python +def calculate_Matrix(n_states, X): + P_counter = np.zeros((n_states, n_states)) + for i in range(len(X)-1): + a_i = X[i] + a_j = X[i+1] + P_counter[a_i, a_j] += 1 + #endfor + # 计算各列之和 + sum = np.sum(P_counter, axis=1, keepdims=True) + print("各个状态出现的次数:\n",sum) + P = P_counter / sum + return P +#enddef + +# 把原始数据中的字符 ABCD 变成 0123 +n_states= 4 +X = [ord(x)-65 for x in data_list] +# 计算转移矩阵 +P = calculate_Matrix(n_states, X) +print("转移概率矩阵:") +print(np.around(P, 1)) # 保留1位小数 +``` +``` +[Out] +各个状态出现的次数: + [[2703.] + [3040.] + [1568.] + [2689.]] +转移概率矩阵: +[[0.1 0.3 0. 0.6] + [0.8 0. 0.2 0. ] + [0. 0.9 0.1 0. ] + [0. 0.3 0.3 0.4]] +``` + + + +用矩阵来表示 + +$$ +P = +\begin{pmatrix} +P_{11} & P_{12} & P_{13} & P_{14} +\\ +P_{21} & P_{22} & P_{23} & P_{24} +\\ +P_{31} & P_{32} & P_{33} & P_{34} +\\ +P_{41} & P_{42} & P_{43} & P_{44} +\end{pmatrix}= +\begin{pmatrix} +0.1 & 0.3 & 0.0 & 0.6 +\\ +0.8 & 0.0 & 0.2 & 0.0 +\\ +0.0 & 0.9 & 0.1 & 0.0 +\\ +0.0 & 0.3 & 0.3 & 0.4 +\end{pmatrix} +\tag{3} +$$ + +而对于一个通用的问题，矩阵形式是： + +$$ +P = +\begin{pmatrix} +P_{11} & \cdots & P_{1n} +\\ +\vdots & \ddots & \vdots +\\ +P_{n1} & \cdots & P_{nn} +\end{pmatrix} +\tag{4} +$$ + +有了矩阵，我们可以方便地使用矩阵运算来代替复杂的循环逻辑 + +首先定义第 0 天的初始向量为： +$$ +X_0=(0,1,0,0) \tag{5} +$$ + +表示豪华宝驴车目前肯定在 B 门店。 + +那么第 1 天在哪个门店的问题可以转换为： + +$$ +X_1 = X_0 P=(0, 1 , 0 , 0) +\begin{pmatrix} +0.1 & 0.3 & 0.0 & 0.6 +\\ +0.8 & 0.0 & 0.2 & 0.0 +\\ +0.0 & 0.9 & 0.1 & 0.0 +\\ +0.0 & 0.3 & 0.3 & 0.4 +\end{pmatrix}= +(0.8,\ 0.0,\ 0.2,\ 0.0) +\tag{6} +$$ + +那么第2天在哪个门店的问题可以递归计算出： + +$$ +X_2 = X_1 P=(0.8,\ 0.0,\ 0.2,\ 0.0) +\begin{pmatrix} +0.1 & 0.3 & 0.0 & 0.6 +\\ +0.8 & 0.0 & 0.2 & 0.0 +\\ +0.0 & 0.9 & 0.1 & 0.0 +\\ +0.0 & 0.3 & 0.3 & 0.4 +\end{pmatrix}= +(0.08, \ 0.42, \ 0.02, \ 0.48) \tag{7} +$$ + +式7的结果和表1完全一致。 + +推广到一般情况，计算第 $n+1$ 天的概率时，需要使用第 $n$ 天的结果 + +$$ +X_{n+1}=X_n P \tag{8} +$$ + + +我们可以很方便地写出代码 + +```Python +P = np.array([ + [0.1, 0.3, 0.0, 0.6], + [0.8, 0.0, 0.2, 0.0], + [0.0, 0.9, 0.1, 0.0], + [0.0, 0.3, 0.3, 0.4] +]) + +def calculate_day(P, day): + X = np.array([0,1,0,0]) + X_curr = X.copy() + for i in range(day): + print(str.format("day {0}: {1} ", i+1, X_curr)) + X_next = np.dot(X_curr, P) + X_curr = X_next +#enddef + +calculate_day(P, 5) # 5表示计算5天的数据 +``` +``` +[Out] +day 0: [0 1 0 0] +day 1: [0.8 0. 0.2 0. ] +day 2: [0.08 0.42 0.02 0.48] +day 3: [0.344 0.186 0.23 0.24 ] +day 4: [0.1832 0.3822 0.1322 0.3024] +day 5: [0.32408 0.26466 0.18038 0.23088] +``` + +OK！经理的问题解决了，在第 5 天的时候，该豪华宝驴车在 4 个门店出现的概率依次是： + +$[0.32408,\ 0.26466,\ 0.18038,\ 0.23088]$ + +### 5 K-步转移概率矩阵 + +从上面的例子我们可以看到，只有一步的转移概率通常是不能满足实际需要的，通常我们需要知道 K 步后的情况如何。 + +比如，5步后的概率应该是： + +$$ +X_5 = X_4 P=(X_3P)P=((X_2P)P)P=(((X_1P)P)P)P=((((X_0P)P)P)P)P=X_0 P^5 \tag{9} +$$ + +式 4 可以定义为一步转移概率矩阵，那么从式9的推导可以得到式 10，定义为 K 步转移概率： + +$$ +P^K = +\begin{pmatrix} +P_{11} & \cdots & P_{1n} +\\ +\vdots & \ddots & \vdots +\\ +P_{n1} & \cdots & P_{nn} +\end{pmatrix}^K +\tag{10} +$$ + +代码如下 + +```Python +# 计算K步转移概率矩阵 +def K_step_matrix(P, K): + Pk=P.copy() + for i in range(K-1): + Pk=np.dot(Pk,P) + return Pk +``` +读者可能会注意到循环次数是 K-1，而不是 K。假设我们计算 K=2 步转移矩阵，那么一步矩阵P和自己做一次矩阵相乘就可以了，而不是 2 次。所以 K 步矩阵做 K-1 次矩阵相乘。另外一个细节是 np.dot(P,Pk) 和 np.dot(Pk,P) 都会得到相同的结果。 + +得到 K 步转移矩阵后，可以用如下代码简单地直接计算第 5 天的情况： + +```Python +if __name__=="__main__": + X = np.array([0,1,0,0]) + P5 = K_step_matrix(P, 5) + print(P5) + X5 = np.dot(X, P5) + print(X5) +``` +``` +[Out] +5步转移矩阵: + [[0.25585 0.31989 0.14028 0.28398] + [0.32408 0.26466 0.18038 0.23088] + [0.19728 0.36459 0.14005 0.29808] + [0.26448 0.29469 0.15843 0.2824 ]] +第 5 天的情况： [0.32408 0.26466 0.18038 0.23088] +``` + +与上面的迭代方法计算结果一致。 + + + +### 6 更多步的情况 + +从 K 步转移可以向更远的地方思考，如果 K 为无穷大时，这个转移概率矩阵是什么情况呢？ + +我们用代码做一个试验 + +```Python +def N_step_matrix(P): + P_curr = P.copy() + for i in range(100000): + P_next=np.dot(P,P_curr) + print("迭代次数=",i+1) + print(P_next) + if np.allclose(P_curr, P_next): + break + P_curr = P_next + return P_next +#enddef +N_step_matrix(P) +``` +注意，在代码中我们使用 np.allclose(P_curr, P_next) 来判断上一次迭代的结果和本次迭代的结果的差值，如果小于1e-6则认为已经收敛，停止循环。运行过程如下所示： +``` +[Out] +迭代次数 = 1 +[[0.25 0.21 0.24 0.3 ] + [0.08 0.42 0.02 0.48] + [0.72 0.09 0.19 0. ] + [0.24 0.39 0.21 0.16]] +迭代次数 = 2 +[[0.193 0.381 0.156 0.27 ] + [0.344 0.186 0.23 0.24 ] + [0.144 0.387 0.037 0.432] + [0.336 0.309 0.147 0.208]] + +...... + +迭代次数 = 27 +[[0.26966338 0.30337037 0.1573036 0.26966265] + [0.26966195 0.30337167 0.15730289 0.26966349] + [0.26966412 0.3033697 0.15730396 0.26966222] + [0.26966285 0.30337085 0.15730334 0.26966296]] +迭代次数 = 28 +[[0.26966264 0.30337105 0.15730323 0.26966309] + [0.26966353 0.30337024 0.15730367 0.26966257] + [0.26966217 0.30337147 0.157303 0.26966336] + [0.26966296 0.30337075 0.15730339 0.2696629 ]] +``` + +我们惊奇地发现，迭代了28次后就已经收敛到很小的误差了。可以认为在经过多次的 “租、还、租、还” 循环后，某辆车在四个门店的出现概率是：$[0.27,\ 0.30,\ 0.16,\ 0.27]$，精确到两位小数。 + +从实际情况来看： +- 可能是因为通州区太远了，所以顾客都偏好与就近还车。 +- 海淀区和朝阳区的停车场要大一些，以便可以停更多的车。 +- 丰台区无所谓，因为属于地广人稀的地段，去那里还车的大概都是去机场顾客。