Pandas 数据框中的排列总和呈超指数增长

我有一个看起来像

import pandas as pd

data = {
  "Race_ID": [2,2,2,2,2,5,5,5,5,5,5],
  "Student_ID": [1,2,3,4,5,9,10,2,3,6,5],
  "theta": [8,9,2,12,4,5,30,3,2,1,50]
}

df = pd.DataFrame(data)

的pandas数据框，我想通过以下方法创建一个新列 df['feature'] ：对于每个 Race_ID ，假设 Student_ID 等于i，那么我们将特征定义为

def f(thetak, thetaj, thetai, *theta):
  prod = 1;
  for t in theta:
    prod = prod * t;
  return ((thetai + thetaj) / (thetai + thetaj + thetai * thetak)) * prod 

其中 k,j,l 是同一个 Student_ID 中的 Race_ID ，使得 k =/= i, j=/=i,k, l=/=k,j,i 且 theta_i 是| ||与 theta 等于i。例如，对于 Student_ID =2， Race_ID =1，我们的特征等于 Student_ID f(2,3,1,4,5)+f(2,3,1,5,4)+ f(2,4,1,3,5)+f(2,4,1,5,3)+f(2,5,1,3,4)+f(2,5,1,4,3 )+f(3,2,1,4,5)+f(3,2,1,5,4)+f(3,4,1,2,5)+f(3,4,1,5 ,2)+f(3,5,1,2,4)+f(3,5,1,4,2)+f(4,2,1,3,5)+f(4,2,1 ,5,3)+f(4,3,1,2,5)+f(4,3,1,5,2)+f(4,5,1,2,3)+f(4,5 ,1,3,2)+f(5,2,1,3,4)+f(5,2,1,4,3)+f(5,3,1,2,4)+f(5 ,3,1,4,2)+f(5,4,1,2,3)+f(5,4,1,3,2)

等于 299.1960138012742。

但是作为 1很快我们就意识到，总和中的项数随着比赛中学生的数量呈超指数增长：如果一场比赛中有 n 名学生，那么就有 (n-1) 个！

总和中的项。幸运的是，由于 f 的对称性，我们可以通过注意以下事项将项数减少到仅仅 (n-1)(n-2) 项：

令 i， j,k 被给定，1,2,3（例如为了缘故）与 i,j,k 不同（即 1,2,3 在 *arg 中）。那么 f(k,j,i,1,2,3) = f(k,j,i,1,3,2) = f(k,j,i,2,1,3) = f(k, j,i,2,3,1) = f(k,j,i,3,1,2) = f(k,j,i,3,2,1)。因此，如果我们只计算任何一项，然后将其乘以 (n-3)，我们就可以减少项数！

因此，例如，对于

=5， Race_ID =9，则有已经有 5!=120 项求和，但是使用上述对称性，我们只需要对 5x4 = 20 项求和（k 有 5 个选择，i 有 4 个选择，l 有 1 个（非唯一选择）），即| ||f(2,3,9,5,6,10)+f(2,5,9,3,6,10)+f(2,6,9,3,5,10)+f(2 ,10,9,3,5,6)+f(3,2,9,5,6,10)+f(3,5,9,3,6,10)+f(3,6,9, 2,5,10)+f(3,10,9,2,5,6)+f(5,2,9,3,6,10)+f(5,3,9,2,6,10 )+f(5,6,9,2,3,10)+f(5,10,9,2,3,6)+f(6,2,9,3,5,10)+f(6 ,3,9,2,5,10)+f(6,5,9,2,3,10)+f(6,10,9,2,3,5)+f(10,2,9, 3,5,6)+f(10,3,9,2,5,6)+f(10,5,9,2,3,6)+f(10,6,9,2,3,5 ) Student_ID 第 5 场比赛中学生 9 的特征将等于上述总和乘以 3！ = 53588.197759

所以问题是：我如何编写上述数据帧的总和？我已经手动计算了这些特征以进行检查，所需的结果如下所示：

非常感谢。

So by question is: how do i write the sum for the above dataframe? I have computed the features by hand for checking and the desired outcome looks like:

import pandas as pd

data = {
  "Race_ID": [2,2,2,2,2,5,5,5,5,5,5],
  "Student_ID": [1,2,3,4,5,9,10,2,3,6,5],
  "theta": [8,9,2,12,4,5,30,3,2,1,50],
  "feature": [299.1960138012742, 268.93506341257876, 634.7909309816431, 204.18901708653254, 483.7234700875771, 53588.197759, 9395.539167178009, 78005.26224935807, 92907.8753942894, 118315.38359654899, 5600.243276203378]
}

df = pd.DataFrame(data)

Thank you so much.

import pandas as pd
from itertools import permutations

def f(thetak, thetaj, thetai, *theta):
  prod = 1
  for t in theta:
    prod = prod * t
  return ((thetai + thetaj) / (thetai + thetaj + thetai * thetak)) * prod

def calculate_feature(df):
    features = []
    for race_id in df['Race_ID'].unique():
        race_df = df[df['Race_ID'] == race_id]
        n = len(race_df)
        for i in range(n):
            thetai = race_df['theta'].iloc[i]
            student_i = race_df['Student_ID'].iloc[i]
            feature = 0
            for j, k in permutations(range(n), 2):
                if j != i and k != i and k != j:
                    thetaj = race_df['theta'].iloc[j]
                    thetak = race_df['theta'].iloc[k]
                    other_thetas = race_df['theta'].iloc[[l for l in range(n) if l not in [i, j, k]]].values
                    feature += f(thetak, thetaj, thetai, *other_thetas)
            features.append(feature * (n-3))  # Multiply by (n-3)!
    return features

data = {
  "Race_ID": [2,2,2,2,2,5,5,5,5,5,5],
  "Student_ID": [1,2,3,4,5,9,10,2,3,6,5],
  "theta": [8,9,2,12,4,5,30,3,2,1,50]
}

df = pd.DataFrame(data)
df['feature'] = calculate_feature(df)
print(df)

This code defines two functions:

1. f(thetak, thetaj, thetai, *theta) : This function calculates the individual term of your sum, as defined in your question.

2. calculate_feature(df) : This function iterates through the dataframe and calculates the feature for each Student_ID within each Race_ID .

   • For each Race_ID , it iterates through each student and calculates the sum of f over all possible unique combinations of j and k , as you described in your optimization.
   • It then multiplies the sum by (n-3)! to account for the symmetry of f .

The final result is stored in a new column called 'feature' in the original dataframe.