最小化受约束的最小二乘法

如何解决最小化受约束的最小二乘法

我试图通过最小化受约束的平方误差总和来对 NFL 球队进行评分。我的数据看起来像：

dat = {"Home_Team": ["KC Chiefs","LA Chargers","Baltimore Ravens"],"Away_Team": ["Houston Texans","Miami Dolphins","KC Chiefs"],"Home_score": [34,20,20],"Away_score": [20,23,34],"Margin": [14,-3,-34]
      }
df = pd.DataFrame(dat)
df

    Home_Team        Away_Team      Home_score  Away_score  Margin
0   KC Chiefs        Houston Texans 34          20          14
1   LA Chargers      Miami Dolphins 20          23          -3
2   Baltimore Ravens KC Chiefs      20          34          -34

保证金是保证金 = Home_score - Away_score。我的目标是为每个团队提出一个数字评级，使得所有团队的评级平均值为零。因此，如果酋长队的评分为 3.0，那么他们比一般球队高 3 分。

根据这些评分，我们以这种方式生成预测：主队的预测胜差为 Home_Edge + Home_rating - Away_rating，其中 Home_Edge 是主场优势（所有主队的常数），Home_rating 是主队的评分，以及 Away_rating 客队的评分。

预测中的误差是预测 - 边际，我想最小化这些误差的平方和。我正在尝试通过以下方式使用 scipy.optimize 执行此操作：

# Our objective function,where x is our array of parameters,# x[0] is the home edge,x[1] the home rating,and x[2] the away rating
# Y is the true,observed margin
def obj_fun(x,Y):
    y = x[0] + x[1] - x[2]
    return np.sum((y - Y)**2)

# Define the constraint function. We have that the ratings average to 0
def con(x):
    return np.mean(x[1])

# Constraint dictionary
cons = {'type': 'eq','fun': con}

# Minimize sum of squared errors
from scipy import optimize

# Initial guesses (numbers I randomly thought of in my head)
home_edge = 0.892
home_ratings = np.array([1.46,9.67,-0.82])
away_ratings = np.array([-3.10,-6.57,1.46])
x_init = [np.repeat(home_edge,3),home_ratings,away_ratings]

# Minimize
results = optimize.minimize(fun = obj_fun,args = (df["Margin"]),x0 = x_init,constraints = cons)

print(results.x)
[-2.9413615   0.          4.72534244  1.46        9.67       -0.82
 -3.1        -6.57        1.46      ]

我不知道这里出了什么问题，但我希望我的输出有 6 个解决方案，而不是 9 个。一个用于主队，其余五个用于每个球队。怎么了？谢谢！