如何计算满足一定条件的每日体重

Question

我有以下 pandas 数据框，它代表 10 个人 (

day_0

) 7 天的消费（

day_-1

是今天，

ids

是昨天等）：

import pandas as pd
import numpy as np

df = pd.DataFrame(np.random.randint(8, 15, size=(10, 7)))
df.columns = ['day_0', 'day_-1', 'day_-2', 'day_-3', 'day_-4', 'day_-5', 'day_-6']
df.index.name = 'id'

print(df.reset_index())

   id  day_0  day_-1  day_-2  day_-3  day_-4  day_-5  day_-6
0   0    10      10      14       8      14      14      14
1   1    10      13      11      11       8      10      10
2   2    10      12       9      12       9      10      10
3   3    12      12       9      11       9      12      13
4   4    12      13       8      12       8      11       9
5   5    13       9       8      13       9      12      10
6   6     8       9       8      14       8      13      14
7   7    13      10      14      12       8       9      11
8   8     8       8      10      12      11      14      14
9   9    14      13      13       9      11      14      13

我想找到每日体重（总共7个体重：

w_0, w_-1, w_-2, w_-3, w_-4, w_-5, w_-6

），它需要具有以下属性：

1. w_0

   >

   w_-1

   >

   w_-2

   > ... >

   w_-6

2. w_0

   +

   w_-1

   +

   w_-2

   + ... +

   w_-6

   = 7
3. 10 个中的 7 个

   ids

   的加权平均值低于阈值（例如 11）

我可以通过使用指数衰减函数并随后进行归一化来实现先决条件 1 和 2：

import numpy as np

n = 7

_lambda = 0.5

# Calculate the weights using exponential decay
weights = np.exp(-_lambda * np.arange(n))

# Normalize the weights so that their sum is equal to the length of the time series
weights *= n / np.sum(weights)

但我不知道如何应用先决条件 3。

这可能吗？我怎样才能在Python中做到这一点？

Answer 1

这是一个疯狂的猜测。这不使用指数衰减，因为这对于满足您的要求似乎不是特别有用。定义具有析取约束的 ILP，即 m ID 中至少有 n 的组合的加权平均值低于阈值：

import io

import numpy as np
import pandas as pd
import scipy.sparse
from scipy.optimize import milp, Bounds, LinearConstraint

with io.StringIO(
'''id,  0, -1, -2, -3, -4, -5, -6
    0, 10, 10, 14,  8, 14, 14, 14
    1, 10, 13, 11, 11,  8, 10, 10
    2, 10, 12,  9, 12,  9, 10, 10
    3, 12, 12,  9, 11,  9, 12, 13
    4, 12, 13,  8, 12,  8, 11,  9
    5, 13,  9,  8, 13,  9, 12, 10
    6,  8,  9,  8, 14,  8, 13, 14
    7, 13, 10, 14, 12,  8,  9, 11
    8,  8,  8, 10, 12, 11, 14, 14
    9, 14, 13, 13,  9, 11, 14, 13
''') as f:
    df = pd.read_csv(f, skipinitialspace=True, index_col=0)
df.columns = pd.Index(name='day', data=df.columns.astype(int))


m, n = df.shape  # number of IDs, days

'''
LP variables:
n weights
m weighted mean threshold binary predicates
'''

# The weight sum must be equal to n
sum_constraint = LinearConstraint(
    A=np.concatenate((
        np.ones(n), np.zeros(m),
    )),
    lb=n, ub=n,
)

# The weights must be strictly decreasing by this amount
min_decrease = 1e-2  # chosen fully arbitrarily
antimonotonic_constraint = LinearConstraint(
    A=scipy.sparse.diags_array(
        (
            np.ones(shape=n - 1),
            np.full(shape=n - 1, fill_value=-1),
        ),
        offsets=(0, 1), shape=(n - 1, n + m), format='csc',
    ),
    lb=min_decrease,
)

'''
For each binary threshold predicate:
pred = 1 iff weights.df_values/n <= threshold
pred <= 2 - (weights.values)/n/threshold
weights.values/threshold + pred*n <= 2*n
'''
threshold = 11
mean_constraint = LinearConstraint(
    A=scipy.sparse.hstack(
        (
            df.values/threshold,
            scipy.sparse.diags_array(
                np.full(shape=m, fill_value=n),
            ),
        ),
        format='csc',
    ),
    ub=2*n,
)

# At least n out of m IDs must be at the threshold or lower
disjunction_constraint = LinearConstraint(
    A=np.concatenate((np.zeros(n), np.ones(m))),
    lb=n,
)

result = milp(
    c=np.zeros(n + m),  # no optimisation objective
    integrality=np.concatenate((
        np.zeros(shape=n, dtype=np.uint8),  # weights are continuous
        np.ones(shape=m, dtype=np.uint8),   # predicates are binary
    )),
    bounds=Bounds(
        lb=np.concatenate((
            np.full(shape=n, fill_value=1e-2),  # minimum weight, arbitrary
            np.zeros(shape=m),  # binary predicate
        )),
        ub=np.concatenate((
            np.full(shape=n, fill_value=np.inf),
            np.ones(shape=m),  # binary predicate
        )),
    ),
    constraints=(
        sum_constraint,
        antimonotonic_constraint,
        mean_constraint,
        disjunction_constraint,
    ),
)
if not result.success:
    raise ValueError(result.message)

weights, threshold_preds = np.split(result.x, (n,))
means = df @ (weights/n)

print('weights =')
print(weights)
print('threshold predicates =')
print(threshold_preds)
print('means =')
print(means)

weights =
[2.89083333 1.16833333 1.15833333 0.88125    0.87125    0.02
 0.01      ]
threshold predicates =
[1. 1. 1. 0. 1. 1. 1. 0. 1. 0.]
means =
id
0    10.925119
1    10.543155
2    10.295655
3    11.005714
4    11.000000
5    11.000000
6     8.945119
7    11.902262
8     9.233631
9    12.663333
dtype: float64