如何在 pandas 数据帧中应用递归数字滤波器？

Question

我有一个像这样的数据框：

days1 = pd.date_range('2020-01-01 01:00:00','2020-01-01 01:19:00',freq='60s')

DF = pd.DataFrame({'Time': days1,
                    'TimeSeries1': [10, 10, 10, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20],
                    'TimeSeries2': [11, 12, 13, 12, 11, 14, 15, 16, 21, 20, 20, 23, 15, 15, 15, 15, 15, 15, 15, 15]})

我想得到以下信息：

对于每个 TimeSeries 列（TimeSeries1 和 TimeSeries2），我想创建一个对应的“_Filtered”列，即： TimeSeries1_Filtered[i] = (1-A)* TimeSeries1_Filtered[i-1] + A*TimeSeries1[i]

“A”是介于 0 和 1 之间的过滤因子。

对于每一列，我需要使用不同的“A”因子。例如：TimeSeries1 的 A1=0.5，TimeSeries1 的 A2=0.8。
我有超过 100 个“TimeSeriesN”列，因此最好以元组或列表的形式传递“A#”参数。

示例 A1=0.5

                      Time  TimeSeries1  TimeSeries1_Filtered
0  2020-01-01 01:00:00           10           10
1  2020-01-01 01:01:00           10           10
2  2020-01-01 01:02:00           10           10
3  2020-01-01 01:03:00           20           15
4  2020-01-01 01:04:00           20           17.5
5  2020-01-01 01:05:00           20           18.75
6  2020-01-01 01:06:00           20           19.375
7  2020-01-01 01:07:00           20           19.6875
8  2020-01-01 01:08:00           20           19.84375
9  2020-01-01 01:09:00           20           19.92188
10 2020-01-01 01:10:00           20           19.96094
11 ...                           ...          ...

谢谢！

编辑：对滤波器符号和方程的更正。感谢@not_speshal 的提醒。

Answer 1

对于第 n 个数据点，递归公式的计算结果为：

filtered[n] = A*(x[n] + (1-A)*x[n-1] + (1-A)**2 * x[n-2] +...) + (1-A)**n * x[0]

您现在可以创建返回上述内容的自定义函数并将其应用于您的数据框：

def ts_filter(srs, A):
    return srs.expanding().apply(lambda x: A*(x*((1-A)**np.arange(len(x))[::-1])).sum() + (1-A)**x.size*x.iat[0])

factors = {"TimeSeries1": 0.5, "TimeSeries2": 0.2}
filtered = df.filter(like="TimeSeries").apply(lambda x: ts_filter(x, A=factors[x.name]))

output = df.join(filtered, rsuffix="_filtered")

输出：

>>> output
                  Time  TimeSeries1  ...  TimeSeries1_filtered  TimeSeries2_filtered
0  2020-01-01 01:00:00           10  ...             10.000000             11.000000
1  2020-01-01 01:01:00           10  ...             10.000000             11.200000
2  2020-01-01 01:02:00           10  ...             10.000000             11.560000
3  2020-01-01 01:03:00           20  ...             15.000000             11.648000
4  2020-01-01 01:04:00           20  ...             17.500000             11.518400
5  2020-01-01 01:05:00           20  ...             18.750000             12.014720
6  2020-01-01 01:06:00           20  ...             19.375000             12.611776
7  2020-01-01 01:07:00           20  ...             19.687500             13.289421
8  2020-01-01 01:08:00           20  ...             19.843750             14.831537
9  2020-01-01 01:09:00           20  ...             19.921875             15.865229
10 2020-01-01 01:10:00           20  ...             19.960938             16.692183
11 2020-01-01 01:11:00           20  ...             19.980469             17.953747
12 2020-01-01 01:12:00           20  ...             19.990234             17.362997
13 2020-01-01 01:13:00           20  ...             19.995117             16.890398
14 2020-01-01 01:14:00           20  ...             19.997559             16.512318
15 2020-01-01 01:15:00           20  ...             19.998779             16.209855
16 2020-01-01 01:16:00           20  ...             19.999390             15.967884
17 2020-01-01 01:17:00           20  ...             19.999695             15.774307
18 2020-01-01 01:18:00           20  ...             19.999847             15.619446
19 2020-01-01 01:19:00           20  ...             19.999924             15.495556

Answer 2

为什么不用像scipy.signal这样的时间序列过滤包呢？

这就是我使用 scipy.signal.lfilter 进行过滤的方式：

（感谢@not_speshal指出OP差分方程中的错误）

from scipy.signal import lfilter

coeffs = {'TimeSeries1': 0.5, 'TimeSeries2': 0.8}
for label, a in coeffs.items():
    DF[f"{label}_Filtered"] = lfilter([a], [1, a-1], DF[label])

但是，您似乎假设初始条件基于每个滤波器在时间 i=0 处于稳态。此解决方案产生您想要的结果：

from scipy.signal import lfilter, lfiltic

coeffs = {'TimeSeries1': 0.5, 'TimeSeries2': 0.8}
for label, a in coeffs.items():
    y_prev = DF[label].iloc[0]  # previous filtered value
    zi = lfiltic([a], [1, a-1], [y_prev])  # initial condition
    DF[f"{label}_Filtered"] = lfilter([a], [1, a-1], DF[label], zi=zi)[0]
print(DF)

输出：

                  Time  TimeSeries1  TimeSeries2  TimeSeries1_Filtered  TimeSeries2_Filtered
0  2020-01-01 01:00:00           10           11             10.000000             11.000000
1  2020-01-01 01:01:00           10           12             10.000000             11.800000
2  2020-01-01 01:02:00           10           13             10.000000             12.760000
3  2020-01-01 01:03:00           20           12             15.000000             12.152000
4  2020-01-01 01:04:00           20           11             17.500000             11.230400
5  2020-01-01 01:05:00           20           14             18.750000             13.446080
...

Answer 3

我刚刚意识到，您使用的自回归过滤器相当于 exponentially-weighted moving average filter 已经存在于 Pandas 中。您只需要关闭前几个采样周期通常使用的调整因子即可。

coeffs = {'TimeSeries1': 0.5, 'TimeSeries2': 0.8}
for label, a in coeffs.items():
    DF[f"{label}_Filtered"] = DF[label].ewm(alpha=a, adjust=False).mean()
print(DF)

输出：

                  Time  TimeSeries1  TimeSeries2  TimeSeries1_Filtered  TimeSeries2_Filtered
0  2020-01-01 01:00:00           10           11             10.000000             11.000000
1  2020-01-01 01:01:00           10           12             10.000000             11.800000
2  2020-01-01 01:02:00           10           13             10.000000             12.760000
3  2020-01-01 01:03:00           20           12             15.000000             12.152000
4  2020-01-01 01:04:00           20           11             17.500000             11.230400
5  2020-01-01 01:05:00           20           14             18.750000             13.446080
...

如何在 pandas 数据帧中应用递归数字滤波器？

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输出：