正态性

在许多情况下，仅最低层次的时间序列（底层时间序列）可用。HierarchicalForecast具有创建所有层次时间序列的工具，并且还允许您计算所有层次的预测区间。在本笔记本中，我们将看到如何做到这一点。

%%capture
!pip install hierarchicalforecast statsforecast

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# 计算基础预测无一致性
from statsforecast.models import AutoARIMA
from statsforecast.core import StatsForecast

#获取分层协调方法及评估
from hierarchicalforecast.methods import BottomUp, MinTrace
from hierarchicalforecast.utils import aggregate, HierarchicalPlot
from hierarchicalforecast.core import HierarchicalReconciliation
from hierarchicalforecast.evaluation import HierarchicalEvaluation

/Users/fedex/miniconda3/envs/hierarchicalforecast/lib/python3.10/site-packages/statsforecast/core.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html
  from tqdm.autonotebook import tqdm

汇总底部时间序列

在这个例子中，我们将使用旅游数据集，该数据集来自预测：原则与实践这本书。该数据集仅包含最低级别的时间序列，因此我们需要为所有层级创建时间序列。

Y_df = pd.read_csv('https://raw.githubusercontent.com/Nixtla/transfer-learning-time-series/main/datasets/tourism.csv')
Y_df = Y_df.rename({'Trips': 'y', 'Quarter': 'ds'}, axis=1)
Y_df.insert(0, 'Country', 'Australia')
Y_df = Y_df[['Country', 'Region', 'State', 'Purpose', 'ds', 'y']]
Y_df['ds'] = Y_df['ds'].str.replace(r'(\d+) (Q\d)', r'\1-\2', regex=True)
Y_df['ds'] = pd.to_datetime(Y_df['ds'])
Y_df.head()

	Country	Region	State	Purpose	ds	y
0	Australia	Adelaide	South Australia	Business	1998-01-01	135.077690
1	Australia	Adelaide	South Australia	Business	1998-04-01	109.987316
2	Australia	Adelaide	South Australia	Business	1998-07-01	166.034687
3	Australia	Adelaide	South Australia	Business	1998-10-01	127.160464
4	Australia	Adelaide	South Australia	Business	1999-01-01	137.448533

数据集可以按以下非严格层次结构进行分组。

spec = [
    ['Country'],
    ['Country', 'State'], 
    ['Country', 'Purpose'], 
    ['Country', 'State', 'Region'], 
    ['Country', 'State', 'Purpose'], 
    ['Country', 'State', 'Region', 'Purpose']
]

使用HierarchicalForecast中的aggregate函数，我们可以生成： 1. Y_df：分层结构系列 \(\mathbf{y}_{[a,b]\tau}\) 2. S_df：包含聚合约束的数据框 \(S_{[a,b]}\) 3. tags：一个包含符合每个聚合级别的’unique_ids’的列表。

Y_df, S_df, tags = aggregate(df=Y_df, spec=spec)
Y_df = Y_df.reset_index()

/Users/fedex/miniconda3/envs/hierarchicalforecast/lib/python3.10/site-packages/sklearn/preprocessing/_encoders.py:828: FutureWarning: `sparse` was renamed to `sparse_output` in version 1.2 and will be removed in 1.4. `sparse_output` is ignored unless you leave `sparse` to its default value.
  warnings.warn(

Y_df.head()

	unique_id	ds	y
0	Australia	1998-01-01	23182.197269
1	Australia	1998-04-01	20323.380067
2	Australia	1998-07-01	19826.640511
3	Australia	1998-10-01	20830.129891
4	Australia	1999-01-01	22087.353380

S_df.iloc[:5, :5]

	Australia/ACT/Canberra/Business	Australia/ACT/Canberra/Holiday	Australia/ACT/Canberra/Other	Australia/ACT/Canberra/Visiting	Australia/New South Wales/Blue Mountains/Business
Australia	1.0	1.0	1.0	1.0	1.0
Australia/ACT	1.0	1.0	1.0	1.0	0.0
Australia/New South Wales	0.0	0.0	0.0	0.0	1.0
Australia/Northern Territory	0.0	0.0	0.0	0.0	0.0
Australia/Queensland	0.0	0.0	0.0	0.0	0.0

tags['Country/Purpose']

array(['Australia/Business', 'Australia/Holiday', 'Australia/Other',
       'Australia/Visiting'], dtype=object)

我们可以使用 HierarchicalPlot 类可视化 S 矩阵和数据，如下所示。

hplot = HierarchicalPlot(S=S_df, tags=tags)

hplot.plot_summing_matrix()

hplot.plot_hierarchically_linked_series(
    bottom_series='Australia/ACT/Canberra/Holiday',
    Y_df=Y_df.set_index('unique_id')
)

划分训练/测试集

我们使用最后两年（8个季度）作为测试集。

Y_test_df = Y_df.groupby('unique_id').tail(8)
Y_train_df = Y_df.drop(Y_test_df.index)

Y_test_df = Y_test_df.set_index('unique_id')
Y_train_df = Y_train_df.set_index('unique_id')

Y_train_df.groupby('unique_id').size()

unique_id
Australia                                                72
Australia/ACT                                            72
Australia/ACT/Business                                   72
Australia/ACT/Canberra                                   72
Australia/ACT/Canberra/Business                          72
                                                         ..
Australia/Western Australia/Experience Perth/Other       72
Australia/Western Australia/Experience Perth/Visiting    72
Australia/Western Australia/Holiday                      72
Australia/Western Australia/Other                        72
Australia/Western Australia/Visiting                     72
Length: 425, dtype: int64

计算基本预测

以下单元计算了 Y_df 中每个时间序列的 基本预测，使用了 AutoARIMA 模型。注意，Y_hat_df 包含了预测结果，但它们并不连贯。为了调整预测区间，我们需要使用 StatsForecast 的 level 参数计算不连贯的区间。

fcst = StatsForecast(df=Y_train_df,
                     models=[AutoARIMA(season_length=4)], 
                     freq='QS', n_jobs=-1)
Y_hat_df = fcst.forecast(h=8, fitted=True, level=[80, 90])
Y_fitted_df = fcst.forecast_fitted_values()

协调预测

下面的单元格使用 HierarchicalReconciliation 类使之前的预测一致。由于层级结构不是严格的，我们不能使用诸如 TopDown 或 MiddleOut 的方法。在此例中，我们使用 BottomUp 和 MinTrace。如果您想计算预测区间，您必须如下使用 level 参数。

reconcilers = [
    BottomUp(),
    MinTrace(method='mint_shrink'),
    MinTrace(method='ols')
]
hrec = HierarchicalReconciliation(reconcilers=reconcilers)
Y_rec_df = hrec.reconcile(Y_hat_df=Y_hat_df, Y_df=Y_fitted_df, 
                          S=S_df, tags=tags, level=[80, 90])

数据框 Y_rec_df 包含了调和后的预测结果。

Y_rec_df.head()

	ds	AutoARIMA	AutoARIMA-lo-90	AutoARIMA-lo-80	AutoARIMA-hi-80	AutoARIMA-hi-90	AutoARIMA/BottomUp	AutoARIMA/BottomUp-lo-90	AutoARIMA/BottomUp-lo-80	AutoARIMA/BottomUp-hi-80	...	AutoARIMA/MinTrace_method-mint_shrink	AutoARIMA/MinTrace_method-mint_shrink-lo-90	AutoARIMA/MinTrace_method-mint_shrink-lo-80	AutoARIMA/MinTrace_method-mint_shrink-hi-80	AutoARIMA/MinTrace_method-mint_shrink-hi-90	AutoARIMA/MinTrace_method-ols	AutoARIMA/MinTrace_method-ols-lo-90	AutoARIMA/MinTrace_method-ols-lo-80	AutoARIMA/MinTrace_method-ols-hi-80	AutoARIMA/MinTrace_method-ols-hi-90
unique_id
Australia	2016-01-01	26212.554688	24694.224609	25029.580078	27395.527344	27730.884766	24368.099609	23674.076441	23827.366706	24908.832513	...	25205.749397	24453.417115	24619.586229	25791.912565	25958.081679	26059.047512	24978.608364	25217.247087	26900.847937	27139.486661
Australia	2016-04-01	25033.667969	23324.066406	23701.669922	26365.666016	26743.269531	22395.921875	21629.482078	21798.767146	22993.076604	...	23720.833190	22915.772233	23093.587632	24348.078748	24525.894148	24769.464257	23554.946551	23823.199470	25715.729045	25983.981963
Australia	2016-07-01	24507.027344	22625.500000	23041.076172	25972.978516	26388.554688	22004.169922	21182.945074	21364.330624	22644.009219	...	23167.123691	22316.298074	22504.221604	23830.025777	24017.949308	24205.855344	22870.661086	23165.568073	25246.142616	25541.049603
Australia	2016-10-01	25598.929688	23559.919922	24010.281250	27187.578125	27637.937500	22325.056641	21456.892977	21648.645996	23001.467285	...	23982.251913	23087.313715	23284.980478	24679.523348	24877.190111	25271.861336	23825.782311	24145.180634	26398.542038	26717.940362
Australia	2017-01-01	26982.578125	24651.535156	25166.396484	28798.757812	29313.619141	23258.001953	22296.178714	22508.618508	24007.385398	...	25002.243615	24016.747195	24234.415731	25770.071498	25987.740034	26611.143736	24959.636647	25324.408272	27897.879201	28262.650825

5 rows × 21 columns

绘制预测图

然后我们可以使用以下函数绘制概率预测。

plot_df = pd.concat([Y_df.set_index(['unique_id', 'ds']), 
                     Y_rec_df.set_index('ds', append=True)], axis=1)
plot_df = plot_df.reset_index('ds')

绘制单个时间序列

hplot.plot_series(
    series='Australia',
    Y_df=plot_df, 
    models=['y', 'AutoARIMA', 'AutoARIMA/MinTrace_method-ols'],
    level=[80]
)

# 由于我们正在绘制一个底部时间序列
# 概率预报和平均预报
# 是相同的
hplot.plot_series(
    series='Australia/Western Australia/Experience Perth/Visiting',
    Y_df=plot_df, 
    models=['y', 'AutoARIMA', 'AutoARIMA/BottomUp'],
    level=[80]
)

绘制层次关联的时间序列

hplot.plot_hierarchically_linked_series(
    bottom_series='Australia/Western Australia/Experience Perth/Visiting',
    Y_df=plot_df, 
    models=['y', 'AutoARIMA', 'AutoARIMA/MinTrace_method-ols', 'AutoARIMA/BottomUp'],
    level=[80]
)

# ACT 仅拥有堪培拉
hplot.plot_hierarchically_linked_series(
    bottom_series='Australia/ACT/Canberra/Other',
    Y_df=plot_df, 
    models=['y', 'AutoARIMA/MinTrace_method-mint_shrink'],
    level=[80, 90]
)

参考文献

Hyndman, R.J., & Athanasopoulos, G. (2021). “预测：原理与实践, 第3版：第11章：预测分层和分组系列。” OTexts: 墨尔本, 澳大利亚. OTexts.com/fpp3 访问于2022年7月。
Shanika L. Wickramasuriya, George Athanasopoulos, 和 Rob J. Hyndman. 通过迹最小化对分层和分组时间序列进行最优预测协调. 美国统计协会杂志, 114(526):804–819, 2019. doi: 10.1080/01621459.2018.1448825. URL https://robjhyndman.com/publications/mint/.

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