📘 基于ARIMA+SARIMA的航空公司 RPM 时间序列预测模型/RPM.ipynb

评估航空公司的运输量和运输需求。因此，对航空公司 RPM 的时间序列预测具有重要的商业意义，可以帮助航空公司进行运输规划、航班安排和市场营销决策。 在航空业中，因素如季节性、节假日、燃油价格、宏观经济状况等都会对乘客需求产生影响，这些影响因素使得航空公司 RPM 的时间序列数据呈现出一定的复杂性和波动性。为了更准确地预测航空公司 RPM，传统的统计方法往往无法很好地处理这些复杂的时间序列特征，因此需要更加先进和灵活的预测模型。 ARIMA (AutoRegressive Integrated Moving Average) 模型是一种经典的时间序列分析方法，可以用于捕捉时间序列数据的自相关和趋势性。SARIMA (Seasonal AutoRegressive Integrated Moving Average) 则是在 ARIMA 模型基础上加入季节性因素的模型，能够更好地处理具有季节性变动的时间序列数据。 基于ARIMA+SARIMA的航空公司 RPM 时间序列预测模型，将结合ARIMA和SARIMA模型的优势，综合考虑航空公司 RPM 时间序列数据的趋势性和季节性变动，从而提高预测的准确性和可靠性。这种模型可以帮助航空公司更好地了解和预测未来乘客需求，有助于他们进行合理的运输资源配置和市场策略制定。

import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.dates as mdates
import seaborn as sns
import numpy as np
import statsmodels.api as sm
import warnings
warnings.filterwarnings("ignore")
df= pd.read_csv('air traffic.csv')
df.head()

	Year	Month	Dom_Pax	Int_Pax	Pax	Dom_Flt	Int_Flt	Flt	Dom_RPM	Int_RPM	RPM	Dom_ASM	Int_ASM	ASM	Dom_LF	Int_LF	LF
0	2003	1	43,032,450	4,905,830	47,938,280	785,160	57,667	842,827	36,211,422	12,885,980	49,097,402	56,191,300	17,968,572	74,159,872	64.44	71.71	66.20
1	2003	2	41,166,780	4,245,366	45,412,146	690,351	51,259	741,610	34,148,439	10,715,468	44,863,907	50,088,434	15,587,880	65,676,314	68.18	68.74	68.31
2	2003	3	49,992,700	5,008,613	55,001,313	797,194	58,926	856,120	41,774,564	12,567,068	54,341,633	57,592,901	17,753,174	75,346,075	72.53	70.79	72.12
3	2003	4	47,033,260	4,345,444	51,378,704	766,260	55,005	821,265	39,465,980	10,370,592	49,836,572	54,639,679	15,528,761	70,168,440	72.23	66.78	71.02
4	2003	5	49,152,352	4,610,834	53,763,186	789,397	55,265	844,662	41,001,934	11,575,026	52,576,960	55,349,897	15,629,821	70,979,718	74.08	74.06	74.07

# 数据类型转换
for col in df.columns:
    if df[col].dtype == 'object':
        df[col] = pd.to_numeric(df[col].str.replace(',', ''), errors='coerce')
df.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 249 entries, 0 to 248
Data columns (total 17 columns):
 #   Column   Non-Null Count  Dtype  
---  ------   --------------  -----  
 0   Year     249 non-null    int64  
 1   Month    249 non-null    int64  
 2   Dom_Pax  249 non-null    int64  
 3   Int_Pax  249 non-null    int64  
 4   Pax      249 non-null    int64  
 5   Dom_Flt  249 non-null    int64  
 6   Int_Flt  249 non-null    int64  
 7   Flt      249 non-null    int64  
 8   Dom_RPM  249 non-null    int64  
 9   Int_RPM  249 non-null    int64  
 10  RPM      249 non-null    int64  
 11  Dom_ASM  249 non-null    int64  
 12  Int_ASM  249 non-null    int64  
 13  ASM      249 non-null    int64  
 14  Dom_LF   249 non-null    float64
 15  Int_LF   249 non-null    float64
 16  LF       249 non-null    float64
dtypes: float64(3), int64(14)
memory usage: 33.2 KB

# 由于我们正在做时间序列预测，现在让我们处理月份和年份列
# 这里我们将合并月份和年份，同时将日期设置为每月的第一天
df['Date'] = pd.to_datetime({'year': df['Year'], 'month': df['Month'], 'day': 1})
# 我们把日期设为索引
df.set_index('Date')

	Year	Month	Dom_Pax	Int_Pax	Pax	Dom_Flt	Int_Flt	Flt	Dom_RPM	Int_RPM	RPM	Dom_ASM	Int_ASM	ASM	Dom_LF	Int_LF	LF
Date
2003-01-01	2003	1	43032450	4905830	47938280	785160	57667	842827	36211422	12885980	49097402	56191300	17968572	74159872	64.44	71.71	66.20
2003-02-01	2003	2	41166780	4245366	45412146	690351	51259	741610	34148439	10715468	44863907	50088434	15587880	65676314	68.18	68.74	68.31
2003-03-01	2003	3	49992700	5008613	55001313	797194	58926	856120	41774564	12567068	54341633	57592901	17753174	75346075	72.53	70.79	72.12
2003-04-01	2003	4	47033260	4345444	51378704	766260	55005	821265	39465980	10370592	49836572	54639679	15528761	70168440	72.23	66.78	71.02
2003-05-01	2003	5	49152352	4610834	53763186	789397	55265	844662	41001934	11575026	52576960	55349897	15629821	70979718	74.08	74.06	74.07
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2023-05-01	2023	5	71423653	10358666	81782319	667331	71924	739255	66743565	26805432	93548998	77821407	31950687	109772094	85.77	83.90	85.22
2023-06-01	2023	6	72482621	11544505	84027126	661293	75279	736572	68789127	29883465	98672591	78058358	33410671	111469028	88.13	89.44	88.52
2023-07-01	2023	7	75378157	12432615	87810772	684939	79738	764677	72267904	31376000	103643904	81986010	35326191	117312202	88.15	88.82	88.35
2023-08-01	2023	8	71477988	11572149	83050137	691482	77137	768619	67933484	29938507	97871992	81997399	34908793	116906192	82.85	85.76	83.72
2023-09-01	2023	9	66858490	9392985	76251475	649308	64241	713549	61777546	26076318	87853864	75748336	31231710	106980046	81.56	83.49	82.12

249 rows × 17 columns

# 让我们看看所有变量相对于日期的趋势
all_columns = [col for col in df.columns if col != 'Date']
for column in all_columns:
    plt.figure(figsize=(10, 4))
    sns.lineplot(data=df, x='Date', y=column)
    plt.title(column)
    plt.xticks(rotation=45)
plt.tight_layout()
plt.show()

新冠疫情的影响非常明显。 对于这个时间序列项目，我们将预测Dom_RPM:国内收入乘客里程数，这是航空公司流量的度量，计算付费乘客在国内航班上飞行的里程数。这是航空公司的一项关键绩效指标，显示了国内付费旅客的数量。

# 我们来做一个新的更简单的df
new_df = df[['Date', 'Dom_RPM']].copy()
new_df['Date'] = pd.to_datetime(new_df['Date'])
new_df.set_index('Date', inplace=True)
# 将数据分成训练集和测试集
split_date = pd.to_datetime('2023-01-01')
train = new_df.loc[:split_date]
test = new_df.loc[split_date:]

plt.figure(figsize=(12,6))
plt.plot(new_df.index, new_df['Dom_RPM'], label='Dom_RPM')
plt.axvline(x=split_date, color='red', linestyle='--', label='Train-Test Split')
plt.legend()
plt.show()

# 平稳性检验
from statsmodels.tsa.stattools import adfuller
st_train= adfuller(train['Dom_RPM'], autolag='AIC')
print(st_train)
"""预差分结果表明，在5%显著性水平下，时间序列可能是平稳的，因为ADF统计量(-3.4227)更接近5%临界值(-2.8744)，p值(0.0102)小于0.05，表明在该水平下可以拒绝单位根的原假设。"""

(-3.422669601629956, 0.010209919011204107, 13, 227, {'1%': -3.4594900381360034, '5%': -2.8743581895178485, '10%': -2.573601605503697}, 7463.150034343683)

'预差分结果表明，在5%显著性水平下，时间序列可能是平稳的，因为ADF统计量(-3.4227)更接近5%临界值(-2.8744)，p值(0.0102)小于0.05，表明在该水平下可以拒绝单位根的原假设。'

plt.figure(figsize=(12, 6))
plt.subplot(2, 1, 1)
pd.plotting.autocorrelation_plot(train['Dom_RPM'])
plt.title(f"Autocorrelation Plot for 'Training Set'")
plt.show()

# 差分训练数据
train['Dom_RPM_diff'] = train['Dom_RPM'].diff().shift(-1)
train.dropna(inplace=True)
# 检查差分后的ADF结果
diff_st_train= adfuller(train['Dom_RPM'], autolag='AIC')
print(diff_st_train)

(-3.534624452976126, 0.007141138016109203, 13, 226, {'1%': -3.4596204846395824, '5%': -2.8744153028455948, '10%': -2.5736320761218576}, 7429.093256516401)

差异后，ADF统计量略负(-3.5346)，p值进一步下降至0.0071，增强了非平稳性原假设的证据。差异后的ADF统计量仍然比5%临界值更负，并且非常接近1%临界值，这表明即使在更严格的显著性水平下，也有更强的平稳性证据。

from statsmodels.tsa.arima.model import ARIMA
import statsmodels.api as sm
plt.plot(train.index, train['Dom_RPM_diff'])
plt.xlabel('Date')
plt.ylabel('Domestic RPM')
plt.xticks(rotation=45)
plt.show()

order = (1, 1, 1)
arima_model = sm.tsa.arima.ARIMA(train['Dom_RPM_diff'].dropna(), order=order)
fitted = arima_model.fit()
print(fitted.summary())

                               SARIMAX Results                                
==============================================================================
Dep. Variable:           Dom_RPM_diff   No. Observations:                  240
Model:                 ARIMA(1, 1, 1)   Log Likelihood               -4058.061
Date:                Sun, 17 Nov 2024   AIC                           8122.122
Time:                        14:23:19   BIC                           8132.552
Sample:                    01-01-2003   HQIC                          8126.325
                         - 12-01-2022                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1         -0.0813      0.052     -1.571      0.116      -0.183       0.020
ma.L1         -0.9967      0.076    -13.055      0.000      -1.146      -0.847
sigma2      4.056e+13    1.3e-15   3.11e+28      0.000    4.06e+13    4.06e+13
===================================================================================
Ljung-Box (L1) (Q):                   0.01   Jarque-Bera (JB):               261.51
Prob(Q):                              0.92   Prob(JB):                         0.00
Heteroskedasticity (H):               2.38   Skew:                            -0.82
Prob(H) (two-sided):                  0.00   Kurtosis:                         7.85
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
[2] Covariance matrix is singular or near-singular, with condition number 1.42e+43. Standard errors may be unstable.

-4058.061的对数似然和AIC(8122.122)、BIC(8132.552)和HQIC(8126.325)等选择标准表明，该模型在捕获动态方面相对有效。自回归项(ar.L1)在-0.0813处没有统计学意义(p值:0.116)，这意味着它可能对模型没有意义，而移动平均项(ma.L1)在-0.9967处非常显著(p值:0.000)，表明对模型有很强的影响。误差方差(sigma2)很大，为4.056e+13，表明存在大量无法解释的变异。 Ljung-Box检验(p值:0.92)表明不存在自相关问题，但Jarque-Bera检验表明残差不遵循正态分布(p值:0.00)，异方差检验指向不同的残差方差(p值:0.00)，表明潜在的模型规范问题。此外，残差的偏度(-0.82)和高峰度(7.85)进一步质疑了模型的假设。

# 分解时间序列以观察季节模式
decomposition = sm.tsa.seasonal_decompose(train['Dom_RPM'], model='additive', period=12)
decomposition.plot()
plt.show()
# ACF和PACF图
fig, (ax1, ax2) = plt.subplots(2,1, figsize=(12,8))
sm.graphics.tsa.plot_acf(train['Dom_RPM'], lags=40, ax=ax1)
sm.graphics.tsa.plot_pacf(train['Dom_RPM'], lags=40, ax=ax2)
plt.show()

# 非季节性组件 
p = 1 # number of lag observations in the model (AR term)
d = 1 # the number of times that the raw observations are differenced (I term)
q = 0  # the size of the moving average window (MA term)
# 季节性组件
P = 0  # seasonal autoregressive order
D = 1  # essasonal differencing order
Q = 1  # seasonal moving average order
s = 12 # the number of periods in a season
seasonal_order = (P, D, Q, s)
sarima_model = sm.tsa.statespace.SARIMAX(train['Dom_RPM'],
order=(p, d, q),
seasonal_order=seasonal_order,
enforce_stationarity=False,
enforce_invertibility=False)
fitted_sarima = sarima_model.fit()
print(fitted_sarima.summary())

                                      SARIMAX Results                                       
============================================================================================
Dep. Variable:                              Dom_RPM   No. Observations:                  240
Model:             SARIMAX(1, 1, 0)x(0, 1, [1], 12)   Log Likelihood               -3547.654
Date:                              Sun, 17 Nov 2024   AIC                           7101.307
Time:                                      14:23:55   BIC                           7111.405
Sample:                                  01-01-2003   HQIC                          7105.388
                                       - 12-01-2022                                         
Covariance Type:                                opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1          0.3927      0.032     12.149      0.000       0.329       0.456
ma.S.L12      -0.6053      0.030    -20.049      0.000      -0.664      -0.546
sigma2      2.006e+13   4.64e-16   4.33e+28      0.000    2.01e+13    2.01e+13
===================================================================================
Ljung-Box (L1) (Q):                   0.82   Jarque-Bera (JB):             14702.13
Prob(Q):                              0.37   Prob(JB):                         0.00
Heteroskedasticity (H):               7.38   Skew:                            -3.66
Prob(H) (two-sided):                  0.00   Kurtosis:                        42.94
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
[2] Covariance matrix is singular or near-singular, with condition number 2.29e+43. Standard errors may be unstable.

使用L-BFGS-B算法对SARIMAX模型进行了成功的优化，经过8次迭代后，模型拟合得到了显著改善，最终函数值为14.78。该SARIMAX模型突出了数据中的非季节性和季节性动态，自回归项(0.3927)和季节移动平均项(-0.6053)的系数都很显著，表明存在显著的自回归效应和强烈的季节性影响。模型的拟合良好度指标，包括对数似然值为-3547.654，以及AIC(7101.307)、BIC(7111.405)和HQIC(7105.388)等标准，为评估模型性能提供了依据，值越低通常表示拟合越好。然而，诊断试验显示残差分布的正态性和异方差的证据。

residuals = fitted_sarima.resid
# 绘制残差
plt.figure(figsize=(10,6))
plt.plot(residuals)
plt.title('Residuals from SARIMA Model')
plt.axhline(y=0, color='black', linestyle='--')
plt.show() # 绘制残差的ACF和PACF
fig, ax = plt.subplots(1,2,figsize=(15,4))
sm.graphics.tsa.plot_acf(residuals, lags=40, ax=ax[0])
sm.graphics.tsa.plot_pacf(residuals, lags=40, ax=ax[1])
plt.show()

# 直方图和核密度估计
residuals.plot(kind='hist', density=True, bins=30, alpha=0.5)
residuals.plot(kind='kde')
plt.title('Histogram and Density Plot of Residuals')
plt.show()

# Q-Q图检查正态性
sm.qqplot(residuals, line='s')
plt.title('Q-Q Plot of Residuals')
plt.show()

# 正态性的统计检验
from scipy import stats
jb_test = stats.jarque_bera(residuals)
print(f'Jarque-Bera test statistics: {jb_test[0]}, p-value: {jb_test[1]}')

Jarque-Bera test statistics: 6792.5399581520915, p-value: 0.0

# 零均值的统计检验
mean_test = stats.ttest_1samp(residuals, 0)
print(f'Test Statistic: {mean_test.statistic}, p-value: {mean_test.pvalue}')

Test Statistic: -0.1012239727859493, p-value: 0.9194575337351738

# 自相关的统计检验
ljung_box_test = sm.stats.acorr_ljungbox(residuals, lags=[40], return_df=True)
print(ljung_box_test)

      lb_stat  lb_pvalue
40  35.096743   0.690385

covid_start = '2020-03-01'
covid_end = '2021-09-01'
# 为covid周期创建虚拟变量
train['covid_dummy'] = np.where((train.index >= covid_start) & (train.index <= covid_end), 1, 0)
# 带有外生变量(哑变量)的SARIMA模型
sarimax_model1 = sm.tsa.statespace.SARIMAX(train['Dom_RPM'],
exog=train['covid_dummy'],
order=(p, d, q),
seasonal_order=(P, D, Q, s),
enforce_stationarity=False,
enforce_invertibility=False)
fitted_sarimax1 = sarimax_model1.fit()
print(fitted_sarimax1.summary())

                                      SARIMAX Results                                       
============================================================================================
Dep. Variable:                              Dom_RPM   No. Observations:                  240
Model:             SARIMAX(1, 1, 0)x(0, 1, [1], 12)   Log Likelihood               -3526.067
Date:                              Sun, 17 Nov 2024   AIC                           7060.134
Time:                                      14:25:03   BIC                           7073.598
Sample:                                  01-01-2003   HQIC                          7065.574
                                       - 12-01-2022                                         
Covariance Type:                                opg                                         
===============================================================================
                  coef    std err          z      P>|z|      [0.025      0.975]
-------------------------------------------------------------------------------
covid_dummy -1.936e+07   6.63e+05    -29.201      0.000   -2.07e+07   -1.81e+07
ar.L1           0.2554      0.034      7.490      0.000       0.189       0.322
ma.S.L12       -0.5813      0.030    -19.413      0.000      -0.640      -0.523
sigma2       1.603e+13      0.071   2.25e+14      0.000     1.6e+13     1.6e+13
===================================================================================
Ljung-Box (L1) (Q):                   0.00   Jarque-Bera (JB):              2602.93
Prob(Q):                              0.98   Prob(JB):                         0.00
Heteroskedasticity (H):               7.57   Skew:                            -1.90
Prob(H) (two-sided):                  0.00   Kurtosis:                        19.66
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
[2] Covariance matrix is singular or near-singular, with condition number 5.19e+29. Standard errors may be unstable.

采用L-BFGS-B算法进行优化，经过9次迭代后呈现收敛性，表明疫情对数据的负面影响显著，如covid dummy系数(-1.936e+07)。自回归分量和季节移动平均分量均具有显著的统计学意义，表明存在重要的时间动态。尽管各种统计指标(如AIC、BIC和对数似然)显示了成功的拟合，但诊断显示残差中没有显著的自相关，但突出了非正态分布和异方差的问题。 这两个代码片段一起工作，构建了一种更复杂的方式来理解covid对数据的影响。在第一个代码片段中，我们计算了两件事:covid dummy，它告诉我们在特定时间是否发生了covid，以及covid impact，它衡量covid -19的影响如何随着时间的推移而减少。 然后，在第二个代码片段中，我们使用这些计算来构建SARIMAX模型。该模型不仅考虑了COVID-19是否存在，还考虑了其影响如何随时间变化。

# 让我们再试一次，但要用衰变
covid_start = pd.to_datetime('2020-03-01')
covid_end = pd.to_datetime('2021-09-01')
# 计算自COVID-19开始以来的天数，并将其除以30转换为月份
train['months_since_covid'] = ((train.index - covid_start) / np.timedelta64(1, 'D') / 30).astype(int)
# 线性衰减
duration_in_months = ((covid_end - covid_start) / np.timedelta64(1, 'D') / 30)
train['covid_impact'] = train['months_since_covid'].apply(
lambda x: max(1 - x / duration_in_months, 0) if x >= 0 else 0
)

sarimax_model_with_decay = sm.tsa.statespace.SARIMAX(
train['Dom_RPM'],
exog=train[['covid_dummy', 'covid_impact']],
order=(p, d, q),
seasonal_order=(P, D, Q, s),
enforce_stationarity=False,
enforce_invertibility=False
)
fitted_sarimax_with_decay = sarimax_model_with_decay.fit()
print(fitted_sarimax_with_decay.summary())

                                      SARIMAX Results                                       
============================================================================================
Dep. Variable:                              Dom_RPM   No. Observations:                  240
Model:             SARIMAX(1, 1, 0)x(0, 1, [1], 12)   Log Likelihood               -3527.070
Date:                              Sun, 17 Nov 2024   AIC                           7064.140
Time:                                      14:25:27   BIC                           7080.969
Sample:                                  01-01-2003   HQIC                          7070.940
                                       - 12-01-2022                                         
Covariance Type:                                opg                                         
================================================================================
                   coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------
covid_dummy  -1.934e+07   6.64e+05    -29.114      0.000   -2.06e+07    -1.8e+07
covid_impact -2.007e+06   8.57e+04    -23.411      0.000   -2.17e+06   -1.84e+06
ar.L1            0.2352      0.035      6.700      0.000       0.166       0.304
ma.S.L12        -0.5838      0.030    -19.598      0.000      -0.642      -0.525
sigma2        1.625e+13      0.027   5.92e+14      0.000    1.63e+13    1.63e+13
===================================================================================
Ljung-Box (L1) (Q):                   0.01   Jarque-Bera (JB):              2841.04
Prob(Q):                              0.93   Prob(JB):                         0.00
Heteroskedasticity (H):               7.63   Skew:                            -2.02
Prob(H) (two-sided):                  0.00   Kurtosis:                        20.38
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
[2] Covariance matrix is singular or near-singular, with condition number 4.74e+30. Standard errors may be unstable.

在解释中，带有这些系数的SARIMAX模型表明，covid -19时期对数据有很大的负面影响(显然……)，covid - dummy表明显著降低。covid - 19影响系数表明，随着时间的推移，影响会减弱，这表明大流行的最初冲击正在逐步恢复。 这是我们对未来12个月的预测。它首先生成一组未来日期，从训练数据中的最后一个日期开始，跨越未来12个月。然后，它创建两个数组，future_covid - dummy和future_covid - impact，来表示预测的外生变量。Future_covid_dummy假设未来不会有covid影响，因此在整个预测期内将所有值设置为0。而future_covid - impact则假设COVID-19的影响逐渐减弱，在12个月的预测期内从0.5下降到0。然后将这些数组组合成df future_exog，它将在进行预测时用作外生变量。

# 预测期数(未来12个月)
n_periods = 12
future_dates = pd.date_range(train.index[-1] + pd.offsets.MonthBegin(), periods=n_periods, freq='M')
# 假设新冠病毒开始消失，那么影响将从0.5降至0
future_covid_dummy = [0] * n_periods
future_covid_impact = np.linspace(start=0.5, stop=0, num=n_periods)  #
# 组合成df
future_exog = pd.DataFrame({'covid_dummy': future_covid_dummy, 'covid_impact': future_covid_impact}, index=future_dates)
n_periods = 12
forecast = fitted_sarimax_with_decay.get_forecast(steps=n_periods, exog=future_exog)
forecast_mean = forecast.predicted_mean
forecast_conf_int = forecast.conf_int()

# 利用历史数据绘制预测图
plt.figure(figsize=(10, 6))
plt.plot(train.index, train['Dom_RPM'], label='Historical')
plt.plot(pd.date_range(train.index[-1], periods=n_periods, freq='M'), forecast_mean, label='Forecast')
plt.fill_between(pd.date_range(train.index[-1], periods=n_periods, freq='M'), forecast_conf_int.iloc[:, 0], forecast_conf_int.iloc[:, 1], color='red', alpha=0.3)
plt.legend()
plt.show()

# 让我们把我们的预测和测试进行比较
plt.figure(figsize=(10, 6))
plt.plot(train.index, train['Dom_RPM'], label='Historical', color='blue')
plt.plot(forecast_mean.index, forecast_mean, label='Forecast', color='red')
plt.fill_between(forecast_conf_int.index, forecast_conf_int.iloc[:, 0], forecast_conf_int.iloc[:, 1], color='red', alpha=0.3)
plt.plot(test.index, test['Dom_RPM'], label='Actual Test Set', color='green')
plt.title('Comparison between the Test and Forecast')
plt.xlabel('Date')
plt.ylabel('Dom_RPM (in millions)')
plt.legend()
plt.show()

预测结果与实际测试数据有些接近。虽然与测试集相比，预测似乎被低估了。