1.原始数据

2020年的LBMA黄金价格数据如下所示：

Date	USD (PM)
2020/01/02	1527.10
2020/01/03	1548.75
2020/01/06	1573.10
2020/01/07	1567.85
2020/01/08	1571.95
2020/01/09	1550.75
2020/01/10	1553.60
2020/01/13	1549.90
2020/01/14	1545.10
2020/01/15	1549.00
2020/01/16	1554.55
2020/01/17	1557.60
2020/01/20	1560.15
2020/01/21	1551.30
2020/01/22	1556.90
2020/01/23	1562.90
2020/01/24	1564.30
2020/01/27	1580.10
2020/01/28	1574.00
2020/01/29	1573.45
2020/01/30	1578.25
2020/01/31	1584.20
2020/02/03	1574.75
2020/02/04	1558.35
2020/02/05	1553.30
2020/02/06	1563.30
2020/02/07	1572.65
2020/02/10	1573.20
2020/02/11	1570.50
2020/02/12	1563.70
2020/02/13	1575.05
2020/02/14	1581.40
2020/02/17	1580.80
2020/02/18	1589.85
2020/02/19	1604.20
2020/02/20	1619.00
2020/02/21	1643.30
2020/02/24	1671.65
2020/02/25	1650.30
2020/02/26	1634.90
2020/02/27	1652.00
2020/02/28	1609.85
2020/03/02	1599.65
2020/03/03	1615.50
2020/03/04	1641.85
2020/03/05	1659.60
2020/03/06	1683.65
2020/03/09	1672.50
2020/03/10	1655.70
2020/03/11	1653.75
2020/03/12	1570.70
2020/03/13	1562.80
2020/03/16	1487.70
2020/03/17	1536.20
2020/03/18	1498.20
2020/03/19	1474.25
2020/03/20	1494.40
2020/03/23	1525.40
2020/03/24	1605.75
2020/03/25	1605.45
2020/03/26	1634.80
2020/03/27	1617.30
2020/03/30	1618.30
2020/03/31	1608.95
2020/04/01	1576.55
2020/04/02	1616.80
2020/04/03	1613.10
2020/04/06	1648.30
2020/04/07	1649.25
2020/04/08	1647.80
2020/04/09	1680.65
2020/04/14	1741.90
2020/04/15	1718.65
2020/04/16	1729.50
2020/04/17	1692.55
2020/04/20	1686.20
2020/04/21	1682.05
2020/04/22	1710.55
2020/04/23	1736.25
2020/04/24	1715.90
2020/04/27	1714.95
2020/04/28	1691.55
2020/04/29	1703.35
2020/04/30	1702.75
2020/05/01	1686.25
2020/05/04	1709.10
2020/05/05	1699.55
2020/05/06	1691.50
2020/05/07	1704.05
2020/05/11	1702.75
2020/05/12	1702.40
2020/05/13	1708.40
2020/05/14	1731.60
2020/05/15	1735.35
2020/05/18	1734.70
2020/05/19	1737.95
2020/05/20	1748.30
2020/05/21	1724.90
2020/05/22	1733.55
2020/05/26	1720.25
2020/05/27	1694.60
2020/05/28	1717.35
2020/05/29	1728.70
2020/06/01	1730.60
2020/06/02	1742.15
2020/06/03	1705.35
2020/06/04	1700.05
2020/06/05	1683.45
2020/06/08	1690.35
2020/06/09	1713.50
2020/06/10	1722.05
2020/06/11	1738.25
2020/06/12	1733.50
2020/06/15	1710.45
2020/06/16	1719.85
2020/06/17	1724.35
2020/06/18	1719.50
2020/06/19	1734.75
2020/06/22	1761.85
2020/06/23	1768.90
2020/06/24	1766.05
2020/06/25	1756.55
2020/06/26	1747.60
2020/06/29	1771.60
2020/06/30	1768.10
2020/07/01	1771.05
2020/07/02	1777.45
2020/07/03	1772.90
2020/07/06	1787.90
2020/07/07	1789.55
2020/07/08	1811.10
2020/07/09	1812.10
2020/07/10	1803.10
2020/07/13	1807.50
2020/07/14	1801.90
2020/07/15	1804.60
2020/07/16	1807.70
2020/07/17	1807.35
2020/07/20	1815.65
2020/07/21	1842.55
2020/07/22	1852.40
2020/07/23	1878.30
2020/07/24	1902.10
2020/07/27	1936.65
2020/07/28	1940.90
2020/07/29	1950.90
2020/07/30	1957.65
2020/07/31	1964.90
2020/08/03	1958.55
2020/08/04	1977.90
2020/08/05	2048.15
2020/08/06	2067.15
2020/08/07	2031.15
2020/08/10	2044.50
2020/08/11	1939.65
2020/08/12	1931.90
2020/08/13	1944.25
2020/08/14	1944.75
2020/08/17	1972.85
2020/08/18	2008.75
2020/08/19	1981.00
2020/08/20	1927.15
2020/08/21	1924.35
2020/08/24	1943.95
2020/08/25	1911.15
2020/08/26	1932.95
2020/08/27	1923.85
2020/08/28	1957.35
2020/09/01	1972.35
2020/09/02	1947.05
2020/09/03	1940.45
2020/09/04	1926.30
2020/09/07	1928.45
2020/09/08	1910.95
2020/09/09	1947.20
2020/09/10	1966.25
2020/09/11	1947.40
2020/09/14	1958.70
2020/09/15	1949.35
2020/09/16	1961.80
2020/09/17	1936.25
2020/09/18	1950.85
2020/09/21	1909.35
2020/09/22	1906.00
2020/09/23	1873.40
2020/09/24	1861.75
2020/09/25	1859.70
2020/09/28	1864.30
2020/09/29	1883.95
2020/09/30	1886.90
2020/10/01	1902.00
2020/10/02	1903.05
2020/10/05	1909.60
2020/10/06	1913.40
2020/10/07	1884.50
2020/10/08	1887.45
2020/10/09	1923.25
2020/10/12	1925.50
2020/10/13	1891.30
2020/10/14	1910.05
2020/10/15	1891.90
2020/10/16	1905.05
2020/10/19	1905.60
2020/10/20	1898.40
2020/10/21	1924.15
2020/10/22	1900.95
2020/10/23	1903.65
2020/10/26	1898.45
2020/10/27	1905.70
2020/10/28	1869.95
2020/10/29	1870.30
2020/10/30	1881.85
2020/11/02	1889.90
2020/11/03	1908.30
2020/11/04	1900.15
2020/11/05	1938.45
2020/11/06	1940.80
2020/11/09	1867.30
2020/11/10	1878.70
2020/11/11	1860.95
2020/11/12	1874.85
2020/11/13	1890.90
2020/11/16	1885.60
2020/11/17	1889.05
2020/11/18	1876.10
2020/11/19	1857.35
2020/11/20	1875.70
2020/11/23	1840.20
2020/11/24	1799.60
2020/11/25	1810.20
2020/11/26	1807.40
2020/11/27	1779.30
2020/11/30	1762.55
2020/12/01	1810.75
2020/12/02	1822.60
2020/12/03	1832.35
2020/12/04	1843.00
2020/12/07	1859.95
2020/12/08	1868.15
2020/12/09	1841.75
2020/12/10	1844.35
2020/12/11	1842.00
2020/12/14	1831.15
2020/12/15	1850.65
2020/12/16	1851.95
2020/12/17	1890.75
2020/12/18	1879.75
2020/12/21	1880.00
2020/12/22	1877.10
2020/12/23	1875.00
2020/12/29	1874.30
2020/12/30	1887.60

2.python程序

import numpy as np
import pandas as pd
from datetime import datetime, timedelta
import statsmodels.api as sm
from arch.unitroot import ADF
from statsmodels.stats.diagnostic import acorr_ljungbox
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
from sklearn.metrics import explained_variance_score, mean_absolute_error, mean_squared_error, r2_score
import matplotlib.pyplot as plt
import warnings

warnings.filterwarnings('ignore')
plt.rcParams['font.sans-serif'] = ['SimSun']
plt.rcParams['axes.unicode_minus'] = False
plt.rc('axes', unicode_minus=False)

dateparse = lambda dates: pd.datetime.strptime(dates, '%Y-%m-%d')
data = pd.read_excel(io='LBMA-GOLD.xlsx', sheet_name='Sheet1', index_col=0, date_parser=dateparse)
print(data)

train = data.iloc[:200, :]
test = data.iloc[200:, :]

print('\n平稳性检验：\n', ADF(train['USD (PM)'], trend='n'))
train['USD-DIFF (PM)'] = train['USD (PM)'].diff(1)
print('\n一阶差分后平稳性检验：\n', ADF(train['USD-DIFF (PM)'].dropna(), trend='n'))

fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(6, 4))
ax[0].plot(train['USD (PM)'])
ax[0].set_title('一阶差分前')
ax[1].plot(train['USD-DIFF (PM)'])
ax[1].axhline(0, color='red')
ax[1].set_title('一阶差分后')
fig.autofmt_xdate(rotation=45)

print('\n白噪声检验：\n', acorr_ljungbox(train['USD (PM)'], lags=20, boxpierce=True))

fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(6, 4))
plot_acf(train['USD (PM)'], ax=ax[0], title='USD (PM)的自相关图')
plot_pacf(train['USD (PM)'], ax=ax[1], method='ywm', title='USD (PM)的偏自相关图')

trend = sm.tsa.arma_order_select_ic(train['USD (PM)'], ic=['aic'], trend='n', max_ar=10, max_ma=10)
d = 1
p = trend.aic_min_order[0]
q = trend.aic_min_order[1]
print(f'\np值：{p}，d值：{d}，q值：{q}')

start_date = datetime(year=2021, month=1, day=1)
end_date = datetime(year=2021, month=1, day=31)

date_list = []
current_date = start_date

while current_date <= end_date:
    date_list.append(current_date.strftime('%Y-%m-%d'))
    current_date += timedelta(days=1)

history = [x for x in train['USD (PM)']]
predictions = list()
model = sm.tsa.ARIMA(history, order=(p, d, q), trend='t')
model_fit = model.fit()
print(model_fit.summary())

yhat = model_fit.forecast()[0]
predictions.append(yhat)
history.append(test['USD (PM)'][0])
for i in range(1, len(test['USD (PM)'])):
    model = sm.tsa.ARIMA(history, order=(p, d, q), trend='t')
    model_fit = model.fit()
    yhat = model_fit.forecast()[0]
    predictions.append(yhat)
    obs = test['USD (PM)'][i]
    history.append(obs)

forecast = model_fit.forecast(steps=len(date_list))
forecast = pd.DataFrame(forecast, index=date_list)
forecast.columns = ['forecast']
forecast.index = pd.to_datetime(forecast.index.values, format='%Y-%m-%d')
print('\n未来一月预测数据：\n', forecast)

ev = explained_variance_score(test['USD (PM)'], predictions)
print(f'\nEV: {ev}')
r2 = r2_score(test['USD (PM)'], predictions)
print(f'R2: {r2}')
mse = mean_squared_error(test['USD (PM)'], predictions)
print(f'MSE: {mse}')
mae = mean_absolute_error(test['USD (PM)'], predictions)
print(f'MAE: {mae}')
rmse = np.sqrt(mean_squared_error(test['USD (PM)'], predictions))
print(f'RMSE: {rmse}')

plt.figure(figsize=(6, 4))
plt.plot(train.index, train['USD (PM)'], color='green', label='训练集的真实数据')
plt.plot(test.index, test['USD (PM)'], color='red', label='测试集的真实数据')
plt.plot(test.index, predictions, color='blue', label='测试集的预测数据')
plt.plot(forecast.index, forecast['forecast'], color='purple', label='未来一月预测数据')
plt.title('ARIMA差分自回归移动平均模型')
plt.xlabel('日期')
plt.ylabel('USD (PM)')
plt.legend()
plt.grid(True)
plt.show()

3.效果展示

            USD (PM)
Date                
2020-01-02   1527.10
2020-01-03   1548.75
2020-01-06   1573.10
2020-01-07   1567.85
2020-01-08   1571.95
...              ...
2020-12-21   1880.00
2020-12-22   1877.10
2020-12-23   1875.00
2020-12-29   1874.30
2020-12-30   1887.60

[252 rows x 1 columns]

平稳性检验：
    Augmented Dickey-Fuller Results   
=====================================
Test Statistic                  1.148
P-value                         0.935
Lags                                4
-------------------------------------

Trend: No Trend
Critical Values: -2.58 (1%), -1.94 (5%), -1.62 (10%)
Null Hypothesis: The process contains a unit root.
Alternative Hypothesis: The process is weakly stationary.

一阶差分后平稳性检验：
    Augmented Dickey-Fuller Results   
=====================================
Test Statistic                 -8.286
P-value                         0.000
Lags                                3
-------------------------------------

Trend: No Trend
Critical Values: -2.58 (1%), -1.94 (5%), -1.62 (10%)
Null Hypothesis: The process contains a unit root.
Alternative Hypothesis: The process is weakly stationary.

白噪声检验：
         lb_stat      lb_pvalue      bp_stat      bp_pvalue
1    194.759300   2.907665e-44   191.866835   1.244089e-43
2    383.039338   6.669059e-84   376.418556   1.827045e-82
3    563.783268  7.149624e-122   552.688626  1.816158e-119
4    737.656023  2.443597e-158   721.396844  8.109841e-155
5    906.832548  8.838113e-194   884.710816  5.419758e-189
6   1071.632944  2.860791e-228  1042.984463  4.507469e-222
7   1232.762834  5.804458e-262  1196.935299  3.248199e-254
8   1389.389918  1.114091e-294  1345.808567  2.944619e-285
9   1541.943662   0.000000e+00  1490.054929   0.000000e+00
10  1689.339573   0.000000e+00  1628.694647   0.000000e+00
11  1831.981022   0.000000e+00  1762.156201   0.000000e+00
12  1970.413004   0.000000e+00  1890.993887   0.000000e+00
13  2104.407330   0.000000e+00  2015.038139   0.000000e+00
14  2233.977639   0.000000e+00  2134.345454   0.000000e+00
15  2359.136348   0.000000e+00  2248.971004   0.000000e+00
16  2480.469071   0.000000e+00  2359.491900   0.000000e+00
17  2597.650304   0.000000e+00  2465.651136   0.000000e+00
18  2711.573499   0.000000e+00  2568.294807   0.000000e+00
19  2821.623435   0.000000e+00  2666.903907   0.000000e+00
20  2928.638565   0.000000e+00  2762.263925   0.000000e+00

p值：8，d值：1，q值：2
                               SARIMAX Results                                
==============================================================================
Dep. Variable:                      y   No. Observations:                  200
Model:                 ARIMA(8, 1, 2)   Log Likelihood                -893.488
Date:                Wed, 16 Jul 2025   AIC                           1810.977
Time:                        23:21:30   BIC                           1850.496
Sample:                             0   HQIC                          1826.971
                                - 200                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
x1             1.8702      1.320      1.417      0.157      -0.717       4.457
ar.L1         -0.8557      0.274     -3.122      0.002      -1.393      -0.319
ar.L2         -0.6625      0.253     -2.621      0.009      -1.158      -0.167
ar.L3          0.0196      0.105      0.186      0.853      -0.187       0.226
ar.L4         -0.1510      0.083     -1.817      0.069      -0.314       0.012
ar.L5         -0.2805      0.113     -2.484      0.013      -0.502      -0.059
ar.L6         -0.2744      0.132     -2.078      0.038      -0.533      -0.016
ar.L7         -0.0822      0.094     -0.874      0.382      -0.267       0.102
ar.L8         -0.1233      0.076     -1.615      0.106      -0.273       0.026
ma.L1          0.8469      0.271      3.120      0.002       0.315       1.379
ma.L2          0.7820      0.262      2.989      0.003       0.269       1.295
sigma2       460.8317     33.910     13.590      0.000     394.370     527.294
===================================================================================
Ljung-Box (L1) (Q):                   0.01   Jarque-Bera (JB):                71.88
Prob(Q):                              0.94   Prob(JB):                         0.00
Heteroskedasticity (H):               1.25   Skew:                            -0.52
Prob(H) (two-sided):                  0.36   Kurtosis:                         5.75
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

未来一月预测数据：
                forecast
2021-01-01  1871.011736
2021-01-02  1878.637793
2021-01-03  1878.726825
2021-01-04  1883.143770
2021-01-05  1884.582227
2021-01-06  1886.011839
2021-01-07  1887.258044
2021-01-08  1887.077311
2021-01-09  1888.905537
2021-01-10  1889.423383
2021-01-11  1891.339065
2021-01-12  1892.732772
2021-01-13  1894.369534
2021-01-14  1895.980334
2021-01-15  1897.192566
2021-01-16  1898.699916
2021-01-17  1899.826056
2021-01-18  1901.236044
2021-01-19  1902.518871
2021-01-20  1903.909368
2021-01-21  1905.330258
2021-01-22  1906.697727
2021-01-23  1908.135360
2021-01-24  1909.480316
2021-01-25  1910.875230
2021-01-26  1912.223809
2021-01-27  1913.591264
2021-01-28  1914.962440
2021-01-29  1916.327573
2021-01-30  1917.713328
2021-01-31  1919.083121

EV: 0.6777102452348833
R2: 0.6737551295319617
MSE: 448.0266135387967
MAE: 14.968920039737192
RMSE: 21.166639164940587