自回归移动平均模型（ARMA）-平稳序列

自回归滑动平均模型（ARMA模型，Auto-Regression and Moving Average Model）是研究时间序列的重要方法，由自回归模型（AR模型）与滑动平均模型（MA模型）为基础“混合”而成，具有适用范围广、预测误差小的特点。ARMA原理分析，见此篇博客。

1. 导入python中的相关模块

import tushare as ts
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.tsa.api as smtsa
from statsmodels.tsa.stattools import adfuller as ADF
from statsmodels.tsa.arima_model import ARMA
from statsmodels.stats.diagnostic import acorr_ljungbox as acorr
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
import warnings
warnings.filterwarnings('ignore')

2. 载入数据集

本模型使用的为某股票公开数据集

ts.set_token('46af5ae83a803c592eeff2620f87402fea47911650b115cf1f671f64')
pro = ts.pro_api()
data = pro.daily(ts_code='000001.SZ', start_date='20100101', end_date='20200101')
data.sort_values(by='trade_date', inplace=True)
data.reset_index(drop='True', inplace=True)

data# 查看data数据

	ts_code	trade_date	open	high	low	close	pre_close	change	pct_chg	vol	amount
0	000001.SZ	20100104	24.52	24.58	23.68	23.71	24.37	-0.66	-2.7100	241922.76	5.802495e+05
1	000001.SZ	20100105	23.75	23.90	22.75	23.30	23.71	-0.41	-1.7300	556499.82	1.293477e+06
2	000001.SZ	20100106	23.25	23.25	22.72	22.90	23.30	-0.40	-1.7200	412143.13	9.444537e+05
3	000001.SZ	20100107	22.90	23.05	22.40	22.65	22.90	-0.25	-1.0900	355336.85	8.041663e+05
4	000001.SZ	20100108	22.50	22.75	22.35	22.60	22.65	-0.05	-0.2200	288543.06	6.506674e+05
...	...	...	...	...	...	...	...	...	...	...	...
2356	000001.SZ	20191225	16.45	16.56	16.24	16.30	16.40	-0.10	-0.6098	414917.98	6.796646e+05
2357	000001.SZ	20191226	16.34	16.48	16.32	16.47	16.30	0.17	1.0429	372033.86	6.103818e+05
2358	000001.SZ	20191227	16.53	16.93	16.43	16.63	16.47	0.16	0.9715	1042574.72	1.741473e+06
2359	000001.SZ	20191230	16.46	16.63	16.10	16.57	16.63	-0.06	-0.3608	976970.31	1.603153e+06
2360	000001.SZ	20191231	16.57	16.63	16.31	16.45	16.57	-0.12	-0.7242	704442.25	1.154704e+06

2361 rows × 11 columns

填补data中的缺失值

data = data.iloc[:, 1:]
data = data.fillna(method='ffill')# 用前一个非缺失值去填充该缺失值
data.head()

	trade_date	open	high	low	close	pre_close	change	pct_chg	vol	amount
0	20100104	24.52	24.58	23.68	23.71	24.37	-0.66	-2.71	241922.76	5.802495e+05
1	20100105	23.75	23.90	22.75	23.30	23.71	-0.41	-1.73	556499.82	1.293477e+06
2	20100106	23.25	23.25	22.72	22.90	23.30	-0.40	-1.72	412143.13	9.444537e+05
3	20100107	22.90	23.05	22.40	22.65	22.90	-0.25	-1.09	355336.85	8.041663e+05
4	20100108	22.50	22.75	22.35	22.60	22.65	-0.05	-0.22	288543.06	6.506674e+05

data = data[['trade_date', 'open', 'close', 'high', 'low']]
data.plot(subplots=True, figsize=(10, 12))
plt.title(' zhangshang stock attributes from 2010-01-01 to 2020-01-01')
plt.show()

3. 平稳性检验

#平稳性检验
adf = ADF(data['close'])
if adf[1] > 0.05:# adf[i]表示对data['close']数据进行1阶差分
    print(u'原始序列经检验不平稳，p值为：%s'%(adf[1]))
else:
    print(u'原始序列经检验平稳，p值为：%s'%(adf[1]))

原始序列经检验平稳，p值为：0.017668059342580877

4. 白噪声检验

p值越大表示数据是随机的可能性越大，随机性很大的数据没有研究意义。

#采用LB统计量的方法进行白噪声检验
p = acorr(data['close'], lags=1)
if p[1] < 0.05:
    print(u'原始序列非白噪声序列，p值为：%s'%p[1])
else:
    print(u'原始序列为白噪声序列，p值为：%s'%p[1])

原始序列非白噪声序列，p值为：[0.]

5. 模型识别

# 定义绘图函数plotds
def plotds (xt, nlag=30, fig_size=(12,8)):
    if not isinstance(xt, pd.Series): #判断xt是否是pd.Series类型数据，不是则转化为该类型数据
        xt = pd.Series(xt)
        
    plt.figure(figsize=fig_size)
    plt.plot(xt)# 原始数据时序图
    plt.title("Time Series")
    plt.show()
    
    plt.figure(figsize=fig_size)
    layout = (2, 2)
    ax_acf = plt.subplot2grid(layout, (1, 0))
    ax_pacf = plt.subplot2grid(layout, (1, 1))
    plot_acf(xt, lags=nlag, ax=ax_acf)# 自相关图
    plot_pacf(xt, lags=nlag, ax=ax_pacf)# 偏自相关图
    plt.show()
    
    return None

plotds(data['close'].dropna(), nlag=50)

#定阶
data_df = data.copy()
aicVal = []
for ari in range(1, 3):
    for maj in range(0,5):
        try:
            arma_obj = smtsa.ARMA(data_df.close.tolist(), order=(ari, maj))\
            .fit(maxlag=30, method='mle', trend='nc')
            aicVal.append([ari, maj, arma_obj.aic])
        except Exception as e:
            print(e)
            

aicVal

[[1, 0, 1958.0143376009719],
 [1, 1, 1958.7504776249589],
 [1, 2, 1960.3388645021796],
 [1, 3, 1962.0299202559945],
 [1, 4, 1963.6365504220475],
 [2, 0, 1958.7847615798437],
 [2, 1, 1960.5080165518111],
 [2, 2, 1957.0524190412552],
 [2, 3, 1959.0187650693679],
 [2, 4, 1961.0165491508224]]

6. 训练模型

从aicVal的结果可以看到模型阶数为（2,2）时，AIC最小为1957.0524190412552，故选择（2,2）阶数的模型。

arma_obj_fin = smtsa.ARMA(data_df['close'].tolist(), order=(2, 2)).fit(maxlag=30, method='mle', trend='nc', disp=False)
arma_obj_fin.summary()

ARMA Model Results

Dep. Variable:	y	No. Observations:	2361
Model:	ARMA(2, 2)	Log Likelihood	-973.526
Method:	mle	S.D. of innovations	0.365
Date:	Thu, 06 May 2021	AIC	1957.052
Time:	19:06:47	BIC	1985.887
Sample:	0	HQIC	1967.551

	coef	std err	z	P>|z|	[0.025	0.975]
ar.L1.y	0.0320	nan	nan	nan	nan	nan
ar.L2.y	0.9676	nan	nan	nan	nan	nan
ma.L1.y	0.9468	nan	nan	nan	nan	nan
ma.L2.y	-0.0329	0.020	-1.628	0.103	-0.073	0.007

Roots

	Real	Imaginary	Modulus	Frequency
AR.1	1.0002	+0.0000j	1.0002	0.0000
AR.2	-1.0333	+0.0000j	1.0333	0.5000
MA.1	-1.0200	+0.0000j	1.0200	0.5000
MA.2	29.7678	+0.0000j	29.7678	0.0000

7.模型拟合效果

#plot the curves
data_df["ARMA"] = arma_obj_fin.predict()
plt.figure(figsize=(10,8))
plt.plot(data_df['close'].iloc[-100:], color='b', label='Actual')
plt.plot(data_df["ARMA"].iloc[-100:], color='r', linestyle='--', label='ARMA(2,2)_pre')
plt.xlabel('index')
plt.ylabel('close price')
plt.legend(loc='best')
plt.show()

8. 模型预测

fig = arma_obj_fin.plot_predict(len(data_df)-50, len(data_df)+10)

predict = arma_obj_fin.predict(start=1, end=len(data_df)+10)
predict[-10:]

array([16.44779647, 16.45696253, 16.44202199, 16.4504122 , 16.43622479,
       16.44388842, 16.43040644, 16.43738965, 16.42456844, 16.43091446])