Understand Python time series forecasting in one article (steps, templates, python code)

Forecasting includes, numerical fitting, linear regression, multiple regression, time series, neural network, etc.

For univariate time series forecasting: models include AR, MA, ARMA, and ARIMA. Generally speaking, ARIMA can represent all.

Data and code links: data and Jupyter files

Take the forecast of changes in U.S. GDP in the next 10 years as a column:

Table of contents

The first step is to import data

The second step is to perform stationary sequence analysis

The third step is to perform the difference operation of the unstable sequence

The fourth step is to perform model ordering, model selection and fitting.

Step 5: Carry out model result analysis and model testing 

Step 6: Make model predictions

PS: Comparison of automated AUTO-ARIMA

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The ARIMA flow chart is as follows:

The first step is to import data:

    Observe the data of the variables to be predicted, remove miscellaneous data, and obtain the attributes of the data itself such as size, type, etc.

import pandas as pd,numpy as np
from matplotlib import pyplot as plt

#加载数据，sheet_name指定excel表的数据页面，header指定指标column属性，loc去除杂数据，可选：parse_dates=['']，index_col='',use_cols=['']
df=pd.read_excel('./data/time1.xls',sheet_name='数据',header=1).loc[1:,:]

#DateFrame索引重置
df=df.set_index('DATE')#df.set_index('DATE',inplace=True)

#查看前5行
print(df.head(5))

#查看列索引
print(df.columns)#print(df.keys())

#查看表的维度
print(df.shape)

#查看行索引
print(df.index)

#np.array()  array1.reshape(,)  df.values.astype(int).tolist() np.vstack((a1,a2))  np.hstack((a1,a2))  round()  iloc

#时间索引拆分
# dates=pd.date_range(start='1991-01-01',end='2007-08-01',freq='MS')#日期取值和格式转换，MS代表每月第一天
# years=[d.strftime('%Y-%m') for d in dates][0:200:25]
# years.append('2007-09')

The result is as follows:

The second step is to perform stationary sequence analysis:

    The time series we study and analyze, that is, panel data, is of research significance only if it is widely stationary. If it is a non-stationary sequence, its difference needs to be converted into a stationary sequence before analysis can be performed. For strictly stationary sequences, the properties do not change, that is, the sequence is white Noise sequence, such sequence has no research significance.
    
    Therefore, after getting the GDP time series data here, we first conduct a white noise test on the original sequence, using the LB statistic. If the p value is less than the significant level a=0.05, the original sequence is considered to be a non-white noise sequence, which is meaningful for research. .
    
    Then draw a time series diagram and subjectively judge the stationarity of the GDP time series. If there is an obvious trend, it is a non-stationary series. If there is no obvious trend, or it is difficult for you to judge the stationarity, then you need to use the ADF unit. Root statistics are used to assist in judging whether it is stationary. For the ADF statistic, you can compare the value of the statistic to judge, or you can compare the p value to judge. For example, when the significance level is a=0.05, if the ADF statistic of your variable is smaller than the ADF statistic for a=0.05, then It indicates that the time series corresponding to the variable is a stationary sequence; a more direct method is to only judge that if the p-value of the variable is less than 0.05, it is also indicated as a stationary sequence.
   

plt.rcParams['axes.unicode_minus'] = False  # 解决保存图像是负号'-'显示为方块的问题
plt.rc('font',family='SimHei')
plt.style.use('ggplot')

df.plot(secondary_y=['CS','INV','P_GDP','GOV_NET'])#单个指标时序图 df['CS'].plot()
plt.xlabel('Date')
plt.ylabel('Value')
plt.title('Time Series Plot')
#plt.grid()
plt.show()

from statsmodels.tsa.stattools import adfuller# ADF检验

for i in df.columns:
    data=df[i]
    print(f'{i}的单位根检验')
    result = adfuller(data)#默认情况下，regression参数为'c'，表示使用包含截距项的回归模型。
    print('ADF Statistic: %f' % result[0])#ADF统计量
    print('p-value: %f' % result[1])#p值
    print('Critical Values:')#在置信水平下的临界值
    for key, value in result[4].items():
        print('\t%s: %.3f' % (key, value))
    print()

Code results:

The third step is to perform the difference operation of the non-stationary sequence:

    Generally, the difference of the time series will not exceed the third order. Perform the difference operation on the original data and repeat the second step of the time series diagram and ADF unit root test operation. If it is found that the differenced sequence is a stationary sequence, record the order of the difference. Here, the difference order of GDP is 1, passing the stationary sequence test. Perform a white noise test on the differentiated sequence again. If it passes as a non-white noise sequence, proceed to the next step.
    

#差分时序图
diff_data = df.diff(periods=1).dropna()# 创建一阶差分的时间序列，加上dropna()后续不需要执行[1:]
#print(diff_data)

diff_data.plot()
plt.xlabel('Date')
plt.ylabel('Value')
plt.title('Time Series Plot')
#plt.grid()
plt.show()

for i in diff_data.columns:
    data=diff_data[i]#Series索引选取，一阶差分第一个数据为NA
    
    print(f'{i}的单位根检验')
    result = adfuller(data)#结果对应位置的数据需要自己判断是什么含义
    print('ADF Statistic: %f' % result[0])#ADF统计量
    print('p-value: %f' % result[1])#p值
    print('Critical Values:')#在置信水平下的临界值
    for key, value in result[4].items():
        print('\t%s: %.3f' % (key, value))
    print()

from statsmodels.stats.diagnostic import acorr_ljungbox

#这里为一阶差分后的平稳序列进行白噪声检验，lags为1，否则lags为0，这里拿上述的GDP指标进行
lags = [1,4,8,16,32]

print('差分序列的白噪声检验结果为：'+'\n',acorr_ljungbox(diff_data['GDP'], lags))  # 返回统计量和p值,这里的lags对应差分阶数

print('原始数据序列的白噪声检验结果为：'+'\n',acorr_ljungbox(df['GDP'], lags))

The code results are as follows:

The fourth step is to perform model ordering, model selection and fitting:

    To determine the order of the model, we first use the autocorrelation diagram and the partial autocorrelation diagram to make a preliminary judgment: Autocorrelation
           coefficient Partial autocorrelation coefficient Difference
    AR Tailing p-order truncation 0
    MA q-order truncation Tailing 0
    ARMA tailing Tailing 0
    ARIMA tailing Tailing d
    In short, the above models can all be implemented using ARIMA (p, d, q). AR, MA, and ARMA are all special cases of ARIMA.
    
    Empirical judgments about tailing and truncation:
    Tailing: negative exponential monotonically converges to 0, or exhibits a cosine decay.
    Truncation: rapidly decays to 0, and fluctuates near 0.
    Truncation approaches 0 more quickly than tailing approaches 0. Truncation will not increase significantly in the later stages
    
       

        After using the chart to make a preliminary judgment, if you still cannot determine the order of the model accurately, you can use the AIC and BIC criteria to assist in judgment, or optimization. For example, I judge that the order of the model may be 2, 1, 4, or 3, 1, 3. At this time, I am not sure. You can do ARIMA (2, 1, 4) first. If you find that the subsequent model is significant If both the test and the parameter test pass, then you can directly use ARIMA(2,1,4) to make predictions and draw conclusions, but you may also try several nearby orders, such as ARIMA(3,1,3), ARIMA (2,1,3) and so on also passed, but the AIC and BIC values ​​of ARIMA (3,1,3) are the smallest, that is, the model has higher fitting accuracy and is a relatively optimal model. , this step is the optimization of the model.

    
    The order setting and optimization here are implemented in python: from q to (data length/10), from p to 0 to (data length/10), and generally the order will not exceed (data length/10) ; Then perform q*p model fitting, and use the model fitting results to compare the values ​​of AIC and BIC. The smaller the BIC value, the smaller the model is the relatively optimal model, and then use the order of the model as the final model. The order of the combination.
    

#生成自相关图，偏自相关图
from statsmodels.graphics.tsaplots import plot_acf,plot_pacf

fig, ax = plt.subplots(figsize=(5, 4))
plot_acf(diff_data['GDP'],ax=ax)#可以换成df的数据，这里用一阶差分数据得到平稳序列
    
ax.set_title("Autocorrelation Plot")
lags=list(range(24))
ax.set_xticks(lags)#这里把x坐标刻度变更精细，加网格图更方便,xticklabels替换标签
ax.set_xlabel("Lag")
ax.set_ylabel("Autocorrelation")
ax.grid(alpha=0.5)
plt.legend(['GDP'])#这里区别直接放入df.columns[i]，如果是多字符如‘CS’，这样会被认为是一个序列，拆成C和S的图例
plt.show()
    
fig, ax = plt.subplots(figsize=(5, 4))
lags=list(range(24))
ax.set_xticks(lags)#这里把x坐标刻度变更精细，加网格图更方便,xticklabels替换标签
plot_pacf(diff_data['GDP'], ax=ax,method='ywm')#ywm替换默认的yw，去除警告
ax.set_xlabel('Lags')
ax.set_ylabel('Partial Autocorrelation')
ax.grid(alpha=0.5)
plt.legend(['GDP'])
plt.show()

import warnings
warnings.filterwarnings("ignore")
import statsmodels.api as sm


#diff_data['GDP'].values.astype(float),这里发现Serise的dtype为object，模型用的应该为float或者int类型，需要注意原数据的数据类型是否一致

Min=float('inf')
for i in range(0,6):#AIC，BIC最小找到p，q阶数来定阶，从0开始定阶是否可行？？
    for j in range(0,6):
        result=sm.tsa.ARIMA(df['GDP'].values.astype(float),order=(i,1,j)).fit()
        print([i,j,result.aic,result.bic])
        if result.bic<Min:
            Min=result.bic
            best_pq=[i,j,result.aic,result.bic]
print(f'最优定阶为{best_pq}')

Code results:

The fifth step is to conduct model result analysis and model testing. 

    Model testing is divided into model parameter testing and model significance testing.

    Model significance test: that is, the white noise test of the residual. If the residual is a white noise sequence, that is, the original sequence information is fully extracted. Look at the LB statistic. The LB statistic of the model result in python is the LB statistic of the residual. , if the p value is less than 0.05, it is a non-white noise sequence. If the p value is greater than 0.05, it means that the residual is a white noise sequence. This is the result we want. Here you can take out the residual value of the model separately, draw the residual time series diagram, QQ diagram, normal distribution diagram yourself, or perform a white noise test yourself to assist in judgment. The white noise sequence obeys the normal distribution, the time series diagram fluctuates smoothly, and the numerical points on the QQ diagram are near the diagonal.
    
    Parameter test of the model: test whether each unknown parameter is significantly 0, and test whether the model is the most streamlined. If the parameter is not significantly non-zero, it can be eliminated from the fitted model and see the t statistic.
    
    Partial explanation of the model results:
    const: constant term       
    ar.L1: autoregressive term coefficient
    ma.L1: moving average term coefficient
    sigma2: variance
    P>|z| under each parameter, if it is less than a=0.05, the null hypothesis is rejected. It is considered that the parameters are significantly non-zero, that is, there is no need to simplify the model.
    Ljung-Box: LB statistic. It should be noted here that the P value of LB needs to be >0.95, that is, the residual sequence is judged to be a white noise sequence, and if it is less than 0.05, it is a non-white noise sequence.
    Jarque-Bera: Results of the JB statistic   
    Heteroskedasticity test indicate a stable case of variance

Result=sm.tsa.ARIMA(df['GDP'].values.astype(float),order=(best_pq[0],1,best_pq[1])).fit()
print(Result.summary())  #显示模型的所有信息

print(len(Result.resid))
#print(Result.resid)这里观察到残差的第一项为原数据的1239.5，即差分数据不管第一项，这里需要调整残差的观测

#这里就可以观察到原始模型的结果LB统计量和这里的白噪声检验是一致的，p>0.05,即认为残差为白噪声序列，原序列信息提取充分。
lags = [1,4,8,16,32]
print('差分序列的白噪声检验结果为：'+'\n',acorr_ljungbox(Result.resid[1:], lags))

## 查看模型的拟合残差分布
fig = plt.figure(figsize=(12,5))
ax = fig.add_subplot(1,2,1)
plt.plot(Result.resid[1:])
plt.title("ARIMA(2,1,1)残差曲线")

## 检查残差是否符合正太分布
ax = fig.add_subplot(1,2,2)
sm.qqplot(Result.resid[1:], line='q', ax=ax)
plt.title("ARIMA(2,1,0)残差Q-Q图")
plt.tight_layout()
plt.show()

fig = plt.figure(figsize=(12,5))
Residual=pd.DataFrame(Result.resid[1:])
Residual.plot(kind='kde', title='密度')
plt.legend('')
plt.show()

The result of the code is:

  

The sixth step is to perform model prediction:

    Model prediction is to use the above-mentioned relatively optimal model to predict the values ​​corresponding to the original data variables at subsequent times.
    Note that in python, if the predict function predicts differential data, pay attention to the beginning of start and the termination of end. For example, the first-order difference data starts from the second observation value, and the corresponding residual error is also the same
    . When drawing later, you need to pay attention to the drawing of the differential model.

#预测，绘制原序列和预测序列值对比图

Predict=Result.predict(start=1, end=len(df['GDP'])-1+1+10); #不加参数默认0到n-1，要加预测个数在end后面N-1+预测n即可
#如果是一阶差分的序列预测，第一个数据已经差分消去了，应该start从第二个观测数据开始，即n=1；如果是0阶，则不需要按默认0到n-1

print(list(zip(range(193,203),Predict[-10:])))#打印预测值

plt.figure()
plt.plot(range(193),df['GDP'].values)#'o-k'
plt.plot(range(193+10),Predict)#'P--'
plt.legend(('原始观测值','预测值'))
plt.xticks(list(range(0,203,10)),rotation=90)
plt.show()

plt.figure()
plt.plot(range(193),df['GDP'].values)#'o-k'
plt.plot(range(192,193+10),Predict[-11:])#'P--'#接着原数据最后一个，进行拟合预测表示
plt.legend(('原始观测值','预测值'))
plt.xticks(list(range(0,203,10)),rotation=90)
plt.show()

Code results:

PS: Comparison of automated AUTO-ARIMA

import pmdarima as pm
# ## 自动搜索合适的参数
model = pm.auto_arima(df['GDP'].values,
                      start_p=1, start_q=1, # p,q的开始值
                      max_p=12, max_q=12, # 最大的p和q
                      d = 0,            # 寻找ARMA模型参数
                      m=1,              # 序列的周期
                      seasonal=False,   # 没有季节性趋势
                      trace=True,error_action='ignore',  
                      suppress_warnings=True, stepwise=True)
 
print(model.summary())

I haven't studied this area in depth. The auto method has advantages and disadvantages, but it can also provide an idea for writing code.