ARIMA模型选择与残差

%load_ext autoreload
%autoreload 2
%matplotlib inline
%config InlineBackend.figure_format='retina'
import pandas as pd
import numpy as np
import statsmodels.api as sm
import statsmodels.formula.api as smf
import statsmodels.tsa.api as smt
#Display and Plotting
import matplotlib.pylab as plt
import seaborn as sns
# pandas与numpy属性设置
pd.set_option('display.float_format',lambda x:'%.5f'%x)#pandas
np.set_printoptions(precision=5,suppress=True) #numpy
pd.set_option('display.max_columns',100)
pd.set_option('display.max_rows',100)
#seaborn.plotting style
sns.set(style='ticks',context='poster')

filename_ts='series1.csv'#数据地址
ts_df=pd.read_csv(filename_ts,index_col=0,parse_dates=[0])#index_col=0设置日期为索引，时间序列模型构建的要求
n_sample=ts_df.shape[0]#获取数据的行数
print(ts_df.shape)
print(ts_df.head())

(120, 1)
value
2006-06-01 0.21507
2006-07-01 1.14225
2006-08-01 0.08077
2006-09-01 -0.73952
2006-10-01 0.53552

#训练样本和测试样本的构建
n_train=int(0.95*n_sample)+1 #训练样本数占95%
n_forecast=n_sample-n_train
#ts_df
ts_train=ts_df.iloc[:n_train]['value']
ts_test=ts_df.iloc[n_train:]['value']
print(ts_train.shape)
print(ts_test.shape)
print("Traing Series:","\n",ts_train.tail(),"\n")
print("Testing Series:","\n",ts_test.head())

(115,)
(5,)
Traing Series:
2015-08-01 0.60371
2015-09-01 -1.27372
2015-10-01 -0.93284
2015-11-01 0.08552
2015-12-01 1.20534
Name: value, dtype: float64

Testing Series:
2016-01-01 2.16411
2016-02-01 0.95226
2016-03-01 0.36485
2016-04-01 -2.26487
2016-05-01 -2.38168
Name: value, dtype: float64

ts_df['diff_1']=ts_df['value'].diff(1)
ts_df['diff_2']=ts_df['diff_1'].diff(1)
ts_df.plot(subplots=True,figsize=(18,12))

def tsplot(y,lags=None,title='',figsize=(14,8)):
    fig=plt.figure(figsize=figsize)
    layout=(2,2)#布局
    ts_ax=plt.subplot2grid(layout,(0,0))# Create an axis at specific location inside a regular grid.
    hist_ax=plt.subplot2grid(layout,(0,1))
    acf_ax=plt.subplot2grid(layout,(1,0))
    pacf_ax=plt.subplot2grid(layout,(1,1))
    y.plot(ax=ts_ax)
    ts_ax.set_title(title)
    y.plot(ax=hist_ax,kind='hist',bins=25)
    hist_ax.set_title('Histogram')
    smt.graphics.plot_acf(y,lags=lags,ax=acf_ax)
    smt.graphics.plot_pacf(y,lags=lags,ax=pacf_ax)
    [ax.set_xlim(0) for ax in [acf_ax, pacf_ax]]
    sns.despine()
    fig.tight_layout()
    return ts_ax,acf_ax,pacf_ax
tsplot(ts_train,title='A Given Training Serries',lags=20)

#Model Estimation
#Fit the model
arima200 = sm.tsa.SARIMAX(ts_train, order=(2,0,0))
model_results = arima200.fit()

import itertools
p_min = 0
d_min = 0
q_min = 0
p_max = 4
d_max = 0
q_max = 4
#初始化一个DF来储存结果
results_bic=pd.DataFrame(index=['AR{}'.format(i) for i in range(p_min,p_max+1)],
                        columns=['MA{}'.format(i) for i in range(q_min,q_max+1)])
#print(results_bic)
for p,d,q in itertools.product(range(p_min,p_max+1),range(d_min,d_max+1),range(q_min,q_max+1)):
    if p==0 and d==0 and q==0:
        results_bic.loc['AR{}'.format(p),'MA{}'.format(q)]=np.nan
        continue
    try:
        model=sm.tsa.SARIMAX(ts_train,order=(p,d,q),
                       # enforce_stationarity=False,
                       # enforce_invertibility=False,
                            )
        results=model.fit()
        results_bic.loc['AR{}'.format(p), 'MA{}'.format(q)]=results.bic
    except:
        continue
results_bic = results_bic[results_bic.columns].astype(float)

#BIC贝叶斯信息准则（Bayesian Information Criterion,BIC）
fig, ax = plt.subplots(figsize=(10, 8))
ax = sns.heatmap(results_bic,
                 mask=results_bic.isnull(),
                 ax=ax,
                 annot=True,
                 fmt='.2f',
                 );
ax.set_title('BIC');

横轴是q值，纵轴是p值，数值越小越好，如图所示，颜色越深数值越小，最小的对应（）（1,1）

#备选模型选择方法，仅限于搜索AR和MA参数

train_results = sm.tsa.arma_order_select_ic(ts_train, ic=['aic', 'bic'], trend='nc', max_ar=4, max_ma=4)

print('AIC', train_results.aic_min_order)
print('BIC', train_results.bic_min_order)

AIC (4, 2)
BIC (1, 1)

模型残差检验：

• ARIMA模型的残差是给是平均值为0 且方差为常数的正态分布
• QQ图：线性即正态分布

#残差分析 正态分布 QQ图线性
model_results.plot_diagnostics(figsize=(16, 12));