python数据分析之时间序列分析详情

## 输出高清图像
%config InlineBackend.figure_fORMat = 'retina'
%matplotlib inline
## 图像显示中文的问题
import matplotlib
matplotlib.rcParams['axes.unicode_minus']=False

import seaborn as sns
sns.set(font= "Kaiti",style="ticks",font_scale=1.4)

## 导入会使用到的相关库
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.stattools import *
import statsmodels.api as sm
import statsmodels.formula.api as smf
from statsmodels.tsa.api import SimpleExpSmoothing,Holt,ExponentialSmoothing,AR,ARIMA,ARMA
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

import pmdarima as pm
from sklearn.metrics import mean_absolute_error

import pyflux as pf
from fbprophet import Prophet
## 忽略提醒
import warnings
warnings.filterwarnings("ignore")

## 读取时间序列数据,该数据包含:X1为飞机乘客数据,X2为一组随机数据
df = pd.read_csv("data/chap6/timeserise.csv")
## 查看数据的变化趋势
df.plot(kind = "line",figsize = (10,6))
plt.grid()
plt.title("时序数据")
plt.show()

## 白噪声检验Ljung-Box检验
## 该检验用来检查序列是否为随机序列，如果是随机序列，那它们的值之间没有任何关系
## 使用LB检验来检验序列是否为白噪声，原假设为在延迟期数内序列之间相互独立。
lags = [4,8,16,32]
LB = sm.stats.diagnostic.acorr_ljungbox(df["X1"],lags = lags,return_df = True)
print("序列X1的检验结果:\n",LB)
LB = sm.stats.diagnostic.acorr_ljungbox(df["X2"],lags = lags,return_df = True)
print("序列X2的检验结果:\n",LB)

## 如果P值小于0.05，说明序列之间不独立，不是白噪声

'''
序列X1的检验结果:
         lb_stat      lb_pvalue
4    427.738684   2.817731e-91
8    709.484498  6.496271e-148
16  1289.037076  1.137910e-264
32  1792.523003   0.000000e+00
序列X2的检验结果:
       lb_stat  lb_pvalue
4    1.822771   0.768314
8    8.452830   0.390531
16  15.508599   0.487750
32  28.717743   0.633459
'''

## 序列的单位根检验，即检验序列的平稳性
dftest = adfuller(df["X2"],autolag='BIC')
dfoutput = pd.Series(dftest[0:4], index=['adf','p-value','usedlag','Number of Observations Used'])
print("X2单位根检验结果:\n",dfoutput)

dftest = adfuller(df["X1"],autolag='BIC')
dfoutput = pd.Series(dftest[0:4], index=['adf','p-value','usedlag','Number of Observations Used'])
print("X1单位根检验结果:\n",dfoutput)

## 对X1进行一阶差分后的序列进行检验
X1diff = df["X1"].diff().dropna()
dftest = adfuller(X1diff,autolag='BIC')
dfoutput = pd.Series(dftest[0:4], index=['adf','p-value','usedlag','Number of Observations Used'])
print("X1一阶差分单位根检验结果:\n",dfoutput)

## 一阶差分后 P值大于0.05, 小于0.1，可以认为其是平稳的
'''
X2单位根检验结果:
 adf                           -1.124298e+01
p-value                        1.788000e-20
usedlag                        0.000000e+00
Number of Observations Used    1.430000e+02
dtype: float64
X1单位根检验结果:
 adf                              0.815369
p-value                          0.991880
usedlag                         13.000000
Number of Observations Used    130.000000
dtype: float64
X1一阶差分单位根检验结果:
 adf                             -2.829267
p-value                          0.054213
usedlag                         12.000000
Number of Observations Used    130.000000
dtype: float64
'''

## KPSS检验的原假设为：序列x是平稳的。

## 对序列X2使用KPSS检验平稳性
dfkpss = kpss(df["X2"])
dfoutput = pd.Series(dfkpss[0:3], index=["kpss_stat"," p-value"," usedlag"])
print("X2 KPSS检验结果:\n",dfoutput)
## 接受序列平稳的原假设
## 对序列X1使用KPSS检验平稳性
dfkpss = kpss(df["X1"])
dfoutput = pd.Series(dfkpss[0:3], index=["kpss_stat"," p-value"," usedlag"])
print("X1 KPSS检验结果:\n",dfoutput)
## 拒绝序列平稳的原假设

## 对序列X1使用KPSS检验平稳性
dfkpss = kpss(X1diff)
dfoutput = pd.Series(dfkpss[0:3], index=["kpss_stat"," p-value"," usedlag"])
print("X1一阶差分KPSS检验结果:\n",dfoutput)
## 接受序列平稳的原假设

'''
X2 KPSS检验结果:
 kpss_stat     0.087559
 p-value      0.100000
 usedlag     14.000000
dtype: float64
X1 KPSS检验结果:
 kpss_stat     1.052175
 p-value      0.010000
 usedlag     14.000000
dtype: float64
X1一阶差分KPSS检验结果:
 kpss_stat     0.05301
 p-value      0.10000
 usedlag     14.00000
dtype: float64
'''

## 检验ARIMA模型的参数d
X1d = pm.arima.ndiffs(df["X1"], alpha=0.05, test="kpss", max_d=3)
print("使用KPSS方法对序列X1的参数d取值进行预测,d = ",X1d)

X1diffd = pm.arima.ndiffs(X1diff, alpha=0.05, test="kpss", max_d=3)
print("使用KPSS方法对序列X1一阶差分后的参数d取值进行预测,d = ",X1diffd)

X2d = pm.arima.ndiffs(df["X2"], alpha=0.05, test="kpss", max_d=3)
print("使用KPSS方法对序列X2的参数d取值进行预测,d = ",X2d)

'''
使用KPSS方法对序列X1的参数d取值进行预测,d =  1
使用KPSS方法对序列X1一阶差分后的参数d取值进行预测,d =  0
使用KPSS方法对序列X1的参数d取值进行预测,d =  0
'''

## 检验SARIMA模型的参数季节阶数D
X1d = pm.arima.nsdiffs(df["X1"], 12, max_D=2)
print("对序列X1的季节阶数D取值进行预测,D = ",X1d)
X1diffd = pm.arima.nsdiffs(X1diff, 12, max_D=2)
print("序列X1一阶差分后的季节阶数D取值进行预测,D = ",X1diffd)

'''
对序列X1的季节阶数D取值进行预测,D =  1
序列X1一阶差分后的季节阶数D取值进行预测,D =  1
'''

## 对随机序列X2进行自相关和偏相关分析可视化
fig = plt.figure(figsize=(16,5))
plt.subplot(1,3,1)
plt.plot(df["X2"],"r-")
plt.grid()
plt.title("X2序列波动")
ax = fig.add_subplot(1,3,2)
plot_acf(df["X2"], lags=60,ax = ax)
plt.grid()
ax = fig.add_subplot(1,3,3)
plot_pacf(df["X2"], lags=60,ax = ax)
plt.grid()
plt.tight_layout()
plt.show()

## 对非随机序列X1进行自相关和偏相关分析可视化
fig = plt.figure(figsize=(16,5))
plt.subplot(1,3,1)
plt.plot(df["X1"],"r-")
plt.grid()
plt.title("X1序列波动")
ax = fig.add_subplot(1,3,2)
plot_acf(df["X1"], lags=60,ax = ax)
plt.grid()
ax = fig.add_subplot(1,3,3)
plot_pacf(df["X1"], lags=60,ax = ax)
plt.ylim([-1,1])
plt.grid()
plt.tight_layout()
plt.show()

## 对非随机序列X1一阶差分后的序列进行自相关和偏相关分析可视化
fig = plt.figure(figsize=(16,5))
plt.subplot(1,3,1)
plt.plot(X1diff,"r-")
plt.grid()
plt.title("X1序列一阶差分后波动")
ax = fig.add_subplot(1,3,2)
plot_acf(X1diff, lags=60,ax = ax)
plt.grid()
ax = fig.add_subplot(1,3,3)
plot_pacf(X1diff, lags=60,ax = ax)
plt.grid()
plt.tight_layout()
plt.show()

## 时间序列的分解
## 通过观察序列X1，可以发现其既有上升的趋势，也有周期性的趋势，所以可以将该序列进行分解
## 使用乘法模型分解结果(通常适用于有增长趋势的序列)
X1decomp = pm.arima.decompose(df["X1"].values,"multiplicative", m=12)
## 可视化出分解的结果
ax = pm.utils.decomposed_plot(X1decomp,figure_kwargs = {"figsize": (10, 6)},
                              show=False)
ax[0].set_title("乘法模型分解结果")
plt.show()

## 使用加法模型分解结果(通常适用于平稳趋势的序列)
X1decomp = pm.arima.decompose(X1diff.values,"additive", m=12)
## 可视化出分解的结果
ax = pm.utils.decomposed_plot(X1decomp,figure_kwargs = {"figsize": (10, 6)},
                              show=False)
ax[0].set_title("加法模型分解结果")
plt.show()

## 数据准备
## 对序列X1进行切分，后面的24个数据用于测试集
train = pd.DataFrame(df["X1"][0:120])
test = pd.DataFrame(df["X1"][120:])
## 可视化切分后的数据
train["X1"].plot(figsize=(14,7), title= "乘客数量数据",label = "X1 train")
test["X1"].plot(label = "X1 test")
plt.legend()
plt.grid()
plt.show()

print(train.shape)
print(test.shape)
df["X1"].shape
'''
(120, 1)
(24, 1)
(144,)
'''

## 简单移动平均进行预测
y_hat_avg = test.copy(deep = False)
y_hat_avg["moving_avg_forecast"] =  train["X1"].rolling(24).mean().iloc[-1]
## 可视化出预测结果
plt.figure(figsize=(14,7))
train["X1"].plot(figsize=(14,7),label = "X1 train")
test["X1"].plot(label = "X1 test")
y_hat_avg["moving_avg_forecast"].plot(style="g--o", lw=2,
                                      label="移动平均预测")
plt.legend()
plt.grid()
plt.title("简单移动平均预测")
plt.show()

## 计算预测结果和真实值的误差
print("预测绝对值误差:",mean_absolute_error(test["X1"],y_hat_avg["moving_avg_forecast"]))
'''
预测绝对值误差: 82.55208333333336
'''

## 数据准备
y_hat_avg = test.copy(deep = False)
## 模型构建
model1 = SimpleExpSmoothing(train["X1"].values).fit(smoothing_level=0.15)
y_hat_avg["exp_smooth_forecast1"] = model1.forecast(len(test))

model2 = SimpleExpSmoothing(train["X1"].values).fit(smoothing_level=0.5)
y_hat_avg["exp_smooth_forecast2"] = model2.forecast(len(test))

## 可视化出预测结果
plt.figure(figsize=(14,7))
train["X1"].plot(figsize=(14,7),label = "X1 train")
test["X1"].plot(label = "X1 test")
y_hat_avg["exp_smooth_forecast1"].plot(style="g--o", lw=2,
                                      label="smoothing_level=0.15")
y_hat_avg["exp_smooth_forecast2"].plot(style="g--s", lw=2,
                                      label="smoothing_level=0.5")
plt.legend()
plt.grid()
plt.title("简单指数平滑预测")
plt.show()

## 计算预测结果和真实值的误差
print("smoothing_level=0.15,预测绝对值误差:",
      mean_absolute_error(test["X1"],y_hat_avg["exp_smooth_forecast1"]))
print("smoothing_level=0.5,预测绝对值误差:",
      mean_absolute_error(test["X1"],y_hat_avg["exp_smooth_forecast2"]))

## 数据准备
y_hat_avg = test.copy(deep = False)
## 模型构建
model1 = Holt(train["X1"].values).fit(smoothing_level=0.1,                                 smoothing_slope = 0.05)
y_hat_avg["holt_forecast1"] = model1.forecast(len(test))
model2 = Holt(train["X1"].values).fit(smoothing_level=0.1,                                 smoothing_slope = 0.25)
y_hat_avg["holt_forecast2"] = model2.forecast(len(test))
## 可视化出预测结果
plt.figure(figsize=(14,7))
train["X1"].plot(figsize=(14,7),label = "X1 train")
test["X1"].plot(label = "X1 test")
y_hat_avg["holt_forecast1"].plot(style="g--o", lw=2,
                                 label="Holt线性趋势法(1)")
y_hat_avg["holt_forecast2"].plot(style="g--s", lw=2,
                                 label="Holt线性趋势法(2)")
plt.legend()
plt.grid()
plt.title("Holt线性趋势法预测")
plt.show()

## 计算预测结果和真实值的误差
print("smoothing_slope = 0.05,预测绝对值误差:",
      mean_absolute_error(test["X1"],y_hat_avg["holt_forecast1"]))
print("smoothing_slope = 0.25,预测绝对值误差:",
      mean_absolute_error(test["X1"],y_hat_avg["holt_forecast2"]))

## 数据准备
y_hat_avg = test.copy(deep = False)
## 模型构建
model1 = ExponentialSmoothing(train["X1"].values,
                              seasonal_periods=12, # 周期性为12  
                              trend="add", seasonal="add").fit()
y_hat_avg["holt_winter_forecast1"] = model1.forecast(len(test))
## 可视化出预测结果
plt.figure(figsize=(14,7))
train["X1"].plot(figsize=(14,7),label = "X1 train")
test["X1"].plot(label = "X1 test")
y_hat_avg["holt_winter_forecast1"].plot(style="g--o", lw=2,
                                 label="Holt-Winters")
plt.legend()
plt.grid()
plt.title("Holt-Winters季节性预测模型")
plt.show()
## 计算预测结果和真实值的误差
print("Holt-Winters季节性预测模型,预测绝对值误差:",
      mean_absolute_error(test["X1"],y_hat_avg["holt_winter_forecast1"]))

## 使用AR模型对乘客数据进行预测 

## 经过前面序列的偏相关系数的可视化结果，使用AR(2)模型可对序列进行建模
## 数据准备
y_hat = test.copy(deep = False)
## 模型构建
ar_model = ARMA(train["X1"].values,order = (2,0)).fit()
## 输出拟合模型的结果
print(ar_model.summary())

## AIC=1141.989；BIC= 1153.138；两个系数是显著的

## 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(ar_model.resid)
plt.title("AR(2)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(ar_model.resid, line='q', ax=ax)
plt.title("AR(2)残差Q-Q图")
plt.tight_layout()
plt.show()

## 预测未来24个数据，并输出95%置信区间
pre, se, conf = ar_model.forecast(24, alpha=0.05)  
## 整理数据
y_hat["ar2_pre"] = pre
y_hat["ar2_pre_lower"] = conf[:,0]
y_hat["ar2_pre_upper"] = conf[:,1]
## 可视化出预测结果
plt.figure(figsize=(14,7))
train["X1"].plot(figsize=(14,7),label = "X1 train")
test["X1"].plot(label = "X1 test")
y_hat["ar2_pre"].plot(style="g--o", lw=2,label="AR(2)")
## 可视化出置信区间
plt.fill_between(y_hat.index, y_hat["ar2_pre_lower"], 
                 y_hat["ar2_pre_upper"],color='k',alpha=.15,
                 label = "95%置信区间")
plt.legend()
plt.grid()
plt.title("AR(2)模型")
plt.show()
# 计算预测结果和真实值的误差
print("AR(2)模型预测的绝对值误差:",
      mean_absolute_error(test["X1"],y_hat["ar2_pre"]))

## 尝试使用ARMA模型进行预测

## 根据前面的自相关系数和偏相关系数,为了降低模型的复杂读,可以使用ARMA(2,1)

## 数据准备
y_hat = test.copy(deep = False)
## 模型构建
arma_model = ARMA(train["X1"].values,order = (2,1)).fit()
## 输出拟合模型的结果
print(arma_model.summary())

## AIC=1141.989；BIC= 1153.138；两个系数是显著的

## 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(arma_model.resid)
plt.title("ARMA(2,1)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(arma_model.resid, line='q', ax=ax)
plt.title("ARMA(2,1)残差Q-Q图")
plt.tight_layout()
plt.show()

## 预测未来24个数据，并输出95%置信区间
pre, se, conf = arma_model.forecast(24, alpha=0.05)
## 整理数据
y_hat["arma_pre"] = pre
y_hat["arma_pre_lower"] = conf[:,0]
y_hat["arma_pre_upper"] = conf[:,1]
## 可视化出预测结果
plt.figure(figsize=(14,7))
train["X1"].plot(figsize=(14,7),label = "X1 train")
test["X1"].plot(label = "X1 test")
y_hat["arma_pre"].plot(style="g--o", lw=2,label="ARMA(2,1)")
## 可视化出置信区间
plt.fill_between(y_hat.index, y_hat["arma_pre_lower"], 
                 y_hat["arma_pre_upper"],color='k',alpha=.15,
                 label = "95%置信区间")
plt.legend()
plt.grid()
plt.title("ARMA(2,1)模型")
plt.show()
# 计算预测结果和真实值的误差
print("ARMA模型预测的绝对值误差:",
      mean_absolute_error(test["X1"],y_hat["arma_pre"]))

## 自动搜索合适的参数
model = pm.auto_arima(train["X1"].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())

## 使用ARMA(3,2)对测试集进行预测
pre, conf = model.predict(n_periods=24, alpha=0.05,
                          return_conf_int=True)
## 可视化ARMA(3,2)的预测结果，整理数据
y_hat["arma_pre"] = pre
y_hat["arma_pre_lower"] = conf[:,0]
y_hat["arma_pre_upper"] = conf[:,1]
## 可视化出预测结果
plt.figure(figsize=(14,7))
train["X1"].plot(figsize=(14,7),label = "X1 train")
test["X1"].plot(label = "X1 test")
y_hat["arma_pre"].plot(style="g--o", lw=2,label="ARMA(3,2)")
## 可视化出置信区间
plt.fill_between(y_hat.index, y_hat["arma_pre_lower"], 
                 y_hat["arma_pre_upper"],color='k',alpha=.15,
                 label = "95%置信区间")
plt.legend()
plt.grid()
plt.title("ARMA(3,2)模型")
plt.show()

# 计算预测结果和真实值的误差
print("ARMA模型预测的绝对值误差:",
      mean_absolute_error(test["X1"],y_hat["arma_pre"]))

## 使用prophet检测时间序列是否有异常值

## 从1991年2月到2005年5月，每周提供美国成品汽车汽油产品的时间序列（每天数千桶）

## 数据准备
data = pm.datasets.load_gasoline()
datadf = pd.DataFrame({"y":data})
datadf["ds"] = pd.date_range(start="1991-2",periods=len(data),freq="W")
## 可视化时间序列的变化情况
datadf.plot(x = "ds",y = "y",style = "b-o",figsize=(14,7))
plt.grid()
plt.title("时间序列数据的波动情况")
plt.show()

## 对该数据建立一个时间序列模型
np.random.seed(1234)  ## 设置随机数种子
model = Prophet(growth="linear",daily_seasonality = False,
                weekly_seasonality=False,
                seasonality_mode = 'multiplicative',
                interval_width = 0.95,   ## 获取95%的置信区间
                )
model = model.fit(datadf)     # 使用数据拟合模型
forecast = model.predict(datadf)  # 使用模型对数据进行预测
forecast["y"] = datadf["y"].reset_index(drop = True)
forecast[["ds","y","yhat","yhat_lower","yhat_upper"]].head()

## 根据模型预测值的置信区间"yhat_lower"和"yhat_upper"判断样本是否为异常值
def outlier_detection(forecast):
    index = np.where((forecast["y"] <= forecast["yhat_lower"])|
                     (forecast["y"] >= forecast["yhat_upper"]),True,False)
    return index
outlier_index = outlier_detection(forecast)
outlier_df = datadf[outlier_index]
print("异常值的数量为:",np.sum(outlier_index))
'''
异常值的数量为: 38
'''

## 可视化异常值的结果
fig, ax = plt.subplots()
## 可视化预测值
forecast.plot(x = "ds",y = "yhat",style = "b-",figsize=(14,7),
              label = "预测值",ax=ax)
## 可视化出置信区间
ax.fill_between(forecast["ds"].values, forecast["yhat_lower"], 
                forecast["yhat_upper"],color='b',alpha=.2,
                label = "95%置信区间")
forecast.plot(kind = "scatter",x = "ds",y = "y",c = "k",
              s = 20,label = "原始数据",ax = ax)
## 可视化出异常值的点
outlier_df.plot(x = "ds",y = "y",style = "rs",ax = ax,
                label = "异常值")
plt.legend(loc = 2)
plt.grid()
plt.title("时间序列异常值检测结果")
plt.show()