Stationary time series analysis

Others 2021-03-22 10:24:11 views: null

Stationary time series analysis

Methodological tools

Difference operation

  • First difference
  • p-th order difference
  • k-step difference

Delay operator

Linear difference equation

ARMA model

AR model

  • AR model stationarity judgment

    • Characteristic root discrimination
    • Stationary domain discrimination

  • Statistical properties of stationary AR models

MA model

  • Statistical properties of MA model

    • Mean
    • variance
    • Auto-covariance q-order truncation
    • Autocorrelation coefficient q-order truncation

  • Reversibility of MA model

  • Partial self-regulating coefficient truncation

ARMA model

  • Stationary condition & reversible condition
  • Transmission form & reversal form
  • Statistical properties of ARMA(p,q) model

Stationary series modeling

Modeling steps

  • Find the sample autocorrelation coefficient ACF and sample partial autocorrelation coefficient PACF value of the sequence
  • According to the properties of the two coefficients, select the appropriate ARMA(p,q) model for fitting
  • Estimate the values ​​of unknown parameters in the model
  • Test the validity of the model. If the fit does not match, return to step 2 to reselect the model fit
  • Model optimization. Pass the test, return to step 2, fully consider various possibilities, establish multiple fitting models, and select the best model
  • Use the fitted model to predict the future trend of the sequence

Sample autocorrelation coefficient & partial autocorrelation coefficient

Model recognition

Parameter Estimation

  • Moment estimation
  • Maximum likelihood estimation
  • Least squares estimation

Model checking

  • Model significance test
  • Parameter significance test

Model optimization

  • Questions raised
  • AIC guidelines
  • SBC (BIC) guidelines

Sequence prediction

Linear prediction function

Principle of Minimal Forecasting Variance

Linear minimum variance and the nature of prediction

  • Conditional unbiased minimum variance estimate
  • AR§ sequence prediction
  • MA(q) sequence prediction
  • ARMA(p,q) sequence prediction

Map

#AR模型平稳性判别
#1、arima.sim函数拟合
#arima.sim(n,list(ar=,ma=,order=),sd=)
#n：拟合序列长度
#list：指定具体的模型参数
#（1）拟合平稳AR(p)模型，要给出自回归系数，如果指定拟合的AR模型为非平稳模型，系统会报错
#（2）拟合MA(q)模型，要给出移动平均系数
#（3）拟合平稳ARMA(p,q)模型需要同时给出自回归系数和移动平均系数，如果指定拟合的ARMA模型为非平稳模型，系统会报错
#（4）拟合ARIMA(p,d,q)模型（第5章介绍），除了需要给出自回归系数和移动平均系数，还需要增加order选项。order=c(p,d,q)
# 其中，p为自回归阶数；d为差分阶数；q为移动平均阶数
#sd：指定序列的标准差，不特殊指定，系统默认 sd=1

#2、filter函数拟合
#filter(e,filter=,method=,circular=)
#e：随机波动序列的变量名
#filter：指定模型系数
#（1）AR(p)模型为filter=c(∅1,∅2,…,∅p)
#（2）MA(q)模型为filter=c(1,-θ1,-θ2,…,-θq)
#method：指定拟合的是AR模型还是MA模型
#（1）method=“recursive”为AR模型
#（2）method=“convolution”为MA模型
#circular：拟合MA模型时专用的一个选项，circular=T可以避免NA数据出现

#3.1
x1=arima.sim(n=100,list(ar=0.8))
x3=arima.sim(n=100,list(ar=c(1,-0.5)))
e=rnorm(100)
x2=filter(e,filter=-1.1,method = "recursive")
x4=filter(e,filter = c(1,0.5),method="recursive")

layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE))

#layout(matrix(c(1,1,2,3), 2, 2, byrow = TRUE))
# layout() 输入一个矩阵，2行2列，
#第1个图占两个1，第2个图占2，第3个图占3

ts.plot(x1) #平稳
ts.plot(x2) #非平稳
ts.plot(x3) #平稳
ts.plot(x4) #非平稳

#3.5
x1=arima.sim(n=1000,list(ar=0.8))
x2=arima.sim(n=1000,list(ar=-0.8))
x3=arima.sim(n=1000,list(ar=c(1,-0.5)))
x4=arima.sim(n=1000,list(ar=c(-1,-0.5)))

#AR模型样本自相关图
layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE))
acf(x1)
acf(x2)
acf(x3)
acf(x4)

#AR模型样本偏自相关图
layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE))
pacf(x1)
pacf(x2)
pacf(x3)
pacf(x4)

#3.6
x1=arima.sim(n=1000,list(ma=-2))
x2=arima.sim(n=1000,list(ma=-0.5))
x3=arima.sim(n=1000,list(ma=c(-4/5,16/25)))
x4=arima.sim(n=1000,list(ma=c(-5/4,25/16)))

layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE))
acf(x1)
acf(x2)
acf(x3)
acf(x4)

layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE))
pacf(x1)
pacf(x2)
pacf(x3)
pacf(x4)

#3.8
x=arima.sim(n=1000,list(ar=0.5,ma=-0.8))
acf(x)
pacf(x)

#3.9
#读入数据，并绘制时序图
layout(matrix(c(1), 1, byrow = TRUE))

a=read.table("E:/data/file8.csv",sep=",",header=T)
x=ts(a$kilometer,start=1950)
plot(x)

#白噪声检验
for(i in 1:2) print(Box.test(x,type="Ljung-Box",lag=6*i))

#绘制自相关图和偏自相关图
acf(x)
pacf(x)

#3.10（有误）
#读入数据，并绘制时序图
overshort<-read.table("E:/data/file9.csv",sep=",",header=T)
overshort<-ts(overshort)
plot(overshort)

#白噪声检验
for(i in 1:2) print(Box.test(overshort,type="Ljung-Box",lag=6*i))

#绘制自相关图和偏自相关图
acf(overshort)  #一阶截尾
pacf(overshort)  #拖尾

#3.11
#读入数据，并绘制时序图
b=read.table("E:/data/file10.csv",sep=",",header=T)
dif_x=ts(diff(b$change_temp),start=1880)
plot(dif_x)

#白噪声检验
for(i in 1:2) print(Box.test(dif_x,type="Ljung-Box",lag=6*1))

#绘制自相关图和偏自相关图
acf(dif_x)
pacf(dif_x)

#auto.arima函数：先安装packages:zoo & forecast，然后用library调用程序包
#auto.arima(x,max.p=5,max.q=,ic=)
#x：需要定阶的序列名
#max.p：自相关系数最高阶数，不特殊指定的话，系统默认值为5
#max.q：移动平均系数最高阶数，不特殊指定的话，系统默认值为5
#ic：指定信息量准则，ic有"aicc","aic"和"bic"三个选项，系统默认AIC准则

library(zoo)
library(forecast)

#3.9 系统自动定阶
auto.arima(x)

#3.10 系统自动定阶
auto.arima(dif_x)

#3.11 系统自动定阶
auto.arima(dif_x)

#arima(x,order=,include.mean=,method=)
#x：要进行模型拟合的序列名
#order：指定模型阶数。order=c(p,d,q)
#（1）p为自回归阶数
#（2）d为差分阶数，本章不涉及缠粉问题，所以d=0
#（3）q为移动平均阶数
#include.mean：要不要包含常数项
#（1）include.mean=T，需要拟合常数项，这也是系统默认设置
#（2）如果不需要拟合常数项，需要特别指定include.mean=F
#method：指定参数估计方法
#（1）method="CSS-ML"，默认的是条件最小二乘与极大似然估计混合方法
#（2）method="ML"，极大似然估计
#（3）mrthod="CSS"，条件最小二乘估计

#3.9续
a=read.table(file="E:/data/file8.csv",sep=",",header=T)
x=ts(a$kilometer,start=1950)
x.fit=arima(x,order=c(2,0,0),method="ML")
x.fit

#3.10续（有误）
overshort=read.table("E:/data/file9.csv",sep=",",header = T)
overshort=ts(overshort)
overshort.fit=arima(overshort,order=c(0,0,1))
overshort.fit

#3.11续
b=read.table("E:/data/file10.csv",sep=",",header = T)
dif_x=ts(diff(b$change_temp),start = 1880)
dif_x.fit=arima(dif_x,order=c(1,0,1))
dif_x.fit

#3.9续
a=read.table(file="E:/data/file8.csv",sep=",",header = T)
x=ts(a$kilometer,start = 1950)
x.fit=arima(x,order=c(2,0,0),method = "ML")
for(i in 1:2) print(Box.test(x.fit$residuals,lag=6*i))

#3.10续（有误）
overshort=read.table("E:/data/file9.csv",sep=",",header = T)
overshort=ts(overshort)
overshort.fit=arima(overshort,order=c(0,0,1))
for(i in 1:2) print(Box.test(overshort.fit$residual,lag=6*i))

#3.11续
b=read.table("E:/data/file10.csv",sep=",",header = T)
dif_x=ts(diff(b$change_temp),start = 1880)
dif_x.fit=arima(dif_x,order=c(1,0,1),method="CSS")
for(i in 1:2) print(Box.test(dif_x.fit$residual,lag=6*i))

#pt(t,df=,lower.tail=)
#t：t统计量的值
#df：自由度
#lower.tail：确定计算概率的方向
#（1）lower.tail=T，计算Pr(X≤x)。对于参数显著性检验，如果参数估计值为负，选择lower.tail=T
#（2）lower.tail=F，计算Pr(X>x)。对于参数显著性检验，如果参数估计值为正，选择lower.tail=F

#3.9续
a=read.table(file="E:/data/file8.csv",sep=",",header = T)
x=ts(a$kilometer,start = 1950)
x.fit=arima(x,order=c(2,0,0),method="ML")
x.fit

#3.15（有误）
#读入数据，绘制时序图
x=read.table(file="E:/data/file11.csv",sep=",",header = T)
x=ts(x)
plot(x)

#序列白噪声检验
for(i in 1:2) print(Box.test(x,lag=6*i))

#绘制自相关图和偏自相关图
acf(x)
pacf(x)

#拟合MA(2)模型
x.fit1=arima(x,order=c(0,0,2))
x.fit1

#MA(2)模型显著性检验
for(i in 1:2) print(Box.test(x,fit1$residual,lag=6*1))

#拟合AR(1)模型
x.fit2=arima(x,order=c(1,0,0))
x.fit2

#AR(1)模型显著性检验
for(i in 1:2) print(Box.test(x.fit2$residual,lag=6*i))


#forecast(objuct,h=,level=)
#object：拟合信息文件名
#h：预测期数
#level：置信区间的置信水平。不特殊指定的话，系统会自动给出置信水平分别为80%和95%的双层置信区间

#3.9续
a=read.table(file="E:/data/file8.csv",sep=",",header = T)
x=ts(a$kilometer,start = 1950)
x.fit=arima(x,order=c(2,0,0))
x.fore=forecast(x.fit,h=5)
x.fore

#系统默认输出预测图
plot(x.fore)

#个性化输出预测图
L1=x.fore$fitted-1.96*sqrt(x.fit$sigma2)
U1=x.fore$fitted+1.96*sqrt(x.fit$sigma2)
L2=ts(x.fore$lower[,2],start = 2009)
U2=ts(x.fore$upper[,2],start = 2009)
c1=min(x,L1,L2)
c2=max(x,L2,U2)
plot(x,type="p",pch=8,xlim=c(1950,2013),ylim=c(c1,c2))
lines(x.fore$fitted,col=2,lwd=2)
lines(x.fore$mean,col=2,lwd=2)
lines(L1,col=4,lty=2)
lines(U1,col=4,lty=2)
lines(L1,col=4,lty=2)
lines(L2,col=4,lty=2)
lines(U2,col=4,lty=2)

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