文章目錄

• 前言
• 一、方差不平穩(wěn)
• 二、具有趨勢性
• • 1、確定性趨勢的刪除——趨勢擬合法
  • 2、隨機趨勢的刪除——差分法
  • 3、未知趨勢的刪除——平滑法
  • • 1. 移動平均法
    • 2. 指數(shù)平滑法
  • 4、季節(jié)性
  • • 1.Method1: 減掉趨勢成分后取平均
    • 2.Method2：使用啞元變量做回歸分析

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前言

時間序列為什么要平穩(wěn)后建模？

• 時間序列是平穩(wěn)的，則沒有任何預測價值，因為均值不變，所以實際上人們只會對非平穩(wěn)時間序列感興趣
• 之所以研究平穩(wěn)時間序列，因為總希望可以通過差分等方式把非平穩(wěn)序列變成平穩(wěn)的，而平穩(wěn)時間序列可以通過數(shù)學公式予以解釋
  

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一、方差不平穩(wěn)

最簡單的是做對數(shù)變換，使得前后方差沒有那么變化。

#Log transform: variance stationary data(AirPassengers) AirPassengers par(mfrow=c(1,2)) plot(AirPassengers) plot(log(AirPassengers))

二、具有趨勢性

1、確定性趨勢的刪除——趨勢擬合法

最簡單的處理方法就是考慮均值函數(shù)可以由一個時間的確定性函數(shù)來描述，比如可以用回歸模型來描述。對于一個有線性趨勢的序列，可以做一個線性模型。

• Linear trend

##read the gasline data ##Abraham and Ledolter (1983) on the ##monthly gasoline demand in Ontario over the period 1960 - 1975. gas = scan("gas.dat") ## note that read.table() would give an error here since the data is not in rectangular format. gas.ts = ts(gas, frequency = 12, start = 1960) gas.ts plot(gas.ts, main = "Gasoline demand in Ontario", ylab = "Million gallons") #fit trend:OLS with constant and trend time = 1:length(gas.ts) fit = lm(gas.ts~time) summary(fit)#R-squared: 0.8225 par(mfrow=c(2,1)) plot(time, gas.ts, type='l', xlab='months', ylab='Million gallons gasoline', main="OLS fit") legend(.1, 256000, c("Time plot", "OLS fit"), col = c(1,2), text.col = "black", merge = TRUE, lty=c(1,1)) abline(fit,col=2,lwd=2) plot(time, fit$resid, type='l', xlab="months", ylab="Residuals",main="Residuals OLS [yt-fit(yt)]") abline(a=0, b=0)#殘差在0附近點波動 #savePlot("plot of Residuals_gas",type="pdf")

• Quadratic trend

fit2 = lm(gas.ts~time+I(time^2)) summary(fit2)#R-squared: 0.8327 names(fit2) par(mfrow=c(2,1)) plot(time, gas.ts, type='l', xlab='months', ylab='Million gallons gasoline',main="OLS fit") legend(.1, 256000, c("Time plot", "OLS fit"), col = c(1,2), text.col = "black",merge = TRUE, lty=c(1,1)) lines(time,fitted(fit2),col=2) plot(time, fit2$resid, type='l', xlab="months", ylab="Residuals",main="Residuals OLS [yt-fit(yt)]") abline(a=0, b=0)

• Polynomial trend

2、隨機趨勢的刪除——差分法

隨機趨勢的刪除——差分法。

• 一階差分可以使線性趨勢的序列實現(xiàn)趨勢平穩(wěn)

Zt = as.ts(rnorm(1000, sd = 20)) RW1 = as.ts(cumsum(Zt)) par(mfrow = c(2,2)) plot(RW1, main = "Random walk") acf(RW1, main = "Correlogram random walk") Yt=as.ts(diff(RW1)) plot(Yt,main='Diff(RW1)') acf(Yt)

• 序列蘊含著曲線趨勢，通常低階（二階或三階）差分可提取出曲線趨勢的影響 ，這里效果不好的話，可以先取對數(shù)，再做差分
• 對于蘊含著固定周期的序列進行步長為周期長度的差分運算，通?？梢暂^好地提取周期信息
• 差分的階數(shù)要適度，避免過差分

3、未知趨勢的刪除——平滑法

平滑法——顧名思義，使序列變平滑。 期望結(jié)果，是顯示出趨勢變化的規(guī)律。

1. 移動平均法

假定在短的時間間隔內(nèi)，序列的取值是比較穩(wěn)定的，序列的大小差異主要是由隨機波動造成的。用一段時間間隔內(nèi)的平均值作為某一期的估計值。

• 簡單移動平均

##Moving average Model Vt5 = filter(Zt, rep(1/5, 5), sides = 2) Vt21 = filter(Zt, rep(1/21, 21), sides = 2) par(mfrow = c(2,2)) plot(Vt5, main = "Moving 5-point average of above white noise series") acf(Vt5, main = "Correlogram 5-point moving average", lag.max = 40, na.action = na.pass) plot(Vt21, main = "Moving 21-point average of above white noise series") acf(Vt21, main = "Correlogram 21-point moving average", lag.max = 40, na.action = na.pass) par(mfrow = c(1,1))

• 權(quán)重移動平均

2. 指數(shù)平滑法

4、季節(jié)性

Seasonality means an effect that happens at the same time and with the same magnitude and direction every year. 從而使得經(jīng)過季節(jié)調(diào)整的序列能夠較好的反應社會經(jīng)濟指標運行基本態(tài)勢。

1.Method1: 減掉趨勢成分后取平均

## Investigating seasonality library(fields) z = matrix(fit$resid, ncol=12, byrow=TRUE) colnames(z) = c("Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec") bplot(z, xlab="Month", ylab="Detrended demand of gasoline", main="Annual seasonality of the detrended demand of gasoline") mean=matrix(0, nrow=ncol(z), ncol=1) #mean for each month for(i in 1:ncol(z)){mean[i,1]=mean(z[,i]) } Seas = as.matrix(rep(mean,16)) # matrix with seasonal term des.gas = fit$resid - Seas # detrended and now deseasonized series par(mfrow=c(2,1)) plot(fit$resid, type='l', xlab="months", ylab="Residuals", main="Residuals OLS [yt-fit(yt)]") abline(a=0, b=0) plot(des.gas, type='l', xlab="months", ylab="Deseasonized residuals", main="Deseasonized ResidualsOLS") abline(a=0, b=0)

2.Method2：使用啞元變量做回歸分析

##seasonal modeling and estimating seasonal trends ##such as for the average monthly temperature data library(TSA) #install package TSA win.graph(width=4.875, height=2.5,pointsize=8) data(tempdub); plot(tempdub,ylab='Temperature',type='o') month.=season(tempdub) # period added to improve table display model2=lm(tempdub~month.) # -1 removes the intercept term summary(model2) Call: lm(formula = tempdub ~ month.)Residuals:Min 1Q Median 3Q Max -8.2750 -2.2479 0.1125 1.8896 9.8250 Coefficients:Estimate Std. Error t value Pr(>|t|) (Intercept) 16.608 0.987 16.828 < 2e-16 *** month.February 4.042 1.396 2.896 0.00443 ** month.March 15.867 1.396 11.368 < 2e-16 *** month.April 29.917 1.396 21.434 < 2e-16 *** month.May 41.483 1.396 29.721 < 2e-16 *** month.June 50.892 1.396 36.461 < 2e-16 *** month.July 55.108 1.396 39.482 < 2e-16 *** month.August 52.725 1.396 37.775 < 2e-16 *** month.September 44.417 1.396 31.822 < 2e-16 *** month.October 34.367 1.396 24.622 < 2e-16 *** month.November 20.042 1.396 14.359 < 2e-16 *** month.December 7.033 1.396 5.039 1.51e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 3.419 on 132 degrees of freedom Multiple R-squared: 0.9712, Adjusted R-squared: 0.9688 F-statistic: 405.1 on 11 and 132 DF, p-value: < 2.2e-16

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