The Box-Cox Transformation and ARIMA Model Fitting

Transcription

1 The Box-Cox Transformation and ARIMA Model Fitting

2 Outline 1 4.3: Variance Stabilizing Transformations 2 6.1: ARIMA Model Identification 3 Homework 3b Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 2/ 18

3 Mathematical Formulation Suppose the variance of a time series Z t satisfies var(z t ) = cf (µ t ) We wish to find a transformation such that,t( ), such that var[t(z t )] is constant. A first-order Taylor series of T(Z t ) about µ t is Now var[t(z t )] is approximated as T(Z t ) T(µ t ) + T (µ t )(Z t µ t ) var [T(Z t )] [ T (µ t ) ] 2 var(zt ) = c [ T (µ t ) ] 2 f (µt ) Therefore T( ) is chosen such that T (µ t ) = 1 f (µt ) which implies T(µ t ) = 1 f (µt ) dmu t Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18

4 Mathematical Formulation Suppose the variance of a time series Z t satisfies var(z t ) = cf (µ t ) We wish to find a transformation such that,t( ), such that var[t(z t )] is constant. A first-order Taylor series of T(Z t ) about µ t is Now var[t(z t )] is approximated as T(Z t ) T(µ t ) + T (µ t )(Z t µ t ) var [T(Z t )] [ T (µ t ) ] 2 var(zt ) = c [ T (µ t ) ] 2 f (µt ) Therefore T( ) is chosen such that T (µ t ) = 1 f (µt ) which implies T(µ t ) = 1 f (µt ) dmu t Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18

5 Mathematical Formulation Suppose the variance of a time series Z t satisfies var(z t ) = cf (µ t ) We wish to find a transformation such that,t( ), such that var[t(z t )] is constant. A first-order Taylor series of T(Z t ) about µ t is Now var[t(z t )] is approximated as T(Z t ) T(µ t ) + T (µ t )(Z t µ t ) var [T(Z t )] [ T (µ t ) ] 2 var(zt ) = c [ T (µ t ) ] 2 f (µt ) Therefore T( ) is chosen such that T (µ t ) = 1 f (µt ) which implies T(µ t ) = 1 f (µt ) dmu t Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18

6 Mathematical Formulation Suppose the variance of a time series Z t satisfies var(z t ) = cf (µ t ) We wish to find a transformation such that,t( ), such that var[t(z t )] is constant. A first-order Taylor series of T(Z t ) about µ t is Now var[t(z t )] is approximated as T(Z t ) T(µ t ) + T (µ t )(Z t µ t ) var [T(Z t )] [ T (µ t ) ] 2 var(zt ) = c [ T (µ t ) ] 2 f (µt ) Therefore T( ) is chosen such that T (µ t ) = 1 f (µt ) which implies T(µ t ) = 1 f (µt ) dmu t Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18

7 Mathematical Formulation Suppose the variance of a time series Z t satisfies var(z t ) = cf (µ t ) We wish to find a transformation such that,t( ), such that var[t(z t )] is constant. A first-order Taylor series of T(Z t ) about µ t is Now var[t(z t )] is approximated as T(Z t ) T(µ t ) + T (µ t )(Z t µ t ) var [T(Z t )] [ T (µ t ) ] 2 var(zt ) = c [ T (µ t ) ] 2 f (µt ) Therefore T( ) is chosen such that T (µ t ) = 1 f (µt ) which implies T(µ t ) = 1 f (µt ) dmu t Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18

8 Mathematical Formulation Suppose the variance of a time series Z t satisfies var(z t ) = cf (µ t ) We wish to find a transformation such that,t( ), such that var[t(z t )] is constant. A first-order Taylor series of T(Z t ) about µ t is Now var[t(z t )] is approximated as T(Z t ) T(µ t ) + T (µ t )(Z t µ t ) var [T(Z t )] [ T (µ t ) ] 2 var(zt ) = c [ T (µ t ) ] 2 f (µt ) Therefore T( ) is chosen such that T (µ t ) = 1 f (µt ) which implies T(µ t ) = 1 f (µt ) dmu t Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18

9 Box-Cox Transformation Transforming the time series can suppress large fluctuations. The most standard transformation is the log transformation where the new series y t is given by y t = log x t An alternative to the log transformation is the Box-Cox transformation: { (xt λ 1)/λ, λ 0 y t = ln x t, λ = 0 Many other transformations are suggested here. Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 4/ 18

10 Box-Cox Transformation Transforming the time series can suppress large fluctuations. The most standard transformation is the log transformation where the new series y t is given by y t = log x t An alternative to the log transformation is the Box-Cox transformation: { (xt λ 1)/λ, λ 0 y t = ln x t, λ = 0 Many other transformations are suggested here. Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 4/ 18

11 Box-Cox Transformation Transforming the time series can suppress large fluctuations. The most standard transformation is the log transformation where the new series y t is given by y t = log x t An alternative to the log transformation is the Box-Cox transformation: { (xt λ 1)/λ, λ 0 y t = ln x t, λ = 0 Many other transformations are suggested here. Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 4/ 18

12 Box-Cox in R > library(mass) > library(forecast) > x<-rnorm(100)^2 > ts.plot(x) > truehist(x) Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 5/ 18

13 Box-Cox in R (II) > bc<-boxcox(x~1) > lam<-bc$x[which.max(bc$y)] > lam [1] > truehist(boxcox(x,lam)) > ts.plot(boxcox(x,lam)) Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 6/ 18

14 Outline 1 4.3: Variance Stabilizing Transformations 2 6.1: ARIMA Model Identification 3 Homework 3b Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 7/ 18

15 Some Very Old Data Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 8/ 18

16 Glacial Varves variation in thickness amount deposited Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 9/ 18

17 Transformed Glacial Varve Series The transformation log(varve) appears appropriate although fractional differencing may be in order. Let s take a closer look at log(varve). > varve = scan("mydata/varve.dat") > varve2=diff(log(varve)) > ts.plot(varve2) > acf(varve2,lwd=5) > pacf(varve2,lwd=5) Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 10/ 18

18 Transformed Glacial Varve Series The transformation log(varve) appears appropriate although fractional differencing may be in order. Let s take a closer look at log(varve). > varve = scan("mydata/varve.dat") > varve2=diff(log(varve)) > ts.plot(varve2) > acf(varve2,lwd=5) > pacf(varve2,lwd=5) Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 10/ 18

19 Transformed Glacial Varve Series The transformation log(varve) appears appropriate although fractional differencing may be in order. Let s take a closer look at log(varve). > varve = scan("mydata/varve.dat") > varve2=diff(log(varve)) > ts.plot(varve2) > acf(varve2,lwd=5) > pacf(varve2,lwd=5) Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 10/ 18

20 Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 11/ 18

21 Diagnostics of ARIMA(0,1,1) on Logged Varve Data > (varve.ma = arima(log(varve), order = c(0, 1, 1))) Call: arima(x = log(varve), order = c(0, 1, 1)) Coefficients: ma s.e sigma^2 estimated as : log likelihood = , aic = > tsdiag(varve.ma) Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 12/ 18

22 Diagnostics of ARIMA(0,1,1) on Logged Varve Data > (varve.ma = arima(log(varve), order = c(0, 1, 1))) Call: arima(x = log(varve), order = c(0, 1, 1)) Coefficients: ma s.e sigma^2 estimated as : log likelihood = , aic = > tsdiag(varve.ma) Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 12/ 18

23 Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 13/ 18

24 Fitting ARIMA(1,1,1) to Logged Varve Data > pacf(varve.ma$resid, lwd=5) > (varve.arma = arima(log(varve), order = c(1, 1, 1))) Call: arima(x = log(varve), order = c(1, 1, 1)) Coefficients: ar1 ma s.e sigma^2 est as : log likelihood = , aic = Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 14/ 18

25 Fitting ARIMA(1,1,1) to Logged Varve Data > pacf(varve.ma$resid, lwd=5) > (varve.arma = arima(log(varve), order = c(1, 1, 1))) Call: arima(x = log(varve), order = c(1, 1, 1)) Coefficients: ar1 ma s.e sigma^2 est as : log likelihood = , aic = Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 14/ 18

26 Varve ARIMA(1,1,1) Diagnostics > tsdiag(varve.arma) Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 15/ 18

27 Watch Out for Overfitting! Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 16/ 18

28 Outline 1 4.3: Variance Stabilizing Transformations 2 6.1: ARIMA Model Identification 3 Homework 3b Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 17/ 18

29 Homework 3b Read 5.1 and 5.2 of the textbook. Do exercise #4.5 on page 87 of the text. Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 18/ 18