The Box-Cox Transformation and ARIMA Model Fitting · §4.3: Variance Stabilizing...
Transcript of The Box-Cox Transformation and ARIMA Model Fitting · §4.3: Variance Stabilizing...
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
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
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Mathematical Formulation
Suppose the variance of a time series Zt satisfies
var(Zt) = cf (µt)
We wish to find a transformation such that,T(·), such that var[T(Zt)] is constant.A first-order Taylor series of T(Zt) about µt is
T(Zt) ≈ T(µt) + T ′(µt)(Zt − µt)
Now var[T(Zt)] is approximated as
var [T(Zt)] ≈[T ′(µt)
]2 var(Zt) = c[T ′(µt)
]2 f (µt)
Therefore T(·) is chosen such that
T ′(µt) =1√
f (µt)
which implies
T(µt) =∫
1√f (µt)
dmut
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Mathematical Formulation
Suppose the variance of a time series Zt satisfies
var(Zt) = cf (µt)
We wish to find a transformation such that,T(·), such that var[T(Zt)] is constant.A first-order Taylor series of T(Zt) about µt is
T(Zt) ≈ T(µt) + T ′(µt)(Zt − µt)
Now var[T(Zt)] is approximated as
var [T(Zt)] ≈[T ′(µt)
]2 var(Zt) = c[T ′(µt)
]2 f (µt)
Therefore T(·) is chosen such that
T ′(µt) =1√
f (µt)
which implies
T(µt) =∫
1√f (µt)
dmut
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Mathematical Formulation
Suppose the variance of a time series Zt satisfies
var(Zt) = cf (µt)
We wish to find a transformation such that,T(·), such that var[T(Zt)] is constant.A first-order Taylor series of T(Zt) about µt is
T(Zt) ≈ T(µt) + T ′(µt)(Zt − µt)
Now var[T(Zt)] is approximated as
var [T(Zt)] ≈[T ′(µt)
]2 var(Zt) = c[T ′(µt)
]2 f (µt)
Therefore T(·) is chosen such that
T ′(µt) =1√
f (µt)
which implies
T(µt) =∫
1√f (µt)
dmut
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Mathematical Formulation
Suppose the variance of a time series Zt satisfies
var(Zt) = cf (µt)
We wish to find a transformation such that,T(·), such that var[T(Zt)] is constant.A first-order Taylor series of T(Zt) about µt is
T(Zt) ≈ T(µt) + T ′(µt)(Zt − µt)
Now var[T(Zt)] is approximated as
var [T(Zt)] ≈[T ′(µt)
]2 var(Zt) = c[T ′(µt)
]2 f (µt)
Therefore T(·) is chosen such that
T ′(µt) =1√
f (µt)
which implies
T(µt) =∫
1√f (µt)
dmut
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Mathematical Formulation
Suppose the variance of a time series Zt satisfies
var(Zt) = cf (µt)
We wish to find a transformation such that,T(·), such that var[T(Zt)] is constant.A first-order Taylor series of T(Zt) about µt is
T(Zt) ≈ T(µt) + T ′(µt)(Zt − µt)
Now var[T(Zt)] is approximated as
var [T(Zt)] ≈[T ′(µt)
]2 var(Zt) = c[T ′(µt)
]2 f (µt)
Therefore T(·) is chosen such that
T ′(µt) =1√
f (µt)
which implies
T(µt) =∫
1√f (µt)
dmut
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Mathematical Formulation
Suppose the variance of a time series Zt satisfies
var(Zt) = cf (µt)
We wish to find a transformation such that,T(·), such that var[T(Zt)] is constant.A first-order Taylor series of T(Zt) about µt is
T(Zt) ≈ T(µt) + T ′(µt)(Zt − µt)
Now var[T(Zt)] is approximated as
var [T(Zt)] ≈[T ′(µt)
]2 var(Zt) = c[T ′(µt)
]2 f (µt)
Therefore T(·) is chosen such that
T ′(µt) =1√
f (µt)
which implies
T(µt) =∫
1√f (µt)
dmut
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 3/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Box-Cox Transformation
Transforming the time series can suppress large fluctuations. The most standardtransformation is the log transformation where the new series yt is given by
yt = log xt
An alternative to the log transformation is the Box-Cox transformation:
yt =
{(xλ
t − 1)/λ, λ 6= 0ln xt, λ = 0
Many other transformations are suggested here.
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 4/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Box-Cox Transformation
Transforming the time series can suppress large fluctuations. The most standardtransformation is the log transformation where the new series yt is given by
yt = log xt
An alternative to the log transformation is the Box-Cox transformation:
yt =
{(xλ
t − 1)/λ, λ 6= 0ln xt, λ = 0
Many other transformations are suggested here.
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 4/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Box-Cox Transformation
Transforming the time series can suppress large fluctuations. The most standardtransformation is the log transformation where the new series yt is given by
yt = log xt
An alternative to the log transformation is the Box-Cox transformation:
yt =
{(xλ
t − 1)/λ, λ 6= 0ln xt, λ = 0
Many other transformations are suggested here.
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 4/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
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
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Box-Cox in R (II)
> bc<-boxcox(x~1)> lam<-bc$x[which.max(bc$y)]> lam
[1] 0.2222222
> truehist(BoxCox(x,lam))
> ts.plot(BoxCox(x,lam))
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 6/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
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
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Some Very Old Data
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§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Glacial Varves
variation in thickness ∝ amount deposited
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§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Transformed Glacial Varve Series
The transformation∇ log(varve) appears appropriate although fractionaldifferencing 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
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Transformed Glacial Varve Series
The transformation∇ log(varve) appears appropriate although fractionaldifferencing 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
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Transformed Glacial Varve Series
The transformation∇ log(varve) appears appropriate although fractionaldifferencing 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
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 11/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
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:ma1
-0.7705s.e. 0.0341
sigma^2 estimated as 0.2353:log likelihood = -440.72, aic = 885.44
> tsdiag(varve.ma)
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 12/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
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:ma1
-0.7705s.e. 0.0341
sigma^2 estimated as 0.2353:log likelihood = -440.72, aic = 885.44
> tsdiag(varve.ma)
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 12/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 13/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Fitting ARIMA(1,1,1) to Logged Varve Data
> pacf(varve.ma$resid, lwd=5)
> (varve.arma = arima(log(varve), order = c(1, 1, 1)))
Call:arima(x = log(varve), order = c(1, 1, 1))
Coefficients:ar1 ma1
0.2330 -0.8858s.e. 0.0518 0.0292
sigma^2 est as 0.2284: log likelihood = -431.44, aic = 868.88Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 14/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Fitting ARIMA(1,1,1) to Logged Varve Data
> pacf(varve.ma$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 ma1
0.2330 -0.8858s.e. 0.0518 0.0292
sigma^2 est as 0.2284: log likelihood = -431.44, aic = 868.88Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 14/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Varve ARIMA(1,1,1) Diagnostics
> tsdiag(varve.arma)
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§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
Watch Out for Overfitting!
Arthur Berg The Box-Cox Transformation and ARIMA Model Fitting 16/ 18
§4.3: Variance Stabilizing Transformations §6.1: ARIMA Model Identification Homework 3b
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