STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive...

STATISTICS OF CLIMATE EXTREMES//TRENDS IN CLIMATE DATASETS

Richard L SmithDepartments of STOR and Biostatistics, University

of North Carolina at Chapel Hilland

Statistical and Applied Mathematical Sciences Institute (SAMSI)

SAMSI Graduate ClassNovember 14, 2017

1

1

EXAMINE SST AS AN ALTERNATIVECOVARIATE

• Define Gulf Coast Region and 80 neighboring precipitation

stations (Houston Hobby in black)

• Calculate monthly mean SST for Gulf Coast Region

2

ESTIMATES FOR

HOUSTON HOBBY

AIRPORT

4

1. Use 7-day annual maximum precipitation (also tried 3, 4, 5,

6, 8 days: similar results)

2. Fit linear trend in GEV location parameter: p ≈ 0.02

3. Examine all possible consecutive sequences of monthly SST

lags from 0–12 months (91 possible covariates)

4. Best SST model uses means of lags 7, 8, 9, p ≈ 0.0007, but

this raises an obvious “selection bias” issue

5. I could not come up with any argument why the SST analysis

was better than the linear trend analysis after taking account

of the selection bias issue

6. Therefore, try a combined analysis over many stations

5

Here is a table of the 10 best models ordered by the deviance-based P-value,with the SST lags that are included in each.

Lags P-value7,8,9 0.00077,8 0.0009

7,8,9,10 0.00155,6,7,8,9,10 0.00256,7,8,9,10 0.0030

6,7,8,9 0.00305,6,7,8,9 0.0031

4,5,6,7,8,9,10 0.00358 0.0038

3,4,5,6,7,8,9,10 0.0043

Conclusion: Lags 6–10 all feature in many models

Difficult to quantify the effect of selection bias — Benjamini-Hochberg testfor the False Discovery Rate not valid because the tests are not independent

Even so, 10 P-values that are less than 0.005 (out of 91 models tested) seemsto be unlikely to occur by chance

6

COMBINED ESTIMATES

FOR 80 GULF COAST

STATIONS

7

1. Initial analysis not promising: many stations showed no statis-

ticially significant trend

2. Definition of SST variable: Settled on averages of lags 7–12

months

3. Based on 7-day maxima, 16 (out of 80) have a statistically

significant linear trend at p = 0.05, 13 have a statistically

significant SST trend at p = 0.005

4. Must be wary of selection bias issues, but these are still

greater numbers than could be explained “just by chance”

5. Therefore, proceed to a full spatial analysis

8

Spatial Analysis

(Similar to: D.M. Holland, V. De Oliveira, L.H. Cox and R.L. Smith (2000), Estimation of

regional trends in sulfur dioxide over the eastern United States. Environmetrics 11, 373-393.)

Let Z be “full spatial field” of the linear trend (Z(s) is true trendat location s).

Let Z be estimated spatial field at stations s1, ..., sn. Model:

Z | Z ∼ N (Z, W) ,

Z ∼ N (0, V(θ))

where W is assumed known and V(θ) is the covariance matrix ofsome spatial field — I use exponential covariance matrix whereθ is range. So

Z ∼ N (0, W + V(θ))

and Z may be reconstructed from Z by kriging

(Estimation of W uses bootstrapping)

9

I estimated this model using

1. Z is vector of linear trend estimates at each station

2. Z is vector of SST trend estimates at each station

3. Also considered adjusted analysis: put in both linear trend

and residuals from SSTs regressed on linear trend (two co-

variates in same model, kriging done separately for each)

10

RESULTS

11

Estimates of Overall Trend Parameter

Model Estimate Standard Error t Statistics P-valueSST alone 0.67 0.23 2.91 0.004

SST adjusted 0.53 0.18 2.86 0.004Linear alone 1.01 0.66 1.52 0.13

Linear adjusted 1.03 0.31 3.345 0.0008

12

−96 −94 −92 −90 −88 −86 −84 −82

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26

27

28

29

30

31

Image Plot Based on SST Trends

lonpred

latp

red

0.0

0.5

1.0

1.5

13

−96 −94 −92 −90 −88 −86 −84 −82

25

26

27

28

29

30

31

Image Plot Based on Adjusted SST Trends

lonpred

latp

red

−0.5

0.0

0.5

1.0

1.5

14

−96 −94 −92 −90 −88 −86 −84 −82

25

26

27

28

29

30

31

Image Plot Based on Linear Trends

lonpred

latp

red

0.5

1.0

1.5

2.0

15

−96 −94 −92 −90 −88 −86 −84 −82

25

26

27

28

29

30

31

Image Plot Based on Adjusted Linear Trends

lonpred

latp

red

0.0

0.5

1.0

1.5

2.0

16

“Noboot” option

Instead of using the bootstrap to estimate the matrix W, we sim-

ply assumed W was diagonal, with diagonal entries corresponding

to the squares of the standard errors in the GEV fitting proce-

dure (in other words: estimates are independent from station to

station)

The spatial model fitting procedure was repeated, and the new

images plotted.

The results were very little different.

17

Estimates of Overall Trend Parameter

Model Estimate Standard Error t Statistics P-valueSST alone 0.61 0.21 2.92 0.003

SST adjusted 0.54 0.21 2.58 0.01Linear alone 0.81 0.80 1.02 0.31

Linear adjusted 1.01 0.47 2.13 0.03

18

−96 −94 −92 −90 −88 −86 −84 −82

25

26

27

28

29

30

31

Image Plot Based on SST Trends (noboot)

lonpred

latp

red

0.0

0.5

1.0

19

−96 −94 −92 −90 −88 −86 −84 −82

25

26

27

28

29

30

31

Image Plot Based on Adjusted SST Trends (noboot)

lonpred

latp

red

−0.5

0.0

0.5

1.0

1.5

20

−96 −94 −92 −90 −88 −86 −84 −82

25

26

27

28

29

30

31

Image Plot Based on Linear Trends (noboot)

lonpred

latp

red

0.5

1.0

1.5

2.0

21

−96 −94 −92 −90 −88 −86 −84 −82

25

26

27

28

29

30

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Image Plot Based on Adjusted Linear Trends (noboot)

lonpred

latp

red

0.5

1.0

1.5

2.0

22

To do:

1. Rerun the bootstrap procedure to correct for a possible error

in constructing the bootstrap samples (discussed during the

talk)

2. Interpolate α, σ, ξ in same way and hence construct esti-

mates of threshold exceedance probabilities at different loca-

tions under the spatially interpolated model

3. Remark: Direct interpolation of threshold probabilities from

site to site doesn’t work; estimates are too variable for the

spatial model to fit

23

CONCLUSIONS

1. “Adjusted” model shows a clear separation in contributionsof the linear and SST components

2. SST component seems particularly concentrated on Houston,whether adjusted or not

3. Linear trend shows a less definitive pattern

4. Future work:

(a) Draw similar plots for estimated probabilities of a Harvey-level exceedance

(b) Compare extreme value probabilities for different dates(e.g. 2017 v. 1950)

(c) Integrate with CMIP5 model projections for future ex-treme probabilities

(d) POT alternative analysis?

24

TIME SERIES ANALYSIS FORCLIMATE DATA

I Overview

II The post-1998 “hiatus” in temperature trends

III NOAA’s record “streak”

IV Trends or nonstationarity?

25


I Overview




26

1900 1920 1940 1960 1980 2000

−0.

4−

0.2

0.0

0.2

0.4

HadCRUT4−gl Global Temperatureswww.cru.uea.ac.uk

Year

Glo

bal T

empe

ratu

re A

nom

aly Slope 0.74 degrees/century

OLS Standard error 0.037

27

What’s wrong with that picture?

• We fitted a linear trend to data which are obviously autocor-related

• OLS estimate 0.74 deg C per century, standard error 0.037

• So it looks statistically significant, but question how standarderror is affected by the autocorrelation

• First and simplest correction to this: assume an AR(1) timeseries model for the residual

• So I calculated the residuals from the linear trend and fittedan AR(1) model, Xn = φ1Xn−1 + εn, estimated φ1 = 0.62with standard error 0.07. With this model, the standard errorof the OLS linear trend becomes 0.057, still making the trendvery highly significant

• But is this an adequate model?

28

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

ACF of Residuals from Linear Trend

Lag

AC

F

Sample ACF

AR(1)

29

Fit AR(p) of various orders p, calculate log likelihood, AIC, and

the standard error of the linear trend.

Model Xn =∑pi=1 φiXn−i + εn, εn ∼ N [0, σ2

ε ] (IID)

AR order LogLik AIC Trend SE0 72.00548 –140.0110 0.0361 99.99997 –193.9999 0.0572 100.13509 –192.2702 0.0603 101.84946 –193.6989 0.0694 105.92796 –199.8559 0.0825 106.12261 –198.2452 0.0796 107.98867 –199.9773 0.0867 108.16547 –198.3309 0.0898 108.16548 –196.3310 0.0899 108.41251 –194.8250 0.086

10 108.48379 –192.9676 0.087

30

Extend the calculation to ARMA(p,q) for various p and q: modelis Xn −

∑pi=1 φiXn−i = εn +

∑qj=1 θjεn−j, εn ∼ N [0, σ2

ε ] (IID)

AR order MA order0 1 2 3 4 5

0 –140.0 –177.2 –188.4 –186.4 –191.5 –192.01 –194.0 –193.0 –197.4 –195.5 –201.7 –199.82 –192.3 –193.0 –195.4 –199.2 –200.8 –199.13 –193.7 –197.2 –200.3 –197.9 –200.8 –200.14 –199.9 –199.6 –199.8 –197.8 –196.8 –197.45 –198.2 –198.8 –197.8 –195.8 –194.8 –192.86 –200.0 –198.3 –196.4 –195.7 –196.5 –199.67 –198.3 –196.3 –200.2 –199.1 –194.6 –197.38 –196.3 –195.8 –194.4 –192.5 –192.8 –196.49 –194.8 –194.4 –197.6 –197.9 –196.2 –194.4

10 –193.0 –192.5 –195.0 –191.2 –194.9 –192.4

SE of trend based on ARMA(1,4) model: 0.087 deg C per cen-tury

31

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

ACF of Residuals from Linear Trend

Lag

AC

F

Sample ACF

AR(6)

ARMA(1,4)

32

Calculating the standard error of the trend

Estimate β =∑ni=1wiXi, variance

σ2ε

n∑i=1

n∑j=1

wiwjρ|i−j|

where ρ is the autocorrelation function of the fitted ARMA model

Alternative formula (Bloomfield and Nychka, 1992)

Variance(β) = 2∫ 1/2

0w(f)s(f)df

where s(f) is the spectral density of the autocovariance functionand

w(f) =

∣∣∣∣∣∣n∑

j=1

wne−2πijf

∣∣∣∣∣∣2

is the transfer function33

Example based on Barnes and Barnes(Journal of Climate, 2015)

• They compared the OLS estimator of a linear regression withthe epoch estimator computed by taking the difference be-tween the first M and last M values of a series of length N ,for some M < N

2 . The epoch estimator is then rescaled sothat it is in the same units as the classical linear regressionestimator.

• Question: Which performs better under various time seriesassumptions on the underlying series?

• The following plot shows the transfer functions for N =100, M = 33, epoch estimator in red, OLS in blue.

• Generally the OLS estimator is better, but not if the serieshas a spectral peak near 0.03.

34

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

Frequency

Tran

sfer

Fun

ctio

n T

imes

100

0

Ratio of Mean Red and Blue Curves is 1.124972

35

What’s better than the OLS linear trendestimator?

Use generalized least squares (GLS)

yn = β0 + β1xn + un,

un ∼ ARMA(p, q)

Repeat same process with AIC: ARMA(1,4) again best

β = 0.73, standard error 0.10.

36

Calculations in R

ip=4

iq=1

ts1=arima(y2,order=c(ip,0,iq),xreg=1:ny,method=’ML’)

Coefficients:

ar1 ar2 ar3 ar4 ma1 intercept 1:ny

0.0058 0.2764 0.0101 0.3313 0.5884 -0.4415 0.0073

s.e. 0.3458 0.2173 0.0919 0.0891 0.3791 0.0681 0.0010

sigma^2 estimated as 0.009061: log likelihood = 106.8,

aic = -197.59

acf1=ARMAacf(ar=ts1$coef[1:ip],ma=ts1$coef[ip+1:iq],lag.max=150)

37


I Overview




38

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

4

HadCRUT4−gl Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

39

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

4

HadCRUT4−gl Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

40

1960 1970 1980 1990 2000 2010

0.0

0.2

0.4

0.6

NOAA Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

41

1960 1970 1980 1990 2000 2010

0.0

0.2

0.4

0.6

NOAA Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

42

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

GISS (NASA) Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

43

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

GISS (NASA) Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

44

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

Berkeley Earth Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

45

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

Berkeley Earth Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

46

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

Cowtan−Way Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

47

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

Cowtan−Way Temperature Anomalies 1960−2014

Year

Glo

bal T

empe

ratu

re A

nom

aly

48

Statistical Models

Let

• t1i: ith year of series

• yi: temperature anomaly in year ti

• t2i = (t1i − 1998)+

• yi = β0 + β1t1i + β2t2i + ui

• Simple linear regression (OLS): ui ∼ N [0, σ2] (IID)

• Time series regression (GLS): ui − φ1ui−1 − ... − φpui−p =

εi + θ1εi−1 + ...+ θqεi−q, εi ∼ N [0, σ2] (IID)

Fit using arima function in R

49

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

4

HadCRUT4−gl Temperature Anomalies 1960−2014OLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly Change of slope 0.85 deg/cen

(SE 0.50 deg/cen)

50

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

4

HadCRUT4−gl Temperature Anomalies 1960−2014GLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom


(SE 0.4 deg/cen)

51

1960 1970 1980 1990 2000 2010

0.0

0.2

0.4

0.6

NOAA Temperature Anomalies 1960−2014GLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly

Change of slope 0.21 deg/cen(SE 0.62 deg/cen)

52

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

GISS (NASA) Temperature Anomalies 1960−2014GLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly


53

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

Berkeley Earth Temperature Anomalies 1960−2014GLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly


54

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

Cowtan−Way Temperature Anomalies 1960−2014GLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly


55

Adjustment for the El Nino Effect

• El Nino is a weather effect caused by circulation changes in

the Pacific Ocean

• 1998 was one of the strongest El Nino years in history

• A common measure of El Nino is the Southern Oscillation

Index (SOI), computed monthly

• Here use SOI with a seven-month lag as an additional co-

variate in the analysis

56

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

4

HadCRUT4−gl With SOI Signal RemovedGLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom


(SE 0.75 deg/cen)

57

1960 1970 1980 1990 2000 2010

0.0

0.2

0.4

0.6

NOAA With SOI Signal RemovedGLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly


58

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

GISS (NASA) With SOI Signal RemovedGLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly


59

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

Berkeley Earth With SOI Signal RemovedGLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly


60

1960 1970 1980 1990 2000 2010

−0.

20.

00.

20.

40.

6

Cowtan−Way With SOI Signal RemovedGLS Fit, Changepoint at 1998

Year

Glo

bal T

empe

ratu

re A

nom

aly


61

Selecting The Changepoint

If we were to select the changepoint through some form of au-

tomated statistical changepoint analysis, where would we put

it?

62

1960 1970 1980 1990 2000 2010

0.00

0.04

0.08

0.12

HadCRUT4−gl Change Point Posterior Probability

Year

Pos

terio

r P

roba

bilit

y of

Cha

ngep

oint

63

Conclusion from Temperature Trend Analysis

• No evidence of decrease post-1998 — if anything, the trend

increases after this time

• After adjusting for El Nino, even stronger evidence for a

continuously increasing trend

• If we were to select the changepoint instead of fixing it at

1998, we would choose some year in the 1970s

• Thus: No statistical evidence to support the “hiatus” hy-

pothesis

64


I Overview




65

Continental US monthly temperatures, Jan 1895–Oct 2012.

For each month between June 2011 and Sep 2012, the monthly

temperature was in the top tercile of all observations for that

month up to that point in the time series. Attention was first

drawn to this in June 2012, at which point the series of top

tercile events was 13 months long, leading to a naıve calculation

that the probability of that event was (1/3)13 = 6.3 × 10−7.

Eventually, the streak extended to 16 months, but ended at that

point, as the temperature for Oct 2012 was not in the top tercile.

In this study, we estimate the probability of either a 13-month or

a 16-month streak of top-tercile events, under various assump-

tions about the monthly temperature time series.

67

Method

• Two issues with NOAA analysis:

– Neglects autocorrelation

– Ignores selection effect

• Solutions:

– Fit time series model – ARMA or long-range dependence

– Use simulation to determine the probability distribution ofthe longest streak in 117 years

• Some of the issues:

– Selection of ARMA model — AR(1) performs poorly

– Variances differ by month — must take that into account

– Choices of estimation methods, e.g. MLE or Bayesian —Bayesian methods allow one to take account of parameterestimation uncertainty

72

Conclusions

• It’s important to take account of monthly varying standarddeviations as well as means.

• Estimation under a high-order ARMA model or fractionaldifferencing lead to very similar results, but don’t use AR(1).

• In a model with no trend, the probability that there is asequence of length 16 consecutive top-tercile observationssomewhere after year 30 in the 117-year time series is ofthe order of 0.01–0.03, depending on the exact model beingfitted. With a linear trend, these probability rise to somethingover .05. Include a nonlinear trend, and the probabilities areeven higher — in other words, not surprising at all.

• Overall, the results may be taken as supporting the over-all anthropogenic influence on temperature, but not to astronger extent than other methods of analysis.

75


I Overview




76

A Parliamentary Question is a device where any member of the

U.K. Parliament can ask a question of the Government on any

topic, and is entitled to expect a full answer.

77

www.parliament.uk, April 22, 2013

78

Essence of the Met Office Response

• Acknowledged that under certain circumstances an ARIMA(3,1,0)

without drift can fit the data better than an AR(1) model

with drift, as measured by likelihood

• The result depends on the start and finish date of the series

• Provides various reasons why this should not be interpreted

as an argument against climate change

• Still, it didn’t seem to me (RLS) to settle the issue beyond

doubt

80

There is a tradition of this kind of research going back some

time

81

Summary So Far

• Integrated or unit root models (e.g. ARIMA(p, d, q) with d =

1) have been proposed for climate models and there is some

statistical support for them

• If these models are accepted, the evidence for a linear trend

is not clear-cut

• Note that we are not talking about fractionally integrated

models (0 < d < 12) for which there is by now a substantial

tradition. These models have slowly decaying autocorrela-

tions but are still stationary

• Integrated models are not physically realistic but this has not

stopped people advocating them

• I see the need for a more definitive statistical rebuttal

83

HadCRUT4 Global Series, 1900–2012

Model I : yt − yt−1 = ARMA(p, q) (mean 0)

Model II : yt = Linear Trend + ARMA(p, q)

Model III : yt − yt−1 = Nonlinear Trend + ARMA(p, q)

Model IV : yt = Nonlinear Trend + ARMA(p, q)

Use AICC as measure of fit

84

Integrated Time Series, No Trend

p q0 1 2 3 4 5

0 –165.4 –178.9 –182.8 –180.7 –187.7 –185.81 –169.2 –181.3 –180.7 –184.1 –186.3 –184.42 –176.0 –182.8 –185.7 –182.7 –184.7 –184.43 –185.5 –184.2 –185.2 –183.0 –184.4 –184.04 –183.5 –181.5 –183.0 –180.7 –181.5 NA5 –185.2 –183.1 –181.0 –185.8 –183.6 –182.5

85

Stationary Time Series, Linear Trend

p q0 1 2 3 4 5

0 –136.1 –168.8 –178.2 –176.2 –180.9 –181.61 –183.1 –183.1 –186.8 –184.5 –190.8 –188.52 –181.3 –181.7 –184.5 –187.4 –189.2 –187.33 –182.6 –186.6 –188.9 –187.1 –189.3 –187.34 –189.7 –188.7 –188.4 –185.4 –185.1 NA5 –187.9 –187.6 –186.0 –183.0 –182.6 –183.8

86

Integrated Time Series, Nonlinear Trend

p q0 1 2 3 4 5

0 –156.8 –195.1 –201.7 –199.4 –207.7 –208.91 –161.4 –199.5 –199.4 –202.3 –210.3 –209.02 –169.9 –202.3 –210.0 –201.4 –209.7 –208.73 –183.2 –201.0 –203.5 –201.2 –207.3 –204.84 –180.9 –199.3 –201.2 –198.7 –205.3 NA5 –186.8 –201.7 –199.4 –207.7 –204.8 –204.8

87

Stationary Time Series, Nonlinear Trend

p q0 1 2 3 4 5

0 –199.1 –204.6 –202.4 –217.8 –216.9 –215.91 –202.6 –202.3 –215.2 –217.7 –216.1 –214.72 –205.2 –217.3 –205.0 –216.6 –214.1 –213.33 –203.8 –205.9 –203.6 –214.3 –211.7 –213.54 –202.2 –203.5 –213.7 –212.0 –227.1 NA5 –205.7 –203.2 –216.2 –233.3 –212.6 –226.5

88

1900 1920 1940 1960 1980 2000

−0.

2−

0.1

0.0

0.1

0.2

0.3

Integrated Mean 0

Year

Ser

ies

1900 1920 1940 1960 1980 2000

−0.

4−

0.2

0.0

0.2

0.4

Stationary Linear Trend

Year

Ser

ies

1900 1920 1940 1960 1980 2000

−0.

2−

0.1

0.0

0.1

0.2

0.3

Integrated Nonlinear Trend

Year

Ser

ies

1900 1920 1940 1960 1980 2000

−0.

4−

0.2

0.0

0.2

0.4

Stationary Nonlinear Trend

Year

Ser

ies

Four Time Series Models with Fitted Trends

89

Integrated Mean 0

Year

Res

idua

l

0 20 40 60 80 100

−0.

2−

0.1

0.0

0.1

0.2


Year

Res

idua

l

0 20 40 60 80 100

−0.

2−

0.1

0.0

0.1

0.2


Year

Res

idua

l

0 20 40 60 80 100

−0.

10.

00.

1


Year

Res

idua

l

0 20 40 60 80 100

−0.

15−

0.05

0.05

0.15

Residuals From Four Time Series Models

90

Integrated Mean 0

Year

Res

idua

l

0 20 40 60 80 100

−0.

2−

0.1

0.0

0.1

0.2


Year

Res

idua

l

0 20 40 60 80 100

−0.

2−

0.1

0.0

0.1

0.2


Year

Res

idua

l

0 20 40 60 80 100

−0.

10.

00.

1


Year

Res

idua

l

0 20 40 60 80 100

−0.

15−

0.05

0.05

0.15

Residuals From Four Time Series Models

91

Conclusions

• If we restrict ourselves to linear trends, there is not a clear-cut preference between integrated time series models withouta trend and stationary models with a trend

• However, if we extend the analysis to include nonlinear trends,there is a very clear preference that the residuals arestationary, not integrated

• Possible extensions:

– Add fractionally integrated models to the comparison– Bring in additional covariates, e.g. circulation indices and

external forcing factors– Consider using a nonlinear trend derived from a climate

model. That would make clear the connection with de-tection and attribution methods which are the preferredtool for attributing climate change used by climatologists.

92

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