statistics of climate extremes// trends in climate … · 3.examine all possible consecutive...
TRANSCRIPT
![Page 1: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/1.jpg)
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
![Page 2: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/2.jpg)
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
![Page 3: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/3.jpg)
3
![Page 4: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/4.jpg)
ESTIMATES FOR
HOUSTON HOBBY
AIRPORT
4
![Page 5: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/5.jpg)
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
![Page 6: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/6.jpg)
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
![Page 7: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/7.jpg)
COMBINED ESTIMATES
FOR 80 GULF COAST
STATIONS
7
![Page 8: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/8.jpg)
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
![Page 9: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/9.jpg)
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
![Page 10: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/10.jpg)
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
![Page 11: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/11.jpg)
RESULTS
11
![Page 12: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/12.jpg)
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
![Page 13: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/13.jpg)
−96 −94 −92 −90 −88 −86 −84 −82
25
26
27
28
29
30
31
Image Plot Based on SST Trends
lonpred
latp
red
0.0
0.5
1.0
1.5
13
![Page 14: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/14.jpg)
−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
![Page 15: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/15.jpg)
−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
![Page 16: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/16.jpg)
−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
![Page 17: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/17.jpg)
“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
![Page 18: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/18.jpg)
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
![Page 19: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/19.jpg)
−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
![Page 20: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/20.jpg)
−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
![Page 21: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/21.jpg)
−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
![Page 22: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/22.jpg)
−96 −94 −92 −90 −88 −86 −84 −82
25
26
27
28
29
30
31
Image Plot Based on Adjusted Linear Trends (noboot)
lonpred
latp
red
0.5
1.0
1.5
2.0
22
![Page 23: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/23.jpg)
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
![Page 24: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/24.jpg)
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
![Page 25: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/25.jpg)
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
![Page 26: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/26.jpg)
TIME SERIES ANALYSIS FORCLIMATE DATA
I Overview
II The post-1998 “hiatus” in temperature trends
III NOAA’s record “streak”
IV Trends or nonstationarity?
26
![Page 27: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/27.jpg)
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
![Page 28: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/28.jpg)
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
![Page 29: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/29.jpg)
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
![Page 30: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/30.jpg)
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
![Page 31: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/31.jpg)
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
![Page 32: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/32.jpg)
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
![Page 33: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/33.jpg)
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
![Page 34: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/34.jpg)
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
![Page 35: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/35.jpg)
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
![Page 36: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/36.jpg)
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
![Page 37: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/37.jpg)
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
![Page 38: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/38.jpg)
TIME SERIES ANALYSIS FORCLIMATE DATA
I Overview
II The post-1998 “hiatus” in temperature trends
III NOAA’s record “streak”
IV Trends or nonstationarity?
38
![Page 39: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/39.jpg)
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
![Page 40: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/40.jpg)
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
![Page 41: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/41.jpg)
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
![Page 42: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/42.jpg)
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
![Page 43: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/43.jpg)
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
![Page 44: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/44.jpg)
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
![Page 45: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/45.jpg)
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
![Page 46: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/46.jpg)
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
![Page 47: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/47.jpg)
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
![Page 48: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/48.jpg)
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
![Page 49: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/49.jpg)
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
![Page 50: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/50.jpg)
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
![Page 51: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/51.jpg)
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
aly Change of slope 1.16 deg/cen
(SE 0.4 deg/cen)
51
![Page 52: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/52.jpg)
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
![Page 53: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/53.jpg)
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
Change of slope 0.29 deg/cen(SE 0.54 deg/cen)
53
![Page 54: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/54.jpg)
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
Change of slope 0.74 deg/cen(SE 0.6 deg/cen)
54
![Page 55: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/55.jpg)
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
Change of slope 0.93 deg/cen(SE 1.24 deg/cen)
55
![Page 56: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/56.jpg)
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
![Page 57: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/57.jpg)
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
aly Change of slope 0.81 deg/cen
(SE 0.75 deg/cen)
57
![Page 58: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/58.jpg)
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
Change of slope 0.18 deg/cen(SE 0.61 deg/cen)
58
![Page 59: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/59.jpg)
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
Change of slope 0.24 deg/cen(SE 0.54 deg/cen)
59
![Page 60: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/60.jpg)
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
Change of slope 0.69 deg/cen(SE 0.58 deg/cen)
60
![Page 61: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/61.jpg)
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
Change of slope 0.99 deg/cen(SE 0.79 deg/cen)
61
![Page 62: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/62.jpg)
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
![Page 63: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/63.jpg)
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
![Page 64: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/64.jpg)
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
![Page 65: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/65.jpg)
TIME SERIES ANALYSIS FORCLIMATE DATA
I Overview
II The post-1998 “hiatus” in temperature trends
III NOAA’s record “streak”
IV Trends or nonstationarity?
65
![Page 66: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/66.jpg)
66
![Page 67: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/67.jpg)
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
![Page 68: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/68.jpg)
68
![Page 69: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/69.jpg)
69
![Page 70: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/70.jpg)
70
![Page 71: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/71.jpg)
71
![Page 72: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/72.jpg)
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
![Page 73: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/73.jpg)
73
![Page 74: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/74.jpg)
74
![Page 75: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/75.jpg)
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
![Page 76: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/76.jpg)
TIME SERIES ANALYSIS FORCLIMATE DATA
I Overview
II The post-1998 “hiatus” in temperature trends
III NOAA’s record “streak”
IV Trends or nonstationarity?
76
![Page 77: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/77.jpg)
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
![Page 78: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/78.jpg)
www.parliament.uk, April 22, 2013
78
![Page 79: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/79.jpg)
79
![Page 80: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/80.jpg)
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
![Page 81: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/81.jpg)
There is a tradition of this kind of research going back some
time
81
![Page 82: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/82.jpg)
82
![Page 83: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/83.jpg)
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
![Page 84: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/84.jpg)
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
![Page 85: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/85.jpg)
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
![Page 86: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/86.jpg)
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
![Page 87: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/87.jpg)
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
![Page 88: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/88.jpg)
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
![Page 89: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/89.jpg)
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
![Page 90: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/90.jpg)
Integrated Mean 0
Year
Res
idua
l
0 20 40 60 80 100
−0.
2−
0.1
0.0
0.1
0.2
Stationary Linear Trend
Year
Res
idua
l
0 20 40 60 80 100
−0.
2−
0.1
0.0
0.1
0.2
Integrated Nonlinear Trend
Year
Res
idua
l
0 20 40 60 80 100
−0.
10.
00.
1
Stationary Nonlinear Trend
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
![Page 91: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/91.jpg)
Integrated Mean 0
Year
Res
idua
l
0 20 40 60 80 100
−0.
2−
0.1
0.0
0.1
0.2
Stationary Linear Trend
Year
Res
idua
l
0 20 40 60 80 100
−0.
2−
0.1
0.0
0.1
0.2
Integrated Nonlinear Trend
Year
Res
idua
l
0 20 40 60 80 100
−0.
10.
00.
1
Stationary Nonlinear Trend
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
![Page 92: STATISTICS OF CLIMATE EXTREMES// TRENDS IN CLIMATE … · 3.Examine all possible consecutive sequences of monthly SST lags from 0{12 months (91 possible covariates) 4.Best SST model](https://reader034.vdocument.in/reader034/viewer/2022050602/5fa9f60c59922e6fd2028a50/html5/thumbnails/92.jpg)
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