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EVSC 495/EVAT 795 Data Analysis & Climate Change. Instructor: Michael E. Mann. Class hours: TuTh 2:00-3:15 pm. EVSC 495/EVAT 795 WEBPAGE http://holocene.evsc.virginia.edu/~mann/COURSES/495homeFall04.html SYLLABUS LECTURES COURSE INFORMATION PROBLEM SETS. - PowerPoint PPT Presentation
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EVSC 495/EVAT 795 Data Analysis & Climate
Change
Class hours: TuTh 2:00-3:15 pm
Instructor: Michael E. Mann
EVSC 495/EVAT 795 WEBPAGE
http://holocene.evsc.virginia.edu/~mann/COURSES/495homeFall04.html
•SYLLABUS•LECTURES
•COURSE INFORMATION•PROBLEM SETS
Although there is no ideal textbook for the class, the following book is helpful as supplementary material (two copies are available on reserve in the Science & Engineering library):
Statistical Methods in the Atmospheric Sciences, D. Wilks, 1995 (Academic Press)
INTRODUCTION Hypothesis testing; Probability; Distributions
LECTURE 1
Supplementary Readings: Wilks, chapters 1-4
•Characterizing spatial and temporal patterns of variation
Applications of statistics to the study of climate variability and climate change
•Comparing theoretical predictions and observations
•Detecting statistically significant trends
Applications of statistics to the study of climate variability and climate change
Combined global land air and sea surface temperatures 1860-1997 (relative to 1961-1990 average)
Global Temperature TrendsThe instrumental surface temperature record is not perfect.
The instrumental surface temperature record is not perfect.
Applications of statistics to the study of climate variability and climate change
Grayshade: 1902-1993
Checkerboard: 1854-1993
Note how much sparser the data is prior to prior to the 20th century...
Applications of statistics to the study of climate variability and climate change
Global Temperature TrendsThe instrumental surface temperature record is not perfect.
Note how much sparser the data is prior to the early 19th century...
Applications of statistics to the study of climate variability and climate change
Global Temperature Trends
Statistical methods can be used to estimate the associated uncertainty
Recent history of ENSO
phenomenon
Multivariate ENSO Index
(“MEI”)
Applications of statistics to the study of climate variability and climate change
“Explains” enhanced warming in certain regions of Northern Hemisphere in past couple decades
For the hemisphere on the whole, the warming or cooling due to the NAO is probably a zero-sum game (note that cooling is expected cooling over Greenland and most of Arctic sea, where no data is available
Applications of statistics to the study of climate variability and climate change
Null Hypothesis (H0)
Test Statistic
Alternative Hypothesis (HA)
Hypothesis Testing
Hypothesis Testing
Statistical Model
[Observed Data] = [Signal] + [Noise]
“noise” has to satisfy certain properties!
If not, we must iterate on this process...
Bayesian/Subjectivist vs Frequentist Approach
Frequency Approach
NasENa 0|}Pr{/|
(ie, fraction occurances/opportunities of an event converges to the probability of the event)
Bayesian Approach
Conditional Probability
21|Pr EE
Bayesian/Subjectivist vs Frequentist Approach
(Probability of E1 given the occurrence of E2)
Consider Mutually Exclusive and Collectively Exhaustive (MECE) set of events {Ei} and an event A
I
i iiEEAA
1}Pr{}|Pr{Pr
Bayes’ Theorem
A
iE
iEA
AEi Pr
Pr|Pr|Pr
Bayesian Approach
Bayesian/Subjectivist vs Frequentist Approach
I
i iiEEAA
1}Pr{}|Pr{Pr
Jj j
Ej
EAiE
iEA
AEi
1}Pr{}|Pr{
Pr|Pr|Pr
Bayes’ Theorem
PriorLikelihoodsPosterior Distribution
Bayesian Approach
Bayes’ Theorem
Prior
Jj j
Ej
EAiE
iEA
AEi
1}Pr{}|Pr{
Pr|Pr|Pr
Bayesian Approach
LikelihoodsPosterior Distribution
The central equation of Bayesian statistics combines the prior distribution and the likelihood function to reach the posterior distribution:
Coin Flipping Example
Binomial Distribution
nNpnpn
NniP
)1()()!(!
!nNn
N
n
N
(Probability Of Heads)
What is “p”?
Consider probability of obtaining seven heads in ten flips:
N=10; n=7 P(7)=0.12
Coin Flipping Example
Binomial Distribution
nNpnpn
NniP
)1()()!(!
!nNn
N
n
N
(Probability Of Heads)
What if the coin is weighted?
How does the frequentist deal with this issue?
Consider probability of obtaining seven heads in ten flips:
Coin Flipping Example
Binomial Distribution
nNpnpn
NniP
)1()()!(!
!nNn
N
n
N
(Probability Of Heads)
Weighted coin
1)1(1)()()()(
qxpxqpqpx
iP nNpnp
n
NpiP
)1()(
Coin Flipping Example
(Probability Of Heads)
Weighted coin
1)1(1)()()()(
qxpxqpqpx
iP
Beta Distribution
Coin Flipping Example
(Probability Of Heads)
Probability density function
nNpnpn
NpiP
)1()(
p
Bayesian analysis of a set of coin flips. The prior density was calculated assuming 20 heads from 40 tosses for a perfect coin (p = 0.5). The likelihood or data density was calculated assuming 7 heads from 10 tosses. The resulting posterior density is also plotted
Coin Flipping Example
(Probability Of Heads)
p
The posterior distribution for Pr(heads) peaks just above 0.5 because of the observed data of 7 heads from 10 tosses. The extent of the shift from the data value (0.7) is incorporated into the analysis by the form of the prior distribution. In any case, as the number of observations, n, increases, the resulting distribution, becomes more concentrated at the observed ratio of heads to tosses.
Probability Distributions and PDFs
•Binomial distribution
•Poisson distribution
•Gaussian distribution
•Chi-squared distribution
•Lognormal distribution
•Gamma Distribution
•Beta Distribution
Binomial Distribution
Describes the probability distribution of multiple independent events characterized
by a fixed rate of occurrence
•Coin Flipping
•Dice Rolling
nNpnpn
NniP
)1()()!(!
!nNn
N
n
N
•Precipitation Occurrence?
Binomial Distribution
nNpnpn
NniP
)1()()!(!
!nNn
N
n
N
Now consider the limit where p<<1 and N>>1
enn
niP
!1)(
Under these circumstances, we have the approximation:
pN(occurrence rate)
“Poisson” Distribution
enn
niP
!1)(
(mean occurrence rate)
pN
)(VAR
HistogramEstimate of PDF
Revisit the Binomial Distribution…
nNpnpn
NniP
)1()()!(!
!nNn
N
n
N
)!2/()!2/(!
snsnN
n
N
Let s=n-N/2
])!2/log[(])!2/log[(!lnln sNsNNn
N
Revisit the Binomial Distribution…
nNpnpn
NniP
)1()()!(!
!nNn
N
n
N
Now consider the limit where N>>1 (but p finite)
We now can make use of Stirling’s approximation:
)exp(2/1)2(! mmmmm
mmmm ln)2/1()2ln(2/1!ln
Revisit the Binomial Distribution…
])!2/ln[(])!2/ln[(!lnln sNsNNn
N
NsNNn
N/222ln)/2ln(2/1ln
)exp(2/1)2(! mmmmm
mmmm ln)2/1()2ln(2/1!ln
)/22exp(22/1)/2( NsNNn
N
Revisit the Binomial Distribution…
]/2)2/(2exp[22/1)/2( NNnNNn
N
])!2/ln[(])!2/ln[(!lnln sNsNNn
N
NsNNn
N/222ln)/2ln(2/1ln
)/22exp(22/1)/2( NsNNn
N
]/)(2exp[22/1)/2( 22
xNNx
N
nNpnpn
NniP
)1()()!(!
!nNn
N
n
N
2
21exp
21)(
xx
iP
Gaussian or “Normal” Distribution
]/)(2exp[22/1)/2( 22
xNNx
N
Gaussian or “Normal” Distribution
2
21exp
21)(
xx
iP
Z
Gaussian or “Normal” Distribution
El Nino, La Nina, and ‘La Nada’
1998 GLOBAL TEMPERATURE PATTERN
Heavily influenced by a huge El Nino
NINO3 (90-150W, 5S-5N)
NINO3 (90-150W, 5S-5N)
Variance
2s
Mean
Ni ix
Nx
11
Ni
xxN
si1
2
11
Standard Deviation
Histogram of Monthly Nino3 Index
Goodness of fit?
Topic of next lecture…