PRINCIPAL COMPONENT ANALYSIS

Global Sea Surface Temperatures
From voluntary ship observations:
Colors show the percentage of months
with at least one observation in a
2 by 2 degree grid box.
From paper in Annual Review
of Marine Science (2010)
PRINCIPAL COMPONENT ANALYSIS

Global Sea Surface Temperatures
Climatology 1982-2008
Red areas mark regions with highest
SST variability
PRINCIPAL COMPONENT ANALYSIS

Global Sea Surface Temperatures
Principal Component Analysis (PCA)
(Empirical Orthogonal Functions (EOF))
The first leading Eigenvector
Eigenvectors form now
geographic pattern. Grids with high
positive values and large negative
values are covarying out of phase
(negative correlation). Green regions
show small variations in this
Eigenvector #1.
The Principal Component is a time series
showing the temporal evolution of the SST
variations. This mode is associated
with the El Niño - Southern Oscillation
ANOTHER EXAMPLE OF PCA:
NORTH ATLANTIC OSCILLATION (NAO)
North Atlantic Sea Level Pressure (SLP) in winter season Dec-Mar
Explained variance
of the first EOF* mode:
41.9% of the gridded SLP
variability is represented
by the first eigenvector.
The eigenvector
The temporal evolution (Principal Component)
Hurrell, James & National Center for
Atmospheric Research Staff (Eds). Last
modified 02 Dec 2013. "The Climate Data
Guide: Hurrell North Atlantic Oscillation
(NAO) Index (station-based)." Retrieved
from
https://climatedataguide.ucar.edu/climat
e-data/hurrell-north-atlantic-oscillationnao-index-station-based
Note: EOF (“Empirical Orthogonal Functions” is the more popular term in atmospheric sciences)= PCA
CLIMATIC EFFECTS OF A THE NORTH ATLANTIC
OSCILLATION (DURING WINTER)
(Timm 2003)
EFFECTS OF NAO ON WINTER TEMPERATURES
A negative phase of the NAO (Weaker
Island Low and weaker Azores High)
causes cooler than normal winter
temperature. But other modes of
variability affect the NA winter climate
(Polar vortex , blocking events etc.)
“This latest cold outbreak was one case where they [the
Arctic Oscillation and NAO] were not in strong alignment;
in early January 2014, the NAO index was near zero.”
Quote from http://www.climate.gov/news-features/eventtracker/how-polar-vortex-related-arctic-oscillation
(retrieved 2014-04-22)
Source:
http://www.climate.gov/news-features/climate-current-events/winter-temperatures-influenced-north-atlanticoscillation-la (retrieved 2014-04-22)
WHAT IS NEEDED TO TEST A ‘HYPOTHESIS’
A scientific proposition / hypothesis
 A ‘controlled’ experiment (if possible)
 A measurement (repeated samples)
 A statistical formalism including a

 Null
hypothesis
 Alternative hypothesis (or hypotheses)
?
THE STATISTICAL PROBLEM:
Common phrases of hypotheses (propositions) : Are they suitable for statistical tests?
“We can expect a100-year extreme rainfall event to happen in the next 100 years!”
“The start of the growing season has shifted over the last decades!”
“Temperatures have gone up in the last 50 years!”
“Man-made climate change is real!”
“Rainfall extreme events have increased in the recent years!”
El Niño has an impact on NY winter climate!
“Smoking causes cancer!”
“Winter temperatures at Albany are more variable than in New York City!”
“There is a 50% chance that global warming is real!”
THE STATISTICAL PROBLEM:
“Statistical tests are designed to disprove a Null hypothesis”
“We can expect a100-year extreme rainfall event to happen in the next 100 years !”
“The start of the growing season has not shifted over the last decades!”
“Temperatures have not changed in the last 50 years!”
“Man-made climate change is real!” One can argue for it based on physical principles.
How strong and what changes to expect is another issue]
“Rainfall extreme events have not changed in the recent years!”
El Niño has no impact on NY winter climate!
“Smoking does not increase the risk of cancer!”
“Winter temperatures at Albany and in New York City have the same
variance”
“There is a 50% chance that global warming is real!”
THE STATISTICAL PROBLEM:
“Statistical tests are designed to disprove a hypothesis”
“We can expect a100-year extreme rainfall event to happen in the next 100 years!”
Let’s see what makes this propositions impossible to falsify statistically:
(1) First of all the phrase “we can expect” signals a subjective statement.
(Like last weeks false fire alarm in the building:
Probably most of us expected a false alarm,
but those who had sincere concerns were
not wrong in ‘expecting’ a real emergency situation.
[Or otherwise the fire alarm would be useless,
if it was not able to set off alarms in case of a fire]
(2) A hundred-year extreme rainfall event does not mean it must happen within 100 years.
(Like you cannot expect while throwing dices that ‘6’ appears on every 6th trial.)
THE STATISTICAL PROBLEM:
“Statistical tests are designed to disprove a hypothesis”
“There is a 99.9% chance that global warming is real!”
Let’s see what makes this propositions impossible to falsify statistically:
(1) First of all the phrase “there is a chance” signals: We do not consider a priori that
one or the other statement is a certain event or fact.
(2) The term ‘is real’ is too vague to express a null hypothesis that we could disprove and
then accept the above hypothesis!
(3) NOTE: Even if we had a 1% or 10 or 50% as a number, the problem would be the same.
But let’s assume we would try to falsify the null hypothesis:
“There is not a 99.9% chance that global warming is real!”
We would start looking for samples from around the world
and find ‘0.1%’ evidence against a global warming signal?
The warming hiatus of the last 10-12 years would be one good piece of evidence.
But is it sufficient? Is enough to argue there is more than 0.1% chance of
global warming not being real? Not with any objective statistical measure.
THE STATISTICAL PROBLEM:
“Statistical tests are designed to disprove a hypothesis”
“There is a 50% (or 99.9%) chance that global warming is real!”
This formulation (no matter how you choose the % number) in this type of proposition
is very hard to approach with statistical methods. The problem is with the word ‘real’.
What is ‘real’ or ‘not real’ is most often tied to a subjective ‘emotional’ a priori opinion.*
Better would be: “There is no warming trend in the global mean temperatures
from 1900-present.”
(* You are welcome to challenge this point of view!)
HYPOTHESIS TESTING
Example:
 http://video.pbs.org/video/2365222887/

From PBS NOVA:
 “Inside Animal Minds: Dogs & Super Senses”
(2014-04-22)
 How do birds avoid collisions with objects?
 (video from minute 27:30 to 33:00)

WHAT IS NEEDED TO TEST A ‘HYPOTHESIS’
Watch the scientific experiment shown in the
clip
 What is the scientific ‘proposition’?
 How is the experimental setup designed?
 What is the measurement?
 How often is the measurement repeated?

?
ANSWERS:
ANSWERS:
Experiment
Average
Speed
Sample size
Standard
Deviation
Horizontal
stripes
16.5ft/s
10
?
???
Vertical
stripes
15.3ft/s
10
?
???
TESTING THE SIGNIFICANCE OF THE DIFFERENCES IN THE SPEED
Experiment
Sample size
n
Standard
Deviation s
(guessed)
H. stripes
16.5ft/s
10
1.5
V. stripes
15.3ft/s
10
1
THE STATISTICAL PROBLEM:
The hypothesis is that there is a systematic difference between
the average flight speeds
in the two environments!
Statistical tests are traditionally formulated
the opposite way:
“There is no difference in the speed”.
All measured values are from the same statistical population
(distribution)
TESTING A NULL HYPOTHESIS
Hypothesis/Conclusion
Null hypothesis H0 true
Null hypothesis H0 false
Null hypothesis
accepcted
Correct decision
False decision
(Type II error)
Null hypothesis
rejected
False decision
(Type I error)
Correct decision