Kolmogorov
distribution
3
2
1
0
Sample Quantiles
4
5
Exponential QQ-plot
with confidence band
0
1
2
3
Exponential Quantiles
4
5
Why a 45° line?
Theoretical Q-Q-plot is
What if F = G?
Climate
The weather aspect of climate:
Climate is the distribution of
weather
Weather is a draw from this
distribution
Thus climate covers extremes,
means, variability, skewness etc.
of weather
The distribution is changing over
time: climate change means a
changing distribution
Data and models
Five major global annual mean
temperature data sets
Differ on how they compute
average, how they deal with
missing data etc.
CMIP5 has 45 different climate
models. Historical runs in 38 of
them, projections for 4 different
scenarios for 2000-2100.
15
14
13
12
Temperature (°C)
16
Climate models
and data
1880
1900
1920
1940
1960
1980
2000
Year
1.0
0.5
-0.5
0.0
NCEI
GISS
Berkeley
Hadley
JMA
-1.0
Temperature anomaly (°C)
Reference period 1960-89
1850
1900
1950
Year
2000
Q-Q-plots
1970-1999
0.4
0.0
0.2
NCEI data
0.0
-0.1
-0.2
-0.2
-0.3
NCEI data
0.1
1940-1969
-0.4
-0.2
0.0
0.2
CMIP5 models
0.4
-0.4
-0.2
0.0
0.2
0.4
CMIP5 models
0.6
0.8
But how can we do this
when climate changes?
If Xi~Fi what can we say about the
edf?
Running value
of the KS-statistic
3
2
1
0
Scaled KS Statistic
4
NCEI
1920
1940
1960
1980
2000
1980
2000
End year
3
2
1
0
Scaled KS Statistic
4
NCEI
1920
1940
1960
End year
The warming “hiatus”
1998 was a year of unusually
warm temperatures.
Looking at a 16-year trend from
then on seems to show a lower
trend than earlier.
But 16 years is a pretty short
period in a climate context (we
usually look at 30 year stretches).
An Act of Dog
Do climate models
reproduce recent data?
1.0
0.5
0.0
Temperature anomaly (°C)
1.5
The models simulated since 2000
1985
1990
1995
2000
Year
2005
2010
2015
Q-Q-plot for
recent data
0.4
0.3
0.2
0.1
NCEI data
0.5
0.6
0.7
RCP 4.5
0.0
0.5
CMIP5 models
1.0
1.5
More about
the Q-Q-plot
Recall thet the graph of a function
f is (x,f(x)) for all values of x in the
domain of f.
The theoretical Q-Q plot is the
graph of G-1(F)
Let X ~ F. What is the distribution
of G-1(F(X))
T(x)=G-1(F(x)) is sometimes called
the treatment function.
X1,X2,...,Xn controls (untreated)
Y1,Y2,...,Ym treated
Location-scale models
Let Z be a random variable. For
real a and positive b let X = a + bZ.
E(X )=
Var(X )=
Cdf of X?
Quantile function of X?
Treatment function?
Location-scale family
b=1
a=0
Residual from Q-Q-plot
is called the shift function.
X + Δ(X) ~
Generalized
location shift
Estimated using edf/eqf.
Confidence set:
where M=mn/(m+n) and Gm-I is the
right inverse.
If F = G we have Δ(x) = 0
0.4
0.3
0.2
0.0
0.1
1970-1999
0.6
0.4
0.2
0.0
-0.2
1970-1999
0.8
Back to climate change
-0.4
-0.2
0.0
0.2
1940-1969
0.4
-0.1
0.0
0.1
1940-1969
0.2
0.3
Looking at pictures
Old equipment
(30 frames/sec)
Analog data
and new
(1000 frames/sec)
Digital data
The image
Nose-bleed pictures
Watching a picture
Eye movement consists of
fixations–stops of the gaze
saccades–jumps from one fixation
to another
Saccades are extremely fast, and
once started cannot be interrupted
Questions of interest:
Difference between experienced
art viewers and novices
Difference between pictures
100 150 200
50
0
-100 -50
Shift towards non-experienced
Comparing fixation
durations for
experienced viewers
and novices
0
100
200
300
Experienced
400
400
500
500
Comparing viewing of
two pictures
Shiftplots for
fixation durations
50
0
Monet
150
Non-novices
0
100
200
300
400
500
400
500
Koli
20
-20
Monet
60
Novices
0
100
200
300
Koli