Sildes buenos aires

Arthur CHARPENTIER - Granularity Issues of Climatic Time Series
Granularity Issues on Climatic Time Series
A. Charpentier (UQAM & Université de Rennes 1)
Buenos Aires, November 2015
Big Data & Environment Workshop
http://freakonometrics.hypotheses.org
@freakonometrics 1

Agenda
• the ‘period of return’ (in the context of climate change)
• ﬂood events dynamics, long or short memory?
• long memory and seasonality on wind speed
• long memory vs. fat tails for temperature time series
• detecting abnormalities and outliers on functional data
@freakonometrics 2

‘Period of Return’ in the context of Climate Data
Gumbel (1958). Statistics of Extremes. Columbia University Press
Let T be the time of ﬁrst success for some events occuring with yearly probabiliy
p, then
P[T = k] = (1 − p)k−1
p so that E[T] =
1
p
(geometric distribution, discrete version of the exponential distribution).
@freakonometrics 3

Models for River Levels and Flood Events
In hydrological papers, huge interest on Annual Maximum time series
• Hurst (1951) observed that annual maximum exhibit long-range dependence
(so called Hurst eﬀect),
• Gumbel (1958) observed that annual maximum were i.id with a similar
distribution (so called Gumbel distribution)
How could it be identical series be at the same time independent and with
long-range dependence? Hurst (1951) used 700 years of data on the Nile,
Gumbel (1958) used European data, over less than a century.
Can’t we use more data to model ﬂood events?
@freakonometrics 4

Flood Events
@freakonometrics 5

High Frequency Models (for Financial Data)
On ﬁnancial data,
• “traditional” approach (time series): consider the closing data price, Xt at
the end of day t, i.e. regularly spaced observations,
• “high frequency daya”: the price X is observed at each transaction: let Ti
denote the data of the ith transaction, and Xi the price paid.
See e.g. ACD - Autoregressive Conditional Duration models, introduced par in
Engle & Russell (1998).
In practice, three information are stored: (1) date of transaction, or time between
two consecutive transactions, on the same stock; (2) the volume, i.e. number of
stocks sold and bought (3) the price, i.e. individual stock price (or total price
exchanged)
@freakonometrics 6

Flood Events
The analogous of a transaction is a flood event, where 4 variables are kept,
• time length of the flood event
• time between two consecutive flood events
• volume Vi
• peak Pi
Remark: see Todorovic & Zelenhasic (1970) and et Todorovic &
Rousselle (1970) where marked Poisson processes were considered.
@freakonometrics 7

Some ‘Optimal’ Threshold
The choice of the threshold is crucial. Standard tradeoff
• should be low to have more events
• should be high to have significant flood events
Standard technique in hydrology: given some
function f (e.g. f affine), solve
u = argmax{P(X > f(u)|X > u)}
or its empirical couterpart
u = argmax
#{Xi > f(u)}
#{Xi > u}
with e.g. f(x) = 1, 5x + 5.
@freakonometrics 8

A Two-Duration Model
Engle & Lunde (2003) in trades and quotes: a bivariate point process, consider
a two duration model, that can be used here.
The two dates are Ti beginning of ith flood, and Ti end of the flood. Set
• Xi = Ti+1 − Ti the time length between the begining of two consecutive
floods
• Yi = Ti+1 − Ti the time length between the end of a flood and the begining
of the next one
@freakonometrics 9

Engle & Russell (1998) ACD(p, q) Model
In the one-duration model, let Xi denote the time lengths (Xi = Ti − Ti−1), and
Hi = {X1, ...., Xi−1}. Then



Xi = Ψi · εi, with (εi) i.i.d. noise
E(Xi|Hi−1) = Ψi = ω +
p
k=1
αkXi−k +
q
k=1
βkΨi−k,
i.e.
Xi = ω +
max{p,q}
k=1
(αk + βk)Xk −
q
k=1
βkηi−k + ηi,
where ηi = Xi − Ψi = Xi − E(Xi|Hi−1) (ARMA(max{p, q}, q) representation of
the ACD(p, q)).
In the Exponential ACD(1,1), (εi) is an exponential noise
E(Xi|Hi−1) = Ψi = θ + αXi + βΨi−1, with α, β ≥ 0 and θ > 0,
@freakonometrics 10

Engle & Russell (1998) ACD(p, q) Model
More generally, the conditional density of Xi is
f(x|Hi) =
1
Ψi(Hi, θ)
· gε
x
Ψi(Hi, θ)
e.g. gε(·) = exp[−·], if ε ∼ E(1).
Inference is very similar to GARCH(1,1), the proof being the same as the one in
Lee & Hansen (1994) and Lumsdaine (1996).
@freakonometrics 11

The Two-Duration Model
As in Engle & Lunde (2003), consider some two-EACD model,
f(xi|Hi) =
1
Ψi(Hi, θ1)
· exp −
xi
Ψi(Hi, θ1)
where
Ψi(Hi, θ1) = exp α + δ log(Ψi−1) + γ
Xi−1
Ψi−1
+ β1Pi−1 + β2Vi−1 ,
while
g(yi|xi, Hi) =
1
Φi(xi, Hi, θ2)
· exp −
yi
Φi(xi, Hi, θ2)
where
Φi(xi, Hi, θ2) = exp µ + ρ log(Φi−1) + γ
Yi−1
Φi−1
+ τ
xi
Ψi
+ η1Pi−1 + η2Vi−1 .
@freakonometrics 12

The Two-Duration Model
Deﬁne residuals
εi =
Xi
Ψi(Hi−1, θ1)
.
Since there are two kinds of ﬂoods, ordinary ones and those related to snow melt,
we should consider a mixture distribution for ε, a mixture of exponentials
f(x) = α · λ1 · e−λ1·x
+ (1 − α) · λ2 · e−λ2·x
, x > 0.
or a mixture of Weibull’s
f(x) = α · λ1 · θ−λ1
1 · xλ1−1
· e−(x/θ1)λ1
+ (1 − α) · λ2 · θ−λ2
2 · xλ2−1
· e−(x/θ2)λ2
@freakonometrics 13

Modeling Marks
Finally,
f(pi, vi, xi, yi|Hi−1) = g(pi, vi|Hi−1, xi, yi) · h(xi, yi|Hi−1).
which can be simplified using a triangle approximation,
Volume = Vi = Pi ·
Xi − Yi
2
=
peak × flood duration
2
,
Modeling Peaks
From the threshod based approach, use Pickands-Balkema-de Haan theorem and
fit a Generalized Pareto distribution
h(pi|Hi−1, xi, yi) = α
pi + b(xi − yi) + d
σ
−(1+α)
.
@freakonometrics 14

Application
In Charpentier & Sibaï, D. (2010), we considered a mixture of Weibull
distribution, ﬁtted using EM algorithm, see (conditional) QQ plot, exponential
vs. mixture of Weibull,
There is some dynamics, but not long memory here (from the EACD(1,1)
processes).
@freakonometrics 15

Long Memory and Wind Speed (very popular application)
Haslett & Raftery (1989). Space-time modelling with long-memory
dependence: assessing Ireland’s wind power resource (with discussion). Applied
Statistics. 38. 1-50.
@freakonometrics 16

Daily Wind Speed in Ireland, long memory, really?
@freakonometrics 17

Modeling Stationary Time Series
Given a stationary time series (Xt) , the autocovariance function, is
h → γX (h) = Cov (Xt, Xt−h) = E (XtXt−h) − E (Xt) · E (Xt−h)
for all h ∈ N, and its Fourier transform is the spectral density of (Xt)
fX (ω) =
1
2π
h∈Z
γX (h) exp (iωh)
for all ω ∈ [0, 2π]. Note that
fX (ω) =
1
2π
+∞
h=−∞
γX (h) cos (ωh)
Let ρX(h) denote the autocorrelation function i.e. ρX(h) = γX(h)/γX(0).
@freakonometrics 18

Long-Range Dependence
Stationary time series (Yt) has long range dependence if
∞
h=1
|ρX(h)| = ∞,
and short range dependence if the sum is bounded. E.g. ARMA processes have
short range dependence since
|ρ(h)| ≤ C · rh
, for h = 1, 2, ...
where r ∈]0, 1[.
A popular class of long memory processes is obtained when
ρ(h) ∼ C · h2d−1
as h → ∞,
where d ∈]0, 1/2[. This can be obtained with fractionary processes
(1 − L)d
Xt = εt,
@freakonometrics 19

where (εt) is some white noise. Here, (1 − L)d
is deﬁned as
(1 − L)d
= 1 − dL −
d(1 − d)
2!
L2
−
d(1 − d)(2 − d)
3!
L3
+ · · · =
∞
j=0
φjLj
,
where
φj =
Γ(j − d)
Γ(j + 1)Γ(−d)
=
0<k≤j
k − 1 − d
k
for j = 0, 1, 2, ...
If V ar(εt) = 1, note that par
γX(h) =
Γ(1 − 2d)Γ(h + d)
Γ(d)Γ(1 − d)Γ(h + 1 − d)
∼
Γ(1 − 2d)
Γ(d)Γ(1 − d)
· h2d−1
as h → ∞, and
fX(ω) = 2 sin
ω
2
−2d
∼ ω−2d
as ω → 0.
See also Mandelbrot et Van Ness (1968) for the continuous time version,
with the fractionary Brownian motion.
@freakonometrics 20

Daily Windspeed Time Series
@freakonometrics 21

Defining Long Range Dependence
Hosking (1981, 1984) suggested another definition of long range dependence:
(Xt) is stationnary, and there is ω0 such that fX(ω) → ∞ as ω → ω0.
Such a ω0 can be related to seasonality
Gray, Zhang & Woodward (1989) defined GARMA(p, d, q) processes,
inspired by Hosking (1981)
Φ(L)(1 − 2uL + L2
)d
Xt = Θ(L)εt
Hosking (1981) did not studied those processes since it is difficult to invert
(1 − 2uL + L2
)d
.
@freakonometrics 22

Deﬁning Long Range Dependence with Seasonality
This can be done using Gegenbauer polynomial: for d = 0, |Z| < 1 and |u| ≤ 1,
(1 − 2uL + L2
)−d
=
∞
i=0
Pi,d(u)Ln
,
where
Pi,d(u) =
[i/2]
k=0
(−1)k Γ(d + n − k)
Γ(d)
(2u)n−2k
[k!(n − 2k)!]
If |u| < 1, the limit of (ω − ω0)2d
f(ω) exists when ω → ω0, where ω0 = cos−1
(u).
Further, if |u| < 1 and 0 < d < 1/2, then
ρ(h) ∼ C · h2d−1
· cos(ω0 · h) as h → ∞.
In Bouëtte et al. (2003) we obtained on daily windspeed d ∼ 0, 18.
@freakonometrics 23

Estimation ‘Return Periods’
Using Gray, Zhang & Woodward (1989), it is possible to simulate GARMA
processes, to estimate probabilities
11 21 31
EXCEEDENCE PROBABILITIES
0.00
0.05
0.10
0.15
0.20
11 21 31
TWO CONSECUTIVE WINDY DAYS
0.00
0.05
0.10
0.15
0.20
GARMA (long memory)
seasonal + ARMA (short memory)
11 21 31
FOUR CONSECUTIVE WINDY DAYS
0.00
0.05
0.10
0.15
0.20
11 21 31
0.00
0.05
0.10
0.15
0.20
THREE CONSECUTIVE WINDY DAYS
FIVE CONSECUTIVE WINDY DAYS
@freakonometrics 24

Spectral Density of Hourly Wind Speed in the Netherlands
Some k factor GARMA should be considered, see (Bouëtte et al. (2003)
@freakonometrics 25

The European heatwave of 2003
Third IPCC Assessment, 2001: treatment of extremes (e.g. trends in extreme
high temperature) is “clearly inadequate”. Karl & Trenberth (2003) noticed
that “the likely outcome is more frequent heat waves”, “more intense and longer
lasting” added Meehl & Tebaldi (2004).
In Nîmes, there were more than 30 days with temperatures higher than 35◦
C
(versus 4 in hot summers, and 12 in the previous heat wave, in 1947).
Similarly, the average maximum (minimum) temperature in Paris peaked over
35◦
C for 10 consecutive days, on 4-13 August. Previous records were 4 days in
1998 (8 to 11 of August), and 5 days in 1911 (8 to 12 of August).
Similar conditions were found in London, where maximum temperatures peaked
above 30◦
C during the period 4-13 August
(see e.g. Burt (2004), Burt & Eden (2004) and Fink et al. (2004).)
@freakonometrics 26

Minimum Daily Temperature in Paris, France
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101520
Temperaturein°C
juil. 02 juil. 12 juil. 22 août 01 août 11 août 21 août 31
@freakonometrics 27

Modelling the Minimum Daily Temperature
Karl & Knight (1997) , modeling of the 1995 heatwave in Chicago: minimum
temperature should be most important for health impact (see also Kovats &
Koppe (2005)), several nights with no relief from very warm nighttime
@freakonometrics 28

Instead of boxplots, consider some quantile regression
@freakonometrics 29

Note that the slope for various probability levels is rather stable
unless we focus on heat-waves,
@freakonometrics 30

Which temperature might be interesting ?
Consider the following decomposition
Yt = µt + Xt
where
• µt is a (linear) general tendency
• Xt is the remaining (stationary) noise
@freakonometrics 31

Nonstationarity and linear trend
Consider a spline and lowess regression
@freakonometrics 32

Nonstationarity and linear trend
or a polynomial regression,and compare local slopes,
@freakonometrics 33

Linear trend, and Gaussian noise
Benestad (2003) or Redner & Petersen (2006) suggested that temperature
for a given (calendar) day is an “independent Gaussian random variable with
constant standard deviation σ and a mean that increases at constant speed ν”
In the U.S., ν = 0.03◦
C per year, and σ = 3.5◦
C
In Paris, ν = 0.027◦
C per year, and σ = 3.23◦
C
@freakonometrics 34

The Seasonal Component
There is a seasonal pattern in the daily temperature
@freakonometrics 35

The Residual Part (or stationary component)
Let Xt = Yt − β0 + β1t + St
@freakonometrics 36

The Residual Part (or stationary component)
Xt might look stationary,
One can consider some short-range dependence (ARMA) model, with either light
or heavy tailed innovation process.
@freakonometrics 37

Long range dependence ?
Smith (1993) “we do not believe that the autoregressive model provides an
acceptable method for assessing theses uncertainties” (on temperature series)
Dempster & Liu (1995) suggested that, on a long period, the average annual
temperature should be decomposed as follows
• an increasing linear trend,
• a random component, with long range dependence.
Consider some GARMA time series, as in Charpentier (2011).
@freakonometrics 38

Long range dependence ?
@freakonometrics 39

On return periods, optimistic scenario
@freakonometrics 40

On return periods, pessimistic scenario
@freakonometrics 41

Long Memory, non Stationarity and Temporal Granularity
Hourly Temperature in Montreal, QC, in January,
@freakonometrics 42

Hourly Temperature as a Random Walk?
Use of various test to test for integrated time series (random walk)
• ADF, Augmented Dickey-Fuller, see Fuller (1976) and Said & Dickey
(1984)
• KPSS, Kwiatkowski–Phillips–Schmidt–Shin, see Kwiatkowski et al. (1992)
• PP, Phillips–Perron, see Phillips & Perron (1988)
where random-walk vs. stationnary
@freakonometrics 43

Detecting Abnormalities and Outliers
Consider the case where Xi,t denote the temperature at date/time t, for year i.
Let ϕ1,t, ϕ2,t, ϕ3,t, · · · denote the principal components, and Yi,1, Yi,2, Yi,3, · · · the
principal component scores.
To detect outliers, as in Jones & Rice (1992), Sood et al. (2009) or Hyndman &
Shang (2010) use a bivariate depth plot on {(Y1,i, Y2,i), i = 1, · · · , n}.
E.g. monthly sea surface temperatures,
from January 1950 to December 2006
@freakonometrics 44

The ﬁrst two components are
2 4 6 8 10 12
−20−1001020
ElNino$x
P$ind$coord[,1]
2 4 6 8 10 12
−4−202
ElNino$x
P$ind$coord[,2]
And we can use a depth plot on the ﬁrst two principal component scores.
@freakonometrics 45

−4 −2 0 2 4 6 8 10
−20246
PC score 1
PCscore2
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1973
1982
1983
1987
1997
1998
2 4 6 8 10 12
2022242628 Month
Seasurfacetemperature
1973
1982
1983
1987
1997
1998
@freakonometrics 46

Depth Set and Bag Plot
Here we use Tukey’s depth set concept. In dimension 1, deﬁne
depth(y) = min{F(y), 1 − F(y)}
and the associated depth set of level α ∈ (0, 1) as
Dα = {y ∈ R : depth(y) ≥ 1 − α}
In higher dimension,
depth(y) = inf
u:u=0
{P[Hu(y)]}
where Hu(y) = {x ∈ Rd
: uT
x ≤ uT
y} and the associated depth set of level
α ∈ (0, 1) as
Dα = {y ∈ Rd
: depth(y) ≥ 1 − α}
@freakonometrics 47

Winter Temperature in Montreal
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1.0 1.5 2.0 2.5 3.0 3.5 4.0
−15−10−50
Month
Temperature(in°Celsius)
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5 10 15
−20−15−10−50510
Week of winter (from Dec. 1st till March 31st)
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0 20 40 60 80 100 120
−30−20−10010
Day of winter (from Dec. 1st till March 31st)
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q
DECEMBER JANUARY FEBRUARY MARCH
Winter temperature in Montréal, from December 1st till March 31st, with
Monthly, Weekly, Daily and Hourly temperatures. Winter 2011 is in red.
@freakonometrics 48

Robust 1 PCA Scores
−6 −4 −2 0 2 4 6 8
−4−202468
Comp.1
Comp.2
q
q
q
q
q
q
q
q
q
qq
q
q
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1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
19731974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996 1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
20102011
Monthly dataset
−10 −5 0 5 10 15
−10−5051015
Comp.1
Comp.2
q
q
q
q
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q
q
q
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1953
1954
1955
1956
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1959
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1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
19971998 1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
Weekly dataset
@freakonometrics 49

Robust 1 PCA Scores
−20 −10 0 10 20 30 40 50
−2002040
Comp.1
Comp.2
q
q
q
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1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975 1976
1977
1978
1979
1980
1981
1982
1983
1984 1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
Daily dataset
−100 0 100 200
−1000100200
Comp.1
Comp.2
q
q q
qq
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qq
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1953
1954
1955
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1958
19591960 1961
1962
1963
1964
1965
1966
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1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
19992000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
Hourly dataset
@freakonometrics 50

Standard 2 PCA Scores
0.2 0.4 0.6 0.8 1.0
−0.50.00.5
Dim.1
Dim.2
q
q
q
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1953
1954
1955
1956
1957
1958
1959
1960
19611962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981 1982
1983
19841985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
20082009
2010
2011
Monthly dataset
0.2 0.4 0.6 0.8
−0.50.00.5
Dim.1
Dim.2
q
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1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
Weekly dataset
@freakonometrics 51

Standard 2 PCA Scores
0.1 0.2 0.3 0.4 0.5 0.6 0.7
−0.6−0.4−0.20.00.20.4
Dim.1
Dim.2
q
q
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1953
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1955
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1961
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1965
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1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
20042005
2006
2007
2008
2009
2010
2011
Daily dataset
0.2 0.3 0.4 0.5 0.6 0.7
−0.4−0.20.00.20.4
Dim.1
Dim.2
q
q
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1953
1954
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1963
1964
1965
1966
1967
1968
1969
1970
1971
19721973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
Hourly dataset
@freakonometrics 52

Robust 1 PCA Principal Components
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0 20 40 60 80 100 120
−50050
PCAComp.1
DECEMBER JANUARY FEBRUARY MARCH
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0 20 40 60 80 100 120
−30−20−100102030
PCAComp.2
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@freakonometrics 53

Standard 2 PCA Principal Components
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q
qqqqq
q
q
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qqq
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q
q
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qq
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qqqq
q
q
q
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qqqq
qq
q
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q
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q
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qqqq
q
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qqqq
q
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qqqq
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qq
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qqq
q
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q
0 20 40 60 80 100 120
−50510
PCAComp.1
q
q
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q
q
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qq
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q
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q
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q
qqqqqqqq
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qq
qq
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qq
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qq
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q
qqqqqqqqqqq
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q
qqq
qqq
qqq
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q
qqq
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qq
qqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqq
qq
qqqqqqq
qqqqqq
qq
qqq
q
qqqqqqq
qqqqqqqqqqqqqqqq
qq
qqq
qqqq
q
q
qq
qqqqqqqq
q
q
q
qqqq
qqqqqqqqqqqq
qq
qqqqq
qq
qqqqqqqq
q
qq
qqqqqqqqqq
qqq
q
qqqqqqq
qqqqq
qq
qqq
qq
qq
qqqqqqqq
q
q
q
qqq
qqqqqqqqqqqqq
q
q
q
qq
q
qq
qq
qqqqqqqq
qqq
qq
qq
qqqqqq
qq
qq
qq
qqqq
qqqq
q
qqqqq
qqq
qqqqqqqqqqqqqqq
q
qq
qq
qqqqq
q
q
q
q
q
qqqqqq
qqqqq
qq
qq
q
qq
q
qq
q
qq
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qq
qqqq
qqq
qqqqq
qqqqqq
qqqqq
qqqqq
q
q
qq
qqqqqqqq
qq
qqqqq
qqqq
qqqqqqqq
qqqqqqqqqqqqqqqq
q
q
q
qqq
qqq
qqq
qqqqqqqqqqqqqqqqqqqqqqq
qqqqqq
qqqq
qqqqqqq
qqq
qqq
qqqqqqqqqqqqqq
qq
qqq
q
qqqq
qq
qqqqqqq
qqqqqq
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qq
qq
qqqq
q
qqqqq
qqq
qqqqqq
q
qqqqqqqqqqqqq
qqqqq
qq
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qqqqqqqq
qqq
qqqq
qqqqqq
qq
q
q
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qqq
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qqq
qqq
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q
qq
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qq
q
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q
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qqqqq
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qqqqq
qq
qqqqq
qqq
qqqqqqqq
qq
qq
qqqqqq
q
q
qq
qqqqqqq
qqqqqqq
qqqqqq
q
qq
qq
qqqqqqq
qqqqqqqqqqqqqq
qqqqq
qqqqqqqqqq
qqqqqqq
qq
qqqqqqqq
qqqqqqqq
0 20 40 60 80 100 120
−4−2024
PCAComp.2
q
q
q
q
q
q
q
q q
q
q
q
q q
q
q
q
q
q
q
q
@freakonometrics 54

Sildes buenos aires

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