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Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
Data Driven, Non-Equilibrium DynamicsHow Warm is it Getting and Other Tales in Uncertainty
JUAN M. RESTREPO
Department of Mathematicsand
Department of Statistics, and Physics of Oceans and Atmospheres
Oregon State University
SIAM Geosciences Meeting, 2017
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THREE TIME-DEPENDENT ESTIMATION PROBLEMSGiven a random time series {X(t) ∈ RN : t ≤ t0} (from models,observations, controls):
I Retrodiction:
X(t) : t ≤ t0.
e.g., paleoclimate reconstruction, polluting sourceidentification.
I Nudiction:X(t) : t = t0.
e.g., initial conditions for weather/geodynamics models.I Prediction (no observations used):
X(t) : t > t0.
e.g., weather forecasting.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DATA ASSIMILATION IN GEOSCIENCES AND
ENGINEERING
Combine information derived from data and models....
Bayes Theorem:
P(X|Y) ∝ Likelihood× Prior
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
LOTS OF DATA IS GOOD!When data fool us...
0 100 200 300 400 500 600−2
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Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
ESTIMATING FROM DATAWhen data fool us...
same data, zoomed in
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Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
ESTIMATING X FROM MODEL
dx = 4x(1− x2)︸ ︷︷ ︸−gradV(x)
dt + κdWt︸ ︷︷ ︸stochasticity
Double Well V(x)
-1 Stationary Distribution P(X)
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DATA IS OFTEN SPARSE IN GEOSCIENCES
The Observations Ym
HOT
COLD
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DATA ASSIMILATION IN GEOSCIENCES AND
ENGINEERINGCombine information derived from data and models....
Bayes Theorem:
P(X|Y) ∝ Likelihood× Prior
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DATA ASSIMILATION IN GEOSCIENCES AND
ENGINEERINGCombine information derived from data and models....
Bayes Theorem:
P(X|Y) ∝ Likelihood× Prior
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DATA ASSIMILATION IN GEOSCIENCES AND
ENGINEERINGCombine information derived from data and models....
Bayes Theorem:
P(X|Y) ∝ Likelihood× Prior
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
REALLY GOOD MODEL...
P(X|Y) ∝ Likelihood× Prior
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
REALLY GOOD DATA...
P(X|Y) ∝ Likelihood× Prior
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
TIME DEPENDENT DATA ASSIMILATION
Bayes Theorem in Time:
P(X(0 ≤ t ≤ t∗)|Y(tm ≤ t0) ∝ Πmp(Ym|Xm)Πt[p(Xt)]
I How to find (at least first) moments ofP(X|Y) := P(X(0 ≤ t ≤ t∗)|Y(tm ≤ t0), whennonlinear/non-Gaussian?
I How to estimate when X has high dimensions?I How do we find distributions of P(Y|X) and P(X)?I How good are the estimates, for general case?I How do we interpret the outcome?
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
NONLINEAR/NON-GAUSSIAN EXAMPLE AND
EXTENDED KALMAN FILTER RESULTS1
Time10% uncertainty, ∆t = 1.
1R. Miller, M. Ghil, P. Gauthiez, Advanced data assimilation in stronglynonlinear dynamical systems, J. Atmo. Sci. 51 1037-1056 (1994)
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE EXTENDED KALMAN FILTER RESULTS
Time
20% uncertainty, ∆t = 1.
R. Miller, M. Ghil, P. Gauthiez, Advanced data assimilation in strongly nonlinear dynamical systems, J. Atmo. Sci. 511037-1056 (1994)
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE EXTENDED KALMAN FILTER RESULTS
Time20% uncertainty, ∆t = 0.25.
R. Miller, M. Ghil, P. Gauthiez, Advanced data assimilation in strongly nonlinear dynamical systems, J. Atmo. Sci. 511037-1056 (1994)
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE EXTENDED KALMAN FILTER RESULTS
Time
10% uncertainty, ∆t = 1.
Time
20% uncertainty, ∆t = 1.
Time
20% uncertainty, ∆t = 0.25.
The Good News: you get an estimate.The Bad News: you get an estimate.
R. Miller, M. Ghil, P. Gauthiez, Advanced data assimilation in strongly nonlinear dynamical systems, J. Atmo. Sci. 511037-1056 (1994)
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
APPROACHES ON NONLINEAR/NON-GAUSSIAN
PROBLEMS
I (Variance-minimizer)I KSP, (Kushner, Stratonovich, Pardoux), early 60’s
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
KSP FILTER AND SMOOTHER RESULTS
G. Eyink, J.M.R., Most Probable Histories, via the Mean Field Variational Approach, J. Stat. Phys. 2001G. Eyink, J.M.R., F. Alexander, A mean field approximation in data assimilation for nonlinear dynamics, Physica D, 2004G. Eyink, J.M.R., F. Alexander, Mean-Field Variational Data Assimilation Using Moment Closures, (unpublished) J. Stat.Phys. 2006
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
APPROACHES ON NONLINEAR/NON-GAUSSIAN
PROBLEMS
I Variance-minimizerI KSP, (Kushner, Stratonovich, Pardoux), early 60’s
I 4D-Var/Adjoint, Lorenc, Talagrand, Courtier, 80’s, Representer (Bennett)
I Extended Kalman Kalman, Bucy 60’s, EnKF (Evensen, ’92) ,Local/Transform EnKF UMD group ’95, Hybrid EnKF Reich ’05
I Sample-BasedI Particle Filters Crisan, Van Leeuwen, Gordon, Del Moral, ’90s
I Mean Stochastic Sampler (Harlim and Majda, ’10)
I Langevin Sampler (A. Stuart, ’05)
I Path Integral Monte Carlo (JMR ’07. Alexander, Eyink & JMR, ’05)
I OtherI Mean Field Variational (Eyink, JMR, ’01)
I Diffusion Kernel Filter (Krause, JMR, ’09)
I Relative Entropy Minimizer, (Eyink, et al, ’05)
Restrepo, Leaf, Griewank, Circumventing storage limitations in variational data assimilation, SIAM J. Sci Comp, ’95
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
PIMC THE PATH INTEGRAL MONTE CARLO2
J. Restrepo, A Path Integral Method for Data Assimilation, Physica D, 2007,F. Alexander, G. Eyink, J. Restrepo, Accelerated Monte-Carlo for Optimal Estimation of Time Series, J. Stat. Phys., 2005
2Cartoon from the Abstruse Goose
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
PIMC THE PATH INTEGRAL MONTE CARLO
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Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE PATH INTEGRAL MONTE CARLO SMOOTHER
Π(Q|Y) ∝ e−Uobs(Y,Q)e−Umodel(Q)
J. M.R., A Path Integral Method for Data Assimilation, Physica D, 2007,J.M.R. A Homotopy Path Integral Filter, J. Stat. Phys. 2017, in preparation.F. Alexander, G. Eyink, J. M.R, Accelerated Monte-Carlo for Optimal Estimation of Time Series, J. Stat. Phys., 2005
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
Π(Q|Y) ∝ e−Uobs(Y,Q)e−Umodel(Q)
If Prob[e−Umodel ] ∼ exp(−Z2/D):
dx− f (x, t)dt = [2D(x, t)]1/2dW
is approximated as
qn+1 − qn −∆tf (qn, tn) = [2D(qn, tn)]1/2[Wn+1 −Wn]
n = 0, 1, ...,T − 1 Hence,
Umodel ≈T∑
n=1
[(qn+1−qn−∆tf (qn, tn))>D(qn, tn)−1 (qn+1−qn−∆tf (qn, tn))],
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
Π(Q|Y) ∝ e−Uobs(Y,Q)e−Umodel(Q)
If Prob[e−Uobs(q,Y)] ∼ exp(−Z2/R).
ym −H(qm) = [2R[qm, tm)]1/2ηm
m = 1, 2, . . . ,M.
Udata =
M∑m=1
[(ym −H(qm))> R(qm, tm)−1 (ym −H(qm))],
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
(GHMC) GENERALIZED HYBRID MARKOV CHAIN
MONTE CARLO
I P(Q|Y) ∝ e−H
I H = V(Q,Y) + K(P).I V(Q,Y) =−Umodel(Q)− Uobs(Q,Y)
I K(P) = −12 P>M−1P
I ∂τQ = G δHδP and
∂τP = −G> δHδQ .
J. M.R., A Path Integral Method for Data Assimilation, Physica D, 2007,J.M.R. A Homotopy Path Integral Filter, J. Stat. Phys. 2017, in preparation.F. Alexander, G. Eyink, J. M.R, Accelerated Monte-Carlo for Optimal Estimation of Time Series, J. Stat. Phys., 2005
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DOUBLE WELL OUTCOME
G. Eyink, J.M.R., Most Probable Histories, via the Mean Field Variational Approach, J. Stat. Phys. 2001G. Eyink, J.M.R., F. Alexander, A mean field approximation in data assimilation for nonlinear dynamics, Physica D, 2004G. Eyink, J.M.R., F. Alexander, Mean-Field Variational Data Assimilation Using Moment Closures, (unpublished) J. Stat.Phys. 2006
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
RESEARCH DIRECTIONS
I Variance Estimation: ”less ad-hoc” estimation, increaseensembles, better experimental design.
I Reduce model uncertainty: better models (couplecomputation/data/model design.
I Scales matter: marginalization, NOT interpolation.
I Forecasting: least-squares is not the only thing we know.I Bias/Trend Errors. trends in multiscale problems are
challenging.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
TREND (BIAS) ERRORS LEAD TOBAD ESTIMATION...
P(X|Y) ∝ Likelihood× Prior
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE PREDICTION PROBLEM
Atmospheric CO2 at Mauna Loa Observatory (D. Keeling, and others, Scripps).
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE PREDICTION PROBLEM
Which estimate do we use in the forecast?
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
FORECASTING USING PIMC
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J. Restrepo, A Path Integral Method for Data Assimilation, Physica D, 2007,F. Alexander, G. Eyink, J. Restrepo, Accelerated Monte-Carlo for Optimal Estimation of Time Series, J. Stat. Phys., 2005
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
FORECASTING USING PIMC
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In ”Prediction Mode, only Model has a Bearing on Results”
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
IMPROVING HURRICANE PREDICTIONS
Property Damage ($USD)I Harvey $190B ?I Katrina $108BI Sandy $65BI Ike $30BI Andrew $27BI · · ·
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DYNAMIC LIKELIHOOD DATA ASSIMILATION
Use a model for the wave and observations...
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DYNAMIC LIKELIHOOD DATA ASSIMILATION
Stochastic One-Way Wave Equation:
ut − C(x, t)ux = F(x, t), t > 0, x ∈ [0,L],
u(x, 0) = U(x), x ∈ [0,L],
F(x, t) = f (x, t) + Nf (t), C(x, t) = c(x, t) + Nc(t)
Φ`(0) = U(x`), ` = 1, 2, ...,N
dΦ = f (Φ)dt + A(t)dW(f )t ,
Φ(0) = U(x`),
dx = c(x, t)dt + B(t)dW(c)t ,
x(0) = x`,J.M.R., Dynamic Likelihood Approach to Filtering, Q. J. Roy. Met. Soc, 2017,P. Krause, J.M.R. Using the Diffusion Kernel Filter in Lagrangian Data Assimilation, Mon. Wea. Rev, 2009
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
KF Likelihood Dynamics Likelihood
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DATA:MEASURED •PROPAGATED •
ζn+1 = ∆tc(ζn, tn) + ζn, tn ≥ tm,
Y(ζn+1, tn+1) = Y(ζn, tn),
Rn+1m ≈ An(t)[An(t)]T∆t + Rn, tn ≥ tm,
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE DYNAMIC LIKELIHOOD FILTER
Forecast (Like Kalman Filter):
V = Ln−1〈V〉n−1 + ∆tfn−1, n = 1, 2, . . . ,Nf − 1.
P = Ln−1Pn−1LTn−1 + Qn−1, n = 1, 2, . . . ,Nf − 1.
Multi-analysis (Dynamic Likelihood): project onto state space...
〈V〉n = Vn +Kn∑
m′∈m
(Hnm′Y
nm′ − Vn)δm′,n,
Kn = Pn[Pn +∑
m′∈m
Hnm′R
nm′ [Hn
m′ ]Tδm′,n]−1,
Pn = (I −Kn)Pn.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DYNAMIC LIKELIHOOD DATA ASSIMILATION
Exact Model
Dynamic Likelihood Kalman
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
FEATURE-BASED, LAGRANGIAN DATA BLENDING
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ClassicCont ourAnalysis
ψ ψ
ψ ψ
(a) (b)
(c) (d)
1 2
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
IMPROVING HURRICANE PREDICTIONS
I SHARPENING: Use an L1 estimator.I DISPLACEMENT CORRECTION: Either by adding
constraints, or by doing assimilation in space/time.
S. Rosenthal, S. Venkataramani, J.M.R., A. Mariano, Displacement Data Assimilation, J. Comp. Phys. 2016E. Chunikhina, J.M.R. Compressed Sensing and Optimal Sensor Placement in Data Assimilation, in preparation
.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DISPLACEMENT MAPS VIA CANONICAL
TRANSFORMATIONS
Find M such that
min ||q(M(x))− q0||22.
here (x, y) M→(X,Y).
In 2-Dimensions, the generating function isG(X, y) = Xy + f (X, y).
x =∂G∂y
= X + fy(X, y)
Y =∂G∂X
= y + fX(X, y).
invertible if fyX > −1.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DISPLACEMENT ASSIMILATION USING EKFTarget: min ||q(M(x))− q0||22.
Analysis
Displacement Map M
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
TAME DISTORTION:
Exploit strain tensor σ:
σ =
[x∆x y∆xx∆y y∆y
]=
11 + fyX
[−fyX −fyyfXX fXy − |H[f ]|
]Diagonals: normal strains, off-diagonals: shear strains.
Minimize instead:
J [f ] =
∫D
[q(f )− q0]2 dx dy
+
∫Dα[(x∆x)2 + (y∆y)2
]+ β
[(y∆x)2 + (x∆y)2
]dx dy
α and β adjustable weights.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
AREA-PRESERVING MAPS
Map Regularized Map
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
DISPLACEMENT ASSIMILATION USING ENKF
Truth, EnKF Truth, EnKF+Displacement
yields up to 70% improvements for small ensembles...
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE IMPORTANCE OF DETERMINING A TREND
Eking out change means determining systematic variability, and thedetermination of a trend.
I Global temperature, CO2, greenhouse gases, oceanacidification, ...
I Mean sea level.I Climate interpretation (Variability of weather and climate).I Many applications in econometrics, geosciences,
engineering.
J.M.R., D. Comeau, H. Flaschka, Defining a Trend using the Intrinsic Time Decomposition, New J. Phys. 2014
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
MATHEMATICAL FACT ABOUT EXTREMES
1880 1900 1920 1940 1960 1980 2000year
5
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30
deg
C
RANDOM DATA
An n random data set has about same number of extreme highs andlows. Their occurrence declines as 1/n, the number of datum.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
NOT SEEN IN TEMPERATURE EXTREMES3
1880 1900 1920 1940 1960 1980 2000year
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RANDOM DATA
1880 1900 1920 1940 1960 1980 2000year
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10
15
20
25
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deg
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MOSCOW JULY DATA
An n random data set has about same number of extreme highs andlows. Their occurrence declines as 1/n, the number of datum.
3S. Rahmstorf, D. Coumou, Increase of Extreme Events in a Warming World,PNAS, 2011.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
IS HARVEY A RARE/EXTREME EVENT?
I Harvey is not a rare event,but a manifestation of weather change.
Extreme, but not rare.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
IS HARVEY A RARE/EXTREME EVENT?
I Harvey is not a rare event, but a manifestation of weatherchange. Harvey is extreme, but not rare.
I
I Establishing the connection between climate change andweather outcomes is ongoing research.
I Basic thermodynamics can be used to establish theoutcomes of climate change.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
HOW WARM IS IT GETTING?
The time rate of change of the temperature
CdTdt
=14
(1− α)S︸ ︷︷ ︸incoming radiation
− σT4︸︷︷︸outgoing radiation
Sun Radiation: S = 1361 W/m2. Earth’s Albedo: α ≈ 0.3Calculated global temperature: -18oC.
Doubling CO2 decreases outgoing IR by 4.2 W/m2 (Held &Soden, 2000).Using the CO2 data, the greenhouse adjusted globaltemperature is 14.5oC.
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
HOW WARM IS IT GETTING?4
4Figure, from K. Emanuel
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
A CLIMATE SIGNAL...
0 100 200 300 400age, Kyr
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Vostok Ice Core data, Temperature
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
Given a finite-time time series Y(i), i = 1, 2, ...,N,
The Tendency T(i) is an Executive Summary of Y(i)
I Captures essentials of histogram in the abscissa of Y(i); andI Most essential multi scale information, derived from
ordinate of Y(i).
The Empirical Uncertainty U(i) := Y(i)− T(i)
I is simple entropically,I The histogram of U(i), is easy to parametrize
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
Given a finite-time time series Y(i), i = 1, 2, ...,N,
The Tendency T(i) is an Executive Summary of Y(i)
I Captures essentials of histogram in the abscissa of Y(i); andI Most essential multi scale information, derived from
ordinate of Y(i).
The Empirical Uncertainty U(i) := Y(i)− T(i)
I is simple entropically,I The histogram of U(i), is easy to parametrize
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE INTRINSIC TIME DECOMPOSITION (ITD)Given a sequence of real numbers {Y(i)}N
i=1,
Y(i) = BD +
D∑j=1
Rj(i)
where
Bj(i) = Bj+1(i) + Rj+1(i), j = 0, ...,D,and
B0(i) : = Y(i).
Bj are called BASELINES, and Rj are called ROTATIONS.
ITD is related to EMD (Empirical Mode Decomposition)
Frei and Osorio, Proc. Roy. Soc. London, (2006).
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE INTRINSIC TIME DECOMPOSITION (ITD)
0 2 4 6 8 10 12 14 16 18 20
i-10
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-6
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0
2
4
6
8
10
B0(i)
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE INTRINSIC TIME DECOMPOSITION (ITD)
0 2 4 6 8 10 12 14 16 18 20
i-10
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-6
-4
-2
0
2
4
6
8
10
B0(i)=B1(i)+
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
THE INTRINSIC TIME DECOMPOSITION (ITD)
0 2 4 6 8 10 12 14 16 18 20
i-10
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-6
-4
-2
0
2
4
6
8
10
B0(i)=B1(i)+R1(i)databaselinerotation
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
How does the ITD (and EMD) work?
E [Bj] := {Sj, bj}.
{Sj}nj1 be locations of extrema of baselines, with values bj.
ITD:{Sj+1, bj+1} = E [(I + Mj)bj].
What is E?
(I + Mj)bj is ForwardTime/CenterDifference approximation of
∂
∂tB =
14wj(x)
∂
∂x
[wj(x)
∂B∂x
],
wj(x) = exp[2∫ x
0 pj(t)dt].
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
EXAMPLE CALCULATION
0 100 200 300 400age, Kyr
-10
-8
-6
-4
-2
0
2
4
tem
p, d
eg C
Vostok Ice Core data, Temperature
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
Y(i) = BD +
D∑k=1
Rk(i), Bj+1(t) + Rj+1(i) = Bj(i).
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Raw Signal
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B1
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−202
Baseline B2
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B3
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B4
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−202
Baseline B5
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B6
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B7
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B8
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Raw Signal
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R1
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R2
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R3
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R4
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R5
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R6
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R7
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
FIND TENDENCYFind ITD:
Y(i) = BD +
D∑j=1
Rj(i),
Bj(i) = Bj+1(i) + Rj+1(i)
Choosing j∗ among the baselines {Bj(i)}Dj=1:
T(i) := Bj∗(i)
The ABSISSA information:I For j = 1, ..,D compute Fj := histogram[Y(i)− Bj(i)]I Determine the Symmetry sj of Fj via percentiles:
sj :=Prj
75 − 2 Prj50 − Prj
25
(Prj75 − Prj
25)
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
Choosing j∗ among the baseline {Bj(i)}Dj=1:
T(i) := Bj∗(i)
The ORDINATE information:Compute the Complexity cj vector cj := corr(Bj,Rj)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Raw Signal
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B1
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−202
Baseline B2
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B3
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B4
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−202
Baseline B5
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B6
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B7
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Baseline B8
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−8−6−4−20
2Raw Signal
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R1
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R2
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R3
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R4
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R5
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R6
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
−4−2
02
proper rotation R7
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
The Tendency T(i), and the Vostok signal Y(i)
0 0.5 1 1.5 2 2.5 3 3.5 4Age #105
-10
-8
-6
-4
-2
0
2
4
degC
Raw Signal
tendency
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
HISTOGRAMS
Vostok: Y(i). U(i) := Y(i)− T(i) Empirical Uncertainty.
-10 -5 0 5 10degC
0
0.1
0.2
0.3
0.4
0.5pdf[Y-mean(Y)]pdf[Y-tendency]
0 0.5 1 1.5 2 2.5 3 3.5 4Age #105
-10
-8
-6
-4
-2
0
2
4
degC
Raw Signal
tendency
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
FINDING TENDENCY: AN APPLICATION TO DATA
ASSIMILATION
I Modeling Tool with which to discern ImportantStructures in Data
I Generates a compact surrogate model of the form
dXt = f (Xt, t)dt + noiset.
I T(i) is the cummulant of the drift term f (·).I Empirical moments would be obtained from hist(Y− T).
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
NON-EQUILIBRIUM DYNAMICS AND ITS DATA
I Data Assimilation (Essential in Geosciences):I high dimensioned, good models, and very sparse data.
I Research challenges:I Focus on Variance/uncertainty estimation.I Better models means less uncertainty.I Beyond Least Squares...I Finding Trends (and bias in time dependent problems).I Better observation networks for better (sparse) data sets.I Efficient marginalization strategies for multiscale problems.I Integration of modeling, data, and computation.
With funding from
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
MY COLLABORATORSI S. Venkataramani (U. Arizona)I G. Eyink (Johns Hopkins)I F. Alexander (LANL)I H. Flaschka (U. Arizona)I J. Ramırez (U. Nacional de Colombia)I A. Mariano (U. of Miami)I C. Dawson (UT Austin)I R. Miller (Oregon State)I S. Rosenthal (PNNL), D. Comeau (LANL), A. Jensen
(OSU), K. Bergstrom (Intel), W. Mayfield (OSU)
Further Informationhttp://www.math.oregonstate.edu/∼restrepo
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
NON-EQUILIBRIUM DYNAMICS AND ITS DATAI Data Assimilation (Essential in Geosciences):
I high dimensioned, good models, and very sparse data.I Research challenges:
I Finding Trends.I Better observation networks for better (sparse) data sets.I Efficient marginalization strategies for multiscale problems.I Focus on Variance/uncertainty estimation.I Better models means less uncertainty.I Integration of modeling, data, and computation.I Beyond Least Squares.
I Current Work in data-driven dynamics and estimation:I Stochastic Parametrization: improving fidelity in models.I Dimension Reduction: glassy systemsI Fidelity Computing: development of an oil-spill model
With funding from
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
NEW: HOMOTOPY PATH INTEGRAL DATA
ASSIMILATIONFind
Z1 =∫
P(Y|X)P(X)dx =∫
M(x)dx,starting from Z0 =
∫P(Y|X)dx =
∫L(x)dx:
Known
Create an optimal schedule S(δs,N), for Zs =∫
MsL1−sdx.A. Jensen, J. M. R., R. Miller, Homotopy Path Integral Data Assimilation, Dynamics & Statistics of the Climate System,2017
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
NEW: DIMENSION REDUCTION,WHEN YOU JUST DON’T HAVE A CHOICE
I Motivation: An ocean oiltransport model:
I 104 chemicals,I 102 droplet sizes,I O(N6) spatial dofs,I O(108) time steps.
I Applications: Chemicalreactions, combustion,Glassy systems.
S. C. Venkataramani, R. Venkataramani, J. M. R. Dimension Reduction for Systems with Slow Relaxation In Memory of LeoP. Kadanoff, J. Stat. Phys. 2017
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
NEW: STOCHASTIC PARAMETRIZATIONMotivation: Model Fidelity
I Data:
I Imperfect Model:dv = F(x, t, α+ δα)dt
I Sensitive-dependence onparameters:var[v] = 1
F2 | dFdα |var[α][〈v〉]2
Historgram of Data
Data Assimilation, usingstochastic model and data
J. M. R., S. Venkatarmani, Stochastic Longshore Model Dynamics, J. Water Res. 2017J.M.R. Wave Breaking Dissipation in the Wave-Driven Ocean Circulation, J. Phys. Ocean. 2007J.M.R, J. Ramirez, Banner, J. C. McWilliams, J. Phys. Ocean. 2010J. Ramirez, J.M.R., Luc Deike, Ken Melville, Stochastic Progressive Wave Breaking, J. Phys. Ocean. in prep. 2017
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
NEW: HIGH FIDELITY COMPUTING
Application of ideas presented here to build an oil transportmodel:
I Dimension reduction and data assimilation.I Stochastic parametrization and filtering ideas.I Couples computational resolution and the physics at
relevant scales.I Modeling, computation, and data assimilation are tightly
coupled, leading to computational efficiency.
J.M. R., J. Ramirez, S. Venkataramani, An Oil Fate Model for Shallow Waters, J. Mar. Sci. Eng, 2015J.M.R., S. Venkataramani, C. Dawson, Nearshore Sticky Waters, Ocean Modelling 2016J. Ramirez, S. Moghimi, J.M.R. Mass Exchange Dynamics of Oceanic Surface and Subsurface Oil, submitted, Bull. Mar.Poll., 2017
Bayes Data Assimilation Research Directions Forecasting New Ideas in Assimilation Computing Trends New Challenges
MY COLLABORATORSI S. Venkataramani (U. Arizona)I G. Eyink (Johns Hopkins)I F. Alexander (LANL)I H. Flaschka (U. Arizona)I J. Ramırez (U. Nacional de Colombia)I A. Mariano (U. of Miami)I C. Dawson (UT Austin)I R. Miller (Oregon State)I S. Rosenthal (PNNL), D. Comeau (LANL), A. Jensen
(OSU), K. Bergstrom (Intel), W. Mayfield (OSU)
Further Informationhttp://www.math.oregonstate.edu/∼restrepo
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