statistical downscaling for bivariate data in climate … 27, 2012 statistical downscaling for...

September 27, 2012

Statistical downscaling for Bivariate Data in ClimateProjections

Xuming HeJoint work with Yunwen Yang and Jingfei Zhang

Department of StatisticsUniversity of Michigan

Motivation Methods Empirical Evidence Conclusion

Outline

1 MotivationStatistical DownscalingAsynchronous RegressionBivariate Downscaling

2 MethodsBivariate QuantilesProposed Bivariate DownscalingComputation

3 Empirical EvidenceSimulationA Real Data Example

4 Conclusion

Bivariate Downscaling for Climate Projections He, Yang, and Zhang DEPARTMENT OF STATISTICS 2 / 34


Global Climate Models

Climate: weather conditions (temperature, humidity, precipitation,sunshine, cloudiness, and winds) throughout the year, averaged over aseries of years.

General Circulation Models (GCMs): a class of computer-drivenphysics-based models for understanding and predicting climate changes.

There are many decent climate models, but with limited resolution(typically 200 by 360).



Global Climate Models



Climate Downscaling

Dynamic downscaling: regional models; expensive to run.

Statistical downscaling: regression-type techniques to relate localmeasurements to the GCM output; cheaper but rely on local station data.



Projection from the North American Regional Climate Change AssessmentProgram



Article from Science, October 14, 2011

www.sciencemag.org SCIENCE VOL 334 14 OCTOBER 2011 173

NEWSFOCUS

Seattle Public Utilities offi cials had a ques-tion for meteorologist Clifford Mass. They were planning to install a quarter-billion dol-lars’ worth of storm-drain pipes that would serve the city for up to 75 years. “Their ques-tion was, what diameter should the pipe be? How will the intensity of extreme precipita-tion change?” Mass says. If global warming means that the past century’s rain records are no guide to how heavy future rains will be, he was asked, what could climate modeling say about adapting to future climate change? “I told them I couldn’t give them an answer,” says the University of Washington (UW), Seattle, researcher.

Climate researchers are quite comfort-able with their projections for the world under a strengthening greenhouse, at least on the broadest scales. Relying heavily on climate modeling, they fi nd that on average the globe will continue warming, more at high northern latitudes than elsewhere. Precipitation will tend to increase at high latitudes and decrease at low latitudes.

But ask researchers what’s in store for the Seattle area, the Pacifi c Northwest, or even the western half of the United States, and they’ll often demur. As Mass notes, “there’s tremen-dous uncertainty here,” and he’s not just talk-ing about the Pacifi c Northwest. Switching from global models to models focusing on a single region creates a more detailed forecast, but it also “piles uncertainty on top of uncer-tainty,” says meteorologist David Battisti of UW Seattle.

First of all, there are the uncertainties inherent in the regional model itself. Then there are the global model’s uncertainties at the regional scale, which it feeds into the regional model. As the saying goes, if the global model gives you garbage, regional modeling will only give you more detailed garbage. And still more uncertainties are cre-ated as data are transferred from the global to the regional model.

Although uncertainties abound, “uncer-tainty tends to be downplayed in a lot of [regional] modeling for adaptation,” says global modeler Christopher Bretherton of UW Seattle. But help is on the way. Regional modelers are well into their first extensive

comparison of global-regional model combi-nations to sort out the uncertainties, although that won’t help Seattle’s storm-drain builders.

Most humble originsPolicymakers have long asked for regional forecasts to help them adapt to climate change, some of which is now unavoidable. Even immediate, rather drastic action to curb emissions of greenhouse gases would not likely limit warming globally to 2°C, gener-ally considered the threshold above which “dangerous” effects set in. And nothing at all can be done to reduce the global warming effects expected in the next several decades.

They are already locked into climate change.So scientists have been doing what they

can for decision-makers. Early on, it wasn’t much. A U.S. government assessment released in 2000, Climate Change Impacts on the United States, relied on the most rudimen-tary regional forecasting technique (Science, 23 June 2000, p. 2113). Expert commit-tee members divided the country into eight regions and then considered what two of their best global climate models had to say about each region over the next century. The two models were somewhat consistent in the far southwest, where the report’s authors found it was likely that warmer and drier condi-tions would eliminate alpine ecosystems and shorten the ski season.

But elsewhere, there was far less consis-tency. Over the eastern two-thirds of the con-tiguous 48 states, for example, the two models couldn’t agree on how much moisture soils would hold in the summer. Kansas corn would either suffer severe droughts more frequently, as one model had it, or enjoy even more mois-ture than it currently does, as the other indi-cated. But at least the uncertainties were plain for all to see.

The uncertainties of regional projec-tions nearly faded from view in the next U.S. effort, Global Climate Change Impacts in the United States. The 2009 study drew on not two but 15 global models melded into single projections. In a technique called statistical downscaling, its authors assumed that local changes would be proportional to changes on the larger scales. And they adjusted regional projections of future climate according to how well model simulations of past climate matched actual climate.

Statistical downscaling yielded a broad warming across the lower 48 states with less warming across the southeast and up the West Coast. Precipitation was mostly down, especially in the southwest. But discussion of uncertainties in the modeling fell largely to a footnote (number 110), in which the authors cite a half-dozen papers to support their assertion that statistical downscaling techniques are “well-documented” and thor-oughly corroborated.

The other sort of downscaling, known as dynamical downscaling or regional model-ing, has yet to be fully incorporated into a U.S. national assessment. But an example of state-of-the-art regional modeling appeared 30 June in Environmental Research Let-ters. To investigate what will happen in the U.S. wine industry, regional modeler Noah Diffenbaugh of Purdue University in West Lafayette, Indiana, and his colleagues embedded a detailed model that spanned

Vital Details of Global Warming Are Eluding ForecastersDecision-makers need to know how to prepare for inevitable climate change, but climate

researchers are still struggling to sharpen their fuzzy picture of what the future holds

P R E D I C T I N G C L I M AT E C H A N G E

Winter Temperature Change

Global Model

Sharp but true? Feeding a global climate model’s prediction for midcentury (top) into a regional model gives more details (bottom), but modelers aren’t sure how accurate the details are.C

RE

DIT

: N

OR

TH

AM

ER

ICA

N R

EG

ION

AL C

LIM

AT

E C

HA

NG

E A

SS

ES

SM

EN

T P

RO

GR

AM

Degrees C

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 4 5 7

Published by AAAS

on

Nove

mbe

r 7, 2

011

www.

scie

ncem

ag.o

rgDo

wnlo

aded

from



About 2009 Climate Change Impacts on the United States

Quote from the Science article:

Statistical downscaling yielded a broad warming across the lower 48 stateswith less warming across the southeast and up the West Coast. Precipitationwas mostly down, especially in the southwest. But discussion of uncertaintiesin the modeling fell largely to a footnote (number 110), in which the authorscite a half-dozen papers to support their assertion that statistical downscalingtechniques are well-documented and thoroughly corroborated.



A basic problem

X(t) is the output of daily maximum temperature from a GCM at a gridcell.

Y(t) is the observed daily maximum temperature at a station inside thegrid cell.

Can you regress Y(t) on X(t)?

A GCM output X(t) is a realization (or a random draw) from theunderlying distribution.



A simple analogy

Yi is the weight measurements obtained from 50 adults.

Xi is the height measurements obtained from a sample of 50 adults fromthe same population.

Can you regress Y(t) on X(t)?

With asynchronous measurements, can we find the relationship to predictY from X?



Statistical Asynchronous Regression (SAR)

SAR (O’Brien, Sornette and McPherro, 2001): simple quantile matchingfor univariate statistical downscaling.

X(t) ∼ F(x) and Y(t) ∼ G(y) for t ∈ T .A monotone matching function µ: G{µ(x)} = F(x).µ(x) = G−1{F(x)}.µ(x) = G−1{F(x)}.

Assumption: the relationship between F(x) and G(y) is stationary overtime.



An illustration of SAR

X(t) = {1, 2, 3, 4, 5, 6, 7, 8, 9}Y(t) = {4, 5, 6, 7, 8, 9, 10, 11, 12}

What if X = 7.5?

X = 7.5 is between the 2nd and the 3rd largest values, we find Y = 10.5.



Bivariate Data

When a bivariate variable is of interest, the SAR can be applied to eachvariable.

However, univariate downscaling cannot capture the association betweenvariables.

In climate studies, some extreme events are characterized by multiplevariables, e.g., the extreme hot and humid days, so it is important not tomess up with associations.



Illustrative example with bivariate data

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

● ●

●

●●

●

●

●●

●

●

●

●

●

●● ●

●●●

●

●

●

●●

●

● ●●

●

●

●

●

●● ●

●

●

●

●

●●

●

●

●

●●

● ●●

●● ●

●

●

● ●●

●

●●

●●

●

●

●

●

●

●

●

●

●

0 20 40 60 80 100

−1

00

01

00

20

0

(a)

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

●

●

● ●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●●

●

●

●

● ● ●

●

●

0 20 40 60 80 100

−1

00

01

00

20

03

00

(b)

Figure:



Basic idea

SAR:

transfer data to ranks;ranks have the same distributions regardless of the initial dataconfigurationswe can match X and Y based on their positions in the rank distributions

Is there such a transformation from data to positions for bivariate data?



Bivariate Pisitions

Definition

The position of z ∈ R2 relative to a bivariate distribution F isP(z) = 4

πE{(z− X)/||z− X||}, where the expectation is taken over X ∼ F.

The sample version: PT(z) = 4πn

∑t∈T

z−X(t)||z−X(t)|| .

Reference: Marden (2004)



Why is this definition of “position” useful?

X(t)→ PT(X(t))→ P2T(X(t)) · · · → PK

T (X(t)), where PkT denotes the

k-th consecutive transformation under the position function PT for thesame set of points.

As k increases the distribution of PkT(X(t)) approaches a fixed stationary

distribution in a circle, regardless of the initial distribution of X(t) .

The fixed stationarity distribution allows us to perform “bivariatequantile matching”.



Stationary distribution

The stationary distribution (Buja et al. 1994)

circularly symmetric around the originradial density r/

√1− r2 on r ∈ (0, 1).

0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

67

r

dens

ity



“Stage-K” position

DefinitionWe define the “stage-K position” of any point z relative to {X(t) : t ∈ T} as

PKT (z) =

4πn

∑t∈T

PK−1T (z)− PK−1

T (X(t))

||PK−1T (z)− PK−1

T (X(t))||.

We say that PKT (z) is (approximately) the stationary position of z for a

sufficiently large K,.

Typically we need to use a small K between 5 and 10.

In practice, we usually transform X and Y to their marginal ranks firstbefore calculating the position.



Illustration of“Stage-K” positions

−10 0 10 20

510

1520

25

(a)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(b)

−1.0 −0.5 0.0 0.5 1.0

−0.

50.

00.

51.

0

(c)

−1.0 −0.5 0.0 0.5 1.0

−1.

0−

0.5

0.0

0.5

1.0

(d)

Figure: (a) shows the original data points, (b) shows the marginal ranks, (c) shows thestage-1 positions, and (d) shows that stage-4 positions. The same two points arespecially marked in each plot.



Another illustration

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

0 20 40 60 80 100

−100

010

020

0

(a)

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

(b)

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

−1.0 −0.5 0.0 0.5 1.0

−1.0

−0.5

0.00.5

1.0

(c)

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

0 20 40 60 80 100

−100

010

020

030

0

(d)

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

(e)

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

−1.0 −0.5 0.0 0.5 1.0

−1.0

−0.5

0.00.5

1.0

(f)

Figure: (a), (d) show the original data points; (b), (e) show the marginal ranks; (c), (f)show the stage-4 positions.



Proposed Bivariate Downscaling Method

{X(t) : t ∈ T} is GCM output of two variables.

{Y(t) : t ∈ T} is the set of station observations of the same variablesover the same period of T .

Initial step: replace data in X and Y by their respective marginal ranks

Target: given a new x, what is the corresponding y?

Calculate u(x) = PKT (x) as the stationary position of x with respect to

{X(t) : t ∈ T}.Find y such that v(y) = PK

T (y) = u(x), where PKT (y) represents the

stationary position of y with respect to {Y(t) : t ∈ T}.



How to solve v(y) = u(x)?

The operation to find y such that v(y) = u(x) requires the inverseoperation of the position function K times.

Given u(K) = PKT (x) w.r.t. {X(t) : t ∈ T}, we need to solve

PT(u(K − 1)) = u(K) w.r.t {PK−1T (Y(t)) : t ∈ T}.

Given u(K − 1), solve PT(u(K − 2)) = u(K − 1) w.r.t{PK−2

T (Y(t)) : t ∈ T}.

Continue until given u(1), solve PT(y) = u(1) w.r.t {Y(t) : t ∈ T}.

WLOG, we will focus on the computation for K = 1 next.



How to solve v(y) = u(x)?

Given u, we need to find the corresponding y satisfying PT(y) = u w.r.t{Y(t) : t ∈ T}.By Chaudhuri (1996), it is equivalent to finding the minimizer of

L(Q) =[∑

t∈T

{||Y(t)− Q||+ (π/4){Y(t)− Q}>u

}].

L(Q) is convex, and the Newton-Raphson method may be used.



A computational problem

The derivative of L(Q) is not defined when Q = Y(t) at some t, but Y(t)could be a solution.

Chaudhuri (1996) suggested to check a degeneracy condition first to seewhether the solution for PT(y) = u is an element of {Y(t) : t ∈ T}.However, its degeneracy condition is a necessary but not a sufficientcondition!

We run into problems using the suggested algorithm.



A modified algorithm

Use the Newton-Raphson method trying to solve PT(y) = u first.

If it fails to converge in a few iterations, we find the minimizer of L(Q)from {Y(t) : t ∈ T}.



Simulation Study

Training data: size of 1000

Xtrain ∼ N((

00

),

(0.25 0.20.2 0.25

)), and Ytrain ∼

((00

),

(0.36 0.10.1 0.49

)).

Testing Data: size of 1000.

Xtest ∼((

0.10.1

),

(0.25 0.20.2 0.25

)).

We apply the univariate downscaling (SAR) and our bivariatedownscaling approach with K = 6.



Downscaling Result

−2 −1 0 1 2

−2−1

01

2

X_train

X1

X2

−2 −1 0 1 2

−2−1

01

2

Y_train

Y1

Y2

−2 −1 0 1 2

−2−1

01

2

Univariate

Y1

Y2

−2 −1 0 1 2−2

−10

12

Bivariate

Y1

Y2

Figure: A simulated data set, where (a) is the scatter plot of Xtrain, (b) is the scatterplot of Ytrain, (c) is of the scatter plot of the univariate downscaling results YUD, and(d) is the scatter plot of the bivariate downsaling results YBD.



Downscaling Result

Bivariate downscaling: the estimated distribution for YBD

N((

0.080.10

),

(0.35 0.100.10 0.50

)),

Univariate downscaling: the estimated distribution for YUD

N((

0.100.11

),

(0.34 0.320.32 0.47

)).

Mean shifts preserved, but correlation missed by SAR.



Downscaling at Elgin, IL

X(t): simulated (TMAX,PRCP) from the GFDL climate model 2.1.

Y(t): observed (TMAX,PRCP) at the Elgin, IL station.

We will focus on the rainy days only (about 30% in Elgin).

Training period 1965-2000: 2965 days in X(t), and 2505 days in Y(t).

Testing period 2001-2010: 1119 days in X(t).



Downscaling at Elgin, IL

0 20 40 60 80 100

01

23

45

Observation:testing

TMAX

PR

CP

−20 0 10 20 30 40

020

4060

80

GCM:testing

TMAX

PR

CP

0 20 40 60 80 100

01

23

45

Univariate Downscaling

TMAX

PR

CP

0 20 40 60 80 1000

12

34

5

bivariate Downscaling

TMAX

PR

CP

Figure: (a) is the scatter plot of the observations in the testing period, (b) is the scatterplot of the GCM output in the testing period, (c) is the scatter plot of the univariatedownscaling results, and (d) is the scatter plot of the bivariate downscaling results.



Some details

When 20 ≤ TMAX ≤ 80, the actual records for the testing period had0.3% of the points above the red line.

Univariate downscaling: 2% records above the red line.Bivariate downscaling: 0.5% records above the red line.

When 80 ≤ TMAX ≤ 100, the actual records for the testing period had11% of the points above the red line.

Univariate downscaling: 3% records above the red line.Bivariate downscaling: 14% records above the red line.



Summary

The proposed bivariate downscaling method inherits the desirableproperties of the SAR.

The bivariate downscaling is able to preserve association betweenvariables.

The merits of the proposed biviariate downscaling method aredemonstrated through numeric experiments and a real data example.

The proposed method is easy to implement, but it will be important tooptimize computer code in any large scale studies.



Extensions

Downscaling at multiple stations within each grid cell.

Downscaling more than two variables.

Downscaling with additional covariates.

Uncertainty assessment?

Thank you!

Work supported in part by the National Science Foundation.


statistical downscaling for bivariate data in climate … 27, 2012 statistical downscaling for...

Documents