Improving an Ecosystem Model Using Earth Science Data

Kazumi Saito (NTT Communication Science Laboratories)Pat Langley (CSLI Stanford University)

Trond E. Grenager (Stanford University)

Introduction• Developing computational methods for

discovering knowledge in communicable forms.

• Improving CASA using observed data. • CASA: an existing computational model of

aspect of the Earth ecosystem developed by Christopher Potter and his colleagues at NASA Ames.

Portion of CASA

Known ConstantObserved VariableVariable

E IPAR

W T2 T1

PET

FPAR_FAS

SR_FAS SRDIFF

sol_conv

srmin

e_max

eet

ahi

tempcA

Topt

PET_TW_M umd_vegFAS_NDVI

solar

NPPc

Some EquationsNPPc: net primary production.

( )max 0,NPPc = E IPAR×

_ max 1 2E = e T T W × × ×

0 5IPAR = FPAR_FAS Solar sol_conv . × × ×

E: value of maximum possible photosynthetic efficiency under temperature and moisture stress scalars.

IPAR: converter for intercepted photosyntheticallyactive radiation by the vegetation cover.

General Problem• Revisions to the model must be consistent

with existing knowledge of Earth science and, ideally, retain similarity to the current model.

• Our research involves attempting to improve the CASA model’s predictive accuracy.

Outline of Approach• Transforming the equations into a neural network• Revising weights in that network• Transforming the network back into equations

( )NPPc= f

L

L

L

originalequations

( )NPPc= g

L

L

L

revisedequations

Some Types of Neural NetworksStandard (sigma-sigma) net:

( )j j jk kw f w x∑ ∑Sigma-pi net (generalized polynomial):

( )exp lnjkwj k j jk kw x w w x=∑ ∑ ∑∏

Pi-sigma net (this talk):

( )j j jk kw f w x∑∏

Transforming Equations

Known ConstantObserved VariableVariable

E IPAR

W T2 T1

PET

FPAR_FAS

SR_FAS SRDIFF

sol_conv

srmin

e_max

eet

ahi

tempcA

Topt

PET_TW_M umd_vegFAS_NDVI

solar

NPPcStress scalars

Intrinsic property

Stress ScalarsOriginal equations:

( )

( )

2

2

1 0 8 0 02 0 00051 ( 0 4472 0 0224 )

12 1 18141 exp 0 2 ( 10 )

11 exp 0 3 ( 10 )

0 5 0 5

T = . + . Topt . Topt= - . . Topt

T = .. Topt tempc

. Topt tempc

eetW = . + .PET

× − ×− + ×

×+ × − + −

×+ × − − +

×

_ max 1 2E = e T T W × × ×

Transformation into Network

( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )( )

( )

( )

21 1 11 12

2

2 21 22

21 21 21 22

3 3 31 32

1 1

1 ( 0 4472 0 0224 )1 12

1 exp 1 exp

11 exp 2 0.2 ( )

0 5 0 5

T = f x x f w w Topt

= - . . Topt

T = f x f x f xx x

f x f w w Topt tempc

Topt tempc

eet eetW = f x x f w w . + .PET PET

= − = + ×

− + ×

= × = ×+ − + −

= + −

=+ − × −

= = + = ×

0_ max 1 2 iE = e T T W w f× × × = ×∏

Intrinsic Values for Vegetation Type

min 0 95

1 ( )

SR_FAS srminFPAR_FAS , .SRDIFF

SR_FAS - srminSRDIFF

− =

≈ ×

FPAR_FAS: fraction of absorbed photosynthetically active radiation by the vegetation cover

SRDIFF: map from the ground cover to an srmax-srmin value

Transformation into Network

( )exp log( ) log( )

SR_FAS - srmin SRDIFF

= SR_FAS srmin SRDIFF− −

1 if_

0 othewiseumd_veg i

umd_veg i=

=

( )13

1log _ _

i- SRDIFF = v_i umd veg i

=

×∑

: weight in neural networkv_i

Revising weights in Networks

supervised learningStep-length

search direction

1st-order2nd-order

BP, etc.Newton method

variablefixed(constant)

Silva-Almeida algorithm,etc.SCG, OSS, BPQ, etc.

2nd-order learning algorithm

Gauss-Newton method

applicability to large-scale problems

×△○

○△×

quasi-Newton methodconjugate gradient method

performance with inaccurate step-length

BPQ Algorithm• The search direction is calculated on the

basis of partial BFGS update.

• The step-length is calculated by using a second-order approximation.

Demonstration ProblemSample set

１

２

３

４

５

y x１

２

３

４

５

0.73

0.88

0.95

0.980.99

1 Hidden unit

Input unit

Output unit

w

x(t)

１

w２

z(t)

1Sample

Learning Neural Network: Result

-0.5

0

0.5

1 -0.5 0 0.5 1

00.10.2

0.3

0.4

0.5

0.6

0.7

2W

1W

BPQ

BP + momentum(1st-order method)

(2nt-order method)Squared error

Experimental Result

360.00

400.00

440.00

480.00

520.00

560.00

RMSE RMSE RMSE

original apparent LOO CV

The RMSE of the original model was reduced by 15 percent, as measured using cross validation.

2o b served p red ic ted

sam p les

(N P P -N P P )R M S E =

n u m b er o f sam ples

∑

Intrinsic Values

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11

initial obtained

The intrinsic values associated with vegetation types obtained in this way were consistently lower

Transforming Network

Step1. Quantize by using a clustering method.

Step2. Determine an adequate number of rulesby using cross-validation.

Step3. Generate nominal conditionby solving a standard classification problem.

( ){ }( )exp : 1,nkl klkl

v q n N=∑ L

Clustering Analysis

380.00

400.00

420.00

440.00

460.00

480.00

500.00

520.00

1 2 3 4 5

apparent RMSE LOO CV RMSE

Number of clusters

Evaluating Experimental Result

380

400

420

440

460

480

500

520

540

before clustering after clustering

initial RMSE appar LOO CV RMSE

Obtained Decision Tree8 = 1 : 0 ( 1 0 . 0 )8 = 0 :

| 9 = 1 : 1 ( 5 8 . 0 )| 9 = 0 :| | 7 = 1 : 1 ( 4 6 . 0 )| | 7 = 0 :| | | 1 1 = 1 : 1 ( 1 1 . 0 )| | | 1 1 = 0 :| | | | 1 = 0 : 2 ( 1 6 8 . 0 / 1 . 0 )| | | | 1 =

tt

tt

tt

tt

tt 1 : 1 ( 1 0 . 0 )

Clustered Intrinsic Values

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11

initial obtained

Conclusion• This talk described an approach to

improving the predictive accuracy of the existing ecosystem model.

• In the experiments, we can reduce the mean squared error of the original model by 15 percent, as measured using cross validation

• In the future, we’ll carry out further experiments along this direction