Gap filling using a Bayesian-regularized neural network

B.H. BraswellUniversity of New Hampshire

MacKay DJC (1992) A practical Bayesian framework for backpropagation networks. Neural Computation, 4, 448-472.

Bishop C (1995) Neural Networks for Pattern Recognition, New York: Oxford University Press.

Nabney I (2002) NETLAB: algorithms for pattern recognition. In: Advances in Pattern Recognition, New York: Springer-Verlag.

Proper Credit

Two-layer ANN is a nonlinear regression

Two-layer ANN is a nonlinear regression

e.g., tanh()usually nonlinear

usually linear

Neural networks are efficient with respect to number of estimated

parameters

Polynomial of order M: Np ~ dM

Consider a problem with d input variables

Neural net with M hidden nodes: Np ~ d∙M

Early stopping

Regularization

Bayesian methods

Avoiding the problem of overfitting:

Avoiding the problem of overfitting:

Early stopping

Regularization

Bayesian methods

Avoiding the problem of overfitting:

Early stopping

Regularization

Bayesian methods

Arti f icial neural net w orks

An ar t ifi c i a l n eura l n e t w o r k ( A NN ) i s a f u n c t i o n a l m app i n g o f a v ec to r x c o n t a i n i n g d i n pu t s , i n to a

v ec to r y c o n t a i n i n g c o u t pu t s . An A NN c o n s i s t s o f “ l a y ers”, eac h h a v i n g M “n o des”. A n o d e i s a

li n ear t ra n s f o r m a t i o n o f i n pu t s f o l l o wed b y app li c at i o n o f a prescr i b ed f u n c t i o n . T h e o u t p ut s o f a l l

n o des i n a l a y er a re c o ll ec t ed i n t o a n ew v ec tor and i n pu t i n t o t h e n e x t l a y er . F or a t w o - l a y er

n e t w o rk,

€

ˆ y k ( x ) = f w kj

( 2 ) ˜ f w ji

(1) x i

i= 0

d

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

j = 0

M

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟, (1)

w h ere f i s a f u n c t i o n , t y p i ca lly f ( a )= a , a n d

€

˜ f i s a d iff e re n t f u n c t i o n t h a t i s usua lly n o n li n ear ( e.g . ,

t a nh ( a )) . T h e t w o m a t r i ces w(1)

a n d w(2 )

represe n t t h e f ree para m e t ers i n t h e regress i o n a n d i n c l ude

b i as t er m s f or j =0 a n d i = 0 .

F o r para m e t e r es t im a t i o n , t h e s t a n da rd b ackpr o pa g a t i o n a l g o r i t hm i s t y p i ca lly used . T h i s m e t h o d

upda t es t h e we i g h t s a n d b i ases w f o r t h e N pa i rs o f o b ser v ed da t a v ec to rs x a n d yi n o rde r t o

mi n imi ze t h e e r r o r:

€

E (w ) =k

( n )y (w ) −

k

( n )ˆ y ( )2

k =1

c

∑n =1

N

∑ (2)

b y es t im a t i n g t h e der i v a t i v e o f E ( w ) w i t h respec t t o w .

1

Bayesian regularization

I n t h e Ba y es i a n f ra m ew o r k , m o de l para m e t ers are t rea t ed as pr o b a b ili t y d i s t r i b u t i o n s , a n d t h e

p o s t er i o r pr o b a b ili t y o f t h e we i g h t s g i v en t h e se t o f o b ser v ed o u t pu ts

€

D ≡ y ( n ); n =1… N , i s:

)( )()|()|( DppDpDp www= (3)

w h ere p ( D | w ) i s t h e pr o b a b il i t y o f t h e o b ser v a t i o n s g i v e n a c h o i ce o f we i g h t s (t h e li ke li h ood ) , p ( w )

i s a pr i or d i s t r i b u t i o n o f we i g h t v a l ues, a n d t h e den o mi n a to r i s a n o r m a li za t i o n c o n s t a n t.

A s su mi n g Gauss i a n d i s t r i b u t i o n s f or b ot h t h e li ke l ih oo d a n d t h e p r i or , t h e p o s t er i o r d i s tr i b u t i o n i s

g i v e n b y

€

p(w | D ) =1

Z D

exp( −β E D ) ⎛

⎝ ⎜

⎞

⎠ ⎟

1

Z W

exp( −α E W ) ⎛

⎝ ⎜

⎞

⎠ ⎟

=1

Z S

exp( −β E D − α E w ) =1

Z S

exp − S (w )( )

(4)

w h ere Z S i s a c o n s t a n t a n d S ca n b e rew r i tt e n as

€

S =β

2y k

( n ) (w ) − ˆ y k( n )

( )2

k =1

c

∑ +α

2n = 1

N

∑ w i

2

i=1

W

∑ ( 5)

T h e para m e t ers a n d represe n t t h e n o i se i n t h e da t a a n d t h e v ar i a n ce o f t h e we i g h t s ,

respec t i v e ly . T h us , a s o l u t i o n t o t h i s pr o b l e m i s f o u n d b y m a x imi z i n g t h e p o s t er i or pr o b a b il i t y w i t h

respec t to w , o r m i n imi z i n g t h e n ega t i v e lo g o f t h e pr o b a b ili t y . T h i s a m o u n t s t o mi n i m i z i n g t h e

m o d ifi ed e r r o r f u n c t i o n S ( w ) i n Equa t i o n 5.

2

Gaussian approximation to the posterior distribution

T o es t im a t e t h e u n cer t a i n t y o f t h e p red i c t i o n s, we use a Gauss i a n appr o x im a t i o n to t h e p o s t er i o r

d i s t r i b u t i o n f o r t h e we i g h t s a n d per f o r m a Ta yl or ser i es e x pa n s i o n ar o u n d t h e m o s t p r o b a b l e v a l ues

w MP ,

)()(21)()( MPTMPMPSS wwAwwww −−+≅ , (6)

w h ere A i s equa l to :

IA α+∇∇=∇∇= )(MPDMP ES (7)

T h i s i s t h e Hess i a n m a t r i x o f t h e err o r f u n c t i o n ( E q ua t i o n 5) , a n d i t s e l e m e n t s ca n b e ca l cu l a t ed

n u m er i ca lly dur i n g t h e b ackpr o paga t i o .n T h e e x pa n s i o n i n Equa t i o n 6 a ll o ws us t o rewr i t e t h e

p o s t er i o r d i s t r i b u t i o n f or t h e we i g h t s :as

€

p(w | D ) =1

Z S

*exp −S (w MP ) −

1

2Δw TAΔw

⎛

⎝ ⎜

⎞

⎠ ⎟ , (8)

w h ere

€

Δw = w − w MP , a n d

€

Z S

* i s g i v e n b y

€

Z S

* (α ,β ) = (2π )W / 2 A−1 / 2

exp S (w MP )( ) . (9)

3

P osterior distribution o f outputs

T h e d i s t r i b u t i o n o f n e t w o rk o u t p ut s (a n d t h us t h e u n cer t a i n t y o f t h e pred i c t i o n) i s es t im a t ed by

assu mi n g t h a t t h e w i d t h o f t h e po s t er i o r d i s tr i b u t i o n i s n ot e x t re m e ly b r o a d , a n d s o t h e pred i c t i o n s

are e x pa n ded as

€

ˆ y (w ) = ˆ y (w MP ) + g T (w − w MP ) (10)

w h ere

€

g ≡ ∇ wˆ y |w MP

. (11)

T h e p o s t er i or d i s t r i b u t i o n o f t h e pred i c t i o n s i s g i v e n b y

€

p( y | D ) = p ( y∫ | w ) p (w | D )dw ( 12)

I n t h e Gauss i a n appr o x im a t i o n f or t h e p o s t er i or o f t h e we i g h t s (Equa t i o n 8 ) , a n d assu mi n g zer o

m ea n Gauss i a n n o i se, t h i s b ec o m es

€

p( y | D ) ∝ exp −β

2y − ˆ y (w )[ ]

2 ⎛

⎝ ⎜

⎞

⎠ ⎟exp −

1

2Δw TAΔw

⎛

⎝ ⎜

⎞

⎠ ⎟∫ dw (13)

T h us, su b s t i t u t i n g Equa t i o n 1 0 , a n d g i v e n Equa t i o n s 7 a n d 9 , t h e p o s t er i o r d i s t r i b u t i o n f o r t h e

o u t pu t s

€

p( y | x , D ) i s n o r m a l , w i t h s t a n dard de v i a t i o n

€

σ y

2 =1

β+ gTA −1g (14)

4

5Determining the regularization coe ff icients

T h e m o s t li ke l y v a l ues o f t h e “h y perpara m e t ers” a n d ca n b e de t e r mi n ed i n a h i erarc h i ca l

f as h i o n . T h e p o s t er i o r f r o m Equa t i o n 3 , n o w a j o i n t d i s t r i b u t i o n c o n t a i n i n g t h ese add i t i o n a l

para m e t ers, i s f i rs t appr o x im a t ed as

€

p(w | D ) ≅ p (w | α MP ,β MP , D ) , (15)

T h us we m us t a l t er n a t i v e l y es t im a t e a n d , t h e n use Equa t i o n 5 t o ca l cu l a t e t h e we i g h t s . Us i n g

Ba y es ' t h e o re m , t h e p o s t er i or d i s t r i b u t i o n f or t h e h y perpara m e t ers i s g i v e n b y

€

p(α ,β | D ) =p(D | α ,β ) p (α ,β )

p (D ). (16)

I n c l ud i n g t h e e x p li c i t depe n de n ce o f t h e h y perpar a m e t ers, t h e n o r m a li za t i o n o f Equa t i o n 3 can b e

wr i tt e n as

€

p(D | α ,β ) = p(D | w,α ,β )∫ p (w | α ,β )dw

= p(D | w,β )∫ p(w | α )dw (17)

Us i n g t h e e x p o n e n t i a l f or m u l a t i o n f o r t h e pr i o r and li ke li h oo d , t h i s e x press i o n i s n o w f ra m ed i n

t er m s o f t h e o r i g i n a l n o r m a li z i n g c o n s t a n t s:

€

p(D | α ,β ) =1

Z D (β )

1

Z W (α )exp( −S (w )) dw∫

=Z S (α ,β )

Z D (β )Z W (α )

(18)

Su b s i t u t i n g t h e Gauss i a n appr o x im a t i o n f o r t h e p o s t er i o r e v a l ua t ed i n t h e n e i g hb o r h ood o f t h e

o p t im a l we i g h t s w M P , (Equa t i o n 9 ) , t h e f u n c t i o n t ha t m us t b e mi n i m i zed n o w i s t h e n ega t i v e log o f

t h e li ke li h oo d p ( D | , ) , w h i ch i s equa l to

€

ln P (D | α ,β ) = α E W

( MP ) + β E D

( MP ) +1

2ln(A ) −

W

2ln( α ) −

N

2ln( β ) +

N

2ln( 2π ) (19)

E v a l ua t i n g t h e d e r i v a t i v e o f t h i s e x press i o n w i t h r e spec t to t h e t w o p a ra m e t ers i nv o l v es ca l cu l a t i n g

t h e e i ge nv a l ues o f A , a n d we ar e l e f t w i t h e x pre s s i o n s f o r t h e o p t im a l v a l ues o f a n d , g i v e n

t h e m o s t pr o b a b l e v a l ues o f t h e we i g h t s w . T h us,

€

α ( n +1) =γ (n)

2 E w

( n ); β ( n +1) =

N - γ (n)

2 E D

( n ) (20 )

w h ere

€

γ ≡λ i

λ i + αi=1

W

∑ (21)

I n prac t i ce , t h e para m e t ers a n d h y perpara m e t ers a r e s o l v ed to ge t h er by a l t er n a t i v e ly upda t i n g w

us i n g s t a n dard b ackpr o paga t i o n , a n d upda t i n g a n d us i n g Equa t i o n s 2 0 a n d 2 1 ( h e n ce t h e i n dex

n +1 a b o v e).

6

Hagen SC, Braswell BH, Frolking, Richardson A, Hollinger D, Linder E (2006) Statistical uncertainty of eddy flux based estimates of gross ecosystem carbon exchange at Howland Forest, Maine. Journal of Geophysical Research, 111.

Braswell BH, Hagen SC, Frolking SE, Salas WE (2003) A multivariable approach for mapping subpixel land cover distributions using MISR and MODIS: An application in the Brazilian Amazon. Remote Sensing of Environment, 87:243-256.

Previous Work

ANN Regression for Land Cover Estimation

Band1

Band2

Band3

Band4

Forest Fraction

Cleared Fraction

Secondary Fraction

Training data suppliedby classified ETM imagery

Forest Secondary Cleared

ETM+observed

MISRpredicted

1.0

0.0

1.0

0.0

0.4

0.0

0.4

0.0

0.6

0.0

0.6

0.0

Mean Val.Error=0.045 km2

Mean Val.Error=0.038 km2

Mean Val.Error=0.025 km2

(R2=.62) (R2=.58) (R2=.47)

ANN Regression for Land Cover Estimation

ANN Estimation of GEE and Resp, with Monte Carlo simulation of Total Prediction uncertainty

Clim Flux

Weekly GEE from Howland Forest, ME based on NEE

ANN Estimation of GEE and Resp, with Monte Carlo simulation of Total Prediction uncertainty

Some demonstrations of the MacKay/BishopANN regression with 1 input and 1 output

Noise=0.10

1.4

Noise=0.10

Linear Regression

Noise=0.10

ANN Regression

Noise=0.02

ANN Regression

Noise=0.20

ANN Regression

Noise=0.20

ANN Regression

Noise=0.10

ANN Regression

Noise=0.05

ANN Regression

Noise=0.05

ANN Regression

Noise=0.05

ANN Regression

Issues associated with multidimensional problems

Sufficient sampling of the the input space

Data normalization (column mean zero and standard deviation one)

Processing time

Algorithm parameter choices

Our gap-filling algorithm

1.Assemble meteorological and flux data in an Nxd table

2.Create five additional columns for sin() and cos() of time of day and day of year, and potential PAR

3.Standardize all columns

4.First iteration: Identify columns with no gaps; use these to fill all the others, one at a time.

5.Create an additional column, NEE(t-1), flux lagged by one time interval

6.Second iteration: Remove filled points from the NEE time series, refill with all other columns

Room for Improvement

1.Don’t extrapolate wildly, revert to time-based filling in areas with low sampling density, especially at the beginning and end of the record

2.Carefully evaluate the sensitivity to internal settings (e.g., alpha, beta, Nnodes)

3.Stepwise analysis for relative importance of driver variables

4.Migrate to C or other faster environment

5.Include uncertainty estimates in the output

6.At least, clean up the code and make it available to others in the project, and/or broader community