Transcript Document

Developing and Testing
Mechanistic Models of
Terrestrial Carbon Cycling
Using Time-Series Data
Ed Rastetter
The Ecosystems Center
Marine Biological Laboratory
Woods Hole, MA USA
Jack Cosby
Environmental Sciences
University of Virginia
Charlottesville, VA USA
Topics:
I. The ANOVA Curse
II. What should be the focus of model
development and testing efforts?
III. Using transfer-function estimations to
identify important system linkages
IV. Using the Extended Kalman Filter as a
test of model adequacy that yields valuable
information on how to improve model structure
The ANOVA Curse
Grant will cover 6
treatment plots, which
we set up in a 2-level
3-replicate
design
Response
30
20
Tells us if a
treatment is
important, but
not how.
10
0
0
10
20
Treatment
30
Response
30
If the same treatments
are spread out
along a
continuum,
20
then we learn a lot more
about how the treatment
is important,
10
but still little about the
system dynamics.
0
0
10
20
Treatment
30
Focus of model development and testing
There has been an emphasis on the individual
processes within models (e.g., photosynthesis,
respiration, transpiration).
But are differences among models because of the
individual processes?
Or is it because of the overall model structure (i.e.,
how the components are linked together)?
Is it the overall structure or the
component processes that matters?
Structure 1:
Structure 2:
F
F
P
P
X
Structure 3:
F
Pb
Y
L
X
U
Pa
L
Ya
Yb
Xa
La
Xb
Lb
R
R
M
Y
R
Rastetter 2003
F
P1   X
F


P2   1  e
 X
F
8
6

P1
P3  Pa  Pb
 Xa
Pa   a 
1 X b
 Xb
Pb   b 
1 X a
Rastetter 2003
4
P
P2



a



b
P3
F
a  F
F
b  F
2
0
0
0.5
1
1.5
F
2
2.5
180
P1S1
Same model structure,
different process equation
P1S2
P2S1
150
X
P2S2
P3S3
120
90
0
50
100
Time
150
200
Response to a ramp in F from time 10 to 100
Rastetter 2003
Its the structure that matters!!!!!!!
(i.e. how the components are linked to one another)
Not the detailed process representation!
Structure 1:
Structure 2:
F
F
P
P
X
Structure 3:
F
Pb
Y
L
X
U
Pa
L
Ya
Yb
Xa
La
Xb
Lb
R
R
M
Y
R
Testing system linkages
ARMA Transfer Function Models
n
x
G
F
r
e
y
+
+
y
yt = b0 xt + b1 xt-1 + ... - a1 yt-1 - a2 yt-2 - ... +
0 nt + 1 nt-1 + ... - 1 rt-1 - 2 rt-2 - ... + et
x - input time series
y - output time series
n - white noise time series
e - error time series
F - Deterministic transfer function
G - Stochastic transfer function
Young 1984
Input Time Series
Output Time Series
PO4
NO3
C B C P N
I O I O O O
C 3 C 2 4 3
P
/
P R R
P F E A
H I N U
Y L C T
R S M F A E E L C C C C H
O T A A L P G A A A A Y E
T E L L O C G R D E T C T
O O O O
O
O O O O O
O O
O O
O O O O
O O A O
X O O O O O O O O O O O O
O O O A A O O O O O O O O
X
X X X X
X X X X
X X X X
X X X X X X X X X X X X X
X X X X X X X X X X X X X
X A
O
O
A X
X
O
X
X
X
O
X
X
O
X
O
O
O
X
O
X
O
O
O
X X
O
X
O O
O
X O
O X
O O
A X
O O
A O
A O
P
R
P/R
L
O
A
A
A
X
X
X
X
X
A
A
A
X
X
A
A
X
X
X
X
X
X
X
X
PHY
FIL
ENC
AUT
A
O
A
X
A
O
A
X
A
O
A
X
A
O
A
X
X
O
X
X
X
O
A
X
A
O
X
X
X
O
A
O
ROT
STE
MAL
FAL
ALO
EPC
EGG
LAR
CAD
CAE
CAT
CYC
HET
O
O
X
O
O
O
X
O
O
O
O
O
O
X
O
A
O
O
O
X
O
X
O
O
O
O
O
O
X
O
O
O
X
O
X
O
O
O
O
X
O
X
O
O
X
X
O
X
O
O
O
O
O
O
X
O
O
O
O
O
X
O
O
O
O
X
O
X
A
A
O
X
O
X
O
X
O
X
O
O
X
O
O
X
X
O
X
X
O
O
O
X
O
X
O
O
O
O
O
X
O
O
O
O
Deterministic
function significant
X
A A
A
O
A
O
X
O
X
O
O
X
O
O
X
O
O
O
O
X
O
X
O
O
X
O
O
X
O
X
O
O
X
O
X
X
O
X
O
O
X
O
O
O
O
X
O
X
X
O
O
X
O
X
O
O
O
O
X
O
A
A
A
O
A
A
X
O
A
A
X
O
X
X
X
O
X
X
X
O
X
O
X O
O O
X
X
O
X
O
O
X
O
X
A
X
O
O
X
O
O
O
X
X
O
O
O
O
X
O
X
O
O
O
O
O
O
X
O
X
X
O
X X
O O
X X
Combined model significant but
deterministic function not significant
X
O
X
X
X
O
X
O
X
O
X
O
O
X
X
O O
X
X
O
X
O
O
A
O
A
X
No significant
pattern
Rastetter 1986
Kalman Filter
• The Kalman Filter is recursive filter that estimates
successive states of a dynamic system from a time
series of noise-corrupted measurements (Data
Assimilation)
•A linear model is used to project the system state
one time step into the future
•Measurements are made after the time step has
elapsed and compared to the model predictions
•Based on this comparison and a recursively updated
assessment of past model performance (estimate
covariance matrix) and past measurement error
(innovations covariance), the Kalman Filter updates,
and hopefully improves, estimates of the modeled
variables
Extended Kalman Filter
• The Extended Kalman Filter (EKF) is essentially
the same as the Kalman filter, but with an
underlying nonlinear model
•To accommodate the nonlinearity, the model must
be linearized at each time step to estimate the
Transition matrix
•This transition matrix is used to update the
estimate covariance
Nonlinear models
Discrete model
xt = f(xt-1, ut, wt)
Ft = J = f
x x
t-1:t-1,ut
Linearized
transition matrix
Continuous model
dx = f(x, u, w)
dt
Ft =exp(JDt)
Linearized
transition matrix
exp(JDt) = I + JDt + (JDt)2/2! +...+ (JDt)n/n! +...
(Continuous) Extended Kalman Filter
Predict
xt:t-1 = xt-1:t-1 +
t
f(x,u,0)dt
predicted state
t-1
Pt:t-1 = Ft Pt-1:t-1 FtT + Qt
Update
yt = zt - Ht xt:t-1
estimate covariance
innovations
St = HtPt:t-1 HtT + Rt
innovations covariance
Kt = Pt:t-1 HtT St-1
Kalman gain
xt:t = xt:t-1 + Kt yt
updated state
Pt:t = (I - Kt Ht) Pt:t-1
updated estimate covariance
Augmented State Vector
x1
x2
x3
x* =
xn
r1
r2
r3
rm
• Once the Kalman Filter has been
extended to incorporate a nonlinear
model, it is easy to augment the
state vector with some or all of the
model parameters
•That is, to treat some or all of the
parameters as if they were state
variables
•This augmented state vector then
serves a the basis for a test of
model adequacy proposed by Cosby
and Hornberger (1984)
EKF Test of Model Adequacy
Cosby & Hornberger 1984
The model embedded in the EKF is
adequate if:
1) Innovations (deviations) are zero mean,
white noise (i.e., no auto-correlation)
2) Parameter estimates (in the augmented
state vector) are fixed mean, white noise
3) There is no cross-correlation between
parameters and state variables or control
(driver) variables
Eight Models Tested by Cosby et al. 1984
O2 concentration in a Danish stream
dc
 K cs  c   R  PS
dt

PS   I
PS   I 
PS   1  e  I
Hyperbolic
I
PS 
 I
I
PS 
2
2
 I

Webb
PS   tanh I 
PS   I e  I
 I

PS   I 1 
 

note 1 model structure, alternate representation of PS
Cosby et al. 1984
Webb
Hyperbolic
mean value
Webb - 1.2
Maximum rate
Hyperbolic - 1.7
Webb - 3.7
Initial slope of PI curve
Hyperbolic - 0.32
both - 0.51
both - 0.94
Cosby et al. 1984
•All 8 models failed in the same way; parameter
controlling initial slope of PI curve had a diel cycle.
•Its not the details of process representation that’s
crucial, its how the processes are linked to one another.
Linear model
“wags” as light
changes
PS (mg O2 L-1 hr -1)
1.5
1.25
All models have
diel hysteresis
1
0.75
0.5
Hyperbolic
Webb
0.25
0
0
0.25
0.5
0.75
Radiation (ly min -1)
1
EKF Test of Model Adequacy
•The EKF can be used as a severe test of model
structure (few models are likely to pass the test)
•More importantly, it yields a great deal of
information on how the model failed that can be
used to improve the model structure
•e.g., the initial-slope parameter in the Cosby
model should be replaced with a variable that
varies on a 24-hour cycle, like a function of CO2
depletion in the water, or C-sink saturation in the
plants
Are we getting the right type of data?
Time series data are extremely expensive and
therefore rare
e.g., eddy flux, hydrographs, chemographs, tree rings,
others?
Their value to understanding of ecosystem dynamics is
definitely worth the expense
The key to good time series data is automation to
assure consistent, regular sampling
There should be a high degree of synchronicity among
time series collected on the same system
Conclusions:
•Time series are far richer in information on system
dynamics and system linkages than data derived from
more conventional experimental designs (e.g., ANOVA)
•Time series provide replication through time, which
allows for statistical rigor without the replication
constraints of more conventional experimental designs
(but perhaps restricts confidence in extrapolation to
other systems)
•The focus of study should be on identifying and
testing the linkages among system components (i.e.,
the system structure) rather than the details of how
the individual processes are represented
Conclusions:
•Transfer-function estimation can be used to identify
links among ecosystem components or test the
importance of postulated linkages
•The Extended Kalman Filter can be used as a severe
test of model adequacy that yields valuable information
on how to improve the model structure
•Unfortunately, high quality time-series data in ecology
are still rare
•However, new expenditures currently proposed for
monitoring the biosphere (e.g., ABACUS, LTER,
NEON, CLEANER, CUAHSI, OOI) may provide the
support to automate time-series sampling of several
important ecosystem properties.
The End