ICES 2010, Kent
by
Tore Schweder and Trond Reitan
University of Oslo
Jorijntje Henderiks
University of Uppsala
Combining statistical timeseries with fossil measurements.
1

1. Introduction:
• Motivating example:
 Phenotypic evolution: irregular time series related by common latent processes
 Coccolith data (microfossils)
 205 data points in space (6 sites) and time (60 million years)
2. Stochastic differential equation vector processes
• Ito representation and diagonalization
• Tracking processes and hidden layers
• Kalman filtering
3. Analysis of coccolith data
• Results for original model
• 723 different models – Bayesian model selection and inference
4. Second application: Phenotypic evolution on a phylogenetic tree
• Primates - preliminary results
5. Conclusion
2

Irregular time series related by latent processes: evolution of body size in Coccolithus
Henderiks – Schweder - Reitan
The size of a single cell algae ( Coccolithus ) is measured by the diameter of its fossilized coccoliths (calcite platelets). Want to model the evolution of a lineage found at six sites.
•In continuous time and ”continuously”.
•By tracking a changing fitness optimum.
•Fitness might be influenced by observed (temperature) and unobserved processes.
•Both fitness and underlying processes might be correlated across sites.
19,899 coccolith measurements, 205 sediment samples (1< n < 400) of body size by site and time (0 to -60 my).
3

205 Sample mean log coccolith size (1 < n < 400) by time and site.
4

Bi-modality rather common.
Speciation? Not studied here!
5

Fitness = expected number of reproducing offspring.
The population tracks the fitness curve (natural selection)
The fitness curve moves about, the population follow.
With a known fitness, µ , the mean phenotype should be an
Ornstein-Uhlenbeck process
(Lande 1976).
dX ( t ) X ( t ) dt dW ( t )
With fitness as a process, µ(t)
,
, we can make a tracking model: dX ( t ) X ( t ) ( t ) dt dW ( t )
6

dX ( t )
X ( t ) dt dW ( t )
1.96 s
Attributes:
• Normally distributed
• Markovian
• long-term level:
• Standard deviation: s= / 2α
• α : pull
• Time for the correlation to drop to 1/e: t =1/α t
-1.96 s
The parameters ( , t, s ) can be estimated from the data. In this case: 1.99, t=1/ α 0.80Myr, s 0.12.
7

dX
1
( t ) dX
2
( t )
1
X
1
( t )
2
X
2
( t )
X
2
( t ) dt dt
1 dW
1
( t )
2 dW
2
( t )
Red process ( t
2
=1/
2
=0.2, s
2
=2 ) tracking black process
( t
1
=1/
1
= 2, s =1 )
Auto-correlation of the upper (black) process, compared to a one-layered
SDE model.
A slow-tracking-fast can always be re-scaled to a fast-tracking slow process.
Impose identifying restriction:
1
≥
2
8

Observations
Layer 1 – local phenotypic expression
Layer 2 – local fitness optimum
T
External series
Layer 3 – hidden climate variations or primary optimum
Fixed layer

X s , l
( t ) state of the l ' th process at site s at time t (log size).
dX s , 1
( t ) s , 1
X s , 1
( t ) X s , 2
( t ) dt noise Population mean dX u ( t ) s , 2
( t ) dX s , 3
( t ) observed s , 2 s , 3
X s , 2
( t )
X s , 3
( t ) covariate
X s , 3
( t ) dt s u ( t ) dt process (temperatu noise re)
Fitness optimum noise Environmen t
0 is the tracking speed of site s belonging to the l ' th layer s , l
0 random walk, s , l
The noise is white, possibly s , l correlated perfect tr acking inside a layer
X
3
, u X
2
X
1
Observatio ns Z s , t are N ( X s , 1
( t ), s , t
2
)
10

X ( t )
W ( t )
( t )
: p
: q -
dimensiona dimensiona u ( t ) : p l vector state process l vector Wi dimensiona l ener process level process,
(Brownian a p motion) r regression matrix
A : p p p dimensiona q dimensiona l pull matrix l diffusion matrix dX ( t ) ( AX ( t ) ( t )) dt dW ( t )
Ito solution :
X ( t ) B ( t )
1
X ( 0 )
0 t
B ( u ) ( u ) du
0 t
B ( u ) dW ( u )
B ( t ) e
At
A n
( t ) n
/ n !
Λ is a diagonal n 0 eigenvalue
V matrix
1 e t
V (when existing) of A , V is the left eigenvecto r matrix.
11

 T he individual size of the algae Coccolithus has evolved over time in the world oceans. What can we say about this evolution?
 How fast do the populations track its fitness optimum?
 Are the fitness optima the same/correlated across oceans, do they vary in concert?
 Does fitness depend on global temperature? How fast does fitness vary over time?
 Are there unmeasured processes influencing fitness?
•
•
•
Model variations:
1, 2 or 3 layers.
Inclusion of external timeseries
In a single layer:



Local or global parameters
Correlation between sites (inter-regional correlation)
Deterministic response to the lower layer
 Random walk (no tracking)
12

Need a linear, normal Markov chain with independent normal observations:
X ( t
1
) X ( t
2
)  X ( t n
) Process z
1 z
The Ito solution
2 z n
Observations t t
X ( t ) B ( t )
1
X ( 0 ) B ( u ) ( u ) du B ( u ) dW ( u )
0 0 gives, together with measurement variances, what is needed to calculate the likelihood using the Kalman filter:
E ( z i
| z
1
,  , z i 1
) m i and Var ( z i
| z
1
,  , z i 1
) v i f ( z | ) n i 1 f
N
( z i
| m i
, v i
, )
13

ML fit of a tracking model with 3 layers
North Atlantic
Red curve: expectancy
Black curve: realization
Green curve: uncertainty
Snapshot:
14

 Exact Gaussian likelihood, multi-modal
 Maximum likelihood by hill climbing from 50 starting points
 BIC for model comparison.
 Bayesian:
 Wide but informative prior distributions respecting identifying restrictions



MCMC (with parallel tempering) +
Importance sampling (for model likelihood)
Bayes factor for model comparison and posterior probabilities
Posterior weight of a property C from posterior model probabilities
W ( C )
C '
1 n
C
1
M C f ( data | M ) f ( data | M ) n
C ' M C '
15

Concerns:


Identification restriction: increasing tracking speed up the layers
In total: 723 models when equivalent models are pruned out)
•
•
Enough data for model selection?
Data Simulated from the ML fit of the original model
Model selection over original model plus 25 likely suspects.
• correct number of layers generally found with the Bayesian approach.
• BIC seems too stingy on the number of layers.
16

3 layers, no regionality, no correlation
17

W ( C )
C '
1 n
C
1
M C f ( data | M ) f ( data | M ) n
C ' M C '
18

19

Summary
 Best 5 models in good agreement.
(together, 19.7% of summed integrated likelihood):






Three layers.
Common expectancy in bottom layer .
No impact of exogenous temperature series.
Lowest layer: Inter-regional correlations, 0.5. Sitespecific pull.
Middle layer: Intermediate tracking.
Upper layer: Very fast tracking.
Middle layers: fitness optima
Top layer: population mean log size
20

21

Measuremen ts i 1 ,  , 209 (92 extant measuremen ts, 117 fossils)
X i , 1
( t )
X i , 2
( t ) mean fitness phenotype optimum
(log mass)
X i , 3
( t )
X i , 3 underlying
X i , 2
X environmen i , 1 same tracking t, primary optimum model as for coccoliths
Phylogenet
T i , j
X i , l
( t )
Time
X
when j , l i split from j .
( t ) when t T i , j
.
Condition ic tree : on these equalities for the given tree .
Observatio be multi ns Z i , k
normally matrix as parametric
X i , k
( t k
) distribute expression
N ( 0 , d with
2 k
) will mean vecto r and covariance s from the SDE model.
22

23

Why linear SDE processes?




Parsimonious: Simplest way of having a stochastic continuous time process that can track something else.
Tractable: The likelihood, L( ) f(Data | ), can be analytically calculated by the Kalman filter or directly by the parameterized multinormal model for the observations. ( = model parameter set)
Some justification from biology, see Lande (1976), Estes and Arnold
(2007), Hansen (1997), Hansen et. al (2008).
Great flexibility, widely applicable...
24

Further comments






Many processes evolve in continuous rather than discrete time. Thinking and modelling might then be more natural in continuous time.
Tracking SDE processes with latent layers allow rather general correlation structure within and between time series. Inference is possible.
Continuous time models allow related processes to be observed at variable frequencies (high-frequency data analyzed along with low frequency data).
Endogeneity, regression structure, co-integration, non- stationarity, causal structure, seasonality …. are possible in SDE processes.
Extensions to nonlinear SDE models…
SDE processes might be driven by non-Gaussian instantaneous stochasticity (jump processes).
25

Bibliography







Commenges D and Gégout-Petit A (2009), A general dynamical statistical model with causal interpretation. J.R. Statist. Soc. B, 71,
719-736
Lande R (1976), Natural Selection and Random Genetic Drift in
Phenotypic Evolution, Evolution 30, 314-334
Hansen TF (1997), Stabilizing Selection and the Comparative
Analysis of Adaptation, Evolution , 51-5, 1341-1351
Estes S, Arnold SJ (2007), Resolving the Paradox of Stasis:
Models with Stabilizing Selection Explain Evolutionary Divergence on All Timescales, The American Naturalist , 169-2, 227-244
Hansen TF, Pienaar J, Orzack SH (2008), A Comparative Method for Studying Adaptation to a Randomly Evolving Environment,
Evolution 62-8, 1965-1977
Schuss Z (1980). Theory and Applications of Stochastic Differential
Equations. John Wiley and Sons, Inc., New York.
Schweder T (1970). Decomposable Markov Processes. J. Applied
Prob. 7, 400 –410
Source codes, examples files and supplementary information can be found at http://folk.uio.no/trondr/layered/.
26