Modelling Phenotypic Evolution by

Stochastic Differential Equations

Combining statistical timeseries with fossil measurements.

Tore Schweder and Trond ReitanUniversity of Oslo

Jorijntje HenderiksUniversity of Uppsala

ICES 2010, Kent

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Overview

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

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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.

Irregular time series related by latent processes: evolution of body size in Coccolithus

Henderiks – Schweder - Reitan

19,899 coccolith measurements, 205 sediment samples (1< n < 400) of body size by site and time (0 to -60 my).

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Our data – Coccolith size measurements

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

Individual samples

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

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Evolution of size distribution

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).

With fitness as a process, µ(t),, we can make a tracking model: 6

)()()( tdWdttXtdX

)()()()( tdWdtttXtdX

The Ornstein-Uhlenbeck process

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

1.96 s

-1.96 st

The parameters (, t, s) can be estimated from the data. In this case: 1.99, t=1/α0.80Myr, s0.12. 7

)()()( tdWdttXtdX

One layer tracking another

Red process (t2=1/2=0.2,

s2=2) tracking black process (t1=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

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Process layers - illustration

Layer 1 – local phenotypic expression

Externalseries

Fixed layer

Observations

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Layer 2 – local fitness optimum

Layer 3 – hidden climate variations or primary optimum

Coccolith model

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Stochastic differential equation (SDE) vector processes

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Model variants for Coccolith evolution

The 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 parametersCorrelation between sites (inter-regional correlation)Deterministic response to the lower layerRandom walk (no tracking)

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Likelihood: Kalman filter

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The Ito solution

gives, together with measurement variances, what is needed to calculate the likelihood using the Kalman filter: and

t t

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iii mzzzE ),,|( 11 iii vzzzVar ),,|( 11

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Observations

Need a linear, normal Markov chain with independent normal observations:

),,|()|( 1 iiiNin vmzfzf

Kalman smoothing (state estimation)ML fit of a tracking model with 3 layers

North AtlanticRed curve: expectancyBlack curve: realizationGreen curve: uncertainty

Snapshot:14

Inference

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

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Model inference concerns

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.

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Results, original model3 layers, no regionality, no correlation

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Bayesian inference on the 723 models

18

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95% credibility bands for the 5 most probable models

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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. Site-specific pull.

Middle layer: Intermediate tracking.

Upper layer: Very fast tracking.

Middle layers: fitness optima

Top layer: population mean log size

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Layered inference and inference uncertainty

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Phenotypic evolution on a phylogenetic tree: Body size of primates

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First results for some primates

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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 multi-normal 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...

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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 non-linear SDE models… SDE processes might be driven by non-Gaussian instantaneous

stochasticity (jump processes).

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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/.

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