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Stochastic motion rate Ornstein-Uhlenbeck mean Continuous state Chasing peaks Brian O’Meara http://www.brianomeara.info

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In what ways can a continuous trait change in an instant of time? Randomly: increase or decrease slightly by chance Directionally: be pulled towards some value and/or

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In what ways can a continuous trait change in an instant of time? Randomly: increase or decrease slightly by chance Directionally: be pulled towards some value and/or

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In what ways can a continuous trait change in an instant of time? Randomly: increase or decrease slightly by chance Directionally: be pulled towards some value and/or

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Randomly: increase or decrease slightly by chance Directionally: be pulled towards some value and/or Rate of wiggle In what ways can a continuous trait change in an instant of time?

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Randomly: increase or decrease slightly by chance Directionally: be pulled towards some value and/or In what ways can a continuous trait change in an instant of time?

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Randomly: increase or decrease slightly by chance Directionally: be pulled towards some value and/or Adds the entire difference In what ways can a continuous trait change in an instant of time?

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Randomly: increase or decrease slightly by chance Directionally: be pulled towards some value and/or Allows directional change less than 100% (even zero) In what ways can a continuous trait change in an instant of time?

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Randomly: increase or decrease slightly by chance Directionally: be pulled towards some value and/or Ornstein-Uhlenbeck process In what ways can a continuous trait change in an instant of time?

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Ornstein-Uhlenbeck process

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1 1 11 1

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2 2 22 2

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3 3 33 3

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3 3 33 3 And so forth...

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Single rate Brownian motion Multiple mean Ornstein- Uhlenbeck Multiple rate Brownian motion Multiple everything all equal some vary 0all equal0some vary NAsome varyNAsome vary Independent contrasts (Felsenstein, 1985), ANCML (Schluter et. al, 1998) Hansen, 1997; OUCH (Butler & King, 2004), SURFACE (Ingram & Mahler, 2012) Brownie (O’Meara et al., 2006, Thomas et al., 2006), MEDUSA (Alfaro et al. 2009) OUwie (Beaulieu et al. 2012) Brownian rateOU attractionOU mean

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Single rate Brownian motion Multiple mean Ornstein- Uhlenbeck Multiple rate Brownian motion Multiple everything all equal some vary 0all equal0some vary NAsome varyNAsome vary Independent contrasts (Felsenstein, 1985), ANCML (Schluter et. al, 1998) Hansen, 1997; OUCH (Butler & King, 2004), SURFACE (Ingram & Mahler, 2012) Brownie (O’Meara et al., 2006, Thomas et al., 2006), AUTEUR (Eastman et al. 2011) OUwie (Beaulieu et al. 2012) Brownian rateOU attractionOU mean

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Single rate Brownian motion Multiple mean Ornstein- Uhlenbeck Multiple rate Brownian motion Multiple everything all equal some vary 0all equal0some vary NAsome varyNAsome vary Independent contrasts (Felsenstein, 1985), ANCML (Schluter et. al, 1998) Hansen, 1997; OUCH (Butler & King, 2004), SURFACE (Ingram & Mahler, 2012) Brownie (O’Meara et al., 2006, Thomas et al., 2006), AUTEUR (Eastman et al. 2011) OUwie (Beaulieu et al. 2012) Brownian rateOU attractionOU mean

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Hansen 1997

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Info and collage from Luke Mahler, http://lukemahler.com/. Photos by J. Losos, B.Falk, M. Landestoy, and L. Mahler

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Butler & King 2004

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7.0311.039.484.470.00 ∆

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Butler & King 2004

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Build up (paint regimes one at a time)Merge SURFACE: Ingram & Mahler 2013

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Mahler, Ingram, Revell, Losos 2013 (Science!)

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Single rate Brownian motion Multiple mean Ornstein- Uhlenbeck Multiple rate Brownian motion Multiple everything all equal some vary 0all equal0some vary NAsome varyNAsome vary Independent contrasts (Felsenstein, 1985), ANCML (Schluter et. al, 1998) Hansen, 1997; OUCH (Butler & King, 2004), SURFACE (Ingram & Mahler, 2012) Brownie (O’Meara et al., 2006, Thomas et al., 2006), AUTEUR (Eastman et al. 2011) OUwie (Beaulieu et al. 2012) Brownian rateOU attractionOU mean

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Single rate Brownian motion Multiple mean Ornstein- Uhlenbeck Multiple rate Brownian motion Multiple everything all equal some vary 0all equal0some vary NAsome varyNAsome vary Independent contrasts (Felsenstein, 1985), ANCML (Schluter et. al, 1998) Hansen, 1997; OUCH (Butler & King, 2004), SURFACE (Ingram & Mahler, 2012) Brownie (O’Meara et al., 2006, Thomas et al., 2006), AUTEUR (Eastman et al. 2011) OUwie (Beaulieu et al. 2012) Brownian rateOU attractionOU mean

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Model Optima (mean = M) Sigma (variance = V) Attraction (A) BM 1 1 BM S ≥2 OU 1 111 OU M ≥211 OU MA ≥21 OU MV ≥2 1 OU MVA ≥2 Beaulieu, Jhwueng, Boettiger, O'Meara 2012

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A priori placement of regimes Model-driven placement of regimes Multiple optima, single value of other parameters OUCH SURFACE, (Auteur [BM only]) Multiple of all parameters OUwiestay tuned

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Hansen & Martins 1996

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Brownian motion OU or OU-like

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PaperSubject Pritchard et al. 1999Population growth of Y chromosome Hickerson et al. 2006 Testing simultaneous divergence Jabot & Chave 2006Hubbell neutral model Fagundes 2007Human evolution Bokma 2010Anagenetic and cladogenetic rates of change Slater et al. 2012Multi-rate BM with incompletely sampled tree Some applications of ABC

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What is the probability of heads for our coin?

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Let p = 0.2 Sim 1: HTTTTHTT Sim 2: TTHHTTTT Sim 3: TTTHHTHH Sim 4: TTTTTTTT Sim 5: TTTTTTTT Sim 6: THHTTTTH Sim 7: TTHHTTHT Sim 8: TTTTHHHT Sim 9: HTTHTTTT Sim 10: THHHTHTT

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What is the probability of heads for our coin? Let p = 0.2 Sim 1: HTTTTHTT Sim 2: TTHHTTTT Sim 3: TTTHHTHH Sim 4: TTTTTTTT Sim 5: TTTTTTTT Sim 6: THHTTTTH Sim 7: TTHHTTHT Sim 8: TTTTHHHT Sim 9: HTTHTTTT Sim 10: THHHTHTT

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What is the probability of heads for our coin? Let p = 0.2 Sim 1: HTTTTHTT Sim 2: TTHHTTTT Sim 3: TTTHHTHH Sim 4: TTTTTTTT Sim 5: TTTTTTTT Sim 6: THHTTTTH Sim 7: TTHHTTHT Sim 8: TTTTHHHT Sim 9: HTTHTTTT Sim 10: THHHTHTT P(data) ≈ 1/10

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What is the probability of heads for our coin? 200 simulations per p True Approximation

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What is the probability of heads for our coin? 2,000 simulations per p True Approximation

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What is the probability of heads for our coin? 20,000 simulations per p True Approximation

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What is the probability of heads for our coin? 200,000 simulations per p True Approximation

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Discrete time taxa[[i]]$nextstates= taxa[[i]]$states + intrinsicFn(…) + extrinsicFn(…)

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Parameter value Distance

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Parameter value Distance

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Parameter value Distance

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Ianaré Sévi

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Character displacement (phenotypic distance at which repulsion is half the maximum repulsion) True value Estimate d value

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