# Stochastic motion rate Ornstein-Uhlenbeck mean Continuous state Chasing peaks Brian O’Meara

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

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

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

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

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?

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?

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?

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?

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?

Ornstein-Uhlenbeck process

1 1 11 1

2 2 22 2

3 3 33 3

3 3 33 3 And so forth...

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

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

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

Hansen 1997

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

Butler & King 2004

7.0311.039.484.470.00 ∆

Butler & King 2004

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

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

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

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

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

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

Hansen & Martins 1996

Brownian motion OU or OU-like

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

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

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

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

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

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

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

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

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

Parameter value Distance

Parameter value Distance

Parameter value Distance

Ianaré Sévi

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

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