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

2. 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

3. 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

4. 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

5. 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?

6. 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?

7. 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?

8. 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?

9. 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?

10. Ornstein-Uhlenbeck process

11. 1 1 11 1

12. 2 2 22 2

13. 3 3 33 3

14. 3 3 33 3 And so forth...

15. 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

16. 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

17. 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

18. Hansen 1997

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

21. Butler & King 2004

23. 7.0311.039.484.470.00 ∆

24. Butler & King 2004

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

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

27. 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

28. 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

29. 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

34. 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

35. Hansen & Martins 1996

36. Brownian motion OU or OU-like

38. 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

39. What is the probability of heads for our coin?

43. 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

44. 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

45. 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

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

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

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

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

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

51. Parameter value Distance

52. Parameter value Distance

53. Parameter value Distance

54. Ianaré Sévi

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

57. END