Particle Markov Chain Monte Carlo Methods in Marine Biogeochemistry. Lawrence Murray and Emlyn Jones CSIRO Mathematics, Informatics and Statistics

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1 Particle Markov Chain Monte Carlo Methods in Marine Biogeochemistry Lawrence Murray and Emlyn Jones CSIRO Mathematics, Informatics and Statistics

2 Outline Case studies in marine biogeochemistry and generic challenges. Conventional state-space models Collapsed state-space models (to deal with the absence of a closed-form transition density) The particle marginal Metropolis-Hastings (PMMH) sampler in this context. Results and implications. Acknowledgements: Eddy Campbell (CSIRO), John Parslow (CSIRO), Nugzar Margvelashvili (CSIRO), Noel Cressie (Ohio State University). Lawrence Murray PMCMC in marine biogeochemistry : 2 of 4

3 Marine biogeochemical model Sea Surface Phytoplankton Growth P (Phytoplankton) N (DIN) Zooplankton Grazing Remineralisation Z (Zooplankton) D (Detritus) Zooplankton Mortality Messy Feeding Base of Mixed Layer Mixing Sinking Lawrence Murray PMCMC in marine biogeochemistry : 3 of 4

4 Issues, issues, issues... These models tend to have long memory and don t mix well. Available observations may be sparse. The model may be strongly nonlinear,......it might even be chaotic. The transition density is unlikely to have a closed form. Lawrence Murray PMCMC in marine biogeochemistry : 4 of 4

5 Conventional state-space model U U 2... U T X X X 2... X T Y Y 2... Y T Lawrence Murray PMCMC in marine biogeochemistry : 5 of 4

6 Conventional state-space model Sampling with sequential Monte Carlo (auxiliary particle filter)... For particle i at time t, extend x i t by drawing xi t q(x t ) and weight with: wt i = p(y t x i t)p(x i t x i t ) q(x i wt. i t) Lawrence Murray PMCMC in marine biogeochemistry : 6 of 4

7 Conventional state-space model Sampling with sequential Monte Carlo (auxiliary particle filter)... For particle i at time t, extend x i t by drawing xi t q(x t ) and weight with: wt i = p(y t x i t)p(x i t x i t ) q(x i wt. i t) We don t have that! Lawrence Murray PMCMC in marine biogeochemistry : 7 of 4

8 Conventional state-space model U U 2... U T X X X 2... X T Y Y 2... Y T Lawrence Murray PMCMC in marine biogeochemistry : 8 of 4

9 Collapsed state-space model... U U 2 U T X Y Y 2... Y T Lawrence Murray PMCMC in marine biogeochemistry : 9 of 4

10 Collapsed state-space model Sampling with sequential Monte Carlo (auxiliary particle filter)... For particle i at time t, extend u i :t by drawing ui t q(u t ) and weight with: wt i = p(y t u i :t, x i )p(u i t u i :t, x i ) q(u i wt. i t) Lawrence Murray PMCMC in marine biogeochemistry : of 4

11 Collapsed state-space model Sampling with sequential Monte Carlo (auxiliary particle filter)... For particle i at time t, extend u i :t by drawing ui t q(u t ) and weight with: wt i = p(y t u i :t, x i )p(u i t) q(u i wt. i t) We do have that! Lawrence Murray PMCMC in marine biogeochemistry : of 4

12 What should q( ) be? PF: bootstrap PF: as PF plus one-step single-pilot lookahead and resample. MUPF: UKF run offline, use time marginals p N (u t y :t ) as proposals. MUPF: as MUPF plus one-step single-pilot lookahead and resample. CUPF: UKF conditioned on each particle, so use p N (u t u :t, y :t ). CUPF: as CUPF plus UKF lookahead and resample. (UKF = Unscented Kalman Filter) Lawrence Murray PMCMC in marine biogeochemistry : 2 of 4

13 Particle marginal Metropolis-Hastings The target (posterior) density is π(u :T, x, θ y :T ), factorised as either: or π (u :T, x θ, y :T )π 2 (θ y :T ) π (u :T x, θ, y :T )π 2 (x, θ y :T ). In either case π is targeted with an auxiliary particle filter, π 2 with Metropolis-Hastings [Andrieu et al., 2, Pitt et al., 2]. The second factorisation should be preferred when prior information over x is scarce, so that importance sampling of it will be degenerate. Lawrence Murray PMCMC in marine biogeochemistry : 3 of 4

14 PZ model Simple phytoplankton-zooplankton model [Jones et al., 2]. Lotka-Volterra with stochastic growth rate and quadratic mortality term. dp dt dz dt α t = α t P cp Z = ecp Z m l Z m q Z 2 N (µ, σ). P is observed with noise. Simulated data used here. Lawrence Murray PMCMC in marine biogeochemistry : 4 of 4

15 PZ model: state estimate Z Prior Posterior Observed Truth P α t t Lawrence Murray PMCMC in marine biogeochemistry : 5 of 4

16 NPZD model The model features nine noise terms, ξ i for i =,..., 9, each coupled to a univariate autoregressive process B i. B i (t + t) = B i (t) ( t/τ P ) + (µ i + P DF σ i ξ i ) t/τ P, where t is a discrete time step (one day), µ i a parameter to be estimated, P DF a common diversity factor to be estimated, σ i a prescribed scaling factor, and τ P a common characteristic time scale, also prescribed. Lawrence Murray PMCMC in marine biogeochemistry : 6 of 4

17 NPZD model A multiplicative temperature correction T c is applied to all rate processes, for which a Q formulation for dependence on temperature, T, is used: T c = Q (T T ref)/, where T ref is a reference temperature, and Q a prescribed constant. Lawrence Murray PMCMC in marine biogeochemistry : 7 of 4

18 NPZD model The zooplankton grazing rate (gr) is dependent on the phytoplankton concentration (zooplankton functional response): gr = T c I Z A υ ( + A υ ), () where υ is a given power. Lawrence Murray PMCMC in marine biogeochemistry : 8 of 4

19 NPZD model The relative availability of phytoplankton, A, is A = Cl Z P I Z ; I Z is the maximum zooplankton ingestion rate (mg P per mg Z per day); Cl Z is the maximum clearance rate (volume in m 3 swept clear per mg Z per day). For υ =, this takes the form of a Type-2 functional response (standard rectangular hyperbola) [Holling, 966], and for υ > a Type-3 sigmoid functional response. Lawrence Murray PMCMC in marine biogeochemistry : 9 of 4

20 NPZD model A quadratic formulation for zooplankton mortality is adopted after Steele [976] and Steele and Henderson [992]: m = T c m Q Z, where the quadratic mortality rate m Q has units of d (mgzm 3 ). Lawrence Murray PMCMC in marine biogeochemistry : 2 of 4

21 NPZD model The detrital remineralization rate is dependent only on temperature: r = T c r D, where r D prescribes the remineralisation rate at a reference temperature. Lawrence Murray PMCMC in marine biogeochemistry : 2 of 4

22 NPZD model The phytoplankton specific growth rate, g, depends on temperature, T, available light or irradiance, E, and dissolved inorganic nutrient, N: g = T c g max h E h N /(h E + h N ). Lawrence Murray PMCMC in marine biogeochemistry : 22 of 4

23 NPZD model The light-limitation factor is given by h E = exp( α λ max E/g max ), where α is the initial slope of the photosynthesis versus irradiance curve (mg C mg Chla mol photon m 2 ), and λ max is the maximum Chla : C ratio (mg Chla mg C ). Lawrence Murray PMCMC in marine biogeochemistry : 23 of 4

24 NPZD model E is the mean photosynthetic available radiation (PAR) in the mixed layer and is given by E = E.( exp( Kz))/Kz, where E is the mean daily photosynthetically available radiation (PAR) just below the air-sea interface and Kz is given by Kz = (K W + a Ch Chla) MLD. (2) K W is attenuation due to the seawater and a Ch. Lawrence Murray PMCMC in marine biogeochemistry : 24 of 4

25 NPZD model The nutrient-limitation factor is given by h N = N (g max T c/a N ) + N, where a N is the maximum specific affinity for nitrogen uptake (d mg N m 3 ). Lawrence Murray PMCMC in marine biogeochemistry : 25 of 4

26 NPZD model The phytoplankton N : C ratio, χ, predicted by the model is given by χ = χmin h E + χ max h N h E + h N, where χ min and χ max are the prescribed minimum and maximum N : C ratios (mg N mg C ). Lawrence Murray PMCMC in marine biogeochemistry : 26 of 4

27 NPZD model The equations governing interactions between the remaining state variables {P, Z, D, N} are: dp = g P gr Z + κ dt MLD (BCP P ) dz = E Z gr Z m Z dt dd dt dn dt = ( E Z ) f D gr Z + m Z r D S D κ (BCD D) MLD = g P + ( E Z ) ( f D ) gr Z + r D + κ (BCN N). MLD D MLD + Lawrence Murray PMCMC in marine biogeochemistry : 27 of 4

28 State estimation (observed) 35 3 DIN (µ g N l ) ln(chla (µ g Chla l )) Prior 95% Posterior 95% Observations Prior median Posterior median Lawrence Murray PMCMC in marine biogeochemistry : 28 of 4

29 State estimation (unobserved) 6 ln(p(µ g N l ) ln(z(µ g N l ) ln(d(µ g N l ) Prior 95% Posterior 95% Prior median Posterior median Lawrence Murray PMCMC in marine biogeochemistry : 29 of 4

30 Parameter estimation ln(k ) W ln(a ) Ch s D ln(f D ) ln(pdf) ln(zdf) ln(µ g max) ln(µ λ max) ln(µ ) RN ln(µ ) an 2 3 ln(µ ) IZ ln(µ ) ClZ ln(µ ) EZ 3 2 ln(µ ) rd ln(µ ) mq Posterior Prior Lawrence Murray PMCMC in marine biogeochemistry : 3 of 4

31 State forecast 25 6 ) DIN(µ g N l )) ln(p(µ g N l /74 2/75 9/75 4 7/74 2/75 9/ )) ln(z(µ g N l /74 2/75 9/75 )) ln(d(µ g N l /74 2/75 9/75 )) ln(chla(µ g N l /74 2/75 9/75 Prior 95% Posterior 95% Forecast 95% Observations Prior median Posterior median Forecast median Lawrence Murray PMCMC in marine biogeochemistry : 3 of 4

32 PZ model: convergence.2. PF MUPF CUPF PF MUPF CUPF.2. PF MUPF CUPF PF MUPF CUPF.8.8 R p Step Step Convergence rates of Markov chains for the PZ case study, (left) particle-matched, and (right) compute-matched. Each line shows the evolution of the ˆR p statistic of Brooks and Gelman [998] for a particular method. Lawrence Murray PMCMC in marine biogeochemistry : 32 of 4

33 NPZD model: convergence PF MUPF CUPF PF MUPF CUPF PF MUPF CUPF PF MUPF CUPF Step Step Convergence rates of Markov chains for the NPZD case study, (left) particle-matched, and (right) compute-matched. Each line shows the evolution of the ˆR p statistic of Brooks and Gelman [998] for a particular method. Lawrence Murray PMCMC in marine biogeochemistry : 33 of 4

34 PZ model: acceptance rates σ PF MUPF CUPF σ PF MUPF CUPF µ µ µ Lawrence Murray PMCMC in marine biogeochemistry : 34 of 4

35 PZ model: acceptance rates, compute-matched σ PF MUPF CUPF σ PF MUPF CUPF µ µ µ Lawrence Murray PMCMC in marine biogeochemistry : 35 of 4

36 NPZD model: acceptance rates.8.8 Cumulative density.6.4 Cumulative density.6.4 PF.2 MUPF CUPF PF MUPF CUPF Conditional acceptance rate (CAR) PF.2 MUPF CUPF PF MUPF CUPF Conditional acceptance rate (CAR) Empirical cdfs of the acceptance rates for each method on the NPZD case study, (left) particle-matched, and (right) compute-matched. Lawrence Murray PMCMC in marine biogeochemistry : 36 of 4

37 NPZD model: parameters vs acceptance rates K W a Ch S D f D PDF ZDF µ g max µ λ max µ RN µ an µ IZ µ ClZ µ EZ µ rd µ mq Lawrence Murray PMCMC in marine biogeochemistry : 37 of 4

38 Lessons for PMMH Initialisation is extremely important, sampler will fail if initialised in a region where the acceptance rate is low. A particularly informative prior distribution may bias the posterior into regions where the acceptance rate is low. Is the rule-of-thumb 23% acceptance rate appropriate in this context? Lawrence Murray PMCMC in marine biogeochemistry : 38 of 4

39 Summary Environmental models such as these in marine biogeochemistry pose significant challenges to Monte Carlo methodology. One particular challenge is the absence of a closed-form transition density. The collapsed state-space model facilitates one means of improving sampler performance without needing this. The particle marginal Metropolis-Hastings sampler can have significant mixing and acceptance rate problems, initialisation and careful consideration of the prior distribution are important. Lawrence Murray PMCMC in marine biogeochemistry : 39 of 4

40 References C. Andrieu, A. Doucet, and R. Holenstein. Particle Markov chain Monte Carlo methods. Journal of the Royal Statistical Society Series B, 72:269 32, 2. S. P. Brooks and A. Gelman. General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7: , 998. C. S. Holling. The functional response of predators to prey density and its role in mimicry and population regulation. Memoirs of the Entomology Society of Canada, 45:6, 966. E. Jones, J. Parslow, and L. M. Murray. A Bayesian approach to state and parameter estimation in a phytoplankton-zooplankton model. Australian Meteorological and Oceanographic Journal, 59(SP):7 6, 2. M. K. Pitt, R. S. Silva, P. Giordani, and R. Kohn. Auxiliary particle filtering within adaptive Metropolis-Hastings sampling. 2. URL J. Steele. Role of predation in ecosystem models. Marine Biology, 35():9, 976. ISSN J. Steele and E. Henderson. The role of predation in plankton models. Journal of Plankton Research, 4():57 72, 992. Lawrence Murray PMCMC in marine biogeochemistry : 4 of 4

41 CSIRO Mathematics, Informatics and Statistics Lawrence Murray Phone: lawrence.murray@csiro.au Web: Contact Us Phone: or enquiries@csiro.au Web: