A nested sampling particle filter for nonlinear data assimilation

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1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14, July 2 A DOI:.2/qj.224 A nested sampling particle filter for nonlinear data assimilation Ahmed H. Elsheikh a,b *, Ibrahim Hoteit c and Mary F. Wheeler a a Institute for Computational Engineering and Sciences (ICES), University of Texas, Austin, TX, USA b Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh, UK c Department of Earth Sciences and Engineering, King Abdullah University, Thuwal, Saudi Arabia *Correspondence to: A. H. Elsheikh, Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh Campus, Edinburgh, EH 1AS, UK. ahmed.elsheikh@pet.hw.ac.uk We present an efficient nonlinear data assimilation filter that combines particle filtering with the nested sampling algorithm. Particle filters (PF) utilize a set of weighted particles as a discrete representation of probability distribution functions (PDF). These particles are propagated through the system dynamics and their weights are sequentially updated based on the likelihood of the observed data. Nested sampling (NS) is an efficient sampling algorithm that iteratively builds a discrete representation of the posterior distributions by focusing a set of particles to high-likelihood regions. This would allow the representation of the posterior PDF with a smaller number of particles and reduce the effects of the curse of dimensionality. The proposed nested sampling particle filter (NSPF) iteratively builds the posterior distribution by applying a constrained sampling from the prior distribution to obtain particles in high-likelihood regions of the search space, resulting in a reduction of the number of particles required for an efficient behaviour of particle filters. Numerical experiments with the 3-dimensional Lorenz3 and the 4-dimensional Lorenz models show that NSPF outperforms PF in accuracy with a relatively smaller number of particles. Key Words: nested sampling; particle filters; sequential data assimilation. Received January 2; Revised 24 July 2; Accepted 2 August 2; Published online in Wiley Online Library April 2 1. Introduction A major task in earth sciences is predicting the future state of a set of variables characterizing atmospheric and/or oceanic flows. The accuracy of the forecast relies on the quality of the physical model as well as the utilized numerical methods. However, the presence of modelling and numerical errors makes a perfect forecast almost impossible. A possible solution to this problem is to integrate observational data with numerical model forecast in a probabilistic inference framework. The goal is then to estimate the posterior probability distribution function (PDF) of the system state variables at a given time while considering all observational data until that time. In this setting, the estimation process involves two steps: the forecast step and the estimation step. The forecast step involves propagating the PDF through the system dynamics to obtain the forecast PDF. The estimation step, referred to as the analysis step in the data assimilation community, involves updating the forecast PDF using the observations by solving a stochastic filtering problem based on Bayes rule (Doucet and Johansen, 2). The Kalman filter (KF) (Kalman, 1) provides an optimal solution to the data assimilation problem when the dynamical model is linear. On the other hand, when the dynamics are nonlinear, the extended Kalman filter (EKF) can be used as it linearizes the problem around the current state estimates (Jazwinski, 1; Anderson and Moore, 1). However, EKF might generate instabilities that lead to filter divergence with strongly nonlinear models (Gauthier et al., ). For such systems, Evensen (14) introduced the ensemble Kalman filter (EnKF) as a sequential Monte Carlo method for nonlinear filtering. In an EnKF, a set of states (ensemble members) are propagated through the nonlinear dynamical model (forecast step), and corrected whenever observations are available (analysis step). The analysis step relies on the discrepancy between the forecast state and observations through a Gaussian approximation of the predicted state distribution. The classical update formula of KF is utilized. Recent advances in KF-based methods include the utilization of semi-parametric PDF representations (e.g. Gaussian Mixtures) (Hoteit et al., 2; Bocquet et al., 2; Hoteit et al., 2; Sondergaard and Lermusiaux, 2a,b) Particle filtering (Gordon et al., ) is a sequential Monte Carlo nonlinear filtering algorithm that has been proposed for object tracking. In particle filters (PF), PDFs are represented by a set of particles (states) with an associated set of weights. These particles, similar to ensemble members in EnKF, are propagated through the system equations in the forecast step. In the analysis step, the weights of the particles are updated based on the deviations of the predicted states from the observed data (Hoteit et al., 2). It was observed that PFs when applied to high-dimensional geophysical models collapse to a point mass after few observation cycles (Anderson and Anderson, 1; Van Leeuwen, 23; Snyder et al., 2). This can be attributed to one of two problems. The first is sample degeneracy, where all but one of the particles have weights close to zero. This problem is avoided by a resampling step, where a particle with an associated c 2 Royal Meteorological Society

2 Nested Sampling 141 high weight is more likely to be sampled, possibly more than once, in the new generated samples, whereas a particle with a weight closetozeroismorelikelynottobedrawnatall.intuitively, the resampling step tries to avoid evaluating the forward model on less likely states (Doucet and Johansen, 2). The second problem with PFs is related to the curse of dimensionality where the required number of particles for a proper representation of a high-dimensional PDF often grows exponentially with the dimensionality of the distribution. Attempts to resolve these issues were reported by Chorin and Tu (2) and Van Leeuwen (2). Chorin and Tu (2) proposed an implicit PF and Van Leeuwen (2) proposed an efficient PF for high-dimensional problems. Both algorithms try to push the particles towards observations to keep track of high-likelihood regions of the search space. These algorithms will be discussed in more detail in section. In this paper, we propose the nested sampling particle filter (NSPF) for nonlinear data assimilation. The proposed filter combines the nested sampling algorithm introduced by Skilling (24, 2) with the standard particle filtering algorithm (Gordon et al., ; Doucet and Johansen, 2). Nested sampling is a highly efficient sampling algorithm that is capable of sampling the posterior distribution and providing an estimate for the marginal likelihood (Bayesian evidence). The nested sampling algorithm efficiently samples high-dimensional spaces where the regions contributing most to the probabilitymass are exponentially localized. Nested sampling has been successfully applied to several cosmological problems (Mukherjee et al., 2; Shaw et al., 2; Vegetti and Koopmans, 2; Bridges et al., 2) and is increasingly applied for Bayesian inference in different scientific fields (Partay et al., 2; Granqvist et al., 2; Mthembu et al., 2; Schuet et al., 2; Burkoff et al., 2; Kügel et al., 2; Elsheikh et al., 2, 2a, 2b). Here, we use nested sampling for updating the forecast PDF to the analysis PDF of the system states. Nested sampling provides a statistically sound yet efficient method for refocusing the particles into high-likelihood regions. The proposed NSPF provides the correct balance between exploration of the posterior PDF for proper uncertainty quantification and the refocusing of particles to high-likelihood regions to reduce the particle degeneracy phenomenon and avoid filter divergence. Through numerical experiments with the Lorenz3 and the Lorenz models, we show that NSPF outperforms the standard PF in terms of accuracy while using significantly smaller number of particles. The remainder of the article is organized as follows: section 2 provides a general introduction to the data assimilation problem followed by a detailed description of particle filters. Section 3 presents the nested sampling algorithm followed by an algorithmic description of the proposed NSPF. In section 4, numerical evaluations of NSPF with the strongly nonlinear Lorenz systems are presented. Following that, a general discussion about NSPF and its relation to recently proposed efficient particle filtering algorithms is presented in section. Finally, the conclusions of our work are drawn in section. 2. Problem statement Data assimilation has two main steps: a forecast step that relies on the model dynamics, and an analysis step where observations are used to correct the most recent forecast. The dynamics of the system are commonly formulated as a Markov model using a time discretized equation of the form x k = F(x k 1 ) + ɛ k, (1) where F( ) :R m R m is the model propagation function, x k R m is the state of the system at time k and ɛ k R m is the noise component representing the modelling errors. This forward propagation equation commonly originates from a set of partial differential equations. If a set of observational data is available at time step k, observations can be related to the state variables by y k = H(x k ) + r k, (2) where y k R n is the vector of observations at time k, H( ) :R m R n is the measurement operator and r k R n is the measurement noise vector. In a Bayesian framework, the distribution of the state variable x k can be fully described by a PDF. The forecast step refers to evolving the state variables using the model in Eq. (1). Formally, the forecast step involves finding the prior PDF p(x k y 1:k 1 )at time k using all the available information up to time k 1, where y 1:k 1 denotes the set of all observations y 1,..., y k 1 up to and including time t k 1. This is done with the model evolution Eq. (1) via the Chapman Kolmogorov formula p(x k y 1:k 1 )= p(x k x k 1 ) p(x k 1 y 1:k 1 )dx k 1. (3) The analysis step involves obtaining the posterior PDF p(x k y 1:k ), which contains all information up to time k using Bayes rule p(x k y 1:k ) = p(y k x k ) p(x k y 1:k 1 ) p(yk x k ) p(x k y 1:k 1 )dx k, (4) where p(y k x k ) is the likelihood function that depends on the measurement Eq. (2) and the statistics of the measurement noise r k. The resulting Bayesian filtering (Eqs (3) and (4)) does not have analytical forms in a general setting, except for a very limited class of problems (Jazwinski, 1). For systems with linear dynamics where only Gaussian PDFs are involved, an optimal filter known as the Kalman filter (KF) exists (Kalman, 1). For nonlinear/non- Gaussian cases, several sub-optimal filters were developed by either linearizing the nonlinear model as in EKF (Jazwinski, 1; Anderson and Moore, 1), or by approximating the evolution of the process by a collection of particles as in EnKF (Evensen, 14). Another method, which provides a direct approximation of the Bayesian filtering equations is the PF (Gordon et al., ; Doucet et al., 21). This will be described in the next subsection Particle filters The general idea of PFs is to approximate the prior and posterior PDFs by a population of states, where each state is called a particle. Each particle is assigned a weight and the PDFs are approximated as a discrete distribution based on these weighted particles. A general PF algorithm can be described by the following steps: Initialization. The filter is initialized by drawing a population of N particles {x 1,..., xn }, from the prior distributions at initial time t =. Each of these particles x i is assigned a weight wi = 1/N as all the particles are initially drawn from the same distribution. The particles and their weights are used as a discrete representation of the prior PDF. Prediction step. Given the prior PDF p( y 1:k 1 ) at time t k 1 represented as a set of weighted particles {xk 1 1,..., xn k 1 } with the weights {w1 k 1,..., wn k 1 }, the predictive PDF (forecast of the current step) is obtained by integrating all the particles with the model Eq. (1) to the time of the next available observation t k. The predictive PDF is represented by the new set of particles {xk 1,..., xn k } and the same weight vector {wk 1 1,..., wn k 1 }. Correction step. The objective of the correction step is to update the prior distribution by accounting for the measurement to obtain the posterior PDF. This is done by updating the weight vector. After a new observation y k has been made, the likelihood of each particle transition from xk 1 i to xi k can be evaluated by p(y k x i 1:k ) = p(y k x i k )p(xi k 1 y 1:k 1), () c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

3 142 A. H. Elsheikh et al. where p(y k xk i ) is easy to evaluate by defining a mismatch function that depends on the statistics of the measurement noise and p(xk 1 i y 1:k 1) is abstracted by the particle weight from the previous iterations. The updated weights can be evaluated by wk i = p(y k x1:k i ) N j=1 p(y k x j () 1:k ). The posterior PDF is then approximated by the propagated particles and their updated weights. Resampling step. The input to this step is the set of particles and their associated weights. The output of the resampling algorithm is another set of particles with equal weights. The details of this optional step will be discussed in the next subsection. After the initialization step, PFs iterate between prediction and correction. As the number of particles N tends to infinity, the discrete distribution weakly converges to the true posterior PDF. However, this process is practically unfeasible for highdimensional systems (Doucet et al., 21; Snyder et al., 2) Resampling techniques A resampling step is most often introduced into PFs to overcome the sample impoverishment problem when most of the particles have weights close to zero. The aim of the resampling step is to favour samples that lie in the high-likelihood regions in order to focus the computational effort on particles that matter most for the estimation. The resampling method has to be carefully chosen. In particular, it should not introduce any bias in the final distribution estimate as highlighted by Doucet et al. (21). During this step, particles are resampled according to their weights. A particle with a high weight is likely to appear in the new generated sample, possibly more than once, whereas a particle with a weight close to zero is likely not to be drawn at all from a given time step to the next. It is important to note that, even though resampling might alleviate the degeneracy problem, it also brings extra random variation to the samples of particles. As a consequence, the filtered quantities of interest (e.g. system state vector x k ) should preferably be computed before resampling and not after. We only present the details of the systematic resampling scheme (Doucet and Johansen, 2) since it is the method utilized in our numerical test cases. However, different resampling schemes are available in the literature (Doucet and Johansen, 2). Suppose that we have a set of particles at time step k denoted by {x 1 k,..., xn k } with the corresponding weights {w1 k,..., wn k }. Systematic resampling relies on the inverse of the cumulative distribution function of the particles weights. Let U(a, b) denote the uniform distribution on the interval [a, b]. To draw M N samples, a single random number u is drawn from the uniform distribution U (, 1) and then samples are iteratively found by using the following simple iteration (Doucet and Johansen, 2): For i = 1,..., M,findj [1, N] such that (i 1 + u) M [ j 1 w l k, l=1 3. Nested sampling particle filter j l=1 w l k ]. Nested sampling (Skilling, 24, 2) is an efficient sampling algorithm that was introduced for general Bayesian inference. The nested sampling algorithm shares many common properties with PFs. In nested sampling, a set of particles is drawn from the prior PDF and is called the active set. The likelihood of these particles is evaluated based on the observations similar to the correction step in PFs. However, in the nested sampling algorithm, an additional step of constrained sampling is performed to add more particles to high-likelihood regions. The final output of the nested sampling algorithm is a set of particles with associated weights, which can be resampled to obtain the posterior distribution. The focusing of particles towards high-likelihood regions (i.e. close to observations) by the constrained sampling step can alleviate the sample impoverishment problem and reduce the effects of the curse of dimensionality, which are common problems in PFs. The next subsection provides the details of the nested sampling algorithm followed by a formulation of a powerful PF based on the application of the nested sampling algorithm to obtain the posterior PDF from the prior PDF at each time step of a standard PF. The proposed NSPF reduces to a standard PF with a resampling step if the number of nested sampling iterations is set to zero Nested sampling algorithm Nested sampling is as an efficient algorithm for evaluating the Bayesian evidence (Skilling, 24, 2), which can be used for model selection and comparison (Gull, 1; Sivia and Skilling, 2). However, similar to PFs, the nested sampling algorithm provides a discrete representation of the posterior distribution in the form of a weighted set of particles as a by-product. The following is a description of the nested sampling algorithm with a focus on efficient evaluation of the Bayesian evidence. Subsequently, we will present a simple method for obtaining a discrete representation of the posterior distribution based on the nested sampling algorithm. In a general Bayesian setting, the posterior PDF is related to the prior PDF by Bayes rule p(d x) p(x) likelihood prior p(x D) = = p(d) normalizing constant, () where x is the vector of unknown states or parameters and D is the observable data. The normalizing constant (marginal likelihood or Bayesian evidence) is defined as p(d) = p(d x) p(x) dx. () The main idea of nested sampling is to convert the highdimensional integral needed for evaluating the evidence into a one-dimensional integral that is easier to evaluate numerically. A change of variable is performed by defining a cumulative prior mass X as X(λ) = p(x)dx, () L(x)>λ where L(x) = p(d x) is the likelihood and λ is a real number from the interval [, 1). This prior mass will have a value of 1 if λ is set to and will decrease to as λ approaches 1. The integration of the Bayesian evidence Z givenbyeq.()canthen be transformed into Z = p(d) = 1 L(X)dX, () with X as the fraction of the total prior mass such that dx = p(x)dx and L(X) is assigned the likelihood value L(x). Thus, if Eq. () can be computed, the problem of calculating the normalizing constant becomes a one-dimensional integral where the integrand is positive and decreasing. If a set of particles x 1, x 2,..., x m is generated such that the likelihood of these particles is an ordered sequence of increasing values L 1 < L 2 < < L m, the corresponding prior masses X 1, X 2,..., X m will be a sequence of decreasing values, < X m < < X 2 < X 1 < 1. () c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

4 Nested Sampling 143 (b) Figure 1. Distribution of samples using the nested sampling algorithm of a two-dimensional problem: particles in the physical space, and (b) the prior masses in the likelihood space. This figure is available in colour online at wileyonlinelibrary.com/journal/qj The one-dimensional integral Z can be estimated using the trapezoidal rule as Z = m Z j, j=1 Z j = Lj 2 ( X j 1 X j). () This equation can be easily evaluated if the difference of the prior masses ( X j 1 X j) can be estimated. To obtain X j, N particles are drawn from the prior such that all the particles satisfy the constraint L(x i ) > L j 1 for all i {1, 2,..., N}. If the particle with the lowest likelihood x worst is selected and L j is set equal to L(x worst ), the prior mass shrinkage factor t j = X j /X j 1,can be estimated statistically from the distribution P(t j ) = Nt N 1 j, where N is the number of particles and t j (, 1). The distribution for t has the following properties, E(log t) = 1/N and σ (log t) = 1/N for the mean and standard deviation, respectively (Skilling, 2; Chopin and Robert, 2). Since each t j is independent, after j iterations the prior volume will shrink down such that log X j (j ± j)/n. As an approximation, X j can be taken as a deterministic value of exp ( j/n) (Skilling, 2). Based on the former description, an iterative algorithm can be formulated starting from the iteration index j = 1and X = 1 using the following steps: (i) Sample the prior distribution using N particles and evaluate the likelihood of each particle. These particles are called the active set; (ii) Select the particle x worst with the lowest likelihood from the active set and set L j = L(x worst ); (iii) Estimate the prior volume X j statistically or use X j = exp ( j/n); (iv) Increment the integral Z by Z j = L j (X j 1 X j )/2; (v) Discard the particle x worst from the active set and replace it with a new particle x new within the remaining likelihood volume by satisfying the hard constraint L(x new ) > L(x worst ); (vi) Increment j and repeat steps (ii) (v) until the change in Z j is small enough. In the previous steps, we did not generate N new particles at each iteration. Instead, the particles were reused between the iterations by replacing the particle having the lowest likelihood (i.e. highest X) with a new particle while keeping the other N 1 particles in the active set. Effectively, the nested sampling algorithm climbs up the likelihood function, by shrinking the parameter space by a factor t j every time a particle is replaced. Ultimately, it samples only the part of the likelihood close to the maximum. The main difficulty of the nested sampling algorithm is to propose a new particle x new that satisfies L(x new ) > L j.this constrained sampling step becomes harder after applying several nested sampling iterations as L j reaches a high value and the parameter space satisfying the lower-bound constraint shrinks to a smaller region. One solution for this problem is to use a Markov Chain Monte-Carlo (MCMC) algorithm to explore the search space and find new particles conforming to the required constraint. Following (Skilling, 2), an efficient constrained sampling can be performed by randomly selecting one particle from the active set s N particles, excluding the particle with the lowest likelihood, to start a short MCMC chain. The starting state of this MCMC chain will satisfy the desired condition of the prior and the likelihood constraint. However, the new particle has to be independent from previous particles. Thus, the objective is to move away from the starting particle without violating the likelihood constraint but far enough to lose memory of the starting particle. Figure 1 shows the action of the nested sampling algorithm on a two-dimensional distribution, where the discarded particles during the nested sampling iterations are shown in the physical space in Figure 1 and on the one-dimensional likelihood space in Figure 1(b). So far, our description of nested sampling was focused on evaluating the Bayesian evidence Z. However, the set of particles discarded during the nested sampling iteration, as well as the particles included in the active set after terminating the algorithm, can be used for estimating the posterior distribution. These particles contribute to the one-dimensional curve L(X) as shown in Figure 1(b). The area under this one-dimensional curve is already decomposed into steps as defined by Eq. (). In other words, the existing sequence of discarded particles x 1, x 2,... during the nested sampling iterations provide a discrete representation of the posterior PDF with an importance weight given by L j (X j 1 X j )/2. To obtain equally weighted posterior samples, we need to resample (with replacement) a subset of these particles with normalized weights obtained from the area under the one-dimensional integral (Skilling, 2). For a complete theoretical analysis of the nested sampling algorithm, interested readers are referred to the original papers by Skilling (24, 2) and further theoretical investigations by Evans (2) and Chopin and Robert (2) Nested sampling particle filter The details of the NSPF are summarized in Algorithm 1. It closely follows the description of PFs presented earlier in this paper, with additional steps from the nested sampling algorithm for adaptive construction of the posterior PDF from the prior PDF. In step 1 of the algorithm, a set-theoretic difference operator \ is used for the difference between two sets (Formally B \ A ={x B x A}). We reiterate that NSPF reduces to a standard PF if the number of nested sampling iterations M is set to zero because the last resampling step will only depend on the likelihood as the particle c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

5 144 A. H. Elsheikh et al. weights. This is clear in Algorithm 1 where steps to 1 will be skipped. In comparison to standard PFs, NSPF has an additional computational cost that is proportional to the number of nested sampling iterations M. These iterations correspond to finding particles in high-likelihood regions. The main computational burden is attributed to the constrained sampling from the prior (step ). This sampling step can be considered as an initial condition constrained sampling step where the new sampled state xk new at time step k has to be propagated through the system dynamics, and the likelihood of the corresponding state xk+1 new after propagation to time step k + 1 has to satisfy the likelihood constraint. As mentioned in the previous subsection, we utilize a short MCMC chain to perform this constrained sampling step. The chain starts from a randomly selected particle from the active set excluding the discarded particle. Thus, the additional cost of each nested sampling iteration will correspond to the number of MCMC steps as each step will require a forward run of the system equations. Fortunately, we do not need to perform long chains as the objective is to slightly move away from the starting state of the chain while satisfying the hard likelihood constraint. Algorithm 1: Nested Sampling Algorithm Input: Draw N particles x 1,..., xn from the prior at t =, Set the number of nested sampling iterations M for time step k > do Propagate Particles: for i = 1,..., N do 4 xk+1 i = f (xi k ) + ɛi k According to the forward model. Initialize Nested Sampling: Set S active ={xk+1 1, x2 k+1,..., xn k+1 }, Set X = 1, S posterior ={}, Nested Sampling Iteration: for j = 1,..., M do Find xk+1 worst = argmin p(y k+1 xk+1 i ) x xi k+1 S active Set L j = p(y k+1 xk+1 worst) Set X j = exp( j/n) Set w j = L ( j X j 1 X j) /2 Set S posterior S posterior {(xk+1 worst, )} Sample xk new p( y 1:k 1 ) such that 1 p(y k+1 xk+1 new) (using Algorithm 2) Set S active ( S active \{xk+1 worst}) {xk+1 new} the active set) 1 1 Set S posterior S posterior {(xk+1 i,(xm /N)L i )} Resampling Step: x i k+1 S active Resample N particles from S posterior for the next time step. Assuming the MCMC chain starts from the particle xk i, where the corresponding state at time k + 1 is xk+1 i and xk+1 i S active, xk+1 i xworst k+1, we utilize a simple random walk proposal distribution q(ξ k xk i ) for the MCMC chain of the form ξ k = x i k + δ w, w N (, I m), () where δ is a scaling factor controlling the update step-size, I m is the identity matrix of size m and N denotes the normal distribution. The details of the MCMC constrained sampling are presented in Algorithm 2. Algorithm 2: Constrained sampling via Metropolis Hastings algorithm Input: Likelihood hard constraint L(xk+1 worst), starting particle of the MCMC chain xk i and its corresponding evolved state xk+1 i, number of local MCMC iterations l Output: xk new and the corresponding xk+1 new 1 Set θk = xi k 2 for r = 1,..., l do 3 Propose ξ k q(. θ r 1 k ) using Eq. () 4 Propagate ξ k through the system dynamics to obtain ξ k+1 Evaluate L(ξ k+1 ) if L(ξ k+1 ) > L(xk+1 worst) then { } Compute α = min 1, L(ξ k+1) L(θ r 1 k+1 ) Generate u U(, 1) if u α then θk r ξ k, θk+1 r ξ k+1 else θk r θ r 1 k, θk+1 r θ r 1 k+1 else θk r θ r 1 k, θk+1 r θ r 1 k+1 Set x new k θ r k, xnew k+1 θ r k+1 The last step of each NSPF iteration is a resampling step similar to PFs. A systematic resampling scheme, detailed earlier, is utilized. However, it should be noticed that the number of samples in S posterior will equal N + M, wheren is the size of the active set and M is the number of nested sampling iterations. In order to avoid a growing active set size, we only resample N particles from S posterior following an overproduce and select by resampling strategy. 4. Numerical evaluation In the numerical evaluations, the time-averaged root mean squared error (RMSE) is used to evaluate the performance of NSPF. Given a set of state vectors {x 1 k, x2 k,..., xn k } corresponding to N particles at time step k, wherek =,..., k max,withk max being the maximum time index, the RMSE ê is defined as k 1 max 1 ê = (k max +1) m k= ( xk x true k ) T ( xk x true k ), () where x k is mean state vector of all particles, m is the state vector size and xk true is the reference model solution. The estimated forecast mean is based on the propagated states using the resampled particles with equal weights. The time average takes into account all the numerical integration time steps. In all the presented results, we estimate RMSE based on the forecast states as that is what matters in real-time predictions. All numerical results are estimated by running assimilation cycles regardless of the frequency of observations. The first cycles are truncated as a burn-in period and the error statistics are evaluated over the last assimilation steps. As described earlier, the constrained sampling step in NSPF is performed using a short MCMC chain starting from one of the particles in the active set. In the numerical testing, we vary the number of local MCMC steps to study the chain length effects. For each step, a random walk proposal centered around the current state is performed using a scaling factor δ =.1 in Eq. (). c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

6 Nested Sampling local MCMC steps (b) local MCMC steps Active set size = Active set size = 4 2 Active set size = Active set size = (c) 4 2 local MCMC steps (d) local MCMC steps Active set size = Active set size = 4 2 Active set size = Active set size = Figure 2. Time average forecast error (RMSE) for three-element Lorenz model with different active set sizes and different number of nested sampling iterations (model error variance = 1., observation error variance =., time step =.1 and assimilation every 2 time steps). This figure is available in colour online at wileyonlinelibrary.com/journal/qj 4.1. Example 1: Three-dimensional Lorenz3 model The Lorenz3 model (Lorenz, ) is a set of three coupled nonlinear ordinary differential equations defined by: dx dy =γ (y x), dt dt =ρx y xz, dz =xy βz, () dt where x, y, z are time-dependent unknown variables. The commonly chosen parameters are γ =, ρ = 2 and β = /3. The Lorenz3 set of ODEs are integrated using a fourth-order Runge Kutta scheme with time step t =.1. The Lorenz3 model has been the subject of many data assimilation studies due to its nonlinear and chaotic nature (e.g. Van Leeuwen, 2). Here, we study the behaviour of NSPF in tracking the state of the Lorenz3 model. A reference solution is generated by integrating the model in the time interval t [, 1] with the initial condition (x ; y ; z ) = (1.; 1.31; 2.41). The reference solution was evolved using a diagonal model error covariance with variances σmodel 2 setto1. corresponding to the error growth over one time unit for x, y and z, respectively. The stochastic term ɛ applied in the model evolution Eq. (1) is set as ɛ = σ model t dξ where σ 2 model is the model error variance and dξ is drawn from unit normal distribution. Observations are collected for all variables by adding Gaussian noise with standard deviation of either 2 or. to the reference solution and the observations are collected every 2 or 4 time steps. At this level of sparsity of observations and measurement noise, it is expected that non-gaussian features become important. For the data assimilation, particles are evolved using the same modelling error covariance as for the reference solution. The particles were initialized using randomly chosen states from a large collection of states obtained from a long model run. Figure 2 shows the time-average RMSE for NSPF versus the number of forward runs per data assimilation cycle when c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

7 14 A. H. Elsheikh et al. 3 local MCMC steps (b) local MCMC steps Active set size = Active set size = Active set size = Active set size = (c) local MCMC steps (d) local MCMC steps Active set size = Active set size = Active set size = Active set size = Figure 3. Time average forecast error (RMSE) for three-element Lorenz model with different active set sizes and different number of nested sampling iterations (model error variance = 1., observation error variance =., time step =.1 and assimilation every 4 time steps). This figure is available in colour online at wileyonlinelibrary.com/journal/qj observations are collected every 2 time steps and the observation variance is set to.. The RMSE of a standard PF with different number of particles (up to 4 particles) is plotted in black as a reference solution. The results shown in each sub-figure correspond to a different run of the PF and a slightly different time average RMSE is expected due to the stochastic nature of the problem. We evaluate the effect of the active set size and the number of local MCMC sampling steps. For zero nested sampling iterations, the errors in the NSPF highly depend on the active set size as the NSPF reduces to the standard particle filter where the performance depends on the number of particles. It is observed that, for an average of forward runs per assimilation cycle, both PF and NSPF have the same performance. We also observe a slight increase in the required number of forward runs per assimilation cycle if a longer local MCMC chain is run. This is attributed to the explorative nature of the MCMC, which is a random search algorithm. We conclude that, for this simple test case, the performance of NSPF is not sensitive to the active set size or to the length of the local MCMC chain. Also, we conclude that the complexity of the NSPF (in terms of required computational resources) is the same as the standard PF algorithm for this relatively simple problem. Figure 3 shows the time-average RMSE for the NSPF versus the number of forward runs per assimilation cycle when observations are collected every 4 time steps and the observation variance is set to.. Similar to the previous test case, the RMSE of a standard PF with different number of particles (up to 4 particles) is plotted as a reference solution. For this problem, we observe a clear difference in the performance of the NSPF with different active set sizes. In all the sub-figures, the NSPF with smaller active set sizes outperformed the NSPF with larger active set sizes. This is attributed to the optimization nature of the nested sampling algorithm. In the limit, with an active set of one particle, the nested sampling iteration performs a local stochastic search where a sample is replaced by a better sample (with higher likelihood) based on random search. If the active set is larger, the performance of the nested sampling as an optimization algorithm deteriorates as the search is more global. For this problem, the numerical results indicate that the optimization component of the nested sampling has a large effect on the time-average RMSE c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

8 Nested Sampling 14 3 local MCMC steps (b) local MCMC steps Active set size = Active set size = Active set size = Active set size = (c) local MCMC steps (d) local MCMC steps Active set size = Active set size = Active set size = Active set size = Figure 4. Time average forecast error (RMSE) for three-element Lorenz model with different active set sizes and different number of nested sampling iterations (model error variance = 1., observation error variance = 2., time step =.1 and assimilation every 2 time steps). This figure is available in colour online at wileyonlinelibrary.com/journal/qj and the local stochastic search, using smaller active sets, never got trapped in local minima. We also observe that the performance of the NSPF is not sensitive to the number of local MCMC steps. The results shown in the different sub-figures indicate that different runs of the PF produced different time-average RMSE. However all the PF runs were outperformed by the NSPF in terms of the time-average forecast RMSE. Figure 4 shows RMSE convergence rates versus the number of forward runs per assimilation cycle for the Lorenz3 when observation variance is set to 2 and the data assimilation is performed every 2 time steps. For this case, the system is less predictable and a larger number of samples has to be used as a discrete representation of the posterior PDF. This is evident when comparing the NSPF results with different active set sizes. The runs with larger active set sizes outperformed the runs with smaller active set sizes. This is also evident in the slight improved performance of PF runs over the NSPF. However, it should be noted that the predictions with the PF relies on the entire number of forward runs. This is to be contrasted with the predictions using NSPF as it relies only on the particles in the active set. Having said that, all the NSPF runs showed error reduction with increasing number of nested sampling iterations, which is evidence of the consistency of the method. Figure shows the RMSE convergence rates versus the number of forward runs when data is assimilated every 4 time steps and the observation variance is set to 2. Due to the large assimilation cycle, the particles need to be pushed vigorously towards the data and the importance of the optimization component of the nested sampling iteration is evident. This is reflected in the better performance of the NSPF with smaller active sets. Similar to the case when the observation variance was set to., the NSPF clearly outperformed the standard PF in terms of the time-average forecast RMSE. Figure shows the forecasted state of the Lorenz3 using the NSPF versus the true evolution of the system state. An active set of 2 particles is used and the number of nested sampling iterations is set to 2 with five local MCMC sampling steps at each nested sampling iterations. The figures show that different particles follow different trajectories and the distribution covers the true states. The presented results show the expected performance c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

9 14 A. H. Elsheikh et al. 3 local MCMC steps (b) local MCMC steps Active set size = Active set size = Active set size = Active set size = (c) local MCMC steps (d) local MCMC steps Active set size = Active set size = Active set size = Active set size = Figure. Time average forecast error (RMSE) for three-element Lorenz model with different active set sizes and different number of nested sampling iterations (model error variance = 1., observation error variance = 2., time step =.1 and assimilation every 4 time steps). This figure is available in colour online at wileyonlinelibrary.com/journal/qj of a general Bayesian data assimilation where the particles provide a discrete representation of the posterior distribution after accounting for the observation Example 2: Forty-dimensional Lorenz model Lorenz model (Lorenz, 1) is a system of coupled nonlinear ordinary differential equations defined by: dx j dt =(x j+1 x j 2 ) x j 1 x j + F, j = 1,, m, (1) with periodic boundary conditions defined by x 1 = x m 1, x = x m and x 1 = x m+1. This model mimics the time evolution of a scalar atmospheric quantity. It is considered a challenging problem from the perspective of data assimilation because of its highly chaotic nature. Lorenz and Emanuel (1) have shown that it behaves chaotically for F greater than 4.andm = 4. Here we set the dimension size m = 4 and the constant F =.. A fourth-order Runge Kutta scheme with a time step of t =.1 is used to integrate the model. It is shown in Lorenz and Emanuel (1) that a time step of. unit in the model corresponds to h in real life. Both the initial particles and the initial true state vector are randomly initialized from a large set of model states obtained at the end of a long run of the system. Observations are collected with two frequencies: time steps and time steps. As for the spatial sparsity, observations are either collected for all the states, or at the odd-numbered states. The observational noise is considered Gaussian with variance σobs 2 =.1. The model error is applied as ɛ = σ model t dξ where the variance σmodel 2 issetto.1anddξ is drawn from unit normal distribution. Figure shows the time-averaged RMSE obtained for the Lorenz model using NSPF with different active set sizes versus the number of forward runs per assimilation cycles. For this case, the observations were collected every five time steps. The results of running the standard PF with up to particles are shown for comparison. Clearly, the standard PF diverged and the estimated time-average forecast RMSEs were large. In c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

10 Nested Sampling 14 Assimilation every 2 time steps (b) 2 Assimilation every 4 time steps Figure. Forecasted states of the three-element Lorenz model (shown vertically) using the NSPF (with model error variance = 1., observation error variance =., and time step =.1), with assimilation every 2 time steps and (b) 4 time steps. The true system is in black, NSPF particles in red and assimilation times are shown as green dots. This figure is available in colour online at wileyonlinelibrary.com/journal/qj contrast, the NSPF managed to reduce the RMSE consistently regardless of the number of local MCMC steps. The reason lies in how NSPF works. The active set size is used to provide a discrete representation of the posterior PDF (prior PDF for the next time step), thus a larger size of the active set will provide more complete representation of the PDF. On the other hand, smaller active sets provide better local search. For example, nested sampling iterations on an active set of particles results in better focusing of the particles into higher-likelihood regions than the case of an active set of 2 particles (for the same number of nested sampling iterations). This is evident in the current setting, where focusing the particles towards observations is more important than covering the posterior PDF. We expect that the outperformance of the runs with small active set over the runs with larger active set sizes will be reversed if the number of nested sampling iterations is increased. We assume that, if the particles are in high-likelihood regions of the search space, the importance of a correct representation of the PDF will be evident in the results. Figure shows the convergence of the time-average RMSE versus the number of forward runs per assimilation cycle when observations are collected every time steps. The results show that more forward runs are needed per assimilation cycle to avoid filter divergence. It is also clear that NSPF with more local MCMC steps outperformed the cases with shorter local random search chains. This is attributed to the fixed nature of the MCMC step size. An adaptive MCMC could be utilized within the nested sampling iteration to increase the robustness of the algorithm. Furthermore, we repeat the same numerical experiment when observations are collected from only the odd-numbered state variables. Figure shows the convergence rates of the timeaverage forecast RMSE using the PF and the NSPF with different active set sizes. Again, the NSPF clearly outperformed the PF and a smooth error reduction is observed by increasing the number of forward runs per assimilation cycle. As highlighted earlier, the NSPF with smaller active set sizes outperformed the runs with the larger active set sizes because of the large evolution of the state variables between the data assimilation steps. Correspondingly, the pushing of the particles towards the high-likelihood regions is the main factor in the success of the NSPF in reducing the time-averaged forecast RMSE.. Discussion The proposed NSPF shares many objectives with the recent work of Chorin and Tu (2) and Van Leeuwen (2). Chorin and Tu (2) proposed an implicit particle filter where implicit sampling is utilized. The idea in implicit sampling is to define probabilities first, and then search for particles that assume them. This leads to sampling ξ from a simple distribution (say a Gaussian), then a different function is defined for each particle to map the samples from the simple distribution to the state distribution ξ x i k+1. This is done by solving an optimization problem. This technique has many difficulties, including defining the mapping function. Morzfeld et al. (2) presented an implementation of implicit filters in which a random map was utilized. However, for each c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

11 1 A. H. Elsheikh et al local MCMC steps (b) local MCMC steps 3 Active set size = Active set size = Active set size = Active set size = (c) local MCMC steps (d) local MCMC steps 3 Active set size = Active set size = Active set size = Active set size = Figure. Time average forecast error (RMSE) for forty-element Lorenz model with different active set sizes and different number of nested sampling iterations (model error variance =.1, observation error variance =.1, time step =.1, observations are collected for all state variables and assimilation every time steps). This figure is available in colour online at wileyonlinelibrary.com/journal/qj particle an optimization problem has to be solved and the Jacobian of the map has to be evaluated at the minimum solution to update the particles weights. Also, the validity of the sampling relies on finding the global minimum, which is hard to ensure. Van Leeuwen (2) exploited the freedom in choosing the proposal density in PFs to obtain a highly efficient PF. In the standard PF, the transition density is translated to propagating the current particle state via the model dynamics and then adding a noise term to represent the model error statistics. Van Leeuwen (2) proposed to make the proposal density dependent on the future observations by using the following equation x k = F(x k 1 ) + ɛ k + K { y k H(x k 1 ) }, (1) where y k is the future observations and K is a Kalman-like gain matrix. The term K { y k H(x k 1 ) } was described as a relaxation term that pulls the particles towards future observations. It was postulated that if K is chosen wisely, particles will end up close to observations. Adding this relaxation term will be reflected in an additional term in the particle weights, which assumes that the whole term ɛ k + K { y k H(x k 1 ) } is a Gaussian term. However, this assumption may not be valid for nonlinear systems. The NSPF can be viewed as a move resample filter in contrast to the resample move PF (Gilks and Berzuini, 21). In the resample move PF, an MCMC step is applied after the resampling step to ensure the diversity of the ensemble as the resampling step might pick the same particle multiple times. However, the MCMC move step in the resample move filter does not push the particles towards the high-likelihood regions. The resample move algorithm was successfully applied by Dowd (2) for data assimilation with a marine ecological model. NSPF tries to refocus the particles into high-likelihood regions similar to the work of Chorin and Tu (2) and Van Leeuwen (2). However, in contrast to implicit PFs, the pushing of the particles in NSPF towards high-likelihood regions is done via the nested sampling algorithm, which is a global search algorithm. In addition, all forward evaluations during this optimization c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

12 Nested Sampling local MCMC steps (b) local MCMC steps 3 Active set size = Active set size = Active set size = Active set size = (c) local MCMC steps (d) local MCMC steps 3 Active set size = Active set size = Active set size = Active set size = Figure. Time average forecast error (RMSE) for forty-element Lorenz model with different active set sizes and different number of nested sampling iterations (model error variance =.1, observation error variance =.1, time step =.1, observations are collected for all state variables and assimilation every time steps). This figure is available in colour online at wileyonlinelibrary.com/journal/qj procedure are utilized in constructing the posterior PDF. In contrast to the work of Van Leeuwen (2), the particles in NSPF are pushed towards high-likelihood by constrained sampling of the prior, and thus are repropagated through the system dynamics before evaluating the likelihood. The proposed algorithm relies on a random sampling step for refocusing the particles towards high-likelihood regions using short MCMC chains. Many improvements are envisioned for this step. A more appropriate choice would be based on gradientbased Markov Chain Monte Carlo methods (Langevin MCMC or Hybrid MCMC). In these methods, the proposal for the jump between the states has a deterministic component based on the gradient of the posterior and a random component similar to the random walk Metropolis Hastings algorithm (Grenander and Miller, 14).. Conclusions Nonlinear data assimilation and uncertainty quantification are challenging problems. The current best-practice data assimilation technique based on the ensemble Kalman filter relies on restrictive normality assumptions. PFs are general Bayesian data assimilation techniques lacking any restrictive distributional assumptions. However, PFs suffer from sample degeneracy when the state and/or observation vectors are high dimensional, rendering PFs impractical for systems with a spatial component (Snyder et al., 2). In the current article, we presented the NSPF for general Bayesian filtering based on PFs and adaptive construction of the posterior PDF using the nested sampling algorithm. Nested sampling can be considered as a global optimization algorithm utilizing an active set of particles that covers the prior distribution. This active set incrementally climbs the likelihood function to higher-likelihood regions without selecting the best particle and thus avoids being trapped in local minima. The climbing process is done iteratively by discarding the members with lowest data fit, one at a time. The posterior distribution is estimated from the set of discarded members and the final active set after terminating the nested sampling iteration. Climbing the likelihood function requires sampling from the prior while satisfying a lower-bound c 2 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 14 (2)

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