Implicit Estimation of Ecological Model Parameters

Abstract

We introduce an implicit method for state and parameter estimation and apply it to a stochastic ecological model. The method uses an ensemble of particles to approximate the distribution of model solutions and parameters conditioned on noisy observations of the state. For each particle, it first determines likely values based on the observations, then samples around those values. This approach has a strong theoretical foundation, applies to nonlinear models and non-Gaussian distributions, and can estimate any number of model parameters, initial conditions, and model error covariances. The method is called implicit because it updates the particles without forming a predictive distribution of forward model integrations. As a point of comparison for different assimilation techniques, we consider examples in which one or more bifurcations separate the true parameter from its initial approximation. The implicit estimator is asymptotically unbiased, has a root-mean-squared error comparable to or less than the other methods, and is accurate even with small ensemble sizes.

Appendix: The Optimization Problem

Essential to the applicability of our implicit sampling methods is the optimization step of Algorithm 1. We use the Levenberg–Marquardt method to search for the minimizer ζ ^∗. This method is part of a wider class of trust-region (or restricted-step) methods—iterative approaches that restrict the next step to a region centered at the current point (see Fig. 14 for a schematic). At each iteration, the method either expands or contracts the region depending on the ratio of the predicted and actual reductions in the cost function. A notable feature of this method is that it safely handles indefinite Hessians. We use Algorithm 7.3.4 from Conn et al. (2000), which is sketched below. The numerical constants that appear in the method are arbitrary, and the stationary point and rate of convergence of the method are theoretically insensitive to their values. In practice, the constants effect the number of iterations until the algorithm terminates.

Fig. 14

Schematic diagram of the cost function in state and parameter space: (solid) the contour lines of the cost function, (dashed) its quadratic approximation at the starting point, (dotted) the trust region centered at that point. The next step in the iteration is always constrained to lie within a trust region. In the case depicted, the constraint is active

Algorithm 3

(Levenberg–Marquardt)

Let the subscript (k) denote the iteration number and suppose we have some initial guess ζ ₍₀₎ and radius ϵ ₍₀₎.

• 1 At the current point ζ _(k), compute the cost function, its gradient, and Hessian:

  $$J_{(k)} = J( \zeta_{(k)} ), \qquad g_{(k)} = \nabla J( \zeta_{(k)} ),\qquad H_{(k)} = H( \zeta_{(k)} ). $$

• 2 Find the proposed increment Δζ that minimizes the quadratic approximation at the current point, defined such that

  $$K(\zeta_{(k)} + \Delta\zeta) = J_{(k)} + g_{(k)}^T \Delta\zeta + \frac{1}{2} (\Delta\zeta)^T H_{(k)} \Delta\zeta, $$

  constrained to the region where

  $$(\Delta\zeta)^T \Delta\zeta\leq\epsilon_{(k)}^2. $$

  For more details on the solution of quadratic programming problems, see the monograph of Conn et al. (2000).

• 3 Compute the ratio of the actual reduction to predicted reduction of the function at the proposed next step,

  $$r = \frac{ J(\zeta_{(k)}) - J(\zeta_{(k)} + \Delta\zeta)}{ K(\zeta_{(k)}) - K(\zeta_{(k)} + \Delta\zeta)}. $$

  Update the trust region size based on r,

  $$\epsilon_{(k+1)} = \left\{ \begin{array}{l@{\quad}l} \epsilon_{(k)}/2 & \text{if $r < 1/4$}, \\ \min(2 \epsilon_{(k)}, \epsilon_M) & \text{if $r > 3/4$}, \\ \epsilon_{(k)} & \text{otherwise}, \end{array} \right. $$

  where ϵ _M is a user specified maximum radius. Move to the next point only if it decreases the cost function, i.e.,

  $$\zeta_{(k+1)} = \left\{ \begin{array}{l@{\quad}l} \zeta_{(k)} + \Delta\zeta & \text{if $0 < r$}, \\ \zeta_{(k)} & \text{otherwise}. \end{array} \right. $$

  Since the proposed increment Δζ minimizes K within the trust region, the denominator of the ratio r is always positive. Hence, the iteration remains at the current step if and only if the predicted step does not decrease the cost function.

• 4 Replace k with k+1, and return to Step 1 until the norm of the gradient g _(k) is less than a given value or we reach an upper bound of iterations.

The Derivatives of the Cost Function

The structure of the Hessian of J makes it possible to do these computations efficiently even when the dimension of ζ is large. Since the residuals only depend on the previous and current model step, the Hessian has a single band running down its diagonal, corresponding to the state derivatives, full columns at its far-right side, and full rows at its bottom, both corresponding to the parameter derivatives. We need only store the diagonal, subdiagonals, and bottom rows because the Hessian is symmetric. This representation grows linearly in the number of variables, as opposed to quadratically for the full representation. If the model equations are a discretization of a partial differential equation, we lose this special structure, but numerous libraries exist for optimization when the Hessian is sparse. Thus, the Hessian has the block form

$$H = \left [ \begin{array}{c@{\quad}c} H_{\mathbf{x}\mathbf{x}} & H_{\mathbf{x}\theta} \\ H_{\theta\mathbf{x}} & H_{\theta\theta} \end{array} \right ] , $$

where H _xx is a band matrix, and $H_{\theta\mathbf{x}} = H_{\mathbf{x}\theta}^{T}$ since we assume J is smooth.

To simplify notation, define the (column) vector of residuals

$$\rho= ( \mathbf{e}_{\mathfrak{m}(l)+1:\mathfrak{m}(l+k)}, \mathbf{d}_{l+1:l+k} ). $$

The cost function, its gradient, and its Hessian are thus

$$J(\zeta) = \frac{1}{2} \rho^T \rho, \quad \nabla J= (\nabla \rho)^T \rho, \qquad H = (\nabla\rho)^T ( \nabla\rho) + \sum_i H_{\rho_i} \rho_i, $$

where $H_{\rho_{i}}$ is the Hessian of the ith element of ρ. We use the Gauss–Newton approximation of the Hessian,

$$H = (\nabla\rho)^T (\nabla\rho). $$

Since the goal of the optimization is to make the norm of the residuals small, we expect the neglected terms to be small as well (Fletcher 1987). The derivatives of the cost function in the unconstrained variables follow from an application of the chain rule. An added benefit of the Gauss–Newton approximation is that it is always positive semi-definite. Hence, we can stop the optimization at any iteration, and it is still possible to sample a Gaussian whose covariance matrix is the Moore–Penrose pseudoinverse of the Hessian.

The nonzero derivatives of the model noise are

and the derivative of the observation noise is

$$\partial\mathbf{d}_n/\partial\mathbf{x}_{\mathfrak{m}(n)} = ( \sqrt{R})^{-1}\partial h(\mathbf{x}_{\mathfrak{m}(n)})/\partial\mathbf {x}_{\mathfrak{m}(n)}. $$

The state derivatives of the Lotka–Volterra model equations (14) are

$$\frac{\partial\phi}{\partial\mathbf{u}} = \left [ \begin{array}{c@{\quad}c} a_1 + 2 a_2 p + a_3 q/(1 + a_7 p)^2 & a_3 p/(1 + a_7 p) \\ a_6 q/(1 + a_7 p)^2 & a_4 + 2 a_5 q + a_6 p/(1 + a_7 p) \end{array} \right ] , $$

and the derivatives for each parameter are

and

$$\frac{\partial\phi}{\partial a_7} = \begin{bmatrix} -a_3 p^2 q/(1 + a_7 p)^2 \\ -a_6 p^2 q/(1 + a_7 p)^2 \end{bmatrix} . $$

Since the observation functions considered are linear, their Jacobians are identical to the functions.