Forecast Bias Correction: A Second Order Method

The difference between a model forecast and actual observations is called forecast bias. This bias is due to either incomplete model assumptions and/or poorly known parameter values and initial/boundary conditions. In this paper we discuss a method for estimating corrections to parameters and initial conditions that would account for the forecast bias. A set of simple experiments with the logistic ordinary differential equation is performed using an iterative version of a first order version of our method to compare with the second order version of the method.Comment: 27 Pages, 3 figures, 8 table

Minimal chaotic models from the Volterra gyrostat

Low-order models obtained through Galerkin projection of several physically important systems (e.g., Rayleigh-B\'enard convection, mid-latitude quasi-geostrophic dynamics, and vorticity dynamics) appear in the form of coupled gyrostats. Forced dissipative chaos is an important phenomenon in these models, and this paper considers the minimal chaotic models, in the sense of having the fewest external forcing and linear dissipation terms, arising from an underlying gyrostat core. It is shown here that a critical distinction is whether the gyrostat core (without forcing or dissipation) conserves energy, depending on whether the sum of the quadratic coefficients is zero. The paper demonstrates that, for the energy-conserving case of the gyrostat core, the requirement of a characteristic pair of fixed points that repel the chaotic flow dictates placement of forcing and dissipation in the minimal chaotic models. In contrast, if the core does not conserve energy, the forcing can be arranged in additional ways for chaos to appear, especially for the cases where linear feedbacks render fewer invariants in the gyrostat core. In all cases, the linear mode must experience dissipation for chaos to arise. Thus, the Volterra gyrostat presents a clear example where the arrangement of fixed points circumscribes more complex dynamics

Invariants and chaos in the Volterra gyrostat without energy conservation

The model of the Volterra gyrostat (VG) has not only played an important role in rigid body dynamics but also served as the foundation of low-order models of many naturally occurring systems. It is well known that VG possesses two invariants, or constants of motion, corresponding to kinetic energy and squared angular momentum, giving oscillatory solutions to its equations of motion. Nine distinct subclasses of the VG have been identified, two of which the Euler gyroscope and Lorenz gyrostat are each known to have two constants. This paper provides a complete characterization of constants of motion of the VG and its subclasses, showing how these enjoy two constants of motion even when rendered in terms of a non-invertible transformation of parameters, leading to a transformed Volterra gyrostat (tVG). If the quadratic coefficients of the tVG sum to zero, as they do for the VG, the system conserves energy. In all of these cases, the flows preserve volume; however, physical models where the quadratic coefficients do not sum to zero are ubiquitous, and characterization of constants of motion and the resulting dynamics for this more general class of models with volume conservation but without energy conservation is lacking. We provide such a characterization for each of the subclasses. Those with three linear feedback terms have no constants of motion, and thereby admit rich dynamics including chaos. This gives rise to a broad class of three-dimensional volume conserving chaotic flows, arising naturally from model reduction techniques

On polynomial digraphs

Let $\Phi(x,y)$ be a bivariate polynomial with complex coefficients. The zeroes of $\Phi(x,y)$ are given a combinatorial structure by considering them as arcs of a directed graph $G(\Phi)$. This paper studies some relationship between the polynomial $\Phi(x,y)$ and the structure of $G(\Phi)$.Comment: 13 pages, 6 figures, See also http://www-ma2.upc.edu/~montes

On the dual advantage of placing observations through forward sensitivity analysis

The four-dimensional variational data assimilation methodology for assimilating noisy observations into a deterministic model has been the workhorse of forecasting centers for over three decades. While this method provides a computationally efficient framework for dynamic data assimilation, it is largely silent on the important question concerning the minimum number and placement of observations. To answer this question, we demonstrate the dual advantage of placing the observations where the square of the sensitivity of the model solution with respect to the unknown control variables, called forward sensitivities, attains its maximum. Therefore, we can force the observability Gramian to be of full rank, which in turn guarantees efficient recovery of the optimal values of the control variables, which is the first of the two advantages of this strategy. We further show that the proposed strategy of placing observations has another inherent optimality: the square of the sensitivity of the optimal estimates of the control with respect to the observations (used to obtain these estimates) attains its minimum value, a second advantage that is a direct consequence of the above strategy for placing observations. Our analytical framework and numerical experiments on linear and nonlinear systems confirm the effectiveness of our proposed strategy

Polynomial-time sortable stacks of burnt pancakes

Pancake flipping, a famous open problem in computer science, can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible. In that context, a prefix reversal of length k reverses the order of the first k elements of the permutation. The burnt variant of pancake flipping involves permutations of signed integers, and reversals in that case not only reverse the order of elements but also invert their signs. Although three decades have now passed since the first works on these problems, neither their computational complexity nor the maximal number of prefix reversals needed to sort a permutation is yet known. In this work, we prove a new lower bound for sorting burnt pancakes, and show that an important class of permutations, known as "simple permutations", can be optimally sorted in polynomial time.Comment: Accepted pending minor revisio

Analyzing Cost of Debt and Credit Spreads Using a Two Factor Model with Multiple Default Thresholds and Varying Covenant Protection

Abstract: The cost of debt capital for corporations depends on credit spreads. Of course, the dramatically greater credit spreads of 2008 greatly increased the cost of debt for the majority of bond issuers. We analyze the shape of credit spread term structures paying special attention to the humps that have been observed by a number of researchers. The shape of credit spreads depends upon the shape of first passage default. Importantly, our work allows separation of default probability due to breach of barrier versus default probability due to assets being less than face value at maturity. We note that in some cases, first passage default has a hump but not in others. It is useful to see when and how first passage default humps contribute to a humped credit spread. The impact of recently popular weak covenants (covenant lite) is shown to play a major role in the shape of credit spreads. The implications of our study are important to such topics as measuring the riskiness of the banking system dependent upon credit spread slopes

An experiment in hurricane track prediction using parallel computing methods

The barotropic model is used to explore the advantages of parallel processing in deterministic forecasting. We apply this model to the track forecasting of hurricane Elena (1985). In this particular application, solutions to systems of elliptic equations are the essence of the computational mechanics. One set of equations is associated with the decomposition of the wind into irrotational and nondivergent components - this determines the initial nondivergent state. Another set is associated with recovery of the streamfunction from the forecasted vorticity. We demonstrate that direct parallel methods based on accelerated block cyclic reduction (BCR) significantly reduce the computational time required to solve the elliptic equations germane to this decomposition and forecast problem. A 72-h track prediction was made using incremental time steps of 16 min on a network of 3000 grid points nominally separated by 100 km. The prediction took 30 sec on the 8-processor Alliant FX/8 computer. This was a speed-up of 3.7 when compared to the one-processor version. The 72-h prediction of Elena's track was made as the storm moved toward Florida's west coast. Approximately 200 km west of Tampa Bay, Elena executed a dramatic recurvature that ultimately changed its course toward the northwest. Although the barotropic track forecast was unable to capture the hurricane's tight cycloidal looping maneuver, the subsequent northwesterly movement was accurately forecasted as was the location and timing of landfall near Mobile Bay

Discretized Bayesian pursuit – A new scheme for reinforcement learning

The success of Learning Automata (LA)-based estimator algorithms over the classical, Linear Reward-Inaction ( L RI )-like schemes, can be explained by their ability to pursue the actions with the highest reward probability estimates. Without access to reward probability estimates, it makes sense for schemes like the L RI to first make large exploring steps, and then to gradually turn exploration into exploitation by making progressively smaller learning steps. However, this behavior becomes counter-intuitive when pursuing actions based on their estimated reward probabilities. Learning should then ideally proceed in progressively larger steps, as the reward probability estimates turn more accurate. This paper introduces a new estimator algorithm, the Discretized Bayesian Pursuit Algorithm (DBPA), that achieves this. The DBPA is implemented by linearly discretizing the action probability space of the Bayesian Pursuit Algorithm (BPA) [1]. The key innovation is that the linear discrete updating rules mitigate the counter-intuitive behavior of the corresponding linear continuous updating rules, by augmenting them with the reward probability estimates. Extensive experimental results show the superiority of DBPA over previous estimator algorithms. Indeed, the DBPA is probably the fastest reported LA to date