Variability of the Atlantic thermohaline circulation described by three-dimensional empirical orthogonal functions

We describe the use of bivariate 3d empirical orthogonal functions (EOFs) in characterising low frequency variability of the Atlantic thermohaline circulation (THC) in the Hadley Centre global climate model, HadCM3. We find that the leading two modes are well correlated with an index of the meridional overturning circulation (MOC) on decadal timescales, with the leading mode alone accounting for 54% of the decadal variance. Episodes of coherent oscillations in the sub-space of the leading EOFs are identified; these episodes are of great interest for the predictability of the THC, and could indicate the existence of different regimes of natural variability. The mechanism identified for the multi-decadal variability is an internal ocean mode, dominated by changes in convection in the Nordic Seas, which lead the changes in the MOC by a few years. Variations in salinity transports from the Arctic and from the North Atlantic are the main feedbacks which control the oscillation. This mode has a weak feedback onto the atmosphere and hence a surface climatic influence. Interestingly, some of these climate impacts lead the changes in the overturning. There are also similarities to observed multi-decadal climate variability.

A new set of orthogonal patterns in weather and climate: Optimally interpolated patterns.

A new spectral-based approach is presented to find orthogonal patterns from gridded weather/climate data. The method is based on optimizing the interpolation error variance. The optimally interpolated patterns (OIP) are then given by the eigenvectors of the interpolation error covariance matrix, obtained using the cross-spectral matrix. The formulation of the approach is presented, and the application to low-dimension stochastic toy models and to various reanalyses datasets is performed. In particular, it is found that the lowest-frequency patterns correspond to largest eigenvalues, that is, variances, of the interpolation error matrix. The approach has been applied to the Northern Hemispheric (NH) and tropical sea level pressure (SLP) and to the Indian Ocean sea surface temperature (SST). Two main OIP patterns are found for the NH SLP representing respectively the North Atlantic Oscillation and the North Pacific pattern. The leading tropical SLP OIP represents the Southern Oscillation. For the Indian Ocean SST, the leading OIP pattern shows a tripole-like structure having one sign over the eastern and north- and southwestern parts and an opposite sign in the remaining parts of the basin. The pattern is also found to have a high lagged correlation with the Niño-3 index with 6-months lag.

Empirical Orthogonal Functions: The Medium is the Message

Empirical orthogonal function (EOF) analysis is a powerful tool for data compression and dimensionality reduction used broadly in meteorology and oceanography. Often in the literature, EOF modes are interpreted individually, independent of other modes. In fact, it can be shown that no such attribution can generally be made. This review demonstrates that in general individual EOF modes (i) will not correspond to individual dynamical modes, (ii) will not correspond to individual kinematic degrees of freedom, (iii) will not be statistically independent of other EOF modes, and (iv) will be strongly influenced by the nonlocal requirement that modes maximize variance over the entire domain. The goal of this review is not to argue against the use of EOF analysis in meteorology and oceanography; rather, it is to demonstrate the care that must be taken in the interpretation of individual modes in order to distinguish the medium from the message.

Some new three-level orthogonal main effects plans robust to model uncertainty

In this paper, we list some new orthogonal main effects plans for three-level designs for 4, 5 and 6 factors in IS runs and compare them with designs obtained from the existing L-18 orthogonal array. We show that these new designs have better projection properties and can provide better parameter estimates for a range of possible models. Additionally, we study designs in other smaller run-sizes when there are insufficient resources to perform an 18-run experiment. Plans for three-level designs for 4, 5 and 6 factors in 13 to 17 runs axe given. We show that the best designs here are efficient and deserve strong consideration in many practical situations.

Probability density function estimation using orthogonal forward regression

Using the classical Parzen window estimate as the target function, the kernel density estimation is formulated as a regression problem and the orthogonal forward regression technique is adopted to construct sparse kernel density estimates. The proposed algorithm incrementally minimises a leave-one-out test error score to select a sparse kernel model, and a local regularisation method is incorporated into the density construction process to further enforce sparsity. The kernel weights are finally updated using the multiplicative nonnegative quadratic programming algorithm, which has the ability to reduce the model size further. Except for the kernel width, the proposed algorithm has no other parameters that need tuning, and the user is not required to specify any additional criterion to terminate the density construction procedure. Two examples are used to demonstrate the ability of this regression-based approach to effectively construct a sparse kernel density estimate with comparable accuracy to that of the full-sample optimised Parzen window density estimate.

Fully complex-valued radial basis function networks for orthogonal least squares regression

We consider a fully complex-valued radial basis function (RBF) network for regression application. The locally regularised orthogonal least squares (LROLS) algorithm with the D-optimality experimental design, originally derived for constructing parsimonious real-valued RBF network models, is extended to the fully complex-valued RBF network. Like its real-valued counterpart, the proposed algorithm aims to achieve maximised model robustness and sparsity by combining two effective and complementary approaches. The LROLS algorithm alone is capable of producing a very parsimonious model with excellent generalisation performance while the D-optimality design criterion further enhances the model efficiency and robustness. By specifying an appropriate weighting for the D-optimality cost in the combined model selecting criterion, the entire model construction procedure becomes automatic. An example of identifying a complex-valued nonlinear channel is used to illustrate the regression application of the proposed fully complex-valued RBF network.

Sparse kernel density estimator using orthogonal regression based on D-optimality experimental design

A novel sparse kernel density estimator is derived based on a regression approach, which selects a very small subset of significant kernels by means of the D-optimality experimental design criterion using an orthogonal forward selection procedure. The weights of the resulting sparse kernel model are calculated using the multiplicative nonnegative quadratic programming algorithm. The proposed method is computationally attractive, in comparison with many existing kernel density estimation algorithms. Our numerical results also show that the proposed method compares favourably with other existing methods, in terms of both test accuracy and model sparsity, for constructing kernel density estimates.

Sparse multioutput radial basis function network construction using combined locally regularised orthogonal least square and D-optimality experimental design

A construction algorithm for multioutput radial basis function (RBF) network modelling is introduced by combining a locally regularised orthogonal least squares (LROLS) model selection with a D-optimality experimental design. The proposed algorithm aims to achieve maximised model robustness and sparsity via two effective and complementary approaches. The LROLS method alone is capable of producing a very parsimonious RBF network model with excellent generalisation performance. The D-optimality design criterion enhances the model efficiency and robustness. A further advantage of the combined approach is that the user only needs to specify a weighting for the D-optimality cost in the combined RBF model selecting criterion and the entire model construction procedure becomes automatic. The value of this weighting does not influence the model selection procedure critically and it can be chosen with ease from a wide range of values.

Sparse kernel regression modeling using combined locally regularized orthogonal least squares and D-optimality experimental design

The note proposes an efficient nonlinear identification algorithm by combining a locally regularized orthogonal least squares (LROLS) model selection with a D-optimality experimental design. The proposed algorithm aims to achieve maximized model robustness and sparsity via two effective and complementary approaches. The LROLS method alone is capable of producing a very parsimonious model with excellent generalization performance. The D-optimality design criterion further enhances the model efficiency and robustness. An added advantage is that the user only needs to specify a weighting for the D-optimality cost in the combined model selecting criterion and the entire model construction procedure becomes automatic. The value of this weighting does not influence the model selection procedure critically and it can be chosen with ease from a wide range of values.

Kernel density construction using orthogonal forward regression

An automatic algorithm is derived for constructing kernel density estimates based on a regression approach that directly optimizes generalization capability. Computational efficiency of the density construction is ensured using an orthogonal forward regression, and the algorithm incrementally minimizes the leave-one-out test score. Local regularization is incorporated into the density construction process to further enforce sparsity. Examples are included to demonstrate the ability of the proposed algorithm to effectively construct a very sparse kernel density estimate with comparable accuracy to that of the full sample Parzen window density estimate.

Sparse kernel density construction using orthogonal forward regression with leave-one-out test score and local regularization

This paper presents an efficient construction algorithm for obtaining sparse kernel density estimates based on a regression approach that directly optimizes model generalization capability. Computational efficiency of the density construction is ensured using an orthogonal forward regression, and the algorithm incrementally minimizes the leave-one-out test score. A local regularization method is incorporated naturally into the density construction process to further enforce sparsity. An additional advantage of the proposed algorithm is that it is fully automatic and the user is not required to specify any criterion to terminate the density construction procedure. This is in contrast to an existing state-of-art kernel density estimation method using the support vector machine (SVM), where the user is required to specify some critical algorithm parameter. Several examples are included to demonstrate the ability of the proposed algorithm to effectively construct a very sparse kernel density estimate with comparable accuracy to that of the full sample optimized Parzen window density estimate. Our experimental results also demonstrate that the proposed algorithm compares favorably with the SVM method, in terms of both test accuracy and sparsity, for constructing kernel density estimates.

Orthogonal forward selection for constructing the radial basis function network with tunable nodes

An orthogonal forward selection (OFS) algorithm based on the leave-one-out (LOO) criterion is proposed for the construction of radial basis function (RBF) networks with tunable nodes. This OFS-LOO algorithm is computationally efficient and is capable of identifying parsimonious RBF networks that generalise well. Moreover, the proposed algorithm is fully automatic and the user does not need to specify a termination criterion for the construction process.