Anders Björkström

A Generalized View on Continuum Regression

In regression with near collinear explanatory variables, the least squares predictor has large variance. Ordinary least squares regression (OLSR) often leads to unrealistic regression coefficients. Several regularized regression methods have been proposed as alternatives. Well-known are principal components regression (PCR), ridge regression (RR) and continuum regression (CR). The latter two involve a continuous metaparameter, offering additional flexibility.For a univariate response variable, CR incorporates OLSR, PLSR, and PCR as special cases, for special values of the metaparameter. CR is also closely related to RR. However, CR can in fact yield regressors that vary discontinuously with the metaparameter. Thus, the relation between CR and RR is not always one-to-one. We develop a new class of regression methods, LSRR, essentially the same as CR, but without discontinuities, and prove that any optimization principle will yield a regressor proportional to a RR, provided only that the principle implies maximizing some function of the regressors sample correlation coefficient and its sample variance. For a multivariate response vector we demonstrate that a number of well-established regression methods are related, in that they are special cases of basically one general procedure. We try a more general method based on this procedure, with two meta-parameters. In a simulation study we compare this method to ridge regression, multivariate PLSR and repeated univariate PLSR. For most types of data studied, all methods do approximately equally well. There are cases where RR and LSRR yield larger errors than the other methods, and we conclude that one-factor methods are not adequate for situations where more than one latent variable are needed to describe the data. Among those based on latent variables, none of the methods tried is superior to the others in any obvious way.

Covariance structure approximation via gLasso in high-dimensional supervised classification

Recent work has shown that the Lasso-based regularization is very useful for estimating the high-dimensional inverse covariance matrix. A particularly useful scheme is based on penalizing the ℓ1 norm of the off-diagonal elements to encourage sparsity. We embed this type of regularization into high-dimensional classification. A two-stage estimation procedure is proposed which first recovers structural zeros of the inverse covariance matrix and then enforces block sparsity by moving non-zeros closer to the main diagonal. We show that the block-diagonal approximation of the inverse covariance matrix leads to an additive classifier, and demonstrate that accounting for the structure can yield better performance accuracy. Effect of the block size on classification is explored, and a class of asymptotically equivalent structure approximations in a high-dimensional setting is specified. We suggest a variable selection at the block level and investigate properties of this procedure in growing dimension asymptotics. We present a consistency result on the feature selection procedure, establish asymptotic lower an upper bounds for the fraction of separative blocks and specify constraints under which the reliable classification with block-wise feature selection can be performed. The relevance and benefits of the proposed approach are illustrated on both simulated and real data.

Ocean Carbon Models and Inverse Methods

Any theoretical treatment of a problem concerning our environment, such as our present concern, namely that of deducing the parameterization of an ocean-carbon model by inverse methods, must be based on some kind of model. In reality of course, the natural phenomena are so complex and our data so few that all such problems are indeterminate (i.e. have more than one solution that is consistent with the data). However, only by adopting models, and thereby, by definition, imposing a reduction on the complexity of reality do overdetermined systems arise, and also only by adoption of models can we hope to make efficient use of the available data and information. It is obvious that whether or not the results will be of interest depends on how well our model captures the essence of the phenomena in nature which it seeks to describe.

Calibrating Ocean Models by the Constrained Inverse Method

From the results of several investigations (Keeling 1973; Bjorkstrom 1980; Killough and Emanuel 1981; Bjorkstrom, this volume; Bolin 1983; Bolin et al. 1983, p. 231; Fiadeiro 1983; Bolin, this volume) the approach of using highly aggregated box models (less than 15 boxes for the world oceans) appears to be inadequate for the task of estimating accurately the current rate at which the ocean is absorbing excess atmospheric CO2. However, general circulation models of the entire ocean are not at hand; therefore, what is needed is a new generation of models that can serve usefully in the interim to study the important question of the current rate of oceanic CO2 uptake. Further, such models may have other applications, not the least of which may be their use as diagnostic tools for general circulation models of the ocean (see also Bryan et al. 1975; Sarmiento, this volume).

Modelling the Oceanic CO2 Uptake and Future CO2 Levels

In 1978, Siegenthaler and Oeschger summarized several predictions about future atmospheric carbon dioxide levels. The authors concluded that of the cumulative inputs one hundred years ahead, between 46 % and 80 % would remain in the atmosphere. One of the causes for the wide range was the uncertainty about which was the most realistic model used in the calculations.

Exploiting Sparse Dependence Structure in Model Based Classification

Sparsity patterns discovered in the data dependence structure were used to reduce the dimensionality and improve performance accuracy of the model based classifier in a high dimensional framework.