Block GTM: Incorporating prior knowledge of covariance structure in data visualisation

Visualising data for exploratory analysis is a big challenge in scientific and engineering domains where there is a need to gain insight into the structure and distribution of the data. Typically, visualisation methods like principal component analysis and multi-dimensional scaling are used, but it is difficult to incorporate prior knowledge about structure of the data into the analysis. In this technical report we discuss a complementary approach based on an extension of a well known non-linear probabilistic model, the Generative Topographic Mapping. We show that by including prior information of the covariance structure into the model, we are able to improve both the data visualisation and the model fit.

Fast algorithms for automatic mapping with space-limited covariance functions

In this paper we discuss a fast Bayesian extension to kriging algorithms which has been used successfully for fast, automatic mapping in emergency conditions in the Spatial Interpolation Comparison 2004 (SIC2004) exercise. The application of kriging to automatic mapping raises several issues such as robustness, scalability, speed and parameter estimation. Various ad-hoc solutions have been proposed and used extensively but they lack a sound theoretical basis. In this paper we show how observations can be projected onto a representative subset of the data, without losing significant information. This allows the complexity of the algorithm to grow as O(n m 2), where n is the total number of observations and m is the size of the subset of the observations retained for prediction. The main contribution of this paper is to further extend this projective method through the application of space-limited covariance functions, which can be used as an alternative to the commonly used covariance models. In many real world applications the correlation between observations essentially vanishes beyond a certain separation distance. Thus it makes sense to use a covariance model that encompasses this belief since this leads to sparse covariance matrices for which optimised sparse matrix techniques can be used. In the presence of extreme values we show that space-limited covariance functions offer an additional benefit, they maintain the smoothness locally but at the same time lead to a more robust, and compact, global model. We show the performance of this technique coupled with the sparse extension to the kriging algorithm on synthetic data and outline a number of computational benefits such an approach brings. To test the relevance to automatic mapping we apply the method to the data used in a recent comparison of interpolation techniques (SIC2004) to map the levels of background ambient gamma radiation. © Springer-Verlag 2007.

Gabor-based region covariance matrices for face recognition

This paper presents a new method for human face recognition by utilizing Gabor-based region covariance matrices as face descriptors. Both pixel locations and Gabor coefficients are employed to form the covariance matrices. Experimental results demonstrate the advantages of this proposed method.

Statnote 31:analysis of covariance

Analysis of covariance (ANCOVA) is a useful method of ‘error control’, i.e., it can reduce the size of the error variance in an experimental or observational study. An initial measure obtained before the experiment, which is closely related to the final measurement, is used to adjust the final measurements, thus reducing the error variance. When this method is used to reduce the error term, the X variable must not itself be affected by the experimental treatments, because part of the treatment effect would then also be removed. Hence, the method can only be safely used when X is measured before an experiment. A further limitation of the analysis is that only the linear effect of Y on X is being removed and it is possible that Y could be a curvilinear function of X. A question often raised is whether ANCOVA should be used routinely in experiments rather than a randomized blocks or split-plot design, which may also reduce the error variance. The answer to this question depends on the relative precision of the difference methods with reference to each scenario. Considerable judgment is often required to select the best experimental design and statistical help should be sought at an early stage of an investigation.