Data visualisation with missing data: A non-linear approach

Exploratory analysis of data in all sciences seeks to find common patterns to gain insights into the structure and distribution of the data. Typically visualisation methods like principal components analysis are used but these methods are not easily able to deal with missing data nor can they capture non-linear structure in the data. One approach to discovering complex, non-linear structure in the data is through the use of linked plots, or brushing, while ignoring the missing data. In this technical report we discuss a complementary approach based on a non-linear probabilistic model. The generative topographic mapping enables the visualisation of the effects of very many variables on a single plot, which is able to incorporate far more structure than a two dimensional principal components plot could, and deal at the same time with missing data. We show that using the generative topographic mapping provides us with an optimal method to explore the data while being able to replace missing values in a dataset, particularly where a large proportion of the data is missing.

Exploratory data analysis with non-linear and missing data in geochemistry

Exploratory analysis of data seeks to find common patterns to gain insights into the structure and distribution of the data. In geochemistry it is a valuable means to gain insights into the complicated processes making up a petroleum system. Typically linear visualisation methods like principal components analysis, linked plots, or brushing are used. These methods can not directly be employed when dealing with missing data and they struggle to capture global non-linear structures in the data, however they can do so locally. This thesis discusses a complementary approach based on a non-linear probabilistic model. The generative topographic mapping (GTM) enables the visualisation of the effects of very many variables on a single plot, which is able to incorporate more structure than a two dimensional principal components plot. The model can deal with uncertainty, missing data and allows for the exploration of the non-linear structure in the data. In this thesis a novel approach to initialise the GTM with arbitrary projections is developed. This makes it possible to combine GTM with algorithms like Isomap and fit complex non-linear structure like the Swiss-roll. Another novel extension is the incorporation of prior knowledge about the structure of the covariance matrix. This extension greatly enhances the modelling capabilities of the algorithm resulting in better fit to the data and better imputation capabilities for missing data. Additionally an extensive benchmark study of the missing data imputation capabilities of GTM is performed. Further a novel approach, based on missing data, will be introduced to benchmark the fit of probabilistic visualisation algorithms on unlabelled data. Finally the work is complemented by evaluating the algorithms on real-life datasets from geochemical projects.

Addressing missing data in geochemistry: a non-linear approach

Exploratory analysis of petroleum geochemical data seeks to find common patterns to help distinguish between different source rocks, oils and gases, and to explain their source, maturity and any intra-reservoir alteration. However, at the outset, one is typically faced with (a) a large matrix of samples, each with a range of molecular and isotopic properties, (b) a spatially and temporally unrepresentative sampling pattern, (c) noisy data and (d) often, a large number of missing values. This inhibits analysis using conventional statistical methods. Typically, visualisation methods like principal components analysis are used, but these methods are not easily able to deal with missing data nor can they capture non-linear structure in the data. One approach to discovering complex, non-linear structure in the data is through the use of linked plots, or brushing, while ignoring the missing data. In this paper we introduce a complementary approach based on a non-linear probabilistic model. Generative topographic mapping enables the visualisation of the effects of very many variables on a single plot, while also dealing with missing data. We show how using generative topographic mapping also provides an optimal method with which to replace missing values in two geochemical datasets, particularly where a large proportion of the data is missing.