Application of Naïve Bayesian sequential analysis to primary care optometry

The objective of this study was to investigate the effects of circularity, comorbidity, prevalence and presentation variation on the accuracy of differential diagnoses made in optometric primary care using a modified form of naïve Bayesian sequential analysis. No such investigation has ever been reported before. Data were collected for 1422 cases seen over one year. Positive test outcomes were recorded for case history (ethnicity, age, symptoms and ocular and medical history) and clinical signs in relation to each diagnosis. For this reason only positive likelihood ratios were used for this modified form of Bayesian analysis that was carried out with Laplacian correction and Chi-square filtration. Accuracy was expressed as the percentage of cases for which the diagnoses made by the clinician appeared at the top of a list generated by Bayesian analysis. Preliminary analyses were carried out on 10 diagnoses and 15 test outcomes. Accuracy of 100% was achieved in the absence of presentation variation but dropped by 6% when variation existed. Circularity artificially elevated accuracy by 0.5%. Surprisingly, removal of Chi-square filtering increased accuracy by 0.4%. Decision tree analysis showed that accuracy was influenced primarily by prevalence followed by presentation variation and comorbidity. Analysis of 35 diagnoses and 105 test outcomes followed. This explored the use of positive likelihood ratios, derived from the case history, to recommend signs to look for. Accuracy of 72% was achieved when all clinical signs were entered. The drop in accuracy, compared to the preliminary analysis, was attributed to the fact that some diagnoses lacked strong diagnostic signs; the accuracy increased by 1% when only recommended signs were entered. Chi-square filtering improved recommended test selection. Decision tree analysis showed that accuracy again influenced primarily by prevalence, followed by comorbidity and presentation variation. Future work will explore the use of likelihood ratios based on positive and negative test findings prior to considering naïve Bayesian analysis as a form of artificial intelligence in optometric practice.

Probabilistic topographic information visualisation

The focus of this thesis is the extension of topographic visualisation mappings to allow for the incorporation of uncertainty. Few visualisation algorithms in the literature are capable of mapping uncertain data with fewer able to represent observation uncertainties in visualisations. As such, modifications are made to NeuroScale, Locally Linear Embedding, Isomap and Laplacian Eigenmaps to incorporate uncertainty in the observation and visualisation spaces. The proposed mappings are then called Normally-distributed NeuroScale (N-NS), T-distributed NeuroScale (T-NS), Probabilistic LLE (PLLE), Probabilistic Isomap (PIso) and Probabilistic Weighted Neighbourhood Mapping (PWNM). These algorithms generate a probabilistic visualisation space with each latent visualised point transformed to a multivariate Gaussian or T-distribution, using a feed-forward RBF network. Two types of uncertainty are then characterised dependent on the data and mapping procedure. Data dependent uncertainty is the inherent observation uncertainty. Whereas, mapping uncertainty is defined by the Fisher Information of a visualised distribution. This indicates how well the data has been interpolated, offering a level of ‘surprise’ for each observation. These new probabilistic mappings are tested on three datasets of vectorial observations and three datasets of real world time series observations for anomaly detection. In order to visualise the time series data, a method for analysing observed signals and noise distributions, Residual Modelling, is introduced. The performance of the new algorithms on the tested datasets is compared qualitatively with the latent space generated by the Gaussian Process Latent Variable Model (GPLVM). A quantitative comparison using existing evaluation measures from the literature allows performance of each mapping function to be compared. Finally, the mapping uncertainty measure is combined with NeuroScale to build a deep learning classifier, the Cascading RBF. This new structure is tested on the MNist dataset achieving world record performance whilst avoiding the flaws seen in other Deep Learning Machines.

Edge centrality via the Holevo quantity

In the study of complex networks, vertex centrality measures are used to identify the most important vertices within a graph. A related problem is that of measuring the centrality of an edge. In this paper, we propose a novel edge centrality index rooted in quantum information. More specifically, we measure the importance of an edge in terms of the contribution that it gives to the Von Neumann entropy of the graph. We show that this can be computed in terms of the Holevo quantity, a well known quantum information theoretical measure. While computing the Von Neumann entropy and hence the Holevo quantity requires computing the spectrum of the graph Laplacian, we show how to obtain a simplified measure through a quadratic approximation of the Shannon entropy. This in turns shows that the proposed centrality measure is strongly correlated with the negative degree centrality on the line graph. We evaluate our centrality measure through an extensive set of experiments on real-world as well as synthetic networks, and we compare it against commonly used alternative measures.

The average mixing matrix signature

Laplacian-based descriptors, such as the Heat Kernel Signature and the Wave Kernel Signature, allow one to embed the vertices of a graph onto a vectorial space, and have been successfully used to find the optimal matching between a pair of input graphs. While the HKS uses a heat di↵usion process to probe the local structure of a graph, the WKS attempts to do the same through wave propagation. In this paper, we propose an alternative structural descriptor that is based on continuoustime quantum walks. More specifically, we characterise the structure of a graph using its average mixing matrix. The average mixing matrix is a doubly-stochastic matrix that encodes the time-averaged behaviour of a continuous-time quantum walk on the graph. We propose to use the rows of the average mixing matrix for increasing stopping times to develop a novel signature, the Average Mixing Matrix Signature (AMMS). We perform an extensive range of experiments and we show that the proposed signature is robust under structural perturbations of the original graphs and it outperforms both the HKS and WKS when used as a node descriptor in a graph matching task.