Fisher's Pioneering work on Discriminant Analysis and its Impact on AI

Fisher opened many new areas in Multivariate Analysis, and the one which we will consider is discriminant analysis. Several papers by Fisher and others followed from his seminal paper in 1936 where he coined the name discrimination function. Historically, his four papers on discriminant analysis during 1936-1940 connect to the contemporaneous pioneering work of Hotelling and Mahalanobis. We revisit the famous iris data which Fisher used in his 1936 paper and in particular, test the hypothesis of multivariate normality for the data which he assumed. Fisher constructed his genetic discriminant motivated by this application and we provide a deeper insight into this construction; however, this construction has not been well understood as far as we know. We also indicate how the subject has developed along with the computer revolution, noting newer methods to carry out discriminant analysis, such as kernel classifiers, classification trees, support vector machines, neural networks, and deep learning. Overall, with computational power, the whole subject of Multivariate Analysis has changed its emphasis but the impact of this Fisher's pioneering work continues as an integral part of supervised learning in Artificial Intelligence.Comment: 33 pages 17Figure

A Fast Algorithm for Sampling from the Posterior of a von Mises distribution

Motivated by molecular biology, there has been an upsurge of research activities in directional statistics in general and its Bayesian aspect in particular. The central distribution for the circular case is von Mises distribution which has two parameters (mean and concentration) akin to the univariate normal distribution. However, there has been a challenge to sample efficiently from the posterior distribution of the concentration parameter. We describe a novel, highly efficient algorithm to sample from the posterior distribution and fill this long-standing gap

Some Fundamental Properties of a Multivariate von Mises Distribution

In application areas like bioinformatics multivariate distributions on angles are encountered which show significant clustering. One approach to statistical modelling of such situations is to use mixtures of unimodal distributions. In the literature (Mardia et al., 2011), the multivariate von Mises distribution, also known as the multivariate sine distribution, has been suggested for components of such models, but work in the area has been hampered by the fact that no good criteria for the von Mises distribution to be unimodal were available. In this article we study the question about when a multivariate von Mises distribution is unimodal. We give sufficient criteria for this to be the case and show examples of distributions with multiple modes when these criteria are violated. In addition, we propose a method to generate samples from the von Mises distribution in the case of high concentration.Comment: fixed a typo in the article title, minor fixes throughou

Recent Trends in Modelling Spatio-Temporal Data

Il lavoro fornisce una disamina delle pi`u recenti metodologie proposte nell’ambito dei modelli spazio-temporali. Nel tentativo di proporre una visione unificata delle metodologie trattate, viene fornita prima una descrizione dei vari tipi di dati spazio-temporali. Successivamente, si procede con la discussione dei modelli per processi spazialmente continui. La modellistica spazio-temporale `e stata largamente utilizzata per affrontare problemi in ambito ambientale, geostatistico, idrologico e meteorologico. Questo articolo fornisce una analisi dei metodi correntemente applicati in molte di queste aree

Exploring Spatio Temporal Variability by Eigen-Decomposition Techniques

Lo studio della variabilità di fenomeni ambientali può essere realizzato utilizzando diverse metodologie. In questo lavoro si propone una visione unificata di una serie di tecniche multivariate che risultano particolarmente utili per l’identificazione del segnale oggetto di interesse. Facendo riferimento a processi spazialmente continui, le tecniche proposte vengono presentate nel contesto della Decomposizione Generalizzata agli Autovalori. Nonostante la metodologia considerata risulta utile per fini esplorativi, il lavoro ne propone un utilizzo anche a fini predittivi