Conformal Prediction offers a powerful framework for quantifying uncertainty in machine learning models, enabling the construction of prediction sets with finite-sample validity guarantees. While easily adaptable to non-probabilistic models, applying conformal prediction to probabilistic generative models, such as Normalising Flows is not straightforward. This work proposes a novel method to conformalise conditional normalising flows, specifically addressing the problem of obtaining prediction regions for multi-step time series forecasting. Our approach leverages the flexibility of normalising flows to generate potentially disjoint prediction regions, leading to improved predictive efficiency in the presence of potential multimodal predictive distributions.

Machine Learning algorithms are notorious for providing point predictions but not prediction intervals. There are many applications where one requires confidence in predictions and prediction intervals. Stringing together, these intervals give rise to joint prediction regions with the desired significance level. It is an easy task to compute Joint Prediction regions (JPR) when the data is IID. However, the task becomes overly difficult when JPR is needed for time series because of the dependence between the observations. This project aims to implement Wolf and Wunderli's method for constructing JPRs and compare it with other methods (e.g. NP heuristic, Joint Marginals). The method under study is based on bootstrapping and is applied to different datasets (Min Temp, Sunspots), using different predictors (e.g. ARIMA and LSTM). One challenge of applying the method under study is to derive prediction standard errors for models, it cannot be obtained analytically. A novel method to estimate prediction standard error for different predictors is also devised. Finally, the method is applied to a synthetic dataset to find empirical averages and empirical widths and the results from the Wolf and Wunderli paper are consolidated. The experimental results show a narrowing of width with strong predictors like neural nets, widening of width with increasing forecast horizon H and decreasing significance level alpha, controlling the width with parameter k in K-FWE, and loss of information using Joint Marginals.

Normalising flows are statistical models that transform a complex density into a simpler density through the use of bijective transformations enabling both density estimation and data generation from a single model. In the context of image modelling, the predominant choice has been the Glow-based architecture, whereas alternative architectures remain largely unexplored in the research community. In this work, we propose a novel architecture called MixerFlow, based on the MLP-Mixer architecture, further unifying the generative and discriminative modelling architectures. MixerFlow offers an effective mechanism for weight sharing for flow-based models. Our results demonstrate better density estimation on image datasets under a fixed computational budget and scales well as the image resolution increases, making MixeFlow a powerful yet simple alternative to the Glow-based architectures. We also show that MixerFlow provides more informative embeddings than Glow-based architectures.

Normalising Flows are non-parametric statistical models characterised by their dual capabilities of density estimation and generation. This duality requires an inherently invertible architecture. However, the requirement of invertibility imposes constraints on their expressiveness, necessitating a large number of parameters and innovative architectural designs to achieve good results. Whilst flow-based models predominantly rely on neural-network-based transformations for expressive designs, alternative transformation methods have received limited attention. In this work, we present Ferumal flow, a novel kernelised normalising flow paradigm that integrates kernels into the framework. Our results demonstrate that a kernelised flow can yield competitive or superior results compared to neural network-based flows whilst maintaining parameter efficiency. Kernelised flows excel especially in the low-data regime, enabling flexible non-parametric density estimation in applications with sparse data availability.