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The paper is concerned with testing normality in samples of curves and error curves estimated from functional regression models. We propose a general paradigm based on the application of multivariate normality tests to vectors of functional principal components scores. We examine finite sample performance of a number of such tests and select the best performing tests. We apply them to several extensively used functional data sets and determine which can be treated as normal, possibly after a suitable transformation. We also offer practical guidance on software implementations of all tests we study and develop large sample justification for tests based on sample skewness and kurtosis of functional principal component scores.

Modern network infrastructures are in a constant state of transformation, in large part due to the exponential growth of Internet of Things (IoT) devices. The unique properties of IoT-connected networks, such as heterogeneity and non-standardized protocol, have created critical security holes and network mismanagement. In this paper we propose a new measurement tool, Intrinsic Dimensionality (ID), to aid in analyzing and classifying network traffic. A proxy for dataset complexity, ID can be used to understand the network as a whole, aiding in tasks such as network management and provisioning. We use ID to evaluate several modern network datasets empirically. Showing that, for network and device-level data, generated using IoT methodologies, the ID of the data fits into a low dimensional representation. Additionally we explore network data complexity at the sample level using Local Intrinsic Dimensionality (LID) and propose a novel unsupervised intrusion detection technique, the Weighted Hamming LID Estimator. We show that the algortihm performs better on IoT network datasets than the Autoencoder, KNN, and Isolation Forests. Finally, we propose the use of synthetic data as an additional tool for both network data measurement as well as intrusion detection. Synthetically generated data can aid in building a more robust network dataset, while also helping in downstream tasks such as machine learning based intrusion detection models. We explore the effects of synthetic data on ID measurements, as well as its role in intrusion detection systems.