Geoffrey I. Webb

Zahra Zamanzadeh Darban, Geoffrey I. Webb, Shirui Pan et al.

Time series anomaly detection has applications in a wide range of research fields and applications, including manufacturing and healthcare. The presence of anomalies can indicate novel or unexpected events, such as production faults, system defects, or heart fluttering, and is therefore of particular interest. The large size and complex patterns of time series have led researchers to develop specialised deep learning models for detecting anomalous patterns. This survey focuses on providing structured and comprehensive state-of-the-art time series anomaly detection models through the use of deep learning. It providing a taxonomy based on the factors that divide anomaly detection models into different categories. Aside from describing the basic anomaly detection technique for each category, the advantages and limitations are also discussed. Furthermore, this study includes examples of deep anomaly detection in time series across various application domains in recent years. It finally summarises open issues in research and challenges faced while adopting deep anomaly detection models.

Ming Jin, Huan Yee Koh, Qingsong Wen et al.

Time series are the primary data type used to record dynamic system measurements and generated in great volume by both physical sensors and online processes (virtual sensors). Time series analytics is therefore crucial to unlocking the wealth of information implicit in available data. With the recent advancements in graph neural networks (GNNs), there has been a surge in GNN-based approaches for time series analysis. These approaches can explicitly model inter-temporal and inter-variable relationships, which traditional and other deep neural network-based methods struggle to do. In this survey, we provide a comprehensive review of graph neural networks for time series analysis (GNN4TS), encompassing four fundamental dimensions: forecasting, classification, anomaly detection, and imputation. Our aim is to guide designers and practitioners to understand, build applications, and advance research of GNN4TS. At first, we provide a comprehensive task-oriented taxonomy of GNN4TS. Then, we present and discuss representative research works and introduce mainstream applications of GNN4TS. A comprehensive discussion of potential future research directions completes the survey. This survey, for the first time, brings together a vast array of knowledge on GNN-based time series research, highlighting foundations, practical applications, and opportunities of graph neural networks for time series analysis.

Angus Dempster, Daniel F. Schmidt, Geoffrey I. Webb

We demonstrate a simple connection between dictionary methods for time series classification, which involve extracting and counting symbolic patterns in time series, and methods based on transforming input time series using convolutional kernels, namely ROCKET and its variants. We show that by adjusting a single hyperparameter it is possible to move by degrees between models resembling dictionary methods and models resembling ROCKET. We present HYDRA, a simple, fast, and accurate dictionary method for time series classification using competing convolutional kernels, combining key aspects of both ROCKET and conventional dictionary methods. HYDRA is faster and more accurate than the most accurate existing dictionary methods, and can be combined with ROCKET and its variants to further improve the accuracy of these methods.

Zahra Zamanzadeh Darban, Geoffrey I. Webb, Shirui Pan et al.

One main challenge in time series anomaly detection (TSAD) is the lack of labelled data in many real-life scenarios. Most of the existing anomaly detection methods focus on learning the normal behaviour of unlabelled time series in an unsupervised manner. The normal boundary is often defined tightly, resulting in slight deviations being classified as anomalies, consequently leading to a high false positive rate and a limited ability to generalise normal patterns. To address this, we introduce a novel end-to-end self-supervised ContrAstive Representation Learning approach for time series Anomaly detection (CARLA). While existing contrastive learning methods assume that augmented time series windows are positive samples and temporally distant windows are negative samples, we argue that these assumptions are limited as augmentation of time series can transform them to negative samples, and a temporally distant window can represent a positive sample. Our contrastive approach leverages existing generic knowledge about time series anomalies and injects various types of anomalies as negative samples. Therefore, CARLA not only learns normal behaviour but also learns deviations indicating anomalies. It creates similar representations for temporally closed windows and distinct ones for anomalies. Additionally, it leverages the information about representations' neighbours through a self-supervised approach to classify windows based on their nearest/furthest neighbours to further enhance the performance of anomaly detection. In extensive tests on seven major real-world time series anomaly detection datasets, CARLA shows superior performance over state-of-the-art self-supervised and unsupervised TSAD methods. Our research shows the potential of contrastive representation learning to advance time series anomaly detection.

Angus Dempster, Daniel F. Schmidt, Geoffrey I. Webb

We show that it is possible to achieve the same accuracy, on average, as the most accurate existing interval methods for time series classification on a standard set of benchmark datasets using a single type of feature (quantiles), fixed intervals, and an 'off the shelf' classifier. This distillation of interval-based approaches represents a fast and accurate method for time series classification, achieving state-of-the-art accuracy on the expanded set of 142 datasets in the UCR archive with a total compute time (training and inference) of less than 15 minutes using a single CPU core.

Navid Mohammadi Foumani, Lynn Miller, Chang Wei Tan et al.

Time Series Classification and Extrinsic Regression are important and challenging machine learning tasks. Deep learning has revolutionized natural language processing and computer vision and holds great promise in other fields such as time series analysis where the relevant features must often be abstracted from the raw data but are not known a priori. This paper surveys the current state of the art in the fast-moving field of deep learning for time series classification and extrinsic regression. We review different network architectures and training methods used for these tasks and discuss the challenges and opportunities when applying deep learning to time series data. We also summarize two critical applications of time series classification and extrinsic regression, human activity recognition and satellite earth observation.

Matthieu Herrmann, Chang Wei Tan, Geoffrey I. Webb

Dynamic Time Warping (DTW) is a popular time series distance measure that aligns the points in two series with one another. These alignments support warping of the time dimension to allow for processes that unfold at differing rates. The distance is the minimum sum of costs of the resulting alignments over any allowable warping of the time dimension. The cost of an alignment of two points is a function of the difference in the values of those points. The original cost function was the absolute value of this difference. Other cost functions have been proposed. A popular alternative is the square of the difference. However, to our knowledge, this is the first investigation of both the relative impacts of using different cost functions and the potential to tune cost functions to different tasks. We do so in this paper by using a tunable cost function λγ with parameter γ. We show that higher values of γ place greater weight on larger pairwise differences, while lower values place greater weight on smaller pairwise differences. We demonstrate that training γ significantly improves the accuracy of both the DTW nearest neighbor and Proximity Forest classifiers.

Matthieu Herrmann, Chang Wei Tan, Mahsa Salehi et al.

Time series classification (TSC) is a challenging task due to the diversity of types of feature that may be relevant for different classification tasks, including trends, variance, frequency, magnitude, and various patterns. To address this challenge, several alternative classes of approach have been developed, including similarity-based, features and intervals, shapelets, dictionary, kernel, neural network, and hybrid approaches. While kernel, neural network, and hybrid approaches perform well overall, some specialized approaches are better suited for specific tasks. In this paper, we propose a new similarity-based classifier, Proximity Forest version 2.0 (PF 2.0), which outperforms previous state-of-the-art similarity-based classifiers across the UCR benchmark and outperforms state-of-the-art kernel, neural network, and hybrid methods on specific datasets in the benchmark that are best addressed by similarity-base methods. PF 2.0 incorporates three recent advances in time series similarity measures -- (1) computationally efficient early abandoning and pruning to speedup elastic similarity computations; (2) a new elastic similarity measure, Amerced Dynamic Time Warping (ADTW); and (3) cost function tuning. It rationalizes the set of similarity measures employed, reducing the eight base measures of the original PF to three and using the first derivative transform with all similarity measures, rather than a limited subset. We have implemented both PF 1.0 and PF 2.0 in a single C++ framework, making the PF framework more efficient.

Yizhen Zheng, Huan Yee Koh, Maddie Yang et al.

The integration of Large Language Models (LLMs) into the drug discovery and development field marks a significant paradigm shift, offering novel methodologies for understanding disease mechanisms, facilitating drug discovery, and optimizing clinical trial processes. This review highlights the expanding role of LLMs in revolutionizing various stages of the drug development pipeline. We investigate how these advanced computational models can uncover target-disease linkage, interpret complex biomedical data, enhance drug molecule design, predict drug efficacy and safety profiles, and facilitate clinical trial processes. Our paper aims to provide a comprehensive overview for researchers and practitioners in computational biology, pharmacology, and AI4Science by offering insights into the potential transformative impact of LLMs on drug discovery and development.

Yizhen Zheng, Huan Yee Koh, Jiaxin Ju et al.

Large language models are a form of artificial intelligence systems whose primary knowledge consists of the statistical patterns, semantic relationships, and syntactical structures of language1. Despite their limited forms of "knowledge", these systems are adept at numerous complex tasks including creative writing, storytelling, translation, question-answering, summarization, and computer code generation. However, they have yet to demonstrate advanced applications in natural science. Here we show how large language models can perform scientific synthesis, inference, and explanation. We present a method for using general-purpose large language models to make inferences from scientific datasets of the form usually associated with special-purpose machine learning algorithms. We show that the large language model can augment this "knowledge" by synthesizing from the scientific literature. When a conventional machine learning system is augmented with this synthesized and inferred knowledge it can outperform the current state of the art across a range of benchmark tasks for predicting molecular properties. This approach has the further advantage that the large language model can explain the machine learning system's predictions. We anticipate that our framework will open new avenues for AI to accelerate the pace of scientific discovery.

Ali Ismail-Fawaz, Hassan Ismail Fawaz, François Petitjean et al.

Time series data can be found in almost every domain, ranging from the medical field to manufacturing and wireless communication. Generating realistic and useful exemplars and prototypes is a fundamental data analysis task. In this paper, we investigate a novel approach to generating realistic and useful exemplars and prototypes for time series data. Our approach uses a new form of time series average, the ShapeDTW Barycentric Average. We therefore turn our attention to accurately generating time series prototypes with a novel approach. The existing time series prototyping approaches rely on the Dynamic Time Warping (DTW) similarity measure such as DTW Barycentering Average (DBA) and SoftDBA. These last approaches suffer from a common problem of generating out-of-distribution artifacts in their prototypes. This is mostly caused by the DTW variant used and its incapability of detecting neighborhood similarities, instead it detects absolute similarities. Our proposed method, ShapeDBA, uses the ShapeDTW variant of DTW, that overcomes this issue. We chose time series clustering, a popular form of time series analysis to evaluate the outcome of ShapeDBA compared to the other prototyping approaches. Coupled with the k-means clustering algorithm, and evaluated on a total of 123 datasets from the UCR archive, our proposed averaging approach is able to achieve new state-of-the-art results in terms of Adjusted Rand Index.

Amr Abourayya, Jens Kleesiek, Kanishka Rao et al.

In many critical applications, sensitive data is inherently distributed and cannot be centralized due to privacy concerns. A wide range of federated learning approaches have been proposed to train models locally at each client without sharing their sensitive data, typically by exchanging model parameters, or probabilistic predictions (soft labels) on a public dataset or a combination of both. However, these methods still disclose private information and restrict local models to those that can be trained using gradient-based methods. We propose a federated co-training (FedCT) approach that improves privacy by sharing only definitive (hard) labels on a public unlabeled dataset. Clients use a consensus of these shared labels as pseudo-labels for local training. This federated co-training approach empirically enhances privacy without compromising model quality. In addition, it allows the use of local models that are not suitable for parameter aggregation in traditional federated learning, such as gradient-boosted decision trees, rule ensembles, and random forests. Furthermore, we observe that FedCT performs effectively in federated fine-tuning of large language models, where its pseudo-labeling mechanism is particularly beneficial. Empirical evaluations and theoretical analyses suggest its applicability across a range of federated learning scenarios.

Navid Mohammadi Foumani, Soheila Ghane, Nam Nguyen et al.

Foundation models for EEG analysis are still in their infancy, limited by two key challenges: (1) variability across datasets caused by differences in recording devices and configurations, and (2) the low signal-to-noise ratio (SNR) of EEG, where brain signals are often buried under artifacts and non-brain sources. To address these challenges, we present EEG-X, a device-agnostic and noise-robust foundation model for EEG representation learning. EEG-X introduces a novel location-based channel embedding that encodes spatial information and improves generalization across domains and tasks by allowing the model to handle varying channel numbers, combinations, and recording lengths. To enhance robustness against noise, EEG-X employs a noise-aware masking and reconstruction strategy in both raw and latent spaces. Unlike previous models that mask and reconstruct raw noisy EEG signals, EEG-X is trained to reconstruct denoised signals obtained through an artifact removal process, ensuring that the learned representations focus on neural activity rather than noise. To further enhance reconstruction-based pretraining, EEG-X introduces a dictionary-inspired convolutional transformation (DiCT) layer that projects signals into a structured feature space before computing reconstruction (MSE) loss, reducing noise sensitivity and capturing frequency- and shape-aware similarities. Experiments on datasets collected from diverse devices show that EEG-X outperforms state-of-the-art methods across multiple downstream EEG tasks and excels in cross-domain settings where pre-trained and downstream datasets differ in electrode layouts. The models and code are available at: https://github.com/Emotiv/EEG-X

Shiyuan Li, Yixin Liu, Qingfeng Chen et al.

Unsupervised graph representation learning (UGRL) based on graph neural networks (GNNs), has received increasing attention owing to its efficacy in handling graph-structured data. However, existing UGRL methods ideally assume that the node features are noise-free, which makes them fail to distinguish between useful information and noise when applied to real data with noisy features, thus affecting the quality of learned representations. This urges us to take node noisy features into account in real-world UGRL. With empirical analysis, we reveal that feature propagation, the essential operation in GNNs, acts as a "double-edged sword" in handling noisy features - it can both denoise and diffuse noise, leading to varying feature quality across nodes, even within the same node at different hops. Building on this insight, we propose a novel UGRL method based on Multi-hop feature Quality Estimation (MQE for short). Unlike most UGRL models that directly utilize propagation-based GNNs to generate representations, our approach aims to learn representations through estimating the quality of propagated features at different hops. Specifically, we introduce a Gaussian model that utilizes a learnable "meta-representation" as a condition to estimate the expectation and variance of multi-hop propagated features via neural networks. In this way, the "meta representation" captures the semantic and structural information underlying multiple propagated features but is naturally less susceptible to interference by noise, thereby serving as high-quality node representations beneficial for downstream tasks. Extensive experiments on multiple real-world datasets demonstrate that MQE in learning reliable node representations in scenarios with diverse types of feature noise.

Loong Kuan Lee, Geoffrey I. Webb, Daniel F. Schmidt et al.

The ability to compute the exact divergence between two high-dimensional distributions is useful in many applications but doing so naively is intractable. Computing the alpha-beta divergence -- a family of divergences that includes the Kullback-Leibler divergence and Hellinger distance -- between the joint distribution of two decomposable models, i.e chordal Markov networks, can be done in time exponential in the treewidth of these models. However, reducing the dissimilarity between two high-dimensional objects to a single scalar value can be uninformative. Furthermore, in applications such as supervised learning, the divergence over a conditional distribution might be of more interest. Therefore, we propose an approach to compute the exact alpha-beta divergence between any marginal or conditional distribution of two decomposable models. Doing so tractably is non-trivial as we need to decompose the divergence between these distributions and therefore, require a decomposition over the marginal and conditional distributions of these models. Consequently, we provide such a decomposition and also extend existing work to compute the marginal and conditional alpha-beta divergence between these decompositions. We then show how our method can be used to analyze distributional changes by first applying it to a benchmark image dataset. Finally, based on our framework, we propose a novel way to quantify the error in contemporary superconducting quantum computers. Code for all experiments is available at: https://lklee.dev/pub/2023-icdm/code

Rakshitha Godahewa, Geoffrey I. Webb, Daniel Schmidt et al.

Threshold Autoregressive (TAR) models have been widely used by statisticians for non-linear time series forecasting during the past few decades, due to their simplicity and mathematical properties. On the other hand, in the forecasting community, general-purpose tree-based regression algorithms (forests, gradient-boosting) have become popular recently due to their ease of use and accuracy. In this paper, we explore the close connections between TAR models and regression trees. These enable us to use the rich methodology from the literature on TAR models to define a hierarchical TAR model as a regression tree that trains globally across series, which we call SETAR-Tree. In contrast to the general-purpose tree-based models that do not primarily focus on forecasting, and calculate averages at the leaf nodes, we introduce a new forecasting-specific tree algorithm that trains global Pooled Regression (PR) models in the leaves allowing the models to learn cross-series information and also uses some time-series-specific splitting and stopping procedures. The depth of the tree is controlled by conducting a statistical linearity test commonly employed in TAR models, as well as measuring the error reduction percentage at each node split. Thus, the proposed tree model requires minimal external hyperparameter tuning and provides competitive results under its default configuration. We also use this tree algorithm to develop a forest where the forecasts provided by a collection of diverse SETAR-Trees are combined during the forecasting process. In our evaluation on eight publicly available datasets, the proposed tree and forest models are able to achieve significantly higher accuracy than a set of state-of-the-art tree-based algorithms and forecasting benchmarks across four evaluation metrics.

Navid Mohammadi Foumani, Chang Wei Tan, Geoffrey I. Webb et al.

We argue that time series analysis is fundamentally different in nature to either vision or natural language processing with respect to the forms of meaningful self-supervised learning tasks that can be defined. Motivated by this insight, we introduce a novel approach called \textit{Series2Vec} for self-supervised representation learning. Unlike other self-supervised methods in time series, which carry the risk of positive sample variants being less similar to the anchor sample than series in the negative set, Series2Vec is trained to predict the similarity between two series in both temporal and spectral domains through a self-supervised task. Series2Vec relies primarily on the consistency of the unsupervised similarity step, rather than the intrinsic quality of the similarity measurement, without the need for hand-crafted data augmentation. To further enforce the network to learn similar representations for similar time series, we propose a novel approach that applies order-invariant attention to each representation within the batch during training. Our evaluation of Series2Vec on nine large real-world datasets, along with the UCR/UEA archive, shows enhanced performance compared to current state-of-the-art self-supervised techniques for time series. Additionally, our extensive experiments show that Series2Vec performs comparably with fully supervised training and offers high efficiency in datasets with limited-labeled data. Finally, we show that the fusion of Series2Vec with other representation learning models leads to enhanced performance for time series classification. Code and models are open-source at \url{https://github.com/Navidfoumani/Series2Vec.}

Zahra Zamanzadeh Darban, Qizhou Wang, Charu C. Aggarwal et al.

Supervised anomaly detection methods perform well in identifying known anomalies that are well represented in the training set. However, they often struggle to generalise beyond the training distribution due to decision boundaries that lack a clear definition of normality. Existing approaches typically address this by regularising the representation space during training, leading to separate optimisation in latent and label spaces. The learned normality is therefore not directly utilised at inference, and their anomaly scores often fall within arbitrary ranges that require explicit mapping or calibration for probabilistic interpretation. To achieve unified learning of geometric normality and label discrimination, we propose Centre-Enhanced Discriminative Learning (CEDL), a novel supervised anomaly detection framework that embeds geometric normality directly into the discriminative objective. CEDL reparameterises the conventional sigmoid-derived prediction logit through a centre-based radial distance function, unifying geometric and discriminative learning in a single end-to-end formulation. This design enables interpretable, geometry-aware anomaly scoring without post-hoc thresholding or reference calibration. Extensive experiments on tabular, time-series, and image data demonstrate that CEDL achieves competitive and balanced performance across diverse real-world anomaly detection tasks, validating its effectiveness and broad applicability.

Qingfeng Chen, Shiyuan Li, Yixin Liu et al.

Graph neural networks (GNNs) excel in graph representation learning by integrating graph structure and node features. Existing GNNs, unfortunately, fail to account for the uncertainty of class probabilities that vary with the depth of the model, leading to unreliable and risky predictions in real-world scenarios. To bridge the gap, in this paper, we propose a novel Evidence Fusing Graph Neural Network (EFGNN for short) to achieve trustworthy prediction, enhance node classification accuracy, and make explicit the risk of wrong predictions. In particular, we integrate the evidence theory with multi-hop propagation-based GNN architecture to quantify the prediction uncertainty of each node with the consideration of multiple receptive fields. Moreover, a parameter-free cumulative belief fusion (CBF) mechanism is developed to leverage the changes in prediction uncertainty and fuse the evidence to improve the trustworthiness of the final prediction. To effectively optimize the EFGNN model, we carefully design a joint learning objective composed of evidence cross-entropy, dissonance coefficient, and false confident penalty. The experimental results on various datasets and theoretical analyses demonstrate the effectiveness of the proposed model in terms of accuracy and trustworthiness, as well as its robustness to potential attacks. The source code of EFGNN is available at https://github.com/Shiy-Li/EFGNN.

Navid Mohammadi Foumani, Chang Wei Tan, Geoffrey I. Webb et al.

Transformers have demonstrated outstanding performance in many applications of deep learning. When applied to time series data, transformers require effective position encoding to capture the ordering of the time series data. The efficacy of position encoding in time series analysis is not well-studied and remains controversial, e.g., whether it is better to inject absolute position encoding or relative position encoding, or a combination of them. In order to clarify this, we first review existing absolute and relative position encoding methods when applied in time series classification. We then proposed a new absolute position encoding method dedicated to time series data called time Absolute Position Encoding (tAPE). Our new method incorporates the series length and input embedding dimension in absolute position encoding. Additionally, we propose computationally Efficient implementation of Relative Position Encoding (eRPE) to improve generalisability for time series. We then propose a novel multivariate time series classification (MTSC) model combining tAPE/eRPE and convolution-based input encoding named ConvTran to improve the position and data embedding of time series data. The proposed absolute and relative position encoding methods are simple and efficient. They can be easily integrated into transformer blocks and used for downstream tasks such as forecasting, extrinsic regression, and anomaly detection. Extensive experiments on 32 multivariate time-series datasets show that our model is significantly more accurate than state-of-the-art convolution and transformer-based models. Code and models are open-sourced at \url{https://github.com/Navidfoumani/ConvTran}.

Lynn Miller, Charlotte Pelletier, Geoffrey I. Webb

Earth observation (EO) satellite missions have been providing detailed images about the state of the Earth and its land cover for over 50 years. Long term missions, such as NASA's Landsat, Terra, and Aqua satellites, and more recently, the ESA's Sentinel missions, record images of the entire world every few days. Although single images provide point-in-time data, repeated images of the same area, or satellite image time series (SITS) provide information about the changing state of vegetation and land use. These SITS are useful for modeling dynamic processes and seasonal changes such as plant phenology. They have potential benefits for many aspects of land and natural resource management, including applications in agricultural, forest, water, and disaster management, urban planning, and mining. However, the resulting satellite image time series (SITS) are complex, incorporating information from the temporal, spatial, and spectral dimensions. Therefore, deep learning methods are often deployed as they can analyze these complex relationships. This review presents a summary of the state-of-the-art methods of modelling environmental, agricultural, and other Earth observation variables from SITS data using deep learning methods. We aim to provide a resource for remote sensing experts interested in using deep learning techniques to enhance Earth observation models with temporal information.

Zahra Zamanzadeh Darban, Yiyuan Yang, Geoffrey I. Webb et al.

In time series anomaly detection (TSAD), the scarcity of labeled data poses a challenge to the development of accurate models. Unsupervised domain adaptation (UDA) offers a solution by leveraging labeled data from a related domain to detect anomalies in an unlabeled target domain. However, existing UDA methods assume consistent anomalous classes across domains. To address this limitation, we propose a novel Domain Adaptation Contrastive learning model for Anomaly Detection in multivariate time series (DACAD), combining UDA with contrastive learning. DACAD utilizes an anomaly injection mechanism that enhances generalization across unseen anomalous classes, improving adaptability and robustness. Additionally, our model employs supervised contrastive loss for the source domain and self-supervised contrastive triplet loss for the target domain, ensuring comprehensive feature representation learning and domain-invariant feature extraction. Finally, an effective Center-based Entropy Classifier (CEC) accurately learns normal boundaries in the source domain. Extensive evaluations on multiple real-world datasets and a synthetic dataset highlight DACAD's superior performance in transferring knowledge across domains and mitigating the challenge of limited labeled data in TSAD.

Zahra Zamanzadeh Darban, Qizhou Wang, Geoffrey I. Webb et al.

A recent and promising approach for building time series anomaly detection (TSAD) models is to inject synthetic samples of anomalies within real data sets. The existing injection mechanisms have significant limitations - most of them rely on ad hoc, hand-crafted strategies which fail to capture the natural diversity of anomalous patterns, or are restricted to univariate time series settings. To address these challenges, we design a generative model for TSAD using a variational autoencoder, which is referred to as a Generator for Instantiating Anomalies in Time Series (GenIAS). GenIAS is designed to produce diverse and realistic synthetic anomalies for TSAD tasks. By employing a novel learned perturbation mechanism in the latent space and injecting the perturbed patterns in different segments of time series, GenIAS can generate anomalies with greater diversity and varying scales. Further, guided by a new triplet loss function, which uses a min-max margin and a new variance-scaling approach to further enforce the learning of compact normal patterns, GenIAS ensures that anomalies are distinct from normal samples while remaining realistic. The approach is effective for both univariate and multivariate time series. We demonstrate the diversity and realism of the generated anomalies. Our extensive experiments demonstrate that GenIAS - when integrated into a TSAD task - consistently outperforms seventeen traditional and deep anomaly detection models, thereby highlighting the potential of generative models for time series anomaly generation.

Shahrzad Behzadimanesh, Pierre Le Bodic, Geoffrey I. Webb et al.

Small additive ensembles of symbolic rules offer interpretable prediction models. Traditionally, these ensembles use rule conditions based on conjunctions of simple threshold propositions $x \geq t$ on a single input variable $x$ and threshold $t$, resulting geometrically in axis-parallel polytopes as decision regions. While this form ensures a high degree of interpretability for individual rules and can be learned efficiently using the gradient boosting approach, it relies on having access to a curated set of expressive and ideally independent input features so that a small ensemble of axis-parallel regions can describe the target variable well. Absent such features, reaching sufficient accuracy requires increasing the number and complexity of individual rules, which diminishes the interpretability of the model. Here, we extend classical rule ensembles by introducing logical propositions with learnable sparse linear transformations of input variables, i.e., propositions of the form $\mathbf{x}^\mathrm{T}\mathbf{w} \geq t$, where $\mathbf{w}$ is a learnable sparse weight vector, enabling decision regions as general polytopes with oblique faces. We propose a learning method using sequential greedy optimization based on an iteratively reweighted formulation of logistic regression. Experimental results demonstrate that the proposed method efficiently constructs rule ensembles with the same test risk as state-of-the-art methods while significantly reducing model complexity across ten benchmark datasets.

Connor Cooper, Geoffrey I. Webb, Daniel F. Schmidt

Bayesian network classifiers (BNCs) possess a number of properties desirable for a modern classifier: They are easily interpretable, highly scalable, and offer adaptable complexity. However, traditional methods for learning BNCs have historically underperformed when compared to leading classification methods such as random forests. Recent parameter smoothing techniques using hierarchical Dirichlet processes (HDPs) have enabled BNCs to achieve performance competitive with random forests on categorical data, but these techniques are relatively inflexible, and require a complicated, specialized sampling process. In this paper, we introduce a novel method for parameter estimation that uses a log-linear regression to approximate the behaviour of HDPs. As a linear model, our method is remarkably flexible and simple to interpret, and can leverage the vast literature on learning linear models. Our experiments show that our method can outperform HDP smoothing while being orders of magnitude faster, remaining competitive with random forests on categorical data.

Angus Dempster, Navid Mohammadi Foumani, Chang Wei Tan et al.

We introduce MONSTER-the MONash Scalable Time Series Evaluation Repository-a collection of large datasets for time series classification. The field of time series classification has benefitted from common benchmarks set by the UCR and UEA time series classification repositories. However, the datasets in these benchmarks are small, with median sizes of 217 and 255 examples, respectively. In consequence they favour a narrow subspace of models that are optimised to achieve low classification error on a wide variety of smaller datasets, that is, models that minimise variance, and give little weight to computational issues such as scalability. Our hope is to diversify the field by introducing benchmarks using larger datasets. We believe that there is enormous potential for new progress in the field by engaging with the theoretical and practical challenges of learning effectively from larger quantities of data.

Angus Dempster, Geoffrey I. Webb, Daniel F. Schmidt

Logistic regression is a ubiquitous method for probabilistic classification. However, the effectiveness of logistic regression depends upon careful and relatively computationally expensive tuning, especially for the regularisation hyperparameter, and especially in the context of high-dimensional data. We present a prevalidated ridge regression model that closely matches logistic regression in terms of classification error and log-loss, particularly for high-dimensional data, while being significantly more computationally efficient and having effectively no hyperparameters beyond regularisation. We scale the coefficients of the model so as to minimise log-loss for a set of prevalidated predictions derived from the estimated leave-one-out cross-validation error. This exploits quantities already computed in the course of fitting the ridge regression model in order to find the scaling parameter with nominal additional computational expense.

Ali Ismail-Fawaz, Angus Dempster, Chang Wei Tan et al.

The measurement of progress using benchmarks evaluations is ubiquitous in computer science and machine learning. However, common approaches to analyzing and presenting the results of benchmark comparisons of multiple algorithms over multiple datasets, such as the critical difference diagram introduced by Demšar (2006), have important shortcomings and, we show, are open to both inadvertent and intentional manipulation. To address these issues, we propose a new approach to presenting the results of benchmark comparisons, the Multiple Comparison Matrix (MCM), that prioritizes pairwise comparisons and precludes the means of manipulating experimental results in existing approaches. MCM can be used to show the results of an all-pairs comparison, or to show the results of a comparison between one or more selected algorithms and the state of the art. MCM is implemented in Python and is publicly available.

Loong Kuan Lee, Nico Piatkowski, François Petitjean et al.

There are many applications that benefit from computing the exact divergence between 2 discrete probability measures, including machine learning. Unfortunately, in the absence of any assumptions on the structure or independencies within these distributions, computing the divergence between them is an intractable problem in high dimensions. We show that we are able to compute a wide family of functionals and divergences, such as the alpha-beta divergence, between two decomposable models, i.e. chordal Markov networks, in time exponential to the treewidth of these models. The alpha-beta divergence is a family of divergences that include popular divergences such as the Kullback-Leibler divergence, the Hellinger distance, and the chi-squared divergence. Thus, we can accurately compute the exact values of any of this broad class of divergences to the extent to which we can accurately model the two distributions using decomposable models.

Matthieu Herrmann, Geoffrey I. Webb

Dynamic Time Warping (DTW), and its constrained (CDTW) and weighted (WDTW) variants, are time series distances with a wide range of applications. They minimize the cost of non-linear alignments between series. CDTW and WDTW have been introduced because DTW is too permissive in its alignments. However, CDTW uses a crude step function, allowing unconstrained flexibility within the window, and none beyond it. WDTW's multiplicative weight is relative to the distances between aligned points along a warped path, rather than being a direct function of the amount of warping that is introduced. In this paper, we introduce Amerced Dynamic Time Warping (ADTW), a new, intuitive, DTW variant that penalizes the act of warping by a fixed additive cost. Like CDTW and WDTW, ADTW constrains the amount of warping. However, it avoids both abrupt discontinuities in the amount of warping allowed and the limitations of a multiplicative penalty. We formally introduce ADTW, prove some of its properties, and discuss its parameterization. We show on a simple example how it can be parameterized to achieve an intuitive outcome, and demonstrate its usefulness on a standard time series classification benchmark. We provide a demonstration application in C++.

Rakshitha Godahewa, Christoph Bergmeir, Geoffrey I. Webb et al.

Many businesses and industries nowadays rely on large quantities of time series data making time series forecasting an important research area. Global forecasting models that are trained across sets of time series have shown a huge potential in providing accurate forecasts compared with the traditional univariate forecasting models that work on isolated series. However, there are currently no comprehensive time series archives for forecasting that contain datasets of time series from similar sources available for the research community to evaluate the performance of new global forecasting algorithms over a wide variety of datasets. In this paper, we present such a comprehensive time series forecasting archive containing 20 publicly available time series datasets from varied domains, with different characteristics in terms of frequency, series lengths, and inclusion of missing values. We also characterise the datasets, and identify similarities and differences among them, by conducting a feature analysis. Furthermore, we present the performance of a set of standard baseline forecasting methods over all datasets across eight error metrics, for the benefit of researchers using the archive to benchmark their forecasting algorithms.

Ahmed Shifaz, Charlotte Pelletier, Francois Petitjean et al.

This paper contributes multivariate versions of seven commonly used elastic similarity and distance measures for time series data analytics. Elastic similarity and distance measures are a class of similarity measures that can compensate for misalignments in the time axis of time series data. We adapt two existing strategies used in a multivariate version of the well-known Dynamic Time Warping (DTW), namely, Independent and Dependent DTW, to these seven measures. While these measures can be applied to various time series analysis tasks, we demonstrate their utility on multivariate time series classification using the nearest neighbor classifier. On 23 well-known datasets, we demonstrate that each of the measures but one achieves the highest accuracy relative to others on at least one dataset, supporting the value of developing a suite of multivariate similarity and distance measures. We also demonstrate that there are datasets for which either the dependent versions of all measures are more accurate than their independent counterparts or vice versa. In addition, we also construct a nearest neighbor-based ensemble of the measures and show that it is competitive to other state-of-the-art single-strategy multivariate time series classifiers.

Geoffrey I. Webb, Francois Petitjean

Dynamic Time Warping (DTW) is a popular similarity measure for aligning and comparing time series. Due to DTW's high computation time, lower bounds are often employed to screen poor matches. Many alternative lower bounds have been proposed, providing a range of different trade-offs between tightness and computational efficiency. LB Keogh provides a useful trade-off in many applications. Two recent lower bounds, LB Improved and LB Enhanced, are substantially tighter than LB Keogh. All three have the same worst case computational complexity - linear with respect to series length and constant with respect to window size. We present four new DTW lower bounds in the same complexity class. LB Petitjean is substantially tighter than LB Improved, with only modest additional computational overhead. LB Webb is more efficient than LB Improved, while often providing a tighter bound. LB Webb is always tighter than LB Keogh. The parameter free LB Webb is usually tighter than LB Enhanced. A parameterized variant, LB Webb Enhanced, is always tighter than LB Enhanced. A further variant, LB Webb*, is useful for some constrained distance functions. In extensive experiments, LB Webb proves to be very effective for nearest neighbor search.

Matthieu Herrmann, Geoffrey I. Webb

Nearest neighbor search under elastic distances is a key tool for time series analysis, supporting many applications. However, straightforward implementations of distances require $O(n^2)$ space and time complexities, preventing these applications from scaling to long series. Much work has been devoted to speeding up the NN search process, mostly with the development of lower bounds, allowing to avoid costly distance computations when a given threshold is exceeded. This threshold, provided by the similarity search process, also allows to early abandon the computation of a distance itself. Another approach, is to prune parts of the computation. All these techniques are othogonal to each other. In this work, we develop a new generic strategy, "EAPruned", that tightly integrates pruning with early abandoning. We apply it to six elastic distance measures: DTW, CDTW, WDTW, ERP, MSM and TWE, showing substantial speedup in NN search applications. Pruning alone also shows substantial speedup for some distances, benefiting applications beyond the scope of NN search (e.g. requiring all pairwise distances), and hence where early abandoning is not applicable. We~release our implementation as part of a new C++ library for time series classification, along with easy to use Python/Numpy bindings.

Chang Wei Tan, Angus Dempster, Christoph Bergmeir et al.

We propose MultiRocket, a fast time series classification (TSC) algorithm that achieves state-of-the-art performance with a tiny fraction of the time and without the complex ensembling structure of many state-of-the-art methods. MultiRocket improves on MiniRocket, one of the fastest TSC algorithms to date, by adding multiple pooling operators and transformations to improve the diversity of the features generated. In addition to processing the raw input series, MultiRocket also applies first order differences to transform the original series. Convolutions are applied to both representations, and four pooling operators are applied to the convolution outputs. When benchmarked using the University of California Riverside TSC benchmark datasets, MultiRocket is significantly more accurate than MiniRocket, and competitive with the best ranked current method in terms of accuracy, HIVE-COTE 2.0, while being orders of magnitude faster.

Mario Boley, Simon Teshuva, Pierre Le Bodic et al.

Rule ensembles are designed to provide a useful trade-off between predictive accuracy and model interpretability. However, the myopic and random search components of current rule ensemble methods can compromise this goal: they often need more rules than necessary to reach a certain accuracy level or can even outright fail to accurately model a distribution that can actually be described well with a few rules. Here, we present a novel approach aiming to fit rule ensembles of maximal predictive power for a given ensemble size (and thus model comprehensibility). In particular, we present an efficient branch-and-bound algorithm that optimally solves the per-rule objective function of the popular second-order gradient boosting framework. Our main insight is that the boosting objective can be tightly bounded in linear time of the number of covered data points. Along with an additional novel pruning technique related to rule redundancy, this leads to a computationally feasible approach for boosting optimal rules that, as we demonstrate on a wide range of common benchmark problems, consistently outperforms the predictive performance of boosting greedy rules.

Rakshitha Godahewa, Kasun Bandara, Geoffrey I. Webb et al.

With large quantities of data typically available nowadays, forecasting models that are trained across sets of time series, known as Global Forecasting Models (GFM), are regularly outperforming traditional univariate forecasting models that work on isolated series. As GFMs usually share the same set of parameters across all time series, they often have the problem of not being localised enough to a particular series, especially in situations where datasets are heterogeneous. We study how ensembling techniques can be used with generic GFMs and univariate models to solve this issue. Our work systematises and compares relevant current approaches, namely clustering series and training separate submodels per cluster, the so-called ensemble of specialists approach, and building heterogeneous ensembles of global and local models. We fill some gaps in the existing GFM localisation approaches, in particular by incorporating varied clustering techniques such as feature-based clustering, distance-based clustering and random clustering, and generalise them to use different underlying GFM model types. We then propose a new methodology of clustered ensembles where we train multiple GFMs on different clusters of series, obtained by changing the number of clusters and cluster seeds. Using Feed-forward Neural Networks, Recurrent Neural Networks, and Pooled Regression models as the underlying GFMs, in our evaluation on eight publicly available datasets, the proposed models are able to achieve significantly higher accuracy than baseline GFM models and univariate forecasting methods.

Yuan Jin, Wray Buntine, Francois Petitjean et al.

Supervised learning, characterized by both discriminative and generative learning, seeks to predict the values of single (or sometimes multiple) predefined target attributes based on a predefined set of predictor attributes. For applications where the information available and predictions to be made may vary from instance to instance, we propose the task of target-agnostic learning where arbitrary disjoint sets of attributes can be used for each of predictors and targets for each to-be-predicted instance. For this task, we survey a wide range of techniques available for handling missing values, self-supervised training and pseudo-likelihood training, and adapt them to a suite of algorithms that are suitable for the task. We conduct extensive experiments on this suite of algorithms on a large collection of categorical, continuous and discretized datasets, and report their performance in terms of both classification and regression errors. We also report the training and prediction time of these algorithms when handling large-scale datasets. Both generative and self-supervised learning models are shown to perform well at the task, although their characteristics towards the different types of data are quite different. Nevertheless, our derived theorem for the pseudo-likelihood theory also shows that they are related for inferring a joint distribution model based on the pseudo-likelihood training.

Chaitanya Manapragada, Heitor M Gomes, Mahsa Salehi et al.

Decision tree ensembles are widely used in practice. In this work, we study in ensemble settings the effectiveness of replacing the split strategy for the state-of-the-art online tree learner, Hoeffding Tree, with a rigorous but more eager splitting strategy that we had previously published as Hoeffding AnyTime Tree. Hoeffding AnyTime Tree (HATT), uses the Hoeffding Test to determine whether the current best candidate split is superior to the current split, with the possibility of revision, while Hoeffding Tree aims to determine whether the top candidate is better than the second best and if a test is selected, fixes it for all posterity. HATT converges to the ideal batch tree while Hoeffding Tree does not. We find that HATT is an efficacious base learner for online bagging and online boosting ensembles. On UCI and synthetic streams, HATT as a base learner outperforms HT within a 0.05 significance level for the majority of tested ensembles on what we believe is the largest and most comprehensive set of testbenches in the online learning literature. Our results indicate that HATT is a superior alternative to Hoeffding Tree in a large number of ensemble settings.

Chaitanya Manapragada, Geoffrey I Webb, Mahsa Salehi et al.

Hoeffding trees are the state-of-the-art methods in decision tree learning for evolving data streams. These very fast decision trees are used in many real applications where data is created in real-time due to their efficiency. In this work, we extricate explanations for why these streaming decision tree algorithms for stationary and nonstationary streams (HoeffdingTree and HoeffdingAdaptiveTree) work as well as they do. In doing so, we identify thirteen unique unspecified design decisions in both the theoretical constructs and their implementations with substantial and consequential effects on predictive accuracy---design decisions that, without necessarily changing the essence of the algorithms, drive algorithm performance. We begin a larger conversation about explainability not just of the model but also of the processes responsible for an algorithm's success.

Rakshitha Godahewa, Christoph Bergmeir, Geoffrey I. Webb et al.

Many businesses and industries require accurate forecasts for weekly time series nowadays. However, the forecasting literature does not currently provide easy-to-use, automatic, reproducible and accurate approaches dedicated to this task. We propose a forecasting method in this domain to fill this gap, leveraging state-of-the-art forecasting techniques, such as forecast combination, meta-learning, and global modelling. We consider different meta-learning architectures, algorithms, and base model pools. Based on all considered model variants, we propose to use a stacking approach with lasso regression which optimally combines the forecasts of four base models: a global Recurrent Neural Network model (RNN), Theta, Trigonometric Box-Cox ARMA Trend Seasonal (TBATS) and Dynamic Harmonic Regression ARIMA (DHR-ARIMA), as it shows the overall best performance across seven experimental weekly datasets on four evaluation metrics. Our proposed method also consistently outperforms a set of benchmarks and state-of-the-art weekly forecasting models by a considerable margin with statistical significance. Our method can produce the most accurate forecasts, in terms of mean sMAPE, for the M4 weekly dataset among all benchmarks and all original competition participants.

Matthieu Herrmann, Geoffrey I. Webb

The Dynamic Time Warping ("DTW") distance is widely used in time series analysis, be it for classification, clustering or similarity search. However, its quadratic time complexity prevents it from scaling. Strategies, based on early abandoning DTW or skipping its computation altogether thanks to lower bounds, have been developed for certain use cases such as nearest neighbour search. But vectorization and approximation aside, no advance was made on DTW itself until recently with the introduction of PrunedDTW. This algorithm, able to prune unpromising alignments, was later fitted with early abandoning. We present a new version of PrunedDTW, "EAPrunedDTW", designed with early abandon in mind from the start, and able to early abandon faster than before. We show that EAPrunedDTW significantly improves the computation time of similarity search in the UCR Suite, and renders lower bounds dispensable.

Chang Wei Tan, Christoph Bergmeir, Francois Petitjean et al.

This paper studies Time Series Extrinsic Regression (TSER): a regression task of which the aim is to learn the relationship between a time series and a continuous scalar variable; a task closely related to time series classification (TSC), which aims to learn the relationship between a time series and a categorical class label. This task generalizes time series forecasting (TSF), relaxing the requirement that the value predicted be a future value of the input series or primarily depend on more recent values. In this paper, we motivate and study this task, and benchmark existing solutions and adaptations of TSC algorithms on a novel archive of 19 TSER datasets which we have assembled. Our results show that the state-of-the-art TSC algorithm Rocket, when adapted for regression, achieves the highest overall accuracy compared to adaptations of other TSC algorithms and state-of-the-art machine learning (ML) algorithms such as XGBoost, Random Forest and Support Vector Regression. More importantly, we show that much research is needed in this field to improve the accuracy of ML models. We also find evidence that further research has excellent prospects of improving upon these straightforward baselines.

Chang Wei Tan, Christoph Bergmeir, Francois Petitjean et al.

Time series research has gathered lots of interests in the last decade, especially for Time Series Classification (TSC) and Time Series Forecasting (TSF). Research in TSC has greatly benefited from the University of California Riverside and University of East Anglia (UCR/UEA) Time Series Archives. On the other hand, the advancement in Time Series Forecasting relies on time series forecasting competitions such as the Makridakis competitions, NN3 and NN5 Neural Network competitions, and a few Kaggle competitions. Each year, thousands of papers proposing new algorithms for TSC and TSF have utilized these benchmarking archives. These algorithms are designed for these specific problems, but may not be useful for tasks such as predicting the heart rate of a person using photoplethysmogram (PPG) and accelerometer data. We refer to this problem as Time Series Extrinsic Regression (TSER), where we are interested in a more general methodology of predicting a single continuous value, from univariate or multivariate time series. This prediction can be from the same time series or not directly related to the predictor time series and does not necessarily need to be a future value or depend heavily on recent values. To the best of our knowledge, research into TSER has received much less attention in the time series research community and there are no models developed for general time series extrinsic regression problems. Most models are developed for a specific problem. Therefore, we aim to motivate and support the research into TSER by introducing the first TSER benchmarking archive. This archive contains 19 datasets from different domains, with varying number of dimensions, unequal length dimensions, and missing values. In this paper, we introduce the datasets in this archive and did an initial benchmark on existing models.

Benjamin Lucas, Charlotte Pelletier, Daniel Schmidt et al.

Land cover maps are a vital input variable to many types of environmental research and management. While they can be produced automatically by machine learning techniques, these techniques require substantial training data to achieve high levels of accuracy, which are not always available. One technique researchers use when labelled training data are scarce is domain adaptation (DA) -- where data from an alternate region, known as the source domain, are used to train a classifier and this model is adapted to map the study region, or target domain. The scenario we address in this paper is known as semi-supervised DA, where some labelled samples are available in the target domain. In this paper we present Sourcerer, a Bayesian-inspired, deep learning-based, semi-supervised DA technique for producing land cover maps from SITS data. The technique takes a convolutional neural network trained on a source domain and then trains further on the available target domain with a novel regularizer applied to the model weights. The regularizer adjusts the degree to which the model is modified to fit the target data, limiting the degree of change when the target data are few in number and increasing it as target data quantity increases. Our experiments on Sentinel-2 time series images compare Sourcerer with two state-of-the-art semi-supervised domain adaptation techniques and four baseline models. We show that on two different source-target domain pairings Sourcerer outperforms all other methods for any quantity of labelled target data available. In fact, the results on the more difficult target domain show that the starting accuracy of Sourcerer (when no labelled target data are available), 74.2%, is greater than the next-best state-of-the-art method trained on 20,000 labelled target instances.

Chang Wei Tan, Francois Petitjean, Eamonn Keogh et al.

Research into time series classification has tended to focus on the case of series of uniform length. However, it is common for real-world time series data to have unequal lengths. Differing time series lengths may arise from a number of fundamentally different mechanisms. In this work, we identify and evaluate two classes of such mechanisms -- variations in sampling rate relative to the relevant signal and variations between the start and end points of one time series relative to one another. We investigate how time series generated by each of these classes of mechanism are best addressed for time series classification. We perform extensive experiments and provide practical recommendations on how variations in length should be handled in time series classification.

Charlotte Pelletier, Geoffrey I. Webb, Francois Petitjean

New remote sensing sensors now acquire high spatial and spectral Satellite Image Time Series (SITS) of the world. These series of images are a key component of classification systems that aim at obtaining up-to-date and accurate land cover maps of the Earth's surfaces. More specifically, the combination of the temporal, spectral and spatial resolutions of new SITS makes possible to monitor vegetation dynamics. Although traditional classification algorithms, such as Random Forest (RF), have been successfully applied for SITS classification, these algorithms do not make the most of the temporal domain. Conversely, some approaches that take into account the temporal dimension have recently been tested, especially Recurrent Neural Networks (RNNs). This paper proposes an exhaustive study of another deep learning approaches, namely Temporal Convolutional Neural Networks (TempCNNs) where convolutions are applied in the temporal dimension. The goal is to quantitatively and qualitatively evaluate the contribution of TempCNNs for SITS classification. This paper proposes a set of experiments performed on one million time series extracted from 46 Formosat-2 images. The experimental results show that TempCNNs are more accurate than RF and RNNs, that are the current state of the art for SITS classification. We also highlight some differences with results obtained in computer vision, e.g. about pooling layers. Moreover, we provide some general guidelines on the network architecture, common regularization mechanisms, and hyper-parameter values such as batch size. Finally, we assess the visual quality of the land cover maps produced by TempCNNs.

Chang Wei Tan, Francois Petitjean, Geoffrey I. Webb

There has been renewed recent interest in developing effective lower bounds for Dynamic Time Warping (DTW) distance between time series. These have many applications in time series indexing, clustering, forecasting, regression and classification. One of the key time series classification algorithms, the nearest neighbor algorithm with DTW distance (NN-DTW) is very expensive to compute, due to the quadratic complexity of DTW. Lower bound search can speed up NN-DTW substantially. An effective and tight lower bound quickly prunes off unpromising nearest neighbor candidates from the search space and minimises the number of the costly DTW computations. The speed up provided by lower bound search becomes increasingly critical as training set size increases. Different lower bounds provide different trade-offs between computation time and tightness. Most existing lower bounds interact with DTW warping window sizes. They are very tight and effective at smaller warping window sizes, but become looser as the warping window increases, thus reducing the pruning effectiveness for NN-DTW. In this work, we present a new class of lower bounds that are tighter than the popular Keogh lower bound, while requiring similar computation time. Our new lower bounds take advantage of the DTW boundary condition, monotonicity and continuity constraints to create a tighter lower bound. Of particular significance, they remain relatively tight even for large windows. A single parameter to these new lower bounds controls the speed-tightness trade-off. We demonstrate that these new lower bounds provide an exceptional balance between computation time and tightness for the NN-DTW time series classification task, resulting in greatly improved efficiency for NN-DTW lower bound search.

Mahardhika Pratama, Witold Pedrycz, Geoffrey I. Webb

Existing FNNs are mostly developed under a shallow network configuration having lower generalization power than those of deep structures. This paper proposes a novel self-organizing deep FNN, namely DEVFNN. Fuzzy rules can be automatically extracted from data streams or removed if they play limited role during their lifespan. The structure of the network can be deepened on demand by stacking additional layers using a drift detection method which not only detects the covariate drift, variations of input space, but also accurately identifies the real drift, dynamic changes of both feature space and target space. DEVFNN is developed under the stacked generalization principle via the feature augmentation concept where a recently developed algorithm, namely gClass, drives the hidden layer. It is equipped by an automatic feature selection method which controls activation and deactivation of input attributes to induce varying subsets of input features. A deep network simplification procedure is put forward using the concept of hidden layer merging to prevent uncontrollable growth of dimensionality of input space due to the nature of feature augmentation approach in building a deep network structure. DEVFNN works in the sample-wise fashion and is compatible for data stream applications. The efficacy of DEVFNN has been thoroughly evaluated using seven datasets with non-stationary properties under the prequential test-then-train protocol. It has been compared with four popular continual learning algorithms and its shallow counterpart where DEVFNN demonstrates improvement of classification accuracy. Moreover, it is also shown that the concept drift detection method is an effective tool to control the depth of network structure while the hidden layer merging scenario is capable of simplifying the network complexity of a deep network with negligible compromise of generalization performance.

Fengxiang He, Tongliang Liu, Geoffrey I Webb et al.

When learning from positive and unlabelled data, it is a strong assumption that the positive observations are randomly sampled from the distribution of $X$ conditional on $Y = 1$, where X stands for the feature and Y the label. Most existing algorithms are optimally designed under the assumption. However, for many real-world applications, the observed positive examples are dependent on the conditional probability $P(Y = 1|X)$ and should be sampled biasedly. In this paper, we assume that a positive example with a higher $P(Y = 1|X)$ is more likely to be labelled and propose a probabilistic-gap based PU learning algorithms. Specifically, by treating the unlabelled data as noisy negative examples, we could automatically label a group positive and negative examples whose labels are identical to the ones assigned by a Bayesian optimal classifier with a consistency guarantee. The relabelled examples have a biased domain, which is remedied by the kernel mean matching technique. The proposed algorithm is model-free and thus do not have any parameters to tune. Experimental results demonstrate that our method works well on both generated and real-world datasets.