Time series analysis is a fundamental component of machine learning, especially in astrophysics and cosmology where temporal data abound. This chapter provides a pedagogical review of time series analysis techniques from a machine learning perspective. We cover the basic concepts of time series (stationarity, autocorrelation, seasonality), classical statistical models (autoregressive, moving average, ARIMA, exponential smoothing, state-space models), and modern machine learning approaches. In particular, we discuss how traditional statistical methods lay the groundwork, and then explore machine learning methods for time series, including feature-based regression, tree-based ensemble methods, hidden Markov models, Gaussian processes, and deep learning models (recurrent neural networks, convolutional networks, transformers). Throughout, we illustrate with examples drawn from multiple domains (e.g. astronomy, weather forecasting, finance) to emphasize common principles. The goal is to equip readers with both the theoretical understanding and practical context to apply machine learning techniques for time series analysis in their research.

Prior-fitted networks (PFNs) are a promising class of tabular foundation models that perform in-context learning, whereby the entire labelled training set is supplied as context, and predictions for test queries are produced in a single forward pass. However, the quadratically scaling self-attention mechanism in many PFN architectures makes inference prohibitive for very large training datasets. We propose CRUMB (Clustered Retrieval Using Minimised-MMD Batching), a three-stage inference wrapper that (i) clusters the test queries, (ii) selects a small, distributionally matched training subset for each cluster by greedily minimising the maximum mean discrepancy (MMD), and (iii) runs exact PFN inference on each reduced-context batch. CRUMB is architecture-agnostic and requires no retraining. On the 51-dataset TabArena benchmark, evaluated across three PFN architectures (TabPFNv2, TabICLv1, TabICLv2), we show that CRUMB outperforms similar state-of-the-art context selection strategies. We also show that CRUMB is resilient to covariate drift, as the MMD-minimisation step naturally helps align the training context distribution to match the current test batch distributions.

Reliable decision making pipelines powered by machine learning models require uncertainty quantification (UQ) methods that come with explicit safety guarantees. Conformal prediction provides such UQ by wrapping ML predictions into prediction sets, and recent work by Kiyani et al. (2025b) established that these sets can be translated into optimal risk-averse decision policies -- yet only inheriting marginal safety guarantees. We generalize and strengthen their results by (i) introducing action-conditional conformal prediction, which yields safety guarantees conditioned explicitly on each action taken by the decision maker, (ii) showing that action-conditional prediction sets serve as a proxy for the feasible decision space for risk-averse decision makers aiming to optimize action-conditional value-at-risk, and (iii) proposing a principled finite-sample algorithm based on pinball-loss minimization, connecting the framework of Gibbs et al. (2025) to action-conditional guarantees. Experiments on two real-world datasets confirm that our approach significantly improves action-conditional performance over conformal baselines.

Tissue graph counterfactuals ask how a cell's expression would change under altered spatial neighbor contexts. Such queries are central to predicting cell behavior in tissues, but lack a unified definition, with existing methods targeting specific intervention types or treating cells as i.i.d. In this work, we first formalize tissue graph counterfactuals as a class of spatial interventions that either rewire connections between cells (edge perturbation) or modify the expression of their neighbors (node perturbation). We then introduce Cellina, a framework that uses supervised disentanglement to decompose a cell's intrinsic state from its spatial context, using the latter as a conditioning input for counterfactual predictions. Across benchmarks spanning over 2.5 million spatially-resolved cells in colorectal cancer and mouse brain, Cellina outperforms spatially-informed and non-spatial competitors in insilico graph perturbations, disentanglement, and scalability. Additionally, we show that Cellina reveals biologically distinct cancer subdomains in an unsupervised manner and enables targeted neighbor perturbation simulations.

Attention is the key mechanism underlying in-context learning in transformers, and attention patterns have been observed empirically to emerge abruptly during training. We present a Bayesian theory of feature learning in attention; we then focus on how the copy subcircuit in the first layer of an induction head is learned by analyzing a single-layer softmax attention network trained on a copy task. We derive a closed-form posterior over the attention matrix and reduce it to a low-dimensional order parameter space. This reduction reveals a phase transition in the amount of training data, which we verify using both Bayesian sampling and standard training with Adam. We contrast our results with linear attention and find that softmax attention exhibits a \emph{first-order phase transition} while in linear attention an initial \emph{second-order phase transition} is followed by a smooth, continuous evolution toward the structured attention pattern (\emph{crossover}). Our work provides a first-principles theoretical account of the abrupt emergence of the copy subcircuit, reminiscent of the one observed in training large language models.

Biotic communities vary continuously across space, yet biome maps impose categorical boundaries that compress this variation, particularly at ecotones where transitional communities are ecologically distinct. Could Earth observation (EO) foundation models, which encode spectral, spatial, and temporal information with dense embeddings, convert discrete biome maps into continuous representations that better capture ecological variation? Here, we fit a linear classifier on Clay v1.5 satellite image embeddings to predict biome labels from a categorical map. The softmax output yields a continuous probability vector whose dimensions correspond to named biome classes. We evaluate this approach using six Brazilian biomes, 1.3 million embeddings, and 10,015 withheld forest inventory plots spanning 4,672 plant species. The continuous biome representation outperforms discrete biome labels for predicting species occurrence (mean per-species AUC 0.618 vs. 0.570 across 10 spatial cross-validation folds). Decomposing this gain shows that continuity in the graded probability output, rather than label reassignment, accounts for the improvement; the pattern holds across all distances from biome boundaries. The raw 1024-dimensional embedding remains the strongest predictor we tested (mean AUC 0.646 vs. 0.618), but the continuous representation recovers most of the embedding's gain over discrete labels. This simple approach provides a probabilistic replacement for categorical map labels, preserving their meaning while encoding graded variation that discrete maps suppress.

Compositional data -- vectors encoding relative proportions -- arise across scientific domains, including ecology, geochemistry, and genomics. The features in these data often come with known hierarchical structure (e.g., taxonomies, phylogenies, ontologies), yet existing methods either ignore this structure, discard the intrinsic Aitchison geometry, are designed for binary trees, or yield incomplete coordinate systems. We describe PolyILR, a canonical orthonormal decomposition of the Aitchison tangent space aligned with any tree topology. Our construction defines a weighted local geometry at each internal node capturing full branching structure, then lifts these to a global orthonormal basis where every coordinate corresponds to a specific tree location. On microbiome and single-cell benchmarks, PolyILR yields stable, interpretable features and enables inference at multiscale tree resolution. We also establish a novel theoretical connection to softmax classifiers, suggesting possible applications to probabilistic modeling.

Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological Representation ANalysis of Diagrams), which treats (collections of) PDs as survival data: each topological feature with persistence value $p = d - b$ is a fully observed time-to-event, and the persistence survival function $S(t) = \mathbb{P}(p > t)$ is the central object for comparing diagrams. From this single representation we derive (i) a non-parametric two-sample test with calibrated Type I error and high power from a small number of diagrams; (ii) interpretable effect sizes; and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. We validate calibration and power on synthetic manifolds with controlled topology, demonstrate competitive vectorisation across 14 graph and 3D point cloud benchmarks, and apply the method to study functional brain connectivity in fMRI/neuroscience data. To our knowledge, STRAND is the first method to provide hypothesis testing and vectorisation for persistence diagrams from a single coherent and interpretable representation.

We study distributed optimization with stochastic gradients and finite-bit communication modeled by random (unbiased) quantization. We propose q-PDGD, a quantized stochastic primal-dual method, and analyze it under relaxed global geometry. Under restricted secant inequality (RSI), a constant step-size yields linear contraction to an explicit neighborhood determined by gradient noise, quantization distortion, and network connectivity, while a diminishing step-size achieves O(1/k) convergence without shared-minimizer assumptions. Under Polyak-Lojasiewicz (PL) inequality, we obtain linear-to-neighborhood convergence in the same stochastic quantized setting. Our results match the best-known centralized stochastic rates in oracle complexity, and are supported by experiments demonstrating the predicted tradeoffs between quantization level, step-size choice, and graph structure.

This paper studies efficient online algorithms for multinomial logistic bandits (MLogB), where the feedback distribution over $K+1$ outcomes follows a multinomial logistic model of $d$-dimensional action vectors. A representative UCB-type algorithm, OFUL-MLogB, achieves a regret bound of $\tilde{\mathcal{O}}(Kd\sqrt{T})$, but still requires $\mathcal{O}(K^3d^3)$ time and $\mathcal{O}(K^2d^2)$ space per round due to parameter estimation and optimistic reward construction, which is prohibitive in high-dimensional settings. To address this limitation, we propose EOFD-MLogB, which integrates frequent directions matrix sketching into OFUL-MLogB. By maintaining a low-rank SVD sketch of the accumulated Hessian, constrained online Newton updates in parameter estimation and $Kd \times K$ spectral-norm computations in the reward bonus are reduced to one-dimensional root-finding tasks and $K \times K$ eigenvalue computations, respectively. This yields dominant per-round time complexity $\mathcal{O}(Kd(m+K)^2)$ and space complexity $\mathcal{O}(Kd(m+K))$, where $m \ll d$ is the sketch size. We further prove a regret bound of $\tilde{\mathcal{O}}(Δ_T(Kd\lnΔ_T+m)\sqrt{T})$, where the sketching error factor $Δ_T$ is controlled by the $m$-truncated spectral tail of the Hessian. Thus, when the Hessian is approximately low-rank, the regret is close to that of OFUL-MLogB. Experiments validate the computational efficiency and competitive performance.