Traditional hidden Markov models have been a useful tool to understand and model stochastic dynamic data; in the case of non-Gaussian data, models such as mixture of Gaussian hidden Markov models can be used. However, these suffer from the computation of precision matrices and have a lot of unnecessary parameters. As a consequence, such models often perform better when it is assumed that all variables are independent, a hypothesis that may be unrealistic. Hidden Markov models based on kernel density estimation are also capable of modeling non-Gaussian data, but they assume independence between variables. In this article, we introduce a new hidden Markov model based on kernel density estimation, which is capable of capturing kernel dependencies using context-specific Bayesian networks. The proposed model is described, together with a learning algorithm based on the expectation-maximization algorithm. Additionally, the model is compared to related HMMs on synthetic and real data. From the results, the benefits in likelihood and classification accuracy from the proposed model are quantified and analyzed.
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The features in many prediction models naturally take the form of a hierarchy. The lower levels represent individuals or events. These units group naturally into locations and intervals or other aggregates, often at multiple levels. Levels of groupings may intersect and join, much as relational database tables do. Besides representing the structure of the data, predictive features in hierarchical models can be assigned to their proper levels. Such models lend themselves to hierarchical Bayes solution methods that ``share'' results of inference between groups by generalizing over the case of individual models for each group versus one model that aggregates all groups into one. In this paper we show our work-in-progress applying a hierarchical Bayesian model to forecast purchases throughout the day at store franchises, with groupings over locations and days of the week. We demonstrate using the \textsf{stan} package on individual sales transaction data collected over the course of a year. We show how this solves the dilemma of having limited data and hence modest accuracy for each day and location, while being able to scale to a large number of locations with improved accuracy.
Generalized Time Warping Invariant Dictionary Learning for Time Series Classification and Clustering
Dictionary learning is an effective tool for pattern recognition and classification of time series data. Among various dictionary learning techniques, the dynamic time warping (DTW) is commonly used for dealing with temporal delays, scaling, transformation, and many other kinds of temporal misalignments issues. However, the DTW suffers overfitting or information loss due to its discrete nature in aligning time series data. To address this issue, we propose a generalized time warping invariant dictionary learning algorithm in this paper. Our approach features a generalized time warping operator, which consists of linear combinations of continuous basis functions for facilitating continuous temporal warping. The integration of the proposed operator and the dictionary learning is formulated as an optimization problem, where the block coordinate descent method is employed to jointly optimize warping paths, dictionaries, and sparseness coefficients. The optimized results are then used as hyperspace distance measures to feed classification and clustering algorithms. The superiority of the proposed method in terms of dictionary learning, classification, and clustering is validated through ten sets of public datasets in comparing with various benchmark methods.
Randomized controlled trials (RCTs) are the gold standard for causal inference, but they are often powered only for average effects, making estimation of heterogeneous treatment effects (HTEs) challenging. Conversely, large-scale observational studies (OS) offer a wealth of data but suffer from confounding bias. Our paper presents a novel framework to leverage OS data for enhancing the efficiency in estimating conditional average treatment effects (CATEs) from RCTs while mitigating common biases. We propose an innovative approach to combine RCTs and OS data, expanding the traditionally used control arms from external sources. The framework relaxes the typical assumption of CATE invariance across populations, acknowledging the often unaccounted systematic differences between RCT and OS participants. We demonstrate this through the special case of a linear outcome model, where the CATE is sparsely different between the two populations. The core of our framework relies on learning potential outcome means from OS data and using them as a nuisance parameter in CATE estimation from RCT data. We further illustrate through experiments that using OS findings reduces the variance of the estimated CATE from RCTs and can decrease the required sample size for detecting HTEs.
In this paper, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 years of the publication of the fundamental theorem for Markov bases by Diaconis and Sturmfels. In addition to a literature review we prove three new results on the complexity of Markov bases in hierarchical models, relaxations of the fibers in log-linear models, and limitations of partial sets of moves in providing an irreducible Markov chain.
Physicists routinely need probabilistic models for a number of tasks such as parameter inference or the generation of new realizations of a field. Establishing such models for highly non-Gaussian fields is a challenge, especially when the number of samples is limited. In this paper, we introduce scattering spectra models for stationary fields and we show that they provide accurate and robust statistical descriptions of a wide range of fields encountered in physics. These models are based on covariances of scattering coefficients, i.e. wavelet decomposition of a field coupled with a point-wise modulus. After introducing useful dimension reductions taking advantage of the regularity of a field under rotation and scaling, we validate these models on various multi-scale physical fields and demonstrate that they reproduce standard statistics, including spatial moments up to 4th order. These scattering spectra provide us with a low-dimensional structured representation that captures key properties encountered in a wide range of physical fields. These generic models can be used for data exploration, classification, parameter inference, symmetry detection, and component separation.
Many data insight questions can be viewed as searching in a large space of tables and finding important ones, where the notion of importance is defined in some adhoc user defined manner. This paper presents Holistic Cube Analysis (HoCA), a framework that augments the capabilities of relational queries for such problems. HoCA first augments the relational data model by defining a new data type AbstractCube, which is defined as a function from RegionFeatures space to relational tables. AbstractCube provides a logical form of data for HoCA programs, abstracting away their exact encoding. With this function-as-data modeling, HoCA operators are thus cube-to-cube transformations. We describe two basic but fundamental HoCA operators, cube crawling and cube join (with many possibilities to extend). Cube crawling explores a region space, and outputs a cube that maps regions to signal vectors. Cube join, in turn, is critical for composition, allowing one to join information from different cubes for deeper analysis. Cube crawling introduces two novel programming features, (programmable) Region Analysis Models (RAMs) and Multi-Model Crawling. Crucially, RAM has a notion of population features, which allows one to go beyond only analyzing local features at a region, and program region-population analysis that compares region and population features, capturing a large class of importance notions in data insights. HoCA poses a rich algorithmic space, and we describe several cube crawling implementations leveraging different foundations, and evaluate their performance. We have implemented and deployed HoCA at Google. Even in this early stage, our HoCA offering has attracted more than 30 teams building data-insight systems with it, with applications in system monitoring, experimentation analysis, and business intelligence, some of which have generated significant revenue uplift.
Inverse protein folding is challenging due to its inherent one-to-many mapping characteristic, where numerous possible amino acid sequences can fold into a single, identical protein backbone. This task involves not only identifying viable sequences but also representing the sheer diversity of potential solutions. However, existing discriminative models, such as transformer-based auto-regressive models, struggle to encapsulate the diverse range of plausible solutions. In contrast, diffusion probabilistic models, as an emerging genre of generative approaches, offer the potential to generate a diverse set of sequence candidates for determined protein backbones. We propose a novel graph denoising diffusion model for inverse protein folding, where a given protein backbone guides the diffusion process on the corresponding amino acid residue types. The model infers the joint distribution of amino acids conditioned on the nodes' physiochemical properties and local environment. Moreover, we utilize amino acid replacement matrices for the diffusion forward process, encoding the biologically-meaningful prior knowledge of amino acids from their spatial and sequential neighbors as well as themselves, which reduces the sampling space of the generative process. Our model achieves state-of-the-art performance over a set of popular baseline methods in sequence recovery and exhibits great potential in generating diverse protein sequences for a determined protein backbone structure.
We develop several provably efficient model-free reinforcement learning (RL) algorithms for infinite-horizon average-reward Markov Decision Processes (MDPs). We consider both online setting and the setting with access to a simulator. In the online setting, we propose model-free RL algorithms based on reference-advantage decomposition. Our algorithm achieves $\widetilde{O}(S^5A^2\mathrm{sp}(h^*)\sqrt{T})$ regret after $T$ steps, where $S\times A$ is the size of state-action space, and $\mathrm{sp}(h^*)$ the span of the optimal bias function. Our results are the first to achieve optimal dependence in $T$ for weakly communicating MDPs. In the simulator setting, we propose a model-free RL algorithm that finds an $\epsilon$-optimal policy using $\widetilde{O} \left(\frac{SA\mathrm{sp}^2(h^*)}{\epsilon^2}+\frac{S^2A\mathrm{sp}(h^*)}{\epsilon} \right)$ samples, whereas the minimax lower bound is $\Omega\left(\frac{SA\mathrm{sp}(h^*)}{\epsilon^2}\right)$. Our results are based on two new techniques that are unique in the average-reward setting: 1) better discounted approximation by value-difference estimation; 2) efficient construction of confidence region for the optimal bias function with space complexity $O(SA)$.
We propose and investigate a hidden Markov model (HMM) for the analysis of dependent, aggregated, superimposed two-state signal recordings. A major motivation for this work is that often these signals cannot be observed individually but only their superposition. Among others, such models are in high demand for the understanding of cross-talk between ion channels, where each single channel cannot be measured separately. As an essential building block, we introduce a parameterized vector norm dependent Markov chain model and characterize it in terms of permutation invariance as well as conditional independence. This building block leads to a hidden Markov chain sum process which can be used for analyzing the dependence structure of superimposed two-state signal observations within an HMM. Notably, the model parameters of the vector norm dependent Markov chain are uniquely determined by the parameters of the sum process and are therefore identifiable. We provide algorithms to estimate the parameters, discuss model selection and apply our methodology to real-world ion channel data from the heart muscle, where we show competitive gating.
Hidden Markov models (HMMs) are flexible tools for clustering dependent data coming from unknown populations, allowing nonparametric modelling of the population densities. Identifiability fails when the data is in fact independent, and we study the frontier between learnable and unlearnable two-state nonparametric HMMs. Interesting new phenomena emerge when the cluster distributions are modelled via density functions (the 'emission' densities) belonging to standard smoothness classes compared to the multinomial setting. Notably, in contrast to the multinomial setting previously considered, the identification of a direction separating the two emission densities becomes a critical, and challenging, issue. Surprisingly, it is possible to "borrow strength" from estimators of the smoother density to improve estimation of the other. We conduct precise analysis of minimax rates, showing a transition depending on the relative smoothnesses of the emission densities.
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