This paper presents a technique for reduced-order Markov modeling for compact representation of time-series data. In this work, symbolic dynamics-based tools have been used to infer an approximate generative Markov model. The time-series data are first symbolized by partitioning the continuous measurement space of the signal and then, the discrete sequential data are modeled using symbolic dynamics. In the proposed approach, the size of temporal memory of the symbol sequence is estimated from spectral properties of the resulting stochastic matrix corresponding to a first-order Markov model of the symbol sequence. Then, hierarchical clustering is used to represent the states of the corresponding full-state Markov model to construct a reduced-order or size Markov model with a non-deterministic algebraic structure. Subsequently, the parameters of the reduced-order Markov model are identified from the original model by making use of a Bayesian inference rule. The final model is selected using information-theoretic criteria. The proposed concept is elucidated and validated on two different data sets as examples. The first example analyzes a set of pressure data from a swirl-stabilized combustor, where controlled protocols are used to induce flame instabilities. Variations in the complexity of the derived Markov model represent how the system operating condition changes from a stable to an unstable combustion regime. In the second example, the data set is taken from NASA's data repository for prognostics of bearings on rotating shafts. We show that, even with a very small state-space, the reduced-order models are able to achieve comparable performance and that the proposed approach provides flexibility in the selection of a final model for representation and learning.

Recurrent neural networks (RNNs) are a vital modeling technique that rely on internal states learned indirectly by optimization of a supervised, unsupervised, or reinforcement training loss. RNNs are used to model dynamic processes that are characterized by underlying latent states whose form is often unknown, precluding its analytic representation inside an RNN. In the Predictive-State Representation (PSR) literature, latent state processes are modeled by an internal state representation that directly models the distribution of future observations, and most recent work in this area has relied on explicitly representing and targeting sufficient statistics of this probability distribution. We seek to combine the advantages of RNNs and PSRs by augmenting existing state-of-the-art recurrent neural networks with Predictive-State Decoders (PSDs), which add supervision to the network's internal state representation to target predicting future observations. Predictive-State Decoders are simple to implement and easily incorporated into existing training pipelines via additional loss regularization. We demonstrate the effectiveness of PSDs with experimental results in three different domains: probabilistic filtering, Imitation Learning, and Reinforcement Learning. In each, our method improves statistical performance of state-of-the-art recurrent baselines and does so with fewer iterations and less data.

We present a factorized hierarchical variational autoencoder, which learns disentangled and interpretable representations from sequential data without supervision. Specifically, we exploit the multi-scale nature of information in sequential data by formulating it explicitly within a factorized hierarchical graphical model that imposes sequence-dependent priors and sequence-independent priors to different sets of latent variables. The model is evaluated on two speech corpora to demonstrate, qualitatively, its ability to transform speakers or linguistic content by manipulating different sets of latent variables; and quantitatively, its ability to outperform an i-vector baseline for speaker verification and reduce the word error rate by as much as 35% in mismatched train/test scenarios for automatic speech recognition tasks.

This paper stands in the context of reinforcement learning with partial observability and limited data. In this setting, we focus on the tradeoff between asymptotic bias (suboptimality with unlimited data) and overfitting (additional suboptimality due to limited data), and theoretically show that while potentially increasing the asymptotic bias, a smaller state representation decreases the risk of overfitting. Our analysis relies on expressing the quality of a state representation by bounding L1 error terms of the associated belief states. Theoretical results are empirically illustrated when the state representation is a truncated history of observations. Finally, we also discuss and empirically illustrate how using function approximators and adapting the discount factor may enhance the tradeoff between asymptotic bias and overfitting.

Finally, a comprehensive hands-on machine learning course with specific focus on classification based models for the investment community and passionate investors. In the past few years, there has been a massive adoption and growth in the use of data science, artificial intelligence and machine learning to find alpha. However, information on and application of machine learning to investment are scarce. This course has been designed to address that. It is meant to spark your creative juices and get you started in this space.

In evolving complex systems such as air traffic and social organizations, collective effects emerge from their many components' dynamic interactions. While the dynamic interactions can be represented by temporal networks with nodes and links that change over time, they remain highly complex. It is therefore often necessary to use methods that extract the temporal networks' large-scale dynamic community structure. However, such methods are subject to overfitting or suffer from effects of arbitrary, a priori imposed timescales, which should instead be extracted from data. Here we simultaneously address both problems and develop a principled data-driven method that determines relevant timescales and identifies patterns of dynamics that take place on networks as well as shape the networks themselves. We base our method on an arbitrary-order Markov chain model with community structure, and develop a nonparametric Bayesian inference framework that identifies the simplest such model that can explain temporal interaction data.

As datasets capturing human choices grow in richness and scale--particularly in online domains--there is an increasing need for choice models that escape traditional choice-theoretic axioms such as regularity, stochastic transitivity, and Luce's choice axiom. In this work we introduce the Pairwise Choice Markov Chain (PCMC) model of discrete choice, an inferentially tractable model that does not assume any of the above axioms while still satisfying the foundational axiom of uniform expansion, a considerably weaker assumption than Luce's choice axiom. We show that the PCMC model significantly outperforms both the Multinomial Logit (MNL) model and a mixed MNL (MMNL) model in prediction tasks on both synthetic and empirical datasets known to exhibit violations of Luce's axiom. Our analysis also synthesizes several recent observations connecting the Multinomial Logit model and Markov chains; the PCMC model retains the Multinomial Logit model as a special case.

Like the course I just released on Hidden Markov Models, Recurrent Neural Networks are all about learning sequences - but whereas Markov Models are limited by the Markov assumption, Recurrent Neural Networks are not - and as a result, they are more expressive, and more powerful than anything we've seen on tasks that we haven't made progress on in decades. So what's going to be in this course and how will it build on the previous neural network courses and Hidden Markov Models? In the first section of the course we are going to add the concept of time to our neural networks. I'll introduce you to the Simple Recurrent Unit, also known as the Elman unit. We are going to revisit the XOR problem, but we're going to extend it so that it becomes the parity problem - you'll see that regular feedforward neural networks will have trouble solving this problem but recurrent networks will work because the key is to treat the input as a sequence.

The partially observable Markov decision process (POMDP) provides a principled general framework for planning under uncertainty, but solving POMDPs optimally is computationally intractable, due to the "curse of dimensionality" and the "curse of history". To overcome these challenges, we introduce the Determinized Sparse Partially Observable Tree (DESPOT), a sparse approximation of the standard belief tree, for online planning under uncertainty. A DESPOT focuses online planning on a set of randomly sampled scenarios and compactly captures the "execution" of all policies under these scenarios. We show that the best policy obtained from a DESPOT is near-optimal, with a regret bound that depends on the representation size of the optimal policy. Leveraging this result, we give an anytime online planning algorithm, which searches a DESPOT for a policy that optimizes a regularized objective function. Regularization balances the estimated value of a policy under the sampled scenarios and the policy size, thus avoiding overfitting. The algorithm demonstrates strong experimental results, compared with some of the best online POMDP algorithms available. It has also been incorporated into an autonomous driving system for real-time vehicle control. The source code for the algorithm is available online.

We consider the problem of active feature acquisition, where we sequentially select the subset of features in order to achieve the maximum prediction performance in the most cost-effective way. In this work, we formulate this active feature acquisition problem as a reinforcement learning problem, and provide a novel framework for jointly learning both the RL agent and the classifier (environment). We also introduce a more systematic way of encoding subsets of features that can properly handle innate challenge with missing entries in active feature acquisition problems, that uses the orderless LSTM-based set encoding mechanism that readily fits in the joint learning framework. We evaluate our model on a carefully designed synthetic dataset for the active feature acquisition as well as several real datasets such as electric health record (EHR) datasets, on which it outperforms all baselines in terms of prediction performance as well feature acquisition cost.