Mathematics

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Author: search.filters.author.Darmon, DavidStart date: 2000

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    Complexity-Regularized Regression for Serially-Correlated Residuals with Applications to Stock Market Data
    (MDPI, 2014-12-23) Darmon, David; Girvan, Michelle
    A popular approach in the investigation of the short-term behavior of a non-stationary time series is to assume that the time series decomposes additively into a long-term trend and short-term fluctuations. A first step towards investigating the short-term behavior requires estimation of the trend, typically via smoothing in the time domain. We propose a method for time-domain smoothing, called complexity-regularized regression (CRR). This method extends recent work, which infers a regression function that makes residuals from a model “look random”. Our approach operationalizes non-randomness in the residuals by applying ideas from computational mechanics, in particular the statistical complexity of the residual process. The method is compared to generalized cross-validation (GCV), a standard approach for inferring regression functions, and shown to outperform GCV when the error terms are serially correlated. Regression under serially-correlated residuals has applications to time series analysis, where the residuals may represent short timescale activity. We apply CRR to a time series drawn from the Dow Jones Industrial Average and examine how both the long-term and short-term behavior of the market have changed over time.
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    Statistical Methods for Analyzing Time Series Data Drawn from Complex Social Systems
    (2015) Darmon, David; Girvan, Michelle; Rand, William; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The rise of human interaction in digital environments has lead to an abundance of behavioral traces. These traces allow for model-based investigation of human-human and human-machine interaction `in the wild.' Stochastic models allow us to both predict and understand human behavior. In this thesis, we present statistical procedures for learning such models from the behavioral traces left in digital environments. First, we develop a non-parametric method for smoothing time series data corrupted by serially correlated noise. The method determines the simplest smoothing of the data that simultaneously gives the simplest residuals, where simplicity of the residuals is measured by their statistical complexity. We find that complexity regularized regression outperforms generalized cross validation in the presence of serially correlated noise. Next, we cast the task of modeling individual-level user behavior on social media into a predictive framework. We demonstrate the performance of two contrasting approaches, computational mechanics and echo state networks, on a heterogeneous data set drawn from user behavior on Twitter. We demonstrate that the behavior of users can be well-modeled as processes with self-feedback. We find that the two modeling approaches perform very similarly for most users, but that users where the two methods differ in performance highlight the challenges faced in applying predictive models to dynamic social data. We then expand the predictive problem of the previous work to modeling the aggregate behavior of large collections of users. We use three models, corresponding to seasonal, aggregate autoregressive, and aggregation-of-individual approaches, and find that the performance of the methods at predicting times of high activity depends strongly on the tradeoff between true and false positives, with no method dominating. Our results highlight the challenges and opportunities involved in modeling complex social systems, and demonstrate how influencers interested in forecasting potential user engagement can use complexity modeling to make better decisions. Finally, we turn from a predictive to a descriptive framework, and investigate how well user behavior can be attributed to time of day, self-memory, and social inputs. The models allow us to describe how a user processes their past behavior and their social inputs. We find that despite the diversity of observed user behavior, most models inferred fall into a small subclass of all possible finitary processes. Thus, our work demonstrates that user behavior, while quite complex, belies simple underlying computational structures.