By comparing a specific redistricting plan to an ensemble of plans, we evaluate whether the plan translates individual votes to election outcomes in an unbiased fashion. Explicitly, we evaluate if a given redistricting plan exhibits extreme statistical properties compared to an ensemble of nonpartisan plans satisfying all legal criteria. Thus, we capture how unbiased redistricting plans interpret individual votes via a state's geopolitical landscape. We generate the ensemble of plans through a Markov chain Monte Carlo algorithm coupled with simulated annealing based on a reference distribution that does not include partisan criteria. Using the ensemble and historical voting data, we create a null hypothesis for various election results, free from partisanship, accounting for the state's geo-politics. We showcase our methods on two recent congressional districting plans of NC, along with a plan drawn by a bipartisan panel of retired judges. We find the enacted plans are extreme outliers whereas the bipartisan judges' plan does not give rise to extreme partisan outcomes. Equally important, we illuminate anomalous structures in the plans of interest by developing graphical representations which help identify and understand instances of cracking and packing associated with gerrymandering. These methods were successfully used in recent court cases. Supplementary materials for this article are available online.
We develop methods to evaluate whether a political districting accurately represents the will of the people. To explore and showcase our ideas, we concentrate on the congressional districts for the U.S. House of Representatives and use the state of North Carolina and its redistrictings since the 2010 census. Using a Monte Carlo algorithm, we randomly generate over 24,000 redistrictings that are non-partisan and adhere to criteria from proposed legislation. Applying historical voting data to these random redistrictings, we find that the number of democratic and republican representatives elected varies drastically depending on how districts are drawn. Some results are more common, and we gain a clear range of expected election outcomes. Using the statistics of our generated redistrictings, we critique the particular congressional districtings used in the 2012 and 2016 NC elections as well as a districting proposed by a bipartisan redistricting commission. We find that the 2012 and 2016 districtings are highly atypical and not representative of the will of the people. On the other hand, our results indicate that a plan produced by a bipartisan panel of retired judges is highly typical and representative. Since our analyses are based on an ensemble of reasonable redistrictings of North Carolina, they provide a baseline for a given election which incorporates the geometry of the state's population distribution.
We consider the problem of distributed online convex optimization, where a group of agents collaborate to track the trajectory of the global minimizers of sums of time-varying objective functions in an online manner. For general convex functions, the theoretical upper bounds of existing methods are given in terms of regularity measures associated with the dynamical system as well as the time horizon. It is thus of interest to determine whether the explicit time horizon dependence can be removed as in the case of centralized optimization. In this work, we propose a novel distributed online gradient descent algorithm and show that the dynamic regret bound of this algorithm has no explicit dependence on the time horizon. Instead, it depends on a new regularity measure quantifying the total change in gradients at the optimal points at each time. The main driving force of our algorithm is an online adaptation of the gradient tracking technique used in static optimization. Since, in many applications, time-varying objective functions and the corresponding optimal points follow a non-adversarial dynamical system, we also consider the role of prediction assuming that the optimal points evolve according to a linear dynamical system. We present numerical experiments that show that our proposed algorithm outperforms the existing distributed mirror descentbased state of the art methods in term of the optimizer tracking performance. We also present an empirical example suggesting that the analysis of our algorithm is optimal in the sense that the regularity measures in the theoretical bounds cannot be removed.
Using an ensemble of redistricting plans, we evaluate whether a given political districting faithfully represents the geo-political landscape. Redistricting plans are sampled by a Monte Carlo algorithm from a probability distribution that adheres to realistic and non-partisan criteria. Using the sampled redistricting plans and historical voting data, we produce an ensemble of elections that reveal geo-political structure within the state. We showcase our methods on the two most recent districtings of NC for the U.S. House of Representatives, as well as a plan drawn by a bipartisan redistricting panel. We find the two state enacted plans are highly atypical outliers whereas the bipartisan plan accurately represents the ensemble both in partisan outcome and in the fine scale structure of district-level results.Gerrymandering | Redistricting | Monte Carlo Sampling I n the 2012 NC congressional election, over half the total votes went to Democratic candidates, yet only four of the thirteen congressional representatives were Democrats. Furthermore, the most Democratic district had 29.63% margin of victory, whereas the most Republican district had a 13.11% margin of victory. These results may be due to political gerrymandering or, alternatively, be natural outcomes of NC's geo-political structure as determined by the spatial distribution of partisan votes.To probe the geo-political structure and its effect on election outcomes we (i) sample from the space of congressional redistricting plans that adheres to non-partisan redistricting criteria; (ii) we simulate an election with each of our sampled redistricting plans using the actual partisan votes cast by North Carolinians in the 2012 and 2016 congressional elections; and (iii) we aggregate election results to construct the distributions of partisan vote balance on each district and of the congressional delegation's partisan composition. Districts that do not respect typical election results are considered gerrymandered. When a districting is gerrymandered, the congressional delegation's partisan composition may be not representative of what is typical.Having probed the impact of the geo-political structure, we analyze three specific districting plans: the two most recent districting plans of NC for the U.S. House of Representatives and a plan proposed by a bipartisan panel of retired NC judges. By situating the election outcomes of these three districting plans in our sampled ensemble, we determine whether the three districting plans contain unlikely partisan favoritism and thwart the underlying geo-political structure, as expressed by the people's votes, by shifting each district's partisan vote balance significantly away from what is typical. MethodsTo sample from the space of congressional redistricting plans, we construct a family of probability distributions that are concentrated on plans adhering to non-partisan design criteria from proposed legislation. The non-partisan design criteria ensures that
Patterns of motor activity can be used to decode behavior state. Precise spike timing encoding is present in many motor systems, but is not frequently utilized to decode behavior or to examine how coordination is achieved across many motor units. Testing whether the same coordinated sets of muscles control different movements is difficult without a complete motor representation at the level of the currency of control – action potentials. Here, we demonstrate nearly perfect decoding of six hawk moth flight behaviors elicited in response to wide-field drifting visual stimuli about the flight axes – pitch, roll, and yaw – using a comprehensive, spike-resolved motor program and a simple linear decoding pipeline. A complex decoding scheme is not necessary, even if the functional patterns of control are nonlinear. We show that muscle covariation present in one pair of visual stimulus conditions can be used to decode behavior in a different pair of visual stimulus conditions, indicating the presence of conserved muscle coordination patterns at the level of motor neuronal timings in functionally distinct behaviors. We also demonstrate that as few as half the muscles can be used to retain decoding performance, linking coordination to redundancy in encoding, if not function, across the entire moth flight motor program.
We develop a framework for estimating unknown partial differential equations (PDEs) from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network, and extracts the PDE by equating derivatives of the neural network approximation. Our method applies to PDEs which are linear combinations of user-defined dictionary functions, and generalizes previous methods that only consider parabolic PDEs. We introduce a regularization scheme that prevents the function approximation from overfitting the data and forces it to be a solution of the underlying PDE. We validate the model on simulated data generated by the known PDEs and added Gaussian noise, and we study our method under different levels of noise. We also compare the error of our method with a Cramer-Rao lower bound for an ordinary differential equation (ODE). Our results indicate that our method outperforms other methods in estimating PDEs, especially in the low signal-to-noise (SNR) regime.Index Terms-Partial differential equations, neural networks, Cramer Rao bound.
Background Lemurs once rivalled the diversity of rest of the primate order despite thier confinement to the island of Madagascar. We test the adaptive radiation model of Malagasy lemur diversity using a novel combination of phylogenetic comparative methods and geometric methods for quantifying tooth shape. Results We apply macroevolutionary model fitting approaches and disparity through time analysis to dental topography metrics associated with dietary adaptation, an aspect of mammalian ecology which appears to be closely related to diversification in many clades. Metrics were also reconstructed at internal nodes of the lemur tree and these reconstructions were combined to generate dietary classification probabilities at internal nodes using discriminant function analysis. We used these reconstructions to calculate rates of transition toward folivory per million-year intervals. Finally, lower second molar shape was reconstructed at internal nodes by modelling the change in shape of 3D meshes using squared change parsimony along the branches of the lemur tree. Our analyses of dental topography metrics do not recover an early burst in rates of change or a pattern of early partitioning of subclade disparity. However, rates of change in adaptations for folivory were highest during the Oligocene, an interval of possible forest expansion on the island. Conclusions There was no clear phylogenetic signal of bursts of morphological evolution early in lemur history. Reconstruction of the molar morphologies corresponding to the ancestral nodes of the lemur tree suggest that this may have been driven by a shift toward defended plant resources, however. This suggests a response to the ecological opportunity offered by expanding forests, but not necessarily a classic adaptive radiation initiated by dispersal to Madagascar.
Many techniques for online optimization problems involve making decisions based solely on presently available information: fewer works take advantage of potential predictions. In this paper, we discuss the problem of online convex optimization for parametrizable objectives, i.e. optimization problems that depend solely on the value of a parameter at a given time. We introduce a new regularity for dynamic regret based on the accuracy of predicted values of the parameters and show that, under mild assumptions, accurate prediction can yield tighter bounds on dynamic regret. Inspired by recent advances on learning how to optimize, we also propose a novel algorithm to simultaneously predict and optimize for parametrizable objectives and study its performance using simulated and real data.
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