Political districting without geography

Geographical considerations such as contiguity and compactness are necessary elements of political districting in practice. Yet an analysis of the problem without such constraints yields mathematical insights that can inform real-world model construction. In particular, it clarifies the sharp contrast between proportionality and competitiveness and how it might be overcome in a properly formulated objective function. It also reveals serious weaknesses of the much-discussed efficiency gap as a criterion for gerrymandering.Published versio

Non-Negative Matrix Factorization for Learning Alignment-Specific Models of Protein Evolution

Models of protein evolution currently come in two flavors: generalist and specialist. Generalist models (e.g. PAM, JTT, WAG) adopt a one-size-fits-all approach, where a single model is estimated from a number of different protein alignments. Specialist models (e.g. mtREV, rtREV, HIVbetween) can be estimated when a large quantity of data are available for a single organism or gene, and are intended for use on that organism or gene only. Unsurprisingly, specialist models outperform generalist models, but in most instances there simply are not enough data available to estimate them. We propose a method for estimating alignment-specific models of protein evolution in which the complexity of the model is adapted to suit the richness of the data. Our method uses non-negative matrix factorization (NNMF) to learn a set of basis matrices from a general dataset containing a large number of alignments of different proteins, thus capturing the dimensions of important variation. It then learns a set of weights that are specific to the organism or gene of interest and for which only a smaller dataset is available. Thus the alignment-specific model is obtained as a weighted sum of the basis matrices. Having been constrained to vary along only as many dimensions as the data justify, the model has far fewer parameters than would be required to estimate a specialist model. We show that our NNMF procedure produces models that outperform existing methods on all but one of 50 test alignments. The basis matrices we obtain confirm the expectation that amino acid properties tend to be conserved, and allow us to quantify, on specific alignments, how the strength of conservation varies across different properties. We also apply our new models to phylogeny inference and show that the resulting phylogenies are different from, and have improved likelihood over, those inferred under standard models

Preference Elicitation For Participatory Budgeting

Participatory budgeting enables the allocation of public funds by collecting and aggregating individual preferences; it has already had a sizable real-world impact. But making the most of this new paradigm requires a rethinking of some of the basics of computational social choice, including the very way in which individuals express their preferences. We analytically compare four preference elicitation methods -- knapsack votes, rankings by value or value for money, and threshold approval votes -- through the lens of implicit utilitarian voting, and find that threshold approval votes are qualitatively superior. This conclusion is supported by experiments using data from real participatory budgeting elections

Non-negative matrix factorization.

<p>Non-negative matrix factorization.</p

scores for all models.

<p>Each table entry is the number of datasets with in that range. For any dataset, the best model has . A model with has essentially no support.</p

NNMF basis matrices.

<p>The set of NNMF basis matrices obtained for ranks ranging from 1 to 5. Amino acids are ordered according to their Stanfel classification <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0028898#pone.0028898-Stanfel1" target="_blank">[25]</a>. Rates are indicated in grayscale, with pure white being a rate of zero and pure black being the maximum rate in the matrix.</p

for all models with gamma rate variation (4 categories).

<p>Each table entry is the number of datasets with in that range. For any dataset, the best model has . A model with has essentially no support.</p

NNMF basis matrices correlate with amino acid properties.

<p>The correlations between amino acid properties and the basis matrices. The horizontal black line (at −0.16867) indicates the threshold for significant negative correlation (, one tailed, ).</p

Selecting the larger Pandit alignments.

<p>Each blue dot represents an alignment in the Pandit database. The green region covers the alignments used in the training set, and the thin red region covers those in the test set.</p