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Usage

problem_structure

Historical Votes

Given R regions and V votes, we store a dataset of historical votes in a vote matrix (or a tensor) consisting of the (fully) observed vote outcomes. In the case of a binary outcome, e.g., a referendum, each entry of the vote corresponds to the fraction of "yes" in the r-th region and v-th vote. Alternatively, in the case of multiple outcomes, e.g., an election, each entry of the vote tensor is a vector that corresponds to the fraction of votes for each party. (We assume that the sum of the entries in each vector equals 1). For K parties, the vote tensor will be of shape R x V x K. In the code, the matrix is denoted by M_historical and is used to initialize models.

Weighting of Regions

Additionally, we support some weighting to account for disparities between regions. For example, the weighting may be the number of valid votes or the population in each region. The weighting vector is called weighting and is required when initializing models, together with the historical outcomes. If no such data is available, one can pass weighting=None to ignore weighting.

Real-Time Predictions

To predict the outcomes of an ongoing election in real-time, the algorithm predicts the unobserved entries of the partially observed vector (or matrix) For binary outcomes, this is an R-dimensional vector, and this is an R x K matrix for multiple outcomes. In the code, the partially observed vectors m_current contains both actual, observed values in entries where the regional was available and np.nan in entries for unobserved regions. These unobserved entries are then filled in by the method of choice by calling model.fit_predict(m_current).

Putting It Together

Assuming you have the above quantities ready for a particular vote (i.e., M_historical, weighting, and m_current), inferring the unobserved entries after observing at least one regional outcome of an ongoing election is easy:

from predikon import LogisticSubSVD
model = LogisticSubSVD(M_historical, weighting)
m_predict = model.fit_predict(m_current)
# All unobserved entries in `m_current` are now filled in.

Choosing the Right Model (Hyperparameters)

For binary outcomes, i.e., for referenda, the preferred method is LogisticSubSVD. For multiple possible outcomes, i.e., for elections, the preferred method is LogisticTensorSubSVD. Both further have two important hyperparameters, l2_reg and n_dim:

  • l2_reg controls the strength of the L2 regularization of the parameters
  • n_dim controls the number of latent dimensions used for the model

In many cases, not much historical data is available and thus the number of dimensions can be set to a low value, e.g., between 5 and 10. The l2_reg parameter defaults to 1e-5 but should be tuned for the problem at hand, e.g., using cross-validation. Gaussian versions of our model (GaussianSubSVD and GaussianTensorSubSVD) are also available, and they require the same hyperparameters.

We provide additionally two baselines for comparison and model selection. The standard (weighted) averaging (WeightedAveraging) and probabilistic matrix factorization (MatrixFactorisation) for binary outcomes.