Ridings ranked by forecast gap between the favourite and the runner-up. Selecting a row or a map cell updates the riding panel.
Four sources feed the model. (i) Provincial voting-intention polls from 2014 to the present, compiled from the public tables of Wikipedia (CC BY-SA), each entry linking to the polling firm's original release; regional breakdowns (Montreal, Quebec City, rest of Quebec) come from the firms' reports when published. (ii) The official results of the 2014, 2018, 2022 and 2026 general elections (Élections Québec), expressed on the 2026 electoral map (127 ridings), earlier elections through a geographic correspondence of polling divisions, as well as the by-elections held since the 2026 general election (DGEQ). (iii) The 2021 census (Statistics Canada), aggregated from dissemination areas to ridings. (iv) The status of incumbent members, retirements and leaders' candidacies, collected from the pages of the National Assembly.
A vector of voting intentions is a composition: its shares are positive and sum to one, so the analysis is carried out in the simplex rather than in the Euclidean space of percentages. All compositional responses and predictors are transformed into isometric log-ratio (ILR) coordinates, an orthonormal basis of dimension k−1 in which linear operations and distances are valid. Multiplicative adjustments (local premiums and penalties) are applied as compositional perturbations, the internal operation of the simplex. Parties not broken out in older polls are handled by amalgamation into the residual share, with multiplicative replacement of zeros.
The trajectory of voting intentions is estimated by Gaussian process regression, one independent process per ILR coordinate, Matérn kernel (ν = 3/2). The observation variance of each poll is computed per coordinate: delta method applied to multinomial sampling, Var(ilr_c) = Σ_p M²_cp / (n x_p), where M is the Jacobian of the ILR transformation. A multiplicative scale per coordinate, estimated on a grid, absorbs overdispersion between polling firms. The hyperparameters (range, noise scale and non-stationarity parameters) are marginalised: the prediction is the average of the grid models weighted by their marginal likelihood, so that hyperparameter uncertainty is included in the predictive intervals.
Two forms of non-stationarity are allowed, each containing the stationary model as a special case so that selection rests on the likelihood rather than on a discretionary choice. First, breakpoints at leadership changes (January 14 and April 12, 2026): the covariance between regimes is multiplied by ρ|Δr|, ρ ∈ [0, 1] estimated by marginal likelihood, ρ = 1 restoring the stationary kernel; a partial break is retained on one coordinate, the stationary model elsewhere. Second, a dilation of the time axis during campaigns (from the writ to election day, 2014, 2018, 2022 and 2026 windows): one campaign day counts as k days of kernel time, k shared across all campaigns (hence identified mainly by past campaigns) and k = 1 restoring the uniform clock. The retained dilation is strong: effective range of about 40 to 190 days during campaigns depending on the coordinate, against 300 to 1500 outside campaigns. The calibration of the predictive intervals is measured by sequential validation: at each monthly cutoff from 2016 to 2026, the model fitted on earlier polls only predicts the next poll. The nominal 90% interval covers 79% of these 675 predictions (95% during campaigns, 78% outside). Random k-fold cross-validation gives 90.5%, a figure that measures interpolation of the series, not prediction.
The regional breakdowns published by polling firms (Montreal metropolitan area, Quebec City metropolitan area, rest of Quebec), all firms, released since the previous election are pooled by region as a weighted mean. The weight of a poll is its regional sample size multiplied by exp(−age/τ), age being measured on the clock of the national trend (section 3): one campaign day counts as D days. τ = 240 days and D = 16 are estimated by one-step-ahead forecasting of the region-province gap of each of the 152 regionally broken-down polls published from 2018 to 2026, using only earlier polls of the same cycle, on grids containing no decay and no dilation. The effective constant is thus 15 days during campaigns and 240 days outside; the squared forecast error is 0.163, against 0.216 for a fixed 14-day window and 0.262 without regional polls. For each region, the gap between the region and the province in these polls, in centred log-ratios of the five main parties, minus the same gap in the results of the previous election, shifts the ridings of the region, multiplied by a coefficient estimated jointly with that of the mother-tongue swing (section 5) on the 2014→2018, 2018→2022 and 2022→2026 transitions: 0.75 for regions, 0.25 for language. With both coefficients chosen on two transitions and evaluated on the third, at known provincial vote, the root mean squared error of riding shares falls from 5.66 to 5.44 points and the number of correct winners rises from 309 to 316 out of 348 ridings evaluated. On the eve of the three elections, national trend included, the absolute seat error of the central riding-level scenario falls from 36 to 30 over the three combined.
The starting point of each riding is its result at the general election of October 5, 2026. No differential-turnout correction is applied: the shrinkage weight of that correction, estimated on a grid against the final polls of 2022 and 2026 combined, is zero (mean absolute error of 1.89 points without correction, 2.06 with). The estimated provincial swing (section 3) is applied in log-ratios. Four estimated adjustments are layered on top, all as compositional perturbations:
No local deviation is predicted from the demographic profile. A multivariate Gaussian process regression (Matérn kernel, one process per ILR coordinate) of each riding's change on its census profile (language, education, immigration, employment structure, age, income, density) and on its previous deviation reaches an R² of 0.44 within a transition, but −0.16 from one transition to the next (training on 2018→2022, predicting 2022→2026). Evaluated in seats on the three transitions 2014→2018, 2018→2022 and 2022→2026, each predicted from the other two, it accumulates 221 seats of error against 40 for carrying results forward without local deviation; on 2026, at known provincial vote, it raises the root mean squared error of riding shares from 4.95 to 5.92 points.
The regional (section 4) and mother-tongue adjustments bear on the gap between the polls published since the 2026 election and the results of that election; they apply as soon as regional or language breakdowns are published after election day. The displayed forecast includes the adjustments for which such breakdowns exist on the date of computation.
The seat distribution is obtained by Monte Carlo simulation (5,000 draws) propagating four sources of uncertainty: (i) the posterior distribution of the national trend; (ii) a systemic shock common to all ridings (collective bias of polling firms and drift of opinion until election day), calibrated on the gap between the last polls and the actual results of 2018 and 2022, isotropic in ILR; (iii) a regional shock, local deviations being spatially correlated; (iv) the idiosyncratic deviation of each riding, resampled from the errors of the riding model (provincial swing and mother-tongue swing) in previous general transitions, 2014→2018, 2018→2022 and 2022→2026 pooled (whole ILR vectors, net of their regional mean). These errors are scaled party by party in centred log-ratios: CAQ ×0.75, QLP ×0.75, CPQ ×0.5, PQ ×1, QS ×0.75. The factors are chosen on a grid containing 1 by maximising the out-of-sample log-score of win probabilities, the errors of each transition predicting the others (log-score from 0.824 to 0.772). Evaluated on the three transitions, each predicted from the other two at known provincial vote, this scaling reduces the total log-score from 0.889 to 0.809 (riding model without regional polls) and improves each of the three. Each component is then bounded at ±0.75 standard deviation of its part in the pool. This threshold is chosen on a grid containing no bound, evaluating each election from the other two, on the eve and national trend included: it is retained in two of the three cases (1 in the third), and reduces the total log-score from 0.964 to 0.915, improving it on each of the three elections. Each draw assigns a winner per riding; all 127 seats are thus allocated in every draw.
Two statistics summarise the joint distribution. The win probability of a riding is the frequency of wins over the draws; by linearity of expectation, its sum over ridings equals the party's expected seat count, and these expectations sum exactly to 127, the only additive riding-level decomposition. The reported central scenario is the medoid of the joint distribution: the draw minimising the mean L1 distance to the other draws. Unlike the vector of marginal medians, which matches no attainable scenario and does not sum to the number of seats, the medoid is an actual realisation, consistent across parties. The reported voting intentions describe the state of opinion on the date of computation; the seat simulation bears on election day, set by law on October 7, 2030. Since each draw carries both a national vote composition and a seat allocation, the distortion of the electoral system is itself a joint distribution: the votes-seats panel reports the expected shares of both per party and the Gallagher least-squares index, √(½Σ(v−s)²), summarised as a median and 90% interval over the draws.
Each draw is assigned to a single government scenario: majority when the first party reaches 64 seats, minority otherwise. For a minority government, the parties whose seats, added to those of the first party, reach a majority are listed, excluding the Official Opposition (second party in seats): these parties hold the balance of power. In the certainty chart, each riding is assigned to its favourite party and classed by that party's probability of winning: safe (95% or more), likely (75 to 95%), lean (60 to 75%), toss-up (under 60%).
The out-of-sample validations of sections 4 to 6 bear on the three transitions 2014→2018, 2018→2022 and 2022→2026, each evaluated with parameters estimated without it. The forecast published on the eve of the October 5, 2026 election identified the winner in 104 of the 127 ridings (106 according to the favourite by win probability; mean multiparty Brier score of 0.264). The absolute seat error of its median scenario is 28 (PQ 49 against 59 elected, CAQ 11 against 0, QS 5 against 9, CPQ 22 against 19, QLP 40 against 40); four of the five per-party 90% seat intervals contain the result, the CAQ's (2 to 32) being the exception. The systemic shock is calibrated on the 2018 and 2022 elections; its isotropy in ILR coordinates was tested against an anisotropic alternative (principal axis of observed drifts, directional share on a grid) and retained by leave-one-out validation on the three available drifts. Differential turnout by age is not modelled (optimal weight of zero on the final polls of 2022 and 2026 combined). Also not modelled: the individual name recognition of candidates (tested through Wikipedia notability, with no out-of-sample gain) and the covariance between parties in the trend. The latter was evaluated twice: by injecting the empirical correlation of poll residuals (rejected, the effect on seats remaining within simulation noise), then by intrinsic coregionalisation, the coupling matrix B being estimated jointly by marginal likelihood on the stacked ILR coordinates; the in-sample likelihood gain (+7 nats over the independent model) does not survive cross-validation (predictive log-likelihood lower by 25 nats over 1,355 observations, coverage unchanged), the signature of overfitting of the coupling hyperparameters. House effects, evaluated by temporal cross-validation, bring no predictive gain and are not retained. The weight of the by-election signal remains a prior that cannot be validated on historical data.
All estimations (compositional transformations, Gaussian process regressions, hyperparameter selection and cross-validation) rely on the tangent computing suite. The complete pipeline is released under the GPL-3 licence and reproducible from thegithub.com/votovision/quebec repository, the data fromgithub.com/votovision/data. The forecast is fully recomputed with each newly detected poll.