CLVAE: A Variational Autoencoder for Long-Term Customer Revenue Forecasting

Longitudinal tumor measurements, dropout information, and genetic covariates provide complementary information about treatment response, but integrating these data sources within a single population modeling framework remains challenging. We extend the empirical Bayes variational autoencoder (EB-VAE) framework to joint longitudinal and time-to-event modeling and evaluate it on tumor growth data. The framework represents inter-individual variability using latent individual effects regularized by a covariate-conditioned empirical Bayes prior, while a decoder maps these latent effects to tumor-volume trajectories. To account for informative dropout, the decoder was augmented with a hazard model, yielding joint predictions of tumor growth and time to dropout. We further compared fully neural and hybrid semi-mechanistic decoder formulations and incorporated genomic covariates through a genetics-conditioned prior adaptation. The hybrid decoder recovered treatment-effect parameters broadly consistent with previously reported nonlinear mixed-effects estimates, while achieving prior predictive performance comparable to the neural decoder. The joint model reproduced both tumor-volume distributions and dropout patterns in held-out individuals, and genetic conditioning improved individual-level prior predictions in both cutaneous melanoma and breast cancer experiments. Stability selection identified several biologically plausible genetic indicators, including alterations in BRAF, NRAS, NF1, and MDM2. These results demonstrate that EB-VAE provides a flexible probabilistic framework for combining neural dynamics, mechanistic structure, time-to-event modeling, and high-dimensional covariates in pharmacometric applications.

Vector autoregressive moving-average (VARMA) models have long been considered impractical beyond moderate dimensions: the likelihood is non-convex, the parametrization is identified only up to equivalence, and every evaluation costs a pass over the entire series. Yet their moving-average term captures with a few parameters what a pure autoregression matches only with many lags. We introduce an estimation framework that removes this computational barrier: each optimization iteration is independent of the series length $T$. The framework combines a partial-autocorrelation reparametrization that guarantees stationarity and invertibility by construction, Gaussian priors on the reparametrized coefficients with separate scales for diagonal and off-diagonal entries, and losses that depend on the data only through fixed-size sufficient statistics, evaluated by a Parseval (Fourier) identity at near-linear cost in the truncation length. This yields two point estimators: a regularized least-squares fit and a covariance-marginalized maximum-a-posteriori estimator. We prove that both recover the infinite-autoregressive representation of the true process at a near-parametric rate in fixed dimension, so the truncation introduces no asymptotic bias. The same machinery extends, at the same leading cost, to seasonal dynamics, exogenous regressors (VARMAX), and rolling-window refits. Empirically, the estimators stay close to the oracle forecast error from $d=10$ to $d=40$ (where classical conditional MLE returns non-invertible fits whose forecasts diverge) and match or beat VAR, Bayesian-VAR, component-wise ARMA, and sparse-VARMA baselines on retail-demand, meteorological, and air-quality data. This brings likelihood-based VARMA estimation, at a per-iteration cost independent of the series length, to the problem sizes where practitioners have so far relied on VAR models.

Long-horizon multivariate time series forecasting (LTSF) remains challenging due to non-stationarity, regime shifts, and error accumulation. The Variability-Aware Recursive Neural Network (VARNN) is designed to track such variability by maintaining a residual-memory state driven by one-step prediction errors. However, its original formulation is limited to one-step sequence regression and does not directly support multi-step forecasting. In this work, we extend VARNN to long-horizon forecasting and introduce StateFlow, a recurrent forecasting framework that uses VARNN as a dual-state recurrent backbone to capture two complementary signals from the lookback sequence: a hidden-state trajectory representing primary temporal dynamics, including trend, seasonality, level changes, and recurring patterns, and a residual-memory trajectory representing structured local prediction deviations, driven from a nonlinear recurrent transformation of errors between one-step base predictions and observed values. A chunk-based decoder separately summarizes these trajectories and maps them to the future horizon for direct multi-step forecasting. We further employ a two-stage optimization strategy that first trains the VARNN encoder through a one-step base prediction objective to optimize the internal representations over the lookback sequence, and then trains a horizon-specific decoder for direct multi-step forecasting. Experiments on standard LTSF benchmarks show that StateFlow achieves competitive performance against strong linear, recurrent, convolutional, and Transformer-based baselines while preserving linear recurrent encoding and a compact model design.