Mortality Forecasting as a Flow Field in Tucker Decomposition Space

Abstract

Mortality forecasting methods in the Lee-Carter tradition extrapolate temporal components via time-series models, producing forecasts that can systematically underpredict life expectancy at long horizons and require ad hoc adjustments for sex coherence. We reframe forecasting as integrating a flow field through the low-dimensional score space of a Tucker tensor decomposition of multi-population mortality data from the Human Mortality Database. PCA reduction of the effective core matrices reveals that the mortality transition is essentially a one-dimensional flow: a scalar speed function advances the level, trajectory functions supply the structural scores, and the Tucker reconstruction produces complete sex-specific mortality schedules at each horizon. An era-weighted speed function adapts to contemporary dynamics at each forecast origin, and empirically calibrated convergence rates control relaxation from country-specific to canonical mortality structure. The system is evaluated by leave-country-out cross-validation with a 50-year horizon against Lee-Carter and Hyndman-Ullah benchmarks.

Figures (17)

• Figure 1: Flow-field structure in Tucker PCA space. Top left: raw year-to-year $e_0$ velocity (forward differences) vs $e_0$ -- the scatter is noisy but the LOWESS trend reveals level-dependent improvement; the production speed function uses per-country smoothed velocities in $s_1$ space for a cleaner estimate (\ref{['fig:speed-diagnostic']}). Top centre and right: derivative correlations $\Delta s_1$ vs $\Delta s_2$ and $\Delta s_3$ (raw forward differences); the tight linear relationship ($r = -0.92$) demonstrates one-dimensional dynamics. Bottom: canonical trajectories $s_k$ vs $e_0$ for PCs 1--3 -- each score is a tight function of mortality level, comprising a continuous model life table system in Tucker coordinates.
• Figure 2: Speed function denoising comparison in $s_1$ space. Left: per-country LOWESS-smoothed forward differences pooled across countries (Method A, production) -- the smoothing reveals the underlying improvement trend. Centre: raw forward differences pooled directly (Method B) -- the cross-country LOWESS alone cannot fully denoise the year-to-year noise. Right: overlay of the two LOWESS estimates, showing that per-country smoothing is essential for a well-behaved speed function.
• Figure 3: $s_1$-to-surface-$e_0$ mapping. Left: raw LOWESS with flat extrapolation -- surface $e_0$ saturates at the frontier. Right: with joint tangent extension from $s_1^* \approx -12$ ($e_0 \approx 78$) -- surface $e_0$ continues to improve monotonically. Pink: female; cyan: male; green: both-sex average.
• Figure 4: Validation of the joint tangent extrapolation in $s_1$ space. Per-component score slopes $\mathrm{d}s_k/\mathrm{d}s_1$ for the LOWESS tangent at $s_1^* \approx -12$ ($e_0 \approx 78$, red) and for five frontier countries (Japan, Sweden, Switzerland, Spain, Italy) over their last 20 years. The cosine similarity between the LOWESS tangent and the frontier average is 0.94, confirming that the extrapolation direction agrees with observed frontier dynamics. The magnitude ratio is 0.59 -- the tangent extrapolation is $\sim$40% conservative in speed relative to frontier countries, producing a modestly cautious long-horizon forecast.
• Figure 5: Forecast $e_0$ diagnostic for six countries under $s_1$-space navigation (all-data flow field). Green: surface-derived $e_0$ (raw, before bias correction). Red dash-dot: bias-corrected $e_0$ (reported forecast). The annotation shows the 30-year $e_0$ gain. Because navigation is in $s_1$ space, there is no separate navigation $e_0$ that can diverge from the surface $e_0$.