Multi-dimensional Mortality (MDMx): Sex-Age-Specific Model Life Tables, Fitting, Prediction from Summary Mortality Indicators, and Forecasting

Abstract

Demographers rely on a variety of tools and methods to work with mortality schedules - model life tables, fitting methods, summary-indicator prediction, and forecasting - largely developed independently and not providing structurally coherent sex-specific outputs. The multi-dimensional mortality model (MDMx) unifies all four within one Tucker tensor decomposition demonstrated using the Human Mortality Database (HMD). Period life tables from the HMD are organized as a four-way tensor of logit(1qx) indexed by sex, age, country, and year. Shared factor matrices for sex and age make every output schedule structurally coherent by construction. From this decomposition four capabilities emerge: model life tables via clustering and smooth within-regime trajectories; life table fitting via a three-stage algorithm with Bayes-factor disruption detection; summary-indicator prediction mapping child or adult mortality to complete schedules, reformulating SVD-Comp in tensor coordinates; and forecasting via a damped local linear trend Kalman filter on PCA-reduced core matrices with hierarchical drift.

Figures (65)

• Figure 1: HMD temporal coverage by population. Each colored cell indicates that a 1$\times$1 period life table is available for that population-year. Populations are sorted by the first year of available data.
• Figure 2: Life expectancy at birth ($e_0$) over time for all HMD populations, separately by sex. Thin lines represent individual countries; the bold line is the cross-country median. Sharp dips from wars and pandemics punctuate the secular upward trend.
• Figure 3: Age-specific mortality (${}_1q_x$) on a log scale for selected countries and years. The characteristic U-shape of human mortality is evident: high infant mortality, a childhood minimum, and exponential increase at older ages.
• Figure 4: Logit-transformed mortality for selected populations. On this scale, the Gompertz law appears as an approximately linear increase above age 40. The logit transform stabilizes variance and enables additive modeling of mortality differences.
• Figure 5: Male--female difference in logit mortality by age, for Sweden across selected years. Values above zero indicate excess male mortality. This quantity -- the log odds ratio -- is additive on the logit scale used throughout.

Theorems & Definitions (5)

• Remark 3.1: Temporal pooling as a modeling choice
• Remark 4.1: Weighted HOSVD
• Remark 4.2: Sex differentials as joint structure
• Remark 6.1: The rotation of mortality and residual level correlation
• Remark 6.2: Validation by hierarchical clustering