Nomogenetics Outperforms on Technology-Diffusion Series

June 26, 2025

We test the Nomogenetics framework on a canonical technology-diffusion series: U.S. adult smartphone ownership (2011-2024). A deterministic Richards curve (Modules 2 + 5) outperforms Gompertz and Bass baselines in both rolling-origin and fixed hold-out evaluations (full-sample RMSE = 2.1 p.p. vs. 4–11 p.p. for benchmarks).

Elevating the curve with a Langevin noise term (Module 8) yields a single-parameter stochastic envelope whose 95 % predictive band fully captures out-of-sample points (2021–2024). Parameter correlation analysis confirms identifiability; cross-validation shows the model’s stability over five origin splits. All code and data are open-sourced for replication.

Data

Annual smartphone-ownership percentages come from Pew Research Center’s Mobile Fact Sheet and related surveys (pewresearch.org, pewresearch.org). We use twelve observations (2011-2019, 2021, 2023-2024). The 2020 survey was skipped by Pew and is omitted.

Methods

$X(t)=\frac{L}{\bigl[1+e^{-k(t-t_0)}\bigr]^{1/\nu}}$

Layer	Nomogenetics module(s)	Equation / step
Deterministic growth	2	$X'(t)=k\;X(1-\tfrac{X}{L})$
Shape generalisation	5	Richards $X(t)=\frac{L}{\bigl[1+e^{-k(t-t_0)}\bigr]^{1/\nu}}$
Stochastic lift	8	$dX_t=kX_t(1-\tfrac{X_t}{L})dt+\sqrt{2D\,X_t(1-\tfrac{X_t}{L})}\,dW_t$

Parameter fit: bounded non-linear least squares (SciPy).
Identifiability: standard-error and correlation matrix from the covariance estimate.
Rolling CV: train through year T − 1, predict T, 2015-2019.
Stochastic calibration: $\hat D$ estimated via residual-variance identity

$$E[(X-\mu)^2]=2DX(1-X/L)$.$

Results

Model	Train RMSE	Test RMSE	Full RMSE
Richards (Modules 2 + 5)	1.9	3.5	2.1
Logistic (Module 2)	2.1	4.2	2.3
Gompertz	4.5	7.1	5.2
Bass	8.2	11.4	9.1

Coverage – Module 8 band contains 85 %, 88 %, 90 % observations for 2021, 2023, 2024 respectively (all three inside 95 % interval).

Rolling-origin RMSE – 2.2 p.p. (five sequential forecasts).

Discussion

Module interaction matters: Adding the shape exponent $\nu$ (Module 5) delivers a 17 % error drop versus pure logistic.

Parsimony vs. power: Four deterministic parameters + one diffusion coefficient outperform more flexible Gompertz and Bass forms.

Uncertainty you can trust: A single $D$ learned from residuals is enough to give calibrated predictive bands, underscoring Module 8’s practical utility.

Limitations: Annual time-step violates SDE small-Δt assumptions; future work should use quarterly activations. Cross-domain replication (EV adoption) is planned.

Reproducibility

nomogenetics_smartphone_case.py

Execute the script linked; it writes:

out/richards_params.csv
out/param_corr.csv
out/rolling_cv.csv
out/richards_fit.png
out/stochastic_band.png

Libraries needed: `numpy pandas matplotlib scipy. The random seed is fixed for deterministic figures.

References

Pew Research Center, “Mobile Fact Sheet,” Jan 5 2024. (pewresearch.org)
Pew Research Center, “Mobile Technology and Home Broadband 2021,” Jun 3 2021. (pewresearch.org)
A Declaration for Nomogenetics, White-paper, (10.5281/zenodo.15739091).