Estimating Life Cycle Sales of Technology Products with Frequent Repeat Purchases: A Fractional Calculus-Based Approach
Bibliographic record
Abstract
Accurately predicting the sales trajectory of a product is critically important for firms’ medium- and long-term planning. However, reliable sales prediction models are very difficult to find when repeat purchases or subscription renewals account for a large proportion of product sales, which is often the case for technological products. This study introduces a new sales growth model, the generalized diffusion model with repeat purchases (GDMR), to address this problem. The GDMR draws upon a branch of mathematics called fractional calculus and formulates a product’s sales growth rate using a novel noninteger-order integral equation. Compared with benchmark methods, the GDMR is simple and easy to implement, is suitable for a wide variety of products, and predicts better than benchmark models such as time series and machine learning models. Furthermore, the GDMR can reliably recover a product’s progress of adoptions even when only sales data are available. Because of these important advantages, the GDMR can help firms better understand their products’ market positions and, subsequently, make more informed decisions in production and inventory planning, transportation and logistics, and sales and marketing, thus improving the effectiveness and efficiency of their business operations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".