Skip to content

Numerical Analysis: Japan PC Shipments Diffusion Model Comparison

Student Name: TAI Yungche
Student ID: 1W23CF13
Course: Numerical Analysis
Date: July 22, 2025

Executive Summary

Diffusion models applied to Japanese personal computer (PC) shipping data from 2007 to 2024 are addressed in this paper. The research examines mathematical models originally applied to the study of epidemiology and how these may be applied to predict technology adoption trends. Three models were considered: the basic SIR (Susceptible-Infected-Recovered) model, the Bass-SIR model with constant initial conditions, and an extended Bass-SIR model where all parameters were estimated. With a sum of squared residuals (SSR) of 0.01623897 versus 1.38112993 for the basic SIR model, findings show that the extended Bass-SIR model with four estimated parameters performed much better than conventional methods.

Introduction and Research Motivation

The diffusion of technology products is complex and can be represented using mathematical concepts from epidemiology and marketing studies. The key question motivating this research is: How do models of diffusion predict Japan's PC shipments? This research is important since it involves Japan, a developed technology market with distinct adoption patterns, thus a perfect place to compare various mathematical theories of technology diffusion.

There are several analytical benefits of using PC shipping data. First, fixed parameter estimates are enabled by the systematic monthly observations in the dataset over a significant temporal span (2007–2024). Second, the data have growth and plateau stages characteristic of seasoned technological industries, enabling one to test the model extensively over different market conditions.

Literature Review and Theoretical Foundation

SIR Model Background

The Susceptible-Infected-Recovered (SIR) model arose from epidemiological studies and was first formulated rigorously in 1927 by Kermack and McKendrick. The SIR model is a simple mathematical model for epidemics, but for several decades it resisted the efforts of the community to find an explicit solution. The model divides a population into three compartments:

  • Susceptible (S): Individuals who have not yet adopted the product
  • Infected (I): Current adopters who have an effect on other adopters
  • Recovered (R): Former adopters who no longer influence adoption

The system is governed by the following differential equations:

dS/dt = -βIS/N
dI/dt = βIS/N - γI
dR/dt = γI

Where β is the rate of transmission and γ is the rate of recovery.

Bass-SIR Model Integration

Bass-SIR is a hybrid model that combines Frank Bass's product diffusion model with the SIR framework. The Bass model was initially developed for consumer durables. It has since been used to forecast market adoption of numerous consumer and industrial products and services, including tangible, intangible, medical, and financial products. The model's underlying premise is that the adoption rate comes from two origins: The inherent tendency of customers to adopt the product independent of social pressures to adopt. The additional tendency to adopt the product because others are adopting it.

The Bass-SIR model restates the original SIR equations as:

dS/dt = -S(p + qI)
dI/dt = S(p + qI) - γI
dR/dt = γI

Where:

  • p: External influence rate ("advertising power")
  • q: Internal influence rate ("word of mouth power")
  • γ: Recovery rate (influence loss with time)

This modification allows the model to separate the effects of peer-to-peer adoption from external marketing effects, and the model provides a more realistic picture of technology diffusion mechanisms.

Methodology

Data Source and Validation

The study employed official shipment data available through the Japan Electronics and Information Technology Industries Association (JEITA), a quasi-government organization with standardized reporting procedures. JEITA represents major Japanese tech companies like Sony, Panasonic, and Toshiba and hence offers authentic and comprehensive data. The data sample consists of monthly PC shipment values from April 2007 through April 2024, which equate to 205 data points with an upper potential market of 228,636 thousand units.

Numerical Implementation

The parameter estimation process employed the Levenberg-Marquardt algorithm, an efficient nonlinear least squares optimization method. Levenberg–Marquardt algorithm (LMA or LM), or damped least-squares (DLS) method, is a method for solving non-linear least squares problems. The problems of minimization are faced especially in least squares curve fitting. LMA is a hybrid of the Gauss–Newton algorithm (GNA) and the gradient descent method.

The algorithm's hybrid nature has several advantages:

  • Stability: Comparable to steepest descent when remote from optimal parameters
  • Speed: Comparable to Gauss-Newton when near minimum
  • Robustness: Performs best when encountering rank-deficient Jacobian matrices

For global optimization assurance, the study employed a large grid search method with 3,125 sets of parameters (5^4 trials). This process has searched for parameters systematically to avoid convergence into local minima that are common in nonlinear optimization problems.

Solving Differential Equations

The differential equations were solved numerically using the LSODA (Livermore Solver for Ordinary Differential Equations with Automatic method switching) method, which automatically switches between:

  1. Adams-Moulton Method: For non-stiff equations with smoothly varying solutions
  2. Backward Differentiation Formula (BDF): For stiff equations with rapidly decreasing components

This adaptive approach balances accuracy with computational efficiency in all regimes of solutions.

Results and Analysis

Model Performance Comparison

The comparative analysis revealed stark differences in model performance:

Model Type SSR Value Parameter Count Key Characteristics
Basic SIR 1.38112993 2 Simple infection-recovery dynamics
Bass-SIR (Fixed I₀) 0.17453400 3 Fixed initial adopter assumption
Bass-SIR (All Parameters) 0.01623897 4 Optimized initial conditions

The Bass-SIR model achieved an 85-fold over the baseline SIR model and an 11-fold over the fixed initial condition variant, demonstrating the importance of precise estimation of initial conditions.

Parameter Meaning and Market Insights

Simple SIR Model Parameters

  • β (Infection Rate): 0.100000 per month, implying high contact adoption probability
  • γ (Recovery Rate): 0.057704 per month, suggesting moderate market exit rates

Bass-SIR Model with Fixed I₀ Parameters

  • p (External Influence): 0.001000 per month, indicating little ad-driven adoption
  • q (Internal Influence): 0.070426 per month, indicating strong word-of-mouth effects
  • r (Recovery Rate): 0.055940 per month, as would be expected from simple SIR findings

Extended Bass-SIR Model Parameters

  • p (External Influence): 0.003568 per month, higher than fixed model
  • q (Internal Influence): 0.004827 per month, much lower than fixed model
  • r (Recovery Rate): 0.000000 per month, indicating zero market exit
  • I₀ (Initial Adopters): 0.000000, near zero initial adopter size

The extended model parameter recalibration yields useful implications about PC adoption dynamics in Japan. The near-zero recovery rate indicates high post-adoption retention of the product, while the reduced internal influence coefficient implies that peer effects may be weaker than previously estimated if initial conditions are also properly dealt with.

Interpretation of Market Dynamics

The results indicate Japan's PC market for 2007-2024 was characterized by:

  1. High Retention: The near-zero recovery rate suggests PC adoption is primarily irreversible, with minimal abandonments post-adoption
  2. Limited New Adoption: Low external influence parameters indicate marketing efforts had minimal impact on triggering new adoptions
  3. Mature Market Characteristics: The overall trend is that there is a saturated market with replacement cycles rather than adding new users driving shipments

AI Use Documentation

Prompt Engineering Strategies

The workflow of the research employed generative AI tools strategically for several critical tasks:

  1. Automation of Data Processing: Python scripts were generated with AI assistance to process and normalize JEITA PC shipment data. Prompts were designed to build robust data cleaning scripts that could handle inconsistent formatting and missing values.

  2. Guidance for Parameter Optimization: AI systems provided structured approaches to iterative parameter adjustment. Prompt design followed a "request-response" pattern:

  3. Initial parameter estimation requests
  4. Convergence criteria specification
  5. Error analysis and adjustment strategies

  6. Suggestions for robustness testing

  7. Code Generation and Debugging: Complex numerical subroutines to realize the Levenberg-Marquardt algorithm were generated using AI, with a focus on:

  8. Jacobian matrix calculations

  9. Convergence monitoring
  10. Grid search implementation
  11. Visualization routines

Mitigating AI Limitations

The following measures were taken to prevent potential AI hallucinations or mistakes:

  1. Cross-Validation: All mathematical relationships were verified by more than one independent source
  2. Convergence Testing: Numerical results were tested for convergence with different starting conditions
  3. Literature Verification: Interpretations generated by AI were validated against existing research

Limitations and Future Research Directions

Current Study Limitations

  1. Model Simplifications: Homogeneous population mixing assumed by the Bass-SIR model may not best represent market segmentation
  2. External Factors: The model does not account for economic shocks, technological disruption, or policy interventions explicitly
  3. Data Granularity: Monthly aggregation may mask important short-run dynamics
  4. Geographic Scope: Results may not generalize to other markets with different cultural or economic profiles

Future Research Opportunities

  1. Multi-Product Extensions: Extension of the model to other technology product classes (electric vehicles, streaming services, smartphones)
  2. Economic Integration: Incorporation of macroeconomic variables (GDP, consumer confidence, interest rates)
  3. Advanced Model Variants: Exploration of SIRS models with temporary immunity or time-changing parameters
  4. Cross-Cultural Validation: Model fit validation in different national markets

Practical Implications

The results of this study have several practical implications for tech companies and market analysts:

  1. Market Timing: Understanding the relative importance of external and internal influence can guide the timing of marketing expenditure
  2. Mature Market Strategy: The low recovery rates suggest that retention strategies may be more important than acquisition in mature markets
  3. Forecasting Accuracy: The superior performance of the extended Bass-SIR model provides more believable prediction capacity for business planning

Reflections

Course Learning Integration

The numerical analysis course provided essential foundations to this research in the following topics:

Differential Equation Solving: Theoretical understanding of numerical integration methods was crucial in applying the LSODA algorithm for solving the Bass-SIR system. Course treatment of stability analysis helped in selecting appropriate step sizes and convergence criteria.

Optimization Theory: The compromise of the Levenberg-Marquardt algorithm between gradient descent and Gauss-Newton methods as used in the course had direct applications of class material on optimization trade-offs between stability and convergence rate.

Error Analysis: The systematic treatment of uncertainty in parameter estimation, as illustrated in the grid search method, showed practical applications of numerical error propagation concepts covered in coursework.

Group Work Experience Analysis

Successful Aspects

A number of aspects made the group work successful:

  1. Diverse Expertise: Members contributed various strengths in mathematical modeling, programming implementation, and economic interpretation
  2. Quality Assurance: Peer review of mathematical derivations and code implementations minimized errors considerably
  3. Workload Distribution: Tasks like data processing, parameter optimization, and visualization were effectively parallelized

Challenging Aspects

There were a number of challenges that arose during the group work:

  1. Coordination Complexity: Coordinating various components of the analysis involved a lot of communication and version control
  2. Standard Differences: Team members' various programming styles and documentation needs required standardization efforts to be made
  3. Responsibility Allocation: Coordinating the balanced contribution to the theoretical development, implementation, and analysis phases was challenging

Lessons Learned

The group work experience provided valuable lessons for future collaborative research:

  1. Early Framework Agreement: Agreements on mathematical notation, coding style, and documentation requirements at the start of the project prevent integration problems later
  2. Incremental Validation: Presenting intermediate results regularly allows early detection of conceptual or implementation errors
  3. Communication Tools: Utilization of collaborative software for code sharing and real-time communication significantly improves coordination efficiency

Personal Development

The research project facilitated development in the following key areas:

Technical Skills: Implementing complex numerical algorithms from theoretical descriptions strengthened practical coding capabilities and improved knowledge in optimization theory applications.

Research Methodology: The systematic approach to comparative models, parameter estimation, and robustness testing provided great experience in rigorous scientific inquiry methods.

Critical Analysis: The comparison of model performance on the basis of complexity versus accuracy honed the skill to balance theoretical elegance and practical feasibility.

Conclusion

This study has successfully illustrated that the principles of epidemiological modeling can be aptly modified for the analysis of technology diffusion. The better performance of the extended Bass-SIR model (SSR: 0.01623897 compared to 1.38112993 for simple SIR) confirms the significance of accurate estimation of initial conditions and parameter optimization in nonlinear dynamical systems.

The results show that the PC market of Japan from 2007-2024 had the nature of a mature technology ecosystem. It had high retention and minimal new user acquisition. This conclusion has real-world implications for tech firms selling in comparable markets in that it implies customer retention initiatives would work better than aggressive acquisition drives.

From a computational standpoint, the study emphasized the critical role of robust optimization techniques, in particular the effectiveness of the Levenberg-Marquardt algorithm for nonlinear parameter estimation problems. The grid search validation technique was extremely useful in confirming global optimization convergence, demonstrating the value of extensive robustness testing in scientific computation.

The successful integration of AI tools into the research process while maintaining stringent verification standards is a model for future scientific research with generative AI capabilities. The step-by-step nature of the prompt engineering, source verification, and hallucination detection provides hands-on instructions for accountable integration of AI in academic research.

Overall, the research offers both methodological improvement in diffusion modeling and substantive understanding of mature technology market behavior, while demonstrating the successful application of numerical analysis techniques to empirical economic problems.

References

  1. Bass, F. M. (1969). A new product growth model for consumer durables. Management Science, 15(5), 215-227.

  2. Fibich, G. (2016). Bass-SIR model for diffusion of new products in social networks. Physical Review E, 94(3), 032305.

  3. Japan Electronics and Information Technology Industries Association (JEITA). (n.d.). パーソナルコンピュータ国内出荷実績. Retrieved from https://www.jeita.or.jp/japanese/stat/pc/

  4. Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, 115(772), 700-721.

  5. Levenberg, K. (1944). A method for the solution of certain non-linear problems in least squares. Quarterly of Applied Mathematics, 2(2), 164-168.

  6. Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics, 11(2), 431-441.

  7. Moré, J. J. (1978). The Levenberg-Marquardt algorithm: Implementation and theory. In G. A. Watson (Ed.), Numerical Analysis (pp. 105-116). Springer-Verlag.

  8. Rodrigues, H. S. (2016). Application of SIR epidemiological model: New trends. International Journal of Applied Mathematics and Informatics, 10, 92-97.