TAD理論とその応用: 履歴を可視化する数理フレームワーク

松井, 泰生

doi:10.51094/jxiv.1593

##article.authors##

松井, 泰生独立研究員 https://researchmap.jp/yasuo-matsui

DOI:

https://doi.org/10.51094/jxiv.1593

キーワード:

履歴依存ダイナミクス、 TAD理論（Traced Allocation Dynamics）、緩和関数・忘却曲線、ロジスティック成長モデル、 Kovacs効果（教育・記憶モデル）、在庫管理・広告効果、疫学モデル（SIR）、経済成長モデル

抄録

本研究は、時間を通じた履歴構造を明示的に取り入れる数理フレームワーク TAD（Traced Allocation Dynamics）理論 を提案する。
TADは、入力時刻 TTT と観測時刻 ttt の二重時間軸を基盤とし、内部状態 g(t,T)g(t,T)g(t,T) の動学を割当関数 μ(t,T)\mu(t,T)μ(t,T)、自然減衰 γ(t,T)\gamma(t,T)γ(t,T)、スケーリング係数 σ(t,T)\sigma(t,T)σ(t,T)、および入力 G(t,T)G(t,T)G(t,T) により統一的に記述する。この最小原理から、従来の多様なモデル（在庫鮮度モデル、Adstockモデル、緩和関数、SIRモデル、Solow型経済成長、忘却曲線など）を再構成できることを示すとともに、それらが内包していたが顕在化していなかった履歴依存構造を可視化する。
さらに、TADは既存理論の単なる再現にとどまらず、新しい知見や予言を導く。例えば、ロジスティック成長モデルにおいて、成長率や収容力を出生率・基礎死亡率・密度依存係数といった内生パラメータに分解できることを示した。また、教育・記憶モデルに対しては「Kovacs効果」に類似するハンプ現象を予言し、数値シミュレーションによって確認した。
本稿は、TADが履歴をもつ諸現象を横断的に統一し、新しい理論的発見と実務的応用の双方に資する汎用的基盤理論となり得ることを総合的に論じる。応用領域としては、在庫管理、広告効果分析、人口動態予測、学習スケジューリング、感染症流行解析、さらには大規模言語モデルにおけるメモリ選別などを含み、TADは「過去が現在にどのように寄与するか」を数理的に可視化する新しい視座を提供する。

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本研究は筆者個人の業務外活動として独立に実施したものであり、勤務先を含むいかなる組織からも資金・データ・設備等の提供を受けていません。本稿は一般理論の提示を目的とし、特定企業の業務・製品・見解を代表するものではありません。利益相反はありません。

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TAD理論とその応用

履歴を可視化する数理フレームワーク