符元序列處理之相關研究

Abstract

一個實值的隨機變量𝖷 是一個從概率空間(𝛺, 𝔼, 𝖯) 到(ℝ, ≤, 𝖽, +, ⋅, ℬ) 的可量度函數映射，其中，ℬ 是在“實數線”(ℝ, ≤, 𝖽, +, ⋅) 上的通常博雷爾𝜎-代數，ℝ 是實數集合，≤ 是定義在ℝ 上的標準線性順序關係，𝖽(𝑥, 𝑦) ≜ |𝑥 − 𝑦| 是在ℝ 上的常規距離量度，而 (ℝ, +, ⋅, 0, 1) 則構成ℝ 的場域(field)。本文以例論述當概率空間到實數線的X 映射與實數線有不相類同的結構時，這個隨機變量的定義常導致不佳的統計計算量。本文因此提出兩種可能的替換統計系統，不像傳統的定義由一個隨機變量𝖷 映射至實數線，𝖷 可更普遍的映射到(1) 加權圖(2) 順序距離線性空間ℝ𝘕 。在以上任一映射 法中，𝖷 形成類似基礎隨機過程之有順序並有度量幾何結構映射。理想情況下，𝖷 的原映射結構與𝖷 的目標映射結構，相對於彼此，都是同構和等距的。

A real-valued random variable 𝖷 is a measurable function that maps from a probability space (𝛺, 𝔼, 𝖯) to (ℝ, ≤, 𝖽, +, ⋅, ℬ) where ℬ is the usual Borel σ-algebra on the “real line” (ℝ, ≤, 𝖽, +, ⋅) and where ℝ is the set of real numbers, ≤ is the standard linear order relation on ℝ, 𝖽(𝑥, 𝑦) ≜ |𝑥 − 𝑦| is the usual metric on ℝ, and (ℝ, +, ⋅, 0, 1) is the standard field on ℝ. This text demonstrates that this definition of random variable is often a poor choice for computing statistics when the probability space that 𝖷 maps from has structure that is dissimilar to that of the real line.

This text proposes two alternative statistical systems that, unlike the traditional method of a random variable 𝖷 mapping exclusively to the real line, 𝖷 instead maps more generally to (1) a weighted graph (2) an ordered distance linear space ℝ^𝘕 . In each mapping method, the structure that 𝖷 maps to is preferably one that has order and metric geometry structures similar to that of the underlying stochastic process. And ideally the structure 𝖷 maps from and the structure 𝖷 maps to are, with respect to each other, both isomorphic and isometric.